Subgroups — #46824921

Order 34432.

1124 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771324	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771313	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771312	C₂	1	1	yes	maximal, characteristic, complemented
4	#31771292	C₂	1	2	yes	maximal, complemented
5	#31771286	C₂	1	2	yes	maximal, complemented
6	#31771281	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#31771261	C₂	1	2	yes	maximal, complemented
9	#31771255	C₂	1	1	yes	maximal, characteristic, complemented
10	#31771254	C₂	1	1	yes	maximal, characteristic, complemented
11	#31771249	C₂	1	1	yes	maximal, characteristic, complemented
12	#31771224	C₂	1	1	yes	maximal, characteristic, complemented
13	#31771222	C₂	1	1	yes	maximal, characteristic, complemented
14	#31771200	C₂	1	2	yes	maximal, complemented
15	#31771199	C₂	1	2	yes	maximal, complemented
16	#31771157	C₂	1	1	yes	maximal, characteristic, complemented
17	#31771135	C₂	1	1	yes	maximal, characteristic, complemented
18	#31771111	C₂	1	2	yes	maximal, complemented
19	C₂×#22575430	C₂	1	2	yes	maximal, complemented
20	#31771088	C₂	1	1	yes	maximal, characteristic, complemented
21	#31771084	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×D₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
23	#546×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#16864401	C₂²	1	1	yes	characteristic
25	C₂×#16864401	C₂²	1	2	yes	complemented
26	C₂²×Dic₅₃₈	C₂²	1	2	yes	complemented
27	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
28	#22575453	C₂²	1	1	yes	characteristic, complemented
29	#22575452	C₂²	1	1	yes	characteristic, complemented
30	#22575452	C₂²	1	1	yes	characteristic, complemented
31	#22575451	C₂²	1	1	yes	characteristic, complemented
32	#22575448	C₂²	1	1	yes	characteristic, complemented
33	#22575446	C₂²	1	2	yes	complemented
34	#22575446	C₂²	1	1	yes	characteristic, complemented
35	#22575446	C₂²	1	2	yes	complemented
36	#22575446	C₂²	1	2	yes	complemented
37	#22575446	C₂²	1	2	yes	complemented
38	#22575444	C₂²	1	2	yes	complemented
39	#22575443	C₂²	1	2	yes	complemented
40	#22575443	C₂²	1	2	yes	complemented
41	#22575443	C₂²	1	1	yes	characteristic, complemented
42	#22575440	C₂²	1	1	yes	characteristic
43	#22575440	C₂²	1	2	yes	complemented
44	#22575439	C₂²	1	2	yes	complemented
45	#22575439	C₂²	1	1	yes	characteristic, complemented
46	#22575437	C₂²	1	1	yes	characteristic
47	#22575437	C₂²	1	2	yes	complemented
48	#22575436	C₂²	1	2	yes	complemented
49	#22575435	C₂²	1	2	yes	complemented
50	#22575434	C₂²	1	2	yes	—
51	#22575434	C₂²	1	1	yes	characteristic
52	#22575434	C₂²	1	2	yes	complemented
53	#22575434	C₂²	1	2	yes	complemented
54	#22575434	C₂²	1	2	yes	complemented
55	#22575433	C₂²	1	2	yes	complemented
56	#22575432	C₂²	1	1	yes	characteristic
57	#22575432	C₂²	1	2	yes	complemented
58	#22575432	C₂²	1	1	yes	characteristic, complemented
59	#22575430	C₂²	1	2	yes	—
60	#22575430	C₂²	1	2	yes	—
61	#22575430	C₂²	1	2	yes	—
62	#22575430	C₂²	1	1	yes	characteristic
63	#22575430	C₂²	1	2	yes	complemented
64	#22575430	C₂²	1	2	yes	complemented
65	#22575430	C₂²	1	2	yes	complemented
66	#22575430	C₂²	1	2	yes	complemented
67	#22575430	C₂²	1	2	yes	complemented
68	#22575430	C₂²	1	2	yes	complemented
69	#22575430	C₂²	1	2	yes	complemented
70	#22575429	C₂²	1	1	yes	characteristic, complemented
71	#22575425	C₂²	1	2	yes	—
72	#22575425	C₂²	1	1	yes	characteristic
73	#22575425	C₂²	1	2	yes	complemented
74	#22575424	C₂²	1	1	yes	characteristic
75	#22575422	C₂²	1	2	yes	—
76	Q₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
77	Q₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
78	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic, complemented
79	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic, complemented
80	#22575418	C₂²	1	1	yes	characteristic, complemented
81	#22575418	C₂²	1	2	yes	complemented
82	D₄×Dic₂₆₉	C₂²	1	2	yes	—
83	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
84	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
85	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
86	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
87	D₄×Dic₂₆₉	C₂²	1	2	yes	—
88	D₄×Dic₂₆₉	C₂²	1	2	yes	—
89	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
90	D₄×Dic₂₆₉	C₂²	1	2	yes	complemented
91	D₄×Dic₂₆₉	C₂²	1	2	yes	complemented
92	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
93	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic, complemented
94	C₄×C₂²⋊D₂₆₉	C₂²	1	2	yes	complemented
95	#22575413	C₂²	1	2	yes	—
96	#22575413	C₂²	1	2	yes	complemented
97	#22575412	C₂²	1	1	yes	characteristic, complemented
98	#22575412	C₂²	1	2	yes	complemented
99	#22575410	C₂²	1	1	yes	characteristic, complemented
100	#22575409	C₂²	1	2	yes	—
101	#22575409	C₂²	1	2	yes	complemented
102	#22575409	C₂²	1	2	yes	complemented
103	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
104	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	complemented
105	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
106	C₂×#16864381	C₂²	1	2	yes	complemented
107	C₂×#16864381	C₂²	1	2	yes	complemented
108	C₄⋊C₄×D₂₆₉	C₂²	1	1	yes	characteristic, complemented
109	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
110	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
111	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
112	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
113	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
114	#124×C₂₆₉	C₂²	1	1	yes	characteristic
115	#123×C₂₆₉	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
117	#121×C₂₆₉	C₂²	1	2	yes	—
118	#121×C₂₆₉	C₂²	1	1	yes	characteristic
119	#120×C₂₆₉	C₂²	1	2	yes	—
120	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
121	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
122	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
123	#16864401	—	2	4	no	—
124	#16864401	—	4	4	no	—
125	C₂×Dic₅₃₈	—	4	4	no	—
126	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
127	C₂×Dic₅₃₈	C₂³	1	2	yes	—
128	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
129	C₂×Dic₅₃₈	C₂³	1	2	yes	—
130	C₂×Dic₅₃₈	C₂³	1	2	yes	—
131	C₂×C₂²⋊D₂₆₉	C₂³	1	1	yes	characteristic
132	C₂×C₂²⋊D₂₆₉	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₂₆₉	C₂³	1	1	yes	characteristic
134	C₂×C₂²⋊D₂₆₉	C₂³	1	2	yes	—
135	C₂²×Dic₂₆₉	—	4	4	no	—
136	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
137	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
138	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
139	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
140	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
141	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
142	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
143	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
144	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
145	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
146	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₂₆₉	C₂³	1	2	yes	—
148	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
149	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
150	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
151	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
152	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
153	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
154	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
155	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
156	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
158	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
159	#16864381	C₂³	1	2	yes	—
160	#16864381	C₂³	1	1	yes	characteristic
161	#16864381	C₂³	1	1	yes	characteristic
162	#16864381	C₂³	1	2	yes	—
163	#16864381	C₂³	1	2	yes	—
164	#16864381	C₂³	1	2	yes	—
165	#16864381	C₂³	1	2	yes	—
166	#16864381	C₂³	1	2	yes	—
167	#16864381	C₂³	1	2	yes	—
168	#16864381	C₂³	1	2	yes	—
169	#16864381	C₂³	1	1	yes	characteristic
170	#16864381	C₂³	1	1	yes	characteristic
171	#16864381	C₂³	1	2	yes	—
172	#16864381	C₂³	1	1	yes	characteristic
173	#16864381	C₂³	1	2	yes	—
174	#16864381	C₂³	1	1	yes	characteristic
175	C₂²⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
176	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
177	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
178	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
179	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
180	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
181	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
182	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
183	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
184	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
185	C₂²⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
186	C₂²⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
187	C₂²⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
188	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
189	#16864379	C₂³	1	1	yes	characteristic
190	#16864379	C₂³	1	1	yes	characteristic
191	#16864379	C₂³	1	2	yes	—
192	#16864379	C₂³	1	1	yes	characteristic
193	#16864379	C₂³	1	1	yes	characteristic
194	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
195	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
196	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
197	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
198	C₄×Dic₂₆₉	C₂³	1	2	yes	—
199	C₄×Dic₂₆₉	C₂³	1	2	yes	—
200	C₄×Dic₂₆₉	C₂³	1	2	yes	—
201	C₄×Dic₂₆₉	C₂³	1	2	yes	—
202	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
203	C₂⁴×C₂₆₉	C₂³	1	2	yes	—
204	#41×C₂₆₉	—	2	4	no	—
205	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
206	C₂×D₄×C₂₆₉	—	2	4	no	—
207	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
208	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
209	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
210	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
211	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
212	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
213	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
214	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
215	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
216	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
217	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
218	C₄⋊C₄×C₂₆₉	C₂³	1	2	yes	—
219	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
220	C₂²⋊C₄×C₂₆₉	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₂₆₉	C₂³	1	2	yes	—
222	C₂²⋊C₄×C₂₆₉	C₂³	1	2	yes	—
223	C₂²⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
224	C₂²⋊C₄×C₂₆₉	C₂³	1	2	yes	—
225	C₂²⋊C₄×C₂₆₉	C₂³	1	2	yes	—
226	C₂²⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
227	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
228	C₂²×D₂₆₉	C₂⁴	1	1	yes	characteristic
229	Dic₅₃₈	—	2	2	no	—
230	Dic₅₃₈	—	2	2	no	—
231	Dic₅₃₈	—	2	4	no	—
232	Dic₅₃₈	—	4	2	no	—
233	Dic₅₃₈	—	4	2	no	—
234	C₂²⋊D₂₆₉	—	2	4	no	—
235	C₂²⋊D₂₆₉	—	2	2	no	—
236	C₂²⋊D₂₆₉	—	2	2	no	—
237	C₂²⋊D₂₆₉	—	4	2	no	—
238	C₂×Dic₂₆₉	—	2	4	no	—
239	C₂×Dic₂₆₉	—	2	2	no	—
240	C₂×Dic₂₆₉	—	2	2	no	—
241	C₂×Dic₂₆₉	—	2	4	no	—
242	C₂×Dic₂₆₉	—	2	4	no	—
243	C₂×Dic₂₆₉	—	4	2	no	—
244	C₂×Dic₂₆₉	—	4	2	no	—
245	C₂×Dic₂₆₉	—	4	2	no	—
246	C₂×Dic₂₆₉	—	4	2	no	—
247	C₂×Dic₂₆₉	—	4	2	no	—
248	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
249	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
251	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
252	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
253	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
254	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
255	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
256	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
257	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
258	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
259	C₄×D₂₆₉	—	2	2	no	—
260	C₄×D₂₆₉	—	4	2	no	—
261	C₂³×C₂₆₉	—	2	4	no	—
262	C₂³×C₂₆₉	—	2	4	no	—
263	C₂³×C₂₆₉	—	4	2	no	—
264	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
265	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
266	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
267	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
268	Q₈×C₂₆₉	—	2	2	no	—
269	D₄×C₂₆₉	—	2	4	no	—
270	D₄×C₂₆₉	—	2	4	no	—
271	D₄×C₂₆₉	—	2	2	no	—
272	D₄×C₂₆₉	—	2	2	no	—
273	D₄×C₂₆₉	—	4	2	no	—
274	D₄×C₂₆₉	#41	1	4	yes	complemented
275	C₂×C₄×C₂₆₉	—	2	4	no	—
276	C₂×C₄×C₂₆₉	—	2	2	no	—
277	C₂×C₄×C₂₆₉	—	2	2	no	—
278	C₂×C₄×C₂₆₉	—	4	2	no	—
279	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
280	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
281	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
282	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
283	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
284	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
285	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
286	C₂²×C₂₆₉	—	2	1	no	—
287	C₂²×C₂₆₉	—	2	2	no	—
288	C₂²×C₂₆₉	—	2	1	no	—
289	C₂²×C₂₆₉	—	2	2	no	—
290	C₂²×C₂₆₉	—	2	1	no	—
291	C₂²×C₂₆₉	—	2	2	no	—
292	C₂²×C₂₆₉	—	2	1	no	—
293	C₂²×C₂₆₉	—	2	2	no	—
294	C₂²×C₂₆₉	—	2	1	no	—
295	C₂²×C₂₆₉	—	2	2	no	—
296	C₂²×C₂₆₉	—	4	8	no	—
297	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
298	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
299	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
300	C₂×D₂₆₉	—	2	1	no	—
301	C₂×D₂₆₉	—	2	1	no	—
302	C₂×D₂₆₉	—	2	1	no	—
303	C₄×C₂₆₉	—	2	1	no	—
304	C₄×C₂₆₉	—	2	1	no	—
305	C₄×C₂₆₉	—	2	2	no	—
306	C₄×C₂₆₉	—	2	1	no	—
307	C₄×C₂₆₉	—	2	2	no	—
308	C₄×C₂₆₉	—	2	1	no	—
309	C₄×C₂₆₉	C₂×#41	1	2	yes	—
310	Dic₂₆₉	—	2	1	no	—
311	Dic₂₆₉	—	2	2	no	—
312	Dic₂₆₉	—	2	2	no	—
313	Dic₂₆₉	—	2	1	no	—
314	Dic₂₆₉	—	2	2	no	—
315	Dic₂₆₉	—	2	2	no	—
316	Dic₂₆₉	—	2	1	no	—
317	Dic₂₆₉	—	2	1	no	—
318	Dic₂₆₉	—	2	1	no	—
319	Dic₂₆₉	—	2	1	no	—
320	Dic₂₆₉	H(5,𝔽₂)	1	2	yes	—
321	C₂×C₂₆₉	—	2	2	no	—
322	C₂×C₂₆₉	—	2	4	no	—
323	C₂×C₂₆₉	—	4	1	no	—
324	C₂×C₂₆₉	—	4	2	no	—
325	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
326	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
327	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
328	D₂₆₉	—	4	1	no	—
329	C₂₆₉	#3520	1	1	yes	characteristic, complemented, Sylow
330	#3520	—	269	1	no	maximal, Sylow
331	#556	—	269	1	no	—
332	#554	—	269	1	no	—
333	#554	—	269	1	no	—
334	#552	—	269	2	no	—
335	#551	—	269	2	no	—
336	#551	—	269	2	no	—
337	D₄×Q₈	—	269	1	no	—
338	#548	—	269	2	no	—
339	#547	—	269	1	no	—
340	#547	—	269	1	no	—
341	#546	—	269	1	no	—
342	#546	D₂₆₉	1	1	yes	characteristic, complemented
343	#542	—	269	1	no	—
344	#542	—	269	1	no	—
345	#539	—	269	2	no	—
346	#539	—	269	2	no	—
347	#533	—	269	1	no	—
348	#529	—	269	1	no	—
349	#525	—	269	2	no	—
350	C₂×C₂²⋊Q₈	—	269	2	no	—
351	#520	—	269	1	no	—
352	#520	—	269	1	no	—
353	C₂×C₄×D₄	—	269	1	no	—
354	C₂×#41	—	269	1	no	—
355	C₂×#41	—	269	2	no	—
356	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
357	C₂²×Q₈	—	269	2	no	—
358	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
359	C₂³×C₄	—	269	2	no	—
360	C₄⋊Q₈	—	269	1	no	—
361	C₄⋊Q₈	—	269	1	no	—
362	C₄⋊Q₈	—	269	1	no	—
363	C₄⋊Q₈	—	269	1	no	—
364	#126	—	269	1	no	—
365	#126	—	269	1	no	—
366	#126	—	269	2	no	—
367	#126	—	269	2	no	—
368	#126	—	269	2	no	—
369	#126	—	269	2	no	—
370	#125	—	269	1	no	—
371	#125	—	269	2	no	—
372	#125	—	269	2	no	—
373	#125	—	269	2	no	—
374	#124	—	269	1	no	—
375	#124	—	269	2	no	—
376	#124	—	269	1	no	—
377	#124	—	269	2	no	—
378	#124	C₂×D₂₆₉	1	1	yes	characteristic
379	#123	—	269	2	no	—
380	#123	—	269	1	no	—
381	#123	—	269	2	no	—
382	#123	—	269	2	no	—
383	#123	—	269	2	no	—
384	#123	—	269	2	no	—
385	#123	—	269	2	no	—
386	#123	—	269	1	no	—
387	#123	—	269	2	no	—
388	#123	C₂×D₂₆₉	1	2	yes	—
389	C₂²⋊Q₈	—	269	1	no	—
390	C₂²⋊Q₈	—	269	1	no	—
391	C₂²⋊Q₈	—	269	2	no	—
392	C₂²⋊Q₈	—	269	1	no	—
393	C₂²⋊Q₈	—	269	2	no	—
394	C₂²⋊Q₈	—	269	1	no	—
395	C₂²⋊Q₈	—	269	2	no	—
396	C₂²⋊Q₈	—	269	2	no	—
397	C₂²⋊Q₈	—	269	2	no	—
398	C₂²⋊Q₈	—	269	2	no	—
399	C₂²⋊Q₈	—	269	2	no	—
400	C₂²⋊Q₈	—	269	2	no	—
401	C₂²⋊Q₈	—	269	2	no	—
402	C₂²⋊Q₈	—	269	2	no	—
403	C₂²⋊Q₈	—	269	2	no	—
404	C₂²⋊Q₈	—	269	2	no	—
405	C₂²⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
406	#121	—	269	1	no	—
407	#121	—	269	2	no	—
408	#121	—	269	2	no	—
409	#121	—	269	1	no	—
410	#121	C₂×D₂₆₉	1	2	yes	—
411	#121	C₂×D₂₆₉	1	1	yes	characteristic
412	#120	—	269	2	no	—
413	#120	C₂×D₂₆₉	1	2	yes	—
414	C₄×Q₈	—	269	1	no	—
415	C₄×Q₈	—	269	1	no	—
416	C₄×Q₈	—	269	1	no	—
417	C₄×Q₈	—	269	2	no	—
418	C₄×Q₈	—	269	1	no	—
419	C₄×Q₈	—	269	1	no	—
420	C₄×D₄	—	269	1	no	—
421	C₄×D₄	—	269	2	no	—
422	C₄×D₄	—	269	1	no	—
423	C₄×D₄	—	269	1	no	—
424	C₄×D₄	—	269	1	no	—
425	C₄×D₄	—	269	2	no	—
426	C₄×D₄	—	269	2	no	—
427	C₄×D₄	—	269	1	no	—
428	C₄×D₄	—	269	2	no	—
429	C₄×D₄	—	269	2	no	—
430	C₄×D₄	—	269	2	no	—
431	C₄×D₄	—	269	1	no	—
432	C₄×D₄	—	269	1	no	—
433	C₄×D₄	—	269	2	no	—
434	C₄×D₄	—	269	2	no	—
435	C₄×D₄	C₂×D₂₆₉	1	1	yes	characteristic
436	C₄×D₄	C₂×D₂₆₉	1	1	yes	characteristic
437	#117	—	269	1	no	—
438	#117	—	269	2	no	—
439	#117	—	269	2	no	—
440	#117	—	269	2	no	—
441	#117	—	269	2	no	—
442	#117	—	269	1	no	—
443	C₂×C₄⋊C₄	—	269	1	no	—
444	C₂×C₄⋊C₄	—	269	2	no	—
445	C₂×C₄⋊C₄	—	269	2	no	—
446	C₂×C₄⋊C₄	—	269	1	no	—
447	C₂×C₄⋊C₄	—	269	2	no	—
448	C₂×C₄⋊C₄	—	269	1	no	—
449	C₂×C₂²⋊C₄	—	269	2	no	—
450	C₂×C₂²⋊C₄	—	269	2	no	—
451	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
452	C₂×C₄²	—	269	1	no	—
453	C₂⁴	C₂²×D₂₆₉	1	2	yes	—
454	#41	—	2	4	no	—
455	#41	—	538	4	no	—
456	#41	—	1076	4	no	—
457	C₂×Q₈	—	269	1	no	—
458	C₂×Q₈	—	269	1	no	—
459	C₂×Q₈	—	269	2	no	—
460	C₂×Q₈	—	269	2	no	—
461	C₂×Q₈	—	269	2	no	—
462	C₂×Q₈	—	1076	4	no	—
463	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	269	1	no	—
466	C₂×D₄	—	269	2	no	—
467	C₂×D₄	—	269	1	no	—
468	C₂×D₄	—	269	2	no	—
469	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
470	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
471	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
472	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
473	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
474	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
475	C₂²×C₄	—	269	1	no	—
476	C₂²×C₄	—	269	1	no	—
477	C₂²×C₄	—	269	2	no	—
478	C₂²×C₄	—	269	2	no	—
479	C₂²×C₄	—	269	2	no	—
480	C₂²×C₄	—	269	2	no	—
481	C₂²×C₄	—	269	2	no	—
482	C₂²×C₄	—	269	2	no	—
483	C₂²×C₄	—	269	2	no	—
484	C₂²×C₄	—	269	2	no	—
485	C₂²×C₄	—	269	1	no	—
486	C₂²×C₄	—	269	2	no	—
487	C₂²×C₄	—	1076	4	no	—
488	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
489	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
490	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
491	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
492	C₄⋊C₄	—	269	1	no	—
493	C₄⋊C₄	—	269	1	no	—
494	C₄⋊C₄	—	269	1	no	—
495	C₄⋊C₄	—	269	2	no	—
496	C₄⋊C₄	—	269	2	no	—
497	C₄⋊C₄	—	269	1	no	—
498	C₄⋊C₄	—	269	2	no	—
499	C₄⋊C₄	—	269	2	no	—
500	C₄⋊C₄	—	269	1	no	—
501	C₄⋊C₄	—	269	1	no	—
502	C₄⋊C₄	—	269	2	no	—
503	C₄⋊C₄	—	269	2	no	—
504	C₄⋊C₄	—	269	2	no	—
505	C₄⋊C₄	—	269	2	no	—
506	C₄⋊C₄	—	269	2	no	—
507	C₄⋊C₄	—	269	2	no	—
508	C₄⋊C₄	—	269	1	no	—
509	C₄⋊C₄	—	269	1	no	—
510	C₄⋊C₄	—	269	1	no	—
511	C₄⋊C₄	—	269	2	no	—
512	C₄⋊C₄	—	269	2	no	—
513	C₄⋊C₄	—	269	2	no	—
514	C₄⋊C₄	—	269	2	no	—
515	C₄⋊C₄	—	269	1	no	—
516	C₄⋊C₄	—	269	2	no	—
517	C₄⋊C₄	—	269	1	no	—
518	C₄⋊C₄	—	269	1	no	—
519	C₄⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
520	C₄⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×D₂₆₉	1	2	yes	—
522	C₂²⋊C₄	—	269	1	no	—
523	C₂²⋊C₄	—	269	2	no	—
524	C₂²⋊C₄	—	269	2	no	—
525	C₂²⋊C₄	—	269	2	no	—
526	C₂²⋊C₄	—	269	2	no	—
527	C₂²⋊C₄	—	269	2	no	—
528	C₂²⋊C₄	—	269	2	no	—
529	C₂²⋊C₄	—	269	2	no	—
530	C₂²⋊C₄	—	269	2	no	—
531	C₂²⋊C₄	—	269	1	no	—
532	C₂²⋊C₄	—	269	1	no	—
533	C₂²⋊C₄	—	269	2	no	—
534	C₂²⋊C₄	—	269	2	no	—
535	C₂²⋊C₄	—	269	1	no	—
536	C₂²⋊C₄	—	269	1	no	—
537	C₂²⋊C₄	—	269	1	no	—
538	C₂²⋊C₄	—	269	1	no	—
539	C₂²⋊C₄	—	269	2	no	—
540	C₂²⋊C₄	—	269	1	no	—
541	C₂²⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
542	C₂²⋊C₄	C₂²×D₂₆₉	1	2	yes	—
543	C₂²⋊C₄	C₂²×D₂₆₉	1	2	yes	—
544	C₂²⋊C₄	C₂²×D₂₆₉	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₂₆₉	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₂₆₉	1	2	yes	—
547	C₂²⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
548	C₄²	—	269	1	no	—
549	C₄²	—	269	1	no	—
550	C₄²	—	269	1	no	—
551	C₄²	—	269	1	no	—
552	C₄²	—	269	2	no	—
553	C₄²	—	269	2	no	—
554	C₄²	—	269	2	no	—
555	C₄²	—	269	2	no	—
556	C₄²	—	269	1	no	—
557	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
558	C₂³	—	2	4	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	269	1	no	—
562	C₂³	C₂³×D₂₆₉	1	2	yes	—
563	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
564	C₂³	C₂³×D₂₆₉	1	2	yes	—
565	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
566	Q₈	—	2	2	no	—
567	Q₈	—	538	2	no	—
568	Q₈	—	538	4	no	—
569	Q₈	—	538	2	no	—
570	Q₈	—	1076	2	no	—
571	Q₈	—	1076	2	no	—
572	D₄	—	2	4	no	—
573	D₄	—	2	4	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	4	2	no	—
577	D₄	—	538	2	no	—
578	D₄	—	538	4	no	—
579	D₄	—	538	2	no	—
580	D₄	—	1076	2	no	—
581	D₄	#16864401	1	4	yes	complemented
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	269	1	no	—
587	C₂×C₄	—	269	1	no	—
588	C₂×C₄	—	269	1	no	—
589	C₂×C₄	—	269	1	no	—
590	C₂×C₄	—	269	2	no	—
591	C₂×C₄	—	269	1	no	—
592	C₂×C₄	—	269	2	no	—
593	C₂×C₄	—	269	1	no	—
594	C₂×C₄	—	269	2	no	—
595	C₂×C₄	—	269	1	no	—
596	C₂×C₄	—	269	2	no	—
597	C₂×C₄	—	538	2	no	—
598	C₂×C₄	—	538	4	no	—
599	C₂×C₄	—	538	4	no	—
600	C₂×C₄	—	538	4	no	—
601	C₂×C₄	—	538	2	no	—
602	C₂×C₄	—	538	2	no	—
603	C₂×C₄	—	1076	2	no	—
604	C₂×C₄	—	1076	2	no	—
605	C₂×C₄	—	1076	2	no	—
606	C₂×C₄	—	1076	2	no	—
607	C₂×C₄	—	1076	2	no	—
608	C₂×C₄	—	1076	2	no	—
609	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
611	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
612	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
613	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
614	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
615	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
616	C₂²	—	2	1	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	4	8	no	—
627	C₂²	—	538	1	no	—
628	C₂²	—	538	1	no	—
629	C₂²	—	538	1	no	—
630	C₂²	C₂×#16864401	1	4	yes	—
631	C₂²	#22575500	1	2	yes	—
632	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	2	no	—
638	C₄	—	2	1	no	—
639	C₄	—	269	2	no	—
640	C₄	—	538	1	no	—
641	C₄	—	538	2	no	—
642	C₄	—	538	1	no	—
643	C₄	—	538	2	no	—
644	C₄	—	538	2	no	—
645	C₄	—	538	2	no	—
646	C₄	—	538	1	no	—
647	C₄	—	538	1	no	—
648	C₄	—	538	1	no	—
649	C₄	—	538	1	no	—
650	C₄	C₂×#16864401	1	2	yes	—
651	C₂	—	2	2	no	—
652	C₂	—	2	4	no	—
653	C₂	—	4	2	no	—
654	C₂	—	4	1	no	—
655	C₂	—	1076	1	no	—
656	C₂	C₂²×#16864401	1	1	yes	characteristic
657	C₂	H(5,𝔽₂)×D₂₆₉	1	1	yes	characteristic
658	C₂	C₂×#22575500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁵
Fitting	#546×C₂₆₉	C₂
hypercenter	#546	D₂₆₉
hypocenter	C₂₆₉	#3520
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #546

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #31771324	C₂	semidirect
2. #31771313	C₂	semidirect
3. #31771312	C₂	semidirect
4. #31771292	C₂	semidirect
5. #31771286	C₂	semidirect
6. #31771281	C₂	semidirect
7. D₄×Dic₅₃₈	C₂	semidirect
8. #31771261	C₂	semidirect
9. #31771255	C₂	semidirect
10. #31771254	C₂	semidirect
11. #31771249	C₂	semidirect
12. #31771224	C₂	semidirect
13. #31771222	C₂	semidirect
14. #31771200	C₂	semidirect
15. #31771199	C₂	semidirect
16. #31771157	C₂	semidirect
17. #31771135	C₂	semidirect
18. #31771111	C₂	semidirect
19. C₂×#22575430	C₂	semidirect
20. #31771088	C₂	semidirect
21. #31771084	C₂	semidirect
22. C₂×D₄×Dic₂₆₉	C₂	semidirect
23. #546×C₂₆₉	C₂	semidirect
25. C₂×#16864401	C₂²	semidirect
26. C₂²×Dic₅₃₈	C₂²	semidirect
28. #22575453	C₂²	semidirect
29. #22575452	C₂²	semidirect
30. #22575452	C₂²	semidirect
31. #22575451	C₂²	semidirect
32. #22575448	C₂²	semidirect
33. #22575446	C₂²	semidirect
34. #22575446	C₂²	semidirect
35. #22575446	C₂²	semidirect
36. #22575446	C₂²	semidirect
37. #22575446	C₂²	semidirect
38. #22575444	C₂²	semidirect
39. #22575443	C₂²	semidirect
40. #22575443	C₂²	semidirect
41. #22575443	C₂²	semidirect
43. #22575440	C₂²	semidirect
44. #22575439	C₂²	semidirect
45. #22575439	C₂²	semidirect
47. #22575437	C₂²	semidirect
48. #22575436	C₂²	semidirect
49. #22575435	C₂²	semidirect
52. #22575434	C₂²	semidirect
53. #22575434	C₂²	semidirect
54. #22575434	C₂²	semidirect
55. #22575433	C₂²	semidirect
57. #22575432	C₂²	semidirect
58. #22575432	C₂²	semidirect
63. #22575430	C₂²	semidirect
64. #22575430	C₂²	semidirect
65. #22575430	C₂²	semidirect
66. #22575430	C₂²	semidirect
67. #22575430	C₂²	semidirect
68. #22575430	C₂²	semidirect
69. #22575430	C₂²	semidirect
70. #22575429	C₂²	semidirect
73. #22575425	C₂²	semidirect
76. Q₈×Dic₂₆₉	C₂²	semidirect
77. Q₈×Dic₂₆₉	C₂²	semidirect
78. C₄×Dic₅₃₈	C₂²	semidirect
79. C₄×Dic₅₃₈	C₂²	semidirect
80. #22575418	C₂²	semidirect
81. #22575418	C₂²	semidirect
90. D₄×Dic₂₆₉	C₂²	semidirect
91. D₄×Dic₂₆₉	C₂²	semidirect
93. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
94. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
96. #22575413	C₂²	semidirect
97. #22575412	C₂²	semidirect
98. #22575412	C₂²	semidirect
99. #22575410	C₂²	semidirect
101. #22575409	C₂²	semidirect
102. #22575409	C₂²	semidirect
104. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
105. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
106. C₂×#16864381	C₂²	semidirect
107. C₂×#16864381	C₂²	semidirect
108. C₄⋊C₄×D₂₆₉	C₂²	semidirect
274. D₄×C₂₆₉	#41	semidirect
329. C₂₆₉	#3520	semidirect
342. #546	D₂₆₉	semidirect
581. D₄	#16864401	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#31771324	2
#31771313	2
#31771312	2
#31771292	2
#31771286	2
#31771281	2
D₄×Dic₅₃₈	2
#31771261	2
#31771255	2
#31771254	2
#31771249	2
#31771224	2
#31771222	2
#31771200	2
#31771199	2
#31771157	2
#31771135	2
#31771111	2
C₂×#22575430	2
#31771088	2
#31771084	2
C₂×D₄×Dic₂₆₉	2
#546×C₂₆₉	2
#3520	269