Subgroups — C₂×#31771188 #46824221

Order 34432.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771188	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575448	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22575447	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#22575441	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22575439	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22575438	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22575436	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#22575430	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#22575427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22575426	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#22575425	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#22575423	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#22575421	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₂²⋊D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#16864379	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂⁴×D₂₆₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
20	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic, complemented
25	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₂₆₉	C₂²	1	1	yes	characteristic, complemented
27	#22575448	C₂²	1	8	yes	complemented
28	#22575447	C₂²	1	8	yes	complemented
29	#22575441	C₂²	1	8	yes	complemented
30	#22575439	C₂²	1	8	yes	complemented
31	#22575438	C₂²	1	8	yes	complemented
32	#22575436	C₂²	1	8	yes	complemented
33	#22575430	C₂²	1	8	yes	complemented
34	#22575427	C₂²	1	8	yes	complemented
35	#22575426	C₂²	1	8	yes	complemented
36	#22575425	C₂²	1	8	yes	complemented
37	#22575423	C₂²	1	8	yes	complemented
38	#22575421	C₂²	1	8	yes	complemented
39	C₄×C₂²⋊D₂₆₉	C₂²	1	8	yes	complemented
40	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
41	C₂×#16864381	C₂²	1	1	yes	characteristic, complemented
42	C₂×#16864381	C₂²	1	1	yes	characteristic, complemented
43	C₂×#16864381	C₂²	1	1	yes	characteristic, complemented
44	C₂×#16864381	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
47	C₂×#16864379	C₂²	1	1	yes	characteristic
48	C₂×#16864379	C₂²	1	1	yes	characteristic
49	C₂×#16864379	C₂²	1	1	yes	characteristic
50	C₂×#16864379	C₂²	1	1	yes	characteristic
51	C₂×#16864379	C₂²	1	1	yes	characteristic
52	C₂×#16864379	C₂²	1	1	yes	characteristic
53	C₂×#16864379	C₂²	1	1	yes	characteristic
54	C₂×#16864379	C₂²	1	1	yes	characteristic
55	C₂×#16864379	C₂²	1	8	yes	complemented
56	C₂×#16864379	C₂²	1	1	yes	characteristic, complemented
57	C₂×#16864379	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
59	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
60	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
61	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
62	C₄×D₄×C₂₆₉	C₂²	1	8	yes	complemented
63	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
64	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
67	C₂³×D₂₆₉	—	2	4	no	—
68	C₂³×D₂₆₉	—	4	2	no	—
69	C₂³×D₂₆₉	—	4	2	no	—
70	C₂³×D₂₆₉	C₂³	1	1	yes	characteristic
71	C₂³×D₂₆₉	C₂³	1	1	yes	characteristic
72	C₂³×D₂₆₉	C₂³	1	1	yes	characteristic
73	C₂³×D₂₆₉	C₂³	1	4	yes	—
74	C₂×Dic₅₃₈	—	4	2	no	—
75	C₂×Dic₅₃₈	C₂³	1	4	yes	complemented
76	C₂×D₁₀₇₆	—	4	2	no	—
77	C₂×D₁₀₇₆	—	4	2	no	—
78	C₂×D₁₀₇₆	C₂³	1	4	yes	—
79	C₂×D₁₀₇₆	C₂³	1	4	yes	—
80	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
81	C₂×C₂²⋊D₂₆₉	—	4	2	no	—
82	C₂×C₂²⋊D₂₆₉	—	4	2	no	—
83	C₂×C₂²⋊D₂₆₉	—	4	2	no	—
84	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	—
85	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	—
86	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	—
87	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	complemented
88	C₂²×Dic₂₆₉	—	4	2	no	—
89	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
90	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
91	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
92	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
93	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
94	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
95	C₂×C₄×D₂₆₉	—	4	2	no	—
96	C₂×C₄×D₂₆₉	C₂³	1	4	yes	complemented
97	C₄⋊Dic₂₆₉	C₂³	1	4	yes	complemented
98	#16864381	C₂³	1	4	yes	complemented
99	#16864381	C₂³	1	4	yes	complemented
100	#16864381	C₂³	1	4	yes	complemented
101	#16864381	C₂³	1	4	yes	complemented
102	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
103	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	complemented
104	#16864379	C₂³	1	4	yes	—
105	#16864379	C₂³	1	4	yes	—
106	#16864379	C₂³	1	4	yes	—
107	#16864379	C₂³	1	4	yes	—
108	#16864379	C₂³	1	4	yes	—
109	#16864379	C₂³	1	4	yes	—
110	#16864379	C₂³	1	4	yes	—
111	#16864379	C₂³	1	4	yes	—
112	#16864379	C₂³	1	4	yes	complemented
113	#16864379	C₂³	1	4	yes	complemented
114	C₄×Dic₂₆₉	C₂³	1	4	yes	complemented
115	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
116	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
117	C₂×D₄×C₂₆₉	—	2	4	no	—
118	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
119	C₂²×C₄×C₂₆₉	—	4	2	no	—
120	C₂²×C₄×C₂₆₉	—	4	2	no	—
121	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
122	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
123	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
124	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
125	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
127	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	complemented
128	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	complemented
129	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	—
130	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	complemented
131	C₄²×C₂₆₉	C₂³	1	4	yes	complemented
132	C₂²×D₂₆₉	—	2	2	no	—
133	C₂²×D₂₆₉	—	2	8	no	—
134	C₂²×D₂₆₉	—	2	2	no	—
135	C₂²×D₂₆₉	—	2	2	no	—
136	C₂²×D₂₆₉	—	2	2	no	—
137	C₂²×D₂₆₉	—	2	2	no	—
138	C₂²×D₂₆₉	—	2	2	no	—
139	C₂²×D₂₆₉	—	2	2	no	—
140	C₂²×D₂₆₉	—	2	2	no	—
141	C₂²×D₂₆₉	—	2	2	no	—
142	C₂²×D₂₆₉	—	4	4	no	—
143	C₂²×D₂₆₉	—	4	4	no	—
144	C₂²×D₂₆₉	—	4	16	no	—
145	C₂²×D₂₆₉	C₂⁴	1	2	yes	—
146	C₂²×D₂₆₉	C₂⁴	1	2	yes	—
147	C₂²×D₂₆₉	C₂⁴	1	2	yes	—
148	Dic₅₃₈	—	4	4	no	—
149	D₁₀₇₆	—	4	4	no	—
150	D₁₀₇₆	—	4	4	no	—
151	C₂²⋊D₂₆₉	—	2	8	no	—
152	C₂²⋊D₂₆₉	—	4	4	no	—
153	C₂²⋊D₂₆₉	—	4	4	no	—
154	C₂²⋊D₂₆₉	—	4	4	no	—
155	C₂×Dic₂₆₉	—	2	2	no	—
156	C₂×Dic₂₆₉	—	2	2	no	—
157	C₂×Dic₂₆₉	—	2	2	no	—
158	C₂×Dic₂₆₉	—	2	2	no	—
159	C₂×Dic₂₆₉	—	2	2	no	—
160	C₂×Dic₂₆₉	—	4	4	no	—
161	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
162	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
163	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
164	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
165	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
166	C₄×D₂₆₉	—	4	4	no	—
167	C₂³×C₂₆₉	—	2	2	no	—
168	C₂³×C₂₆₉	—	2	2	no	—
169	C₂³×C₂₆₉	—	2	2	no	—
170	C₂³×C₂₆₉	—	2	2	no	—
171	C₂³×C₂₆₉	—	2	2	no	—
172	C₂³×C₂₆₉	#41	1	4	yes	—
173	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
174	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
175	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
176	D₄×C₂₆₉	—	2	8	no	—
177	C₂×C₄×C₂₆₉	—	2	2	no	—
178	C₂×C₄×C₂₆₉	—	2	2	no	—
179	C₂×C₄×C₂₆₉	—	2	2	no	—
180	C₂×C₄×C₂₆₉	—	2	2	no	—
181	C₂×C₄×C₂₆₉	—	2	2	no	—
182	C₂×C₄×C₂₆₉	—	4	4	no	—
183	C₂×C₄×C₂₆₉	—	4	4	no	—
184	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
185	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
186	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
187	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
188	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
189	C₂²×C₂₆₉	—	2	8	no	—
190	C₂²×C₂₆₉	—	2	2	no	—
191	C₂²×C₂₆₉	—	2	2	no	—
192	C₂²×C₂₆₉	—	2	2	no	—
193	C₂²×C₂₆₉	—	2	2	no	—
194	C₂²×C₂₆₉	—	2	2	no	—
195	C₂²×C₂₆₉	—	4	4	no	—
196	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
197	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
198	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
199	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
200	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
201	C₂×D₂₆₉	—	2	2	no	—
202	C₂×D₂₆₉	—	2	2	no	—
203	C₂×D₂₆₉	—	2	2	no	—
204	C₂×D₂₆₉	—	2	2	no	—
205	C₂×D₂₆₉	—	2	2	no	—
206	C₂×D₂₆₉	—	2	2	no	—
207	C₂×D₂₆₉	—	2	2	no	—
208	C₂×D₂₆₉	—	2	2	no	—
209	C₂×D₂₆₉	—	2	2	no	—
210	C₂×D₂₆₉	—	4	16	no	—
211	C₂×D₂₆₉	—	4	4	no	—
212	C₂×D₂₆₉	—	4	4	no	—
213	C₂×D₂₆₉	—	4	4	no	—
214	C₄×C₂₆₉	—	2	2	no	—
215	C₄×C₂₆₉	—	2	2	no	—
216	C₄×C₂₆₉	—	2	2	no	—
217	C₄×C₂₆₉	—	2	2	no	—
218	C₄×C₂₆₉	—	2	2	no	—
219	Dic₂₆₉	—	2	2	no	—
220	Dic₂₆₉	—	2	2	no	—
221	Dic₂₆₉	—	2	2	no	—
222	Dic₂₆₉	—	2	2	no	—
223	Dic₂₆₉	—	2	2	no	—
224	C₂×C₂₆₉	—	2	4	no	—
225	C₂×C₂₆₉	—	4	2	no	—
226	C₂×C₂₆₉	#538	1	4	yes	complemented
227	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
228	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
230	D₂₆₉	—	4	2	no	—
231	D₂₆₉	—	4	2	no	—
232	D₂₆₉	—	4	2	no	—
233	C₂₆₉	C₂×#538	1	1	yes	characteristic, complemented, Sylow
234	C₂×#538	—	269	1	no	maximal, Sylow
235	#538	—	269	16	no	—
236	C₂×#126	—	269	1	no	—
237	C₂×#126	—	269	1	no	—
238	C₂×#124	—	269	1	no	—
239	C₂×#124	—	269	1	no	—
240	C₂×#123	—	269	1	no	—
241	C₂×#123	—	269	1	no	—
242	C₂×C₂²⋊Q₈	—	269	1	no	—
243	C₂×#121	—	269	1	no	—
244	C₂×#121	—	269	1	no	—
245	C₂×#121	—	269	1	no	—
246	C₂×#120	—	269	1	no	—
247	C₂×#120	—	269	1	no	—
248	C₂×C₄×D₄	—	269	1	no	—
249	C₂×C₄×D₄	D₂₆₉	1	1	yes	characteristic, complemented
250	C₂²×C₂²⋊C₄	—	269	1	no	—
251	C₂⁵	—	269	1	no	—
252	C₂²×Q₈	—	269	1	no	—
253	C₂²×D₄	—	269	1	no	—
254	C₂²×D₄	—	269	1	no	—
255	C₂²×D₄	—	269	1	no	—
256	C₂²×D₄	—	269	1	no	—
257	C₂²×D₄	—	269	1	no	—
258	C₂²×D₄	—	269	1	no	—
259	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
260	C₂³×C₄	—	269	1	no	—
261	C₂³×C₄	—	269	1	no	—
262	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
263	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
264	#126	—	269	8	no	—
265	#126	—	269	8	no	—
266	#124	—	269	8	no	—
267	#124	—	269	8	no	—
268	#123	—	269	8	no	—
269	#123	—	269	8	no	—
270	C₂²⋊Q₈	—	269	8	no	—
271	#121	—	269	8	no	—
272	#121	—	269	8	no	—
273	#121	—	269	8	no	—
274	#120	—	269	8	no	—
275	#120	—	269	8	no	—
276	C₄×D₄	—	269	8	no	—
277	C₄×D₄	C₂×D₂₆₉	1	8	yes	complemented
278	C₂×C₄⋊C₄	—	269	1	no	—
279	C₂×C₄⋊C₄	—	269	1	no	—
280	C₂×C₄⋊C₄	—	269	1	no	—
281	C₂×C₄⋊C₄	—	269	1	no	—
282	C₂×C₄⋊C₄	—	269	1	no	—
283	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
284	C₂×C₂²⋊C₄	—	269	1	no	—
285	C₂×C₂²⋊C₄	—	269	1	no	—
286	C₂×C₂²⋊C₄	—	269	1	no	—
287	C₂×C₂²⋊C₄	—	269	1	no	—
288	C₂×C₂²⋊C₄	—	269	1	no	—
289	C₂×C₂²⋊C₄	—	269	1	no	—
290	C₂×C₂²⋊C₄	—	269	8	no	—
291	C₂×C₂²⋊C₄	—	269	1	no	—
292	C₂×C₂²⋊C₄	—	269	1	no	—
293	C₂×C₂²⋊C₄	—	269	1	no	—
294	C₂×C₂²⋊C₄	—	269	1	no	—
295	C₂×C₂²⋊C₄	—	269	1	no	—
296	C₂×C₂²⋊C₄	—	269	1	no	—
297	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
299	C₂×C₄²	—	269	1	no	—
300	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic, complemented
301	C₂⁴	—	269	1	no	—
302	C₂⁴	—	269	1	no	—
303	C₂⁴	—	269	1	no	—
304	C₂⁴	—	269	4	no	—
305	C₂⁴	—	538	4	no	—
306	C₂⁴	—	1076	2	no	—
307	C₂⁴	—	1076	2	no	—
308	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
309	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
310	C₂×Q₈	—	269	4	no	—
311	C₂×Q₈	—	1076	2	no	—
312	C₂×D₄	—	2	4	no	—
313	C₂×D₄	—	269	4	no	—
314	C₂×D₄	—	269	4	no	—
315	C₂×D₄	—	269	4	no	—
316	C₂×D₄	—	269	4	no	—
317	C₂×D₄	—	269	4	no	—
318	C₂×D₄	—	269	4	no	—
319	C₂×D₄	—	538	4	no	—
320	C₂×D₄	—	1076	2	no	—
321	C₂×D₄	—	1076	2	no	—
322	C₂×D₄	—	1076	2	no	—
323	C₂×D₄	—	1076	2	no	—
324	C₂×D₄	—	1076	2	no	—
325	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
326	C₂²×C₄	—	4	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	269	1	no	—
329	C₂²×C₄	—	269	1	no	—
330	C₂²×C₄	—	269	1	no	—
331	C₂²×C₄	—	269	1	no	—
332	C₂²×C₄	—	269	1	no	—
333	C₂²×C₄	—	269	4	no	—
334	C₂²×C₄	—	269	4	no	—
335	C₂²×C₄	—	1076	2	no	—
336	C₂²×C₄	—	1076	2	no	—
337	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
338	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
339	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
340	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
341	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
342	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
343	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	complemented
344	C₄⋊C₄	—	269	4	no	—
345	C₄⋊C₄	—	269	4	no	—
346	C₄⋊C₄	—	269	4	no	—
347	C₄⋊C₄	—	269	4	no	—
348	C₄⋊C₄	—	269	4	no	—
349	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	complemented
350	C₂²⋊C₄	—	269	4	no	—
351	C₂²⋊C₄	—	269	4	no	—
352	C₂²⋊C₄	—	269	4	no	—
353	C₂²⋊C₄	—	269	4	no	—
354	C₂²⋊C₄	—	269	4	no	—
355	C₂²⋊C₄	—	269	4	no	—
356	C₂²⋊C₄	—	269	4	no	—
357	C₂²⋊C₄	—	269	4	no	—
358	C₂²⋊C₄	—	269	4	no	—
359	C₂²⋊C₄	—	269	4	no	—
360	C₂²⋊C₄	—	269	4	no	—
361	C₂²⋊C₄	—	269	4	no	—
362	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	—
363	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	complemented
364	C₄²	—	269	4	no	—
365	C₄²	C₂²×D₂₆₉	1	4	yes	complemented
366	C₂³	—	2	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	269	2	no	—
372	C₂³	—	269	2	no	—
373	C₂³	—	269	2	no	—
374	C₂³	—	538	2	no	—
375	C₂³	—	538	8	no	—
376	C₂³	—	538	2	no	—
377	C₂³	—	538	2	no	—
378	C₂³	—	538	2	no	—
379	C₂³	—	538	2	no	—
380	C₂³	—	538	2	no	—
381	C₂³	—	538	2	no	—
382	C₂³	—	538	2	no	—
383	C₂³	—	538	2	no	—
384	C₂³	—	1076	4	no	—
385	C₂³	—	1076	4	no	—
386	C₂³	—	1076	16	no	—
387	C₂³	#16864402	1	4	yes	—
388	C₂³	C₂³×D₂₆₉	1	2	yes	—
389	C₂³	C₂³×D₂₆₉	1	2	yes	—
390	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
391	Q₈	—	1076	4	no	—
392	D₄	—	2	8	no	—
393	D₄	—	538	8	no	—
394	D₄	—	1076	4	no	—
395	D₄	—	1076	4	no	—
396	D₄	—	1076	4	no	—
397	D₄	—	1076	4	no	—
398	D₄	—	1076	4	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	269	2	no	—
407	C₂×C₄	—	269	2	no	—
408	C₂×C₄	—	269	2	no	—
409	C₂×C₄	—	269	2	no	—
410	C₂×C₄	—	269	2	no	—
411	C₂×C₄	—	538	2	no	—
412	C₂×C₄	—	538	2	no	—
413	C₂×C₄	—	538	2	no	—
414	C₂×C₄	—	538	2	no	—
415	C₂×C₄	—	538	2	no	—
416	C₂×C₄	—	1076	4	no	—
417	C₂×C₄	—	1076	4	no	—
418	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
419	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
420	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
421	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
422	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	8	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	538	2	no	—
431	C₂²	—	538	2	no	—
432	C₂²	—	538	2	no	—
433	C₂²	—	538	2	no	—
434	C₂²	—	538	2	no	—
435	C₂²	—	538	2	no	—
436	C₂²	—	538	2	no	—
437	C₂²	—	538	2	no	—
438	C₂²	—	538	2	no	—
439	C₂²	—	1076	16	no	—
440	C₂²	—	1076	4	no	—
441	C₂²	—	1076	4	no	—
442	C₂²	—	1076	4	no	—
443	C₂²	C₂×#16864402	1	4	yes	—
444	C₂²	C₂×#16864402	1	2	yes	—
445	C₂²	#22575498	1	2	yes	—
446	C₂²	#22575499	1	2	yes	—
447	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	538	2	no	—
454	C₄	—	538	2	no	—
455	C₄	—	538	2	no	—
456	C₄	—	538	2	no	—
457	C₄	—	538	2	no	—
458	C₂	—	2	4	no	—
459	C₂	—	4	2	no	—
460	C₂	—	1076	2	no	—
461	C₂	—	1076	2	no	—
462	C₂	—	1076	2	no	—
463	C₂	#31771188	1	4	yes	direct product factor
464	C₂	C₂²×#16864402	1	1	yes	characteristic
465	C₂	C₂×#22575498	1	1	yes	characteristic
466	C₂	C₂×#22575499	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×C₄×D₄×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₆₉
hypocenter	C₂₆₉	C₂×#538
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #31771188	C₂	direct (N × H)
2. C₂×#22575448	C₂	semidirect
3. C₂×#22575447	C₂	semidirect
4. C₂×#22575441	C₂	semidirect
5. C₂×#22575439	C₂	semidirect
6. C₂×#22575438	C₂	semidirect
7. C₂×#22575436	C₂	semidirect
8. C₂×#22575430	C₂	semidirect
9. C₂×#22575427	C₂	semidirect
10. C₂×#22575426	C₂	semidirect
11. C₂×#22575425	C₂	semidirect
12. C₂×#22575423	C₂	semidirect
13. C₂×#22575421	C₂	semidirect
14. C₂×C₄×C₂²⋊D₂₆₉	C₂	semidirect
15. C₂²×#16864379	C₂	semidirect
16. C₂×C₄×D₄×C₂₆₉	C₂	semidirect
18. C₂²×Dic₅₃₈	C₂²	semidirect
24. C₂²×C₂²⋊D₂₆₉	C₂²	semidirect
26. C₂²×C₄×D₂₆₉	C₂²	semidirect
27. #22575448	C₂²	semidirect
28. #22575447	C₂²	semidirect
29. #22575441	C₂²	semidirect
30. #22575439	C₂²	semidirect
31. #22575438	C₂²	semidirect
32. #22575436	C₂²	semidirect
33. #22575430	C₂²	semidirect
34. #22575427	C₂²	semidirect
35. #22575426	C₂²	semidirect
36. #22575425	C₂²	semidirect
37. #22575423	C₂²	semidirect
38. #22575421	C₂²	semidirect
39. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
40. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
41. C₂×#16864381	C₂²	semidirect
42. C₂×#16864381	C₂²	semidirect
43. C₂×#16864381	C₂²	semidirect
44. C₂×#16864381	C₂²	semidirect
46. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
55. C₂×#16864379	C₂²	semidirect
56. C₂×#16864379	C₂²	semidirect
57. C₂×#16864379	C₂²	semidirect
58. C₂×C₄×Dic₂₆₉	C₂²	semidirect
61. C₂³×C₄×C₂₆₉	C₂²	semidirect
62. C₄×D₄×C₂₆₉	C₂²	semidirect
63. C₂×C₄⋊C₄×C₂₆₉	C₂²	semidirect
65. C₂×C₂²⋊C₄×C₂₆₉	C₂²	semidirect
66. C₂×C₄²×C₂₆₉	C₂²	semidirect
75. C₂×Dic₅₃₈	C₂³	semidirect
87. C₂×C₂²⋊D₂₆₉	C₂³	semidirect
96. C₂×C₄×D₂₆₉	C₂³	semidirect
97. C₄⋊Dic₂₆₉	C₂³	semidirect
98. #16864381	C₂³	semidirect
99. #16864381	C₂³	semidirect
100. #16864381	C₂³	semidirect
101. #16864381	C₂³	semidirect
103. C₂²⋊Dic₂₆₉	C₂³	semidirect
112. #16864379	C₂³	semidirect
113. #16864379	C₂³	semidirect
114. C₄×Dic₂₆₉	C₂³	semidirect
127. C₂²×C₄×C₂₆₉	C₂³	semidirect
128. C₄⋊C₄×C₂₆₉	C₂³	semidirect
130. C₂²⋊C₄×C₂₆₉	C₂³	semidirect
131. C₄²×C₂₆₉	C₂³	semidirect
226. C₂×C₂₆₉	#538	semidirect
233. C₂₆₉	C₂×#538	semidirect
249. C₂×C₄×D₄	D₂₆₉	semidirect
263. C₂³×C₄	C₂×D₂₆₉	semidirect
277. C₄×D₄	C₂×D₂₆₉	semidirect
283. C₂×C₄⋊C₄	C₂×D₂₆₉	semidirect
298. C₂×C₂²⋊C₄	C₂×D₂₆₉	semidirect
300. C₂×C₄²	C₂×D₂₆₉	semidirect
343. C₂²×C₄	C₂²×D₂₆₉	semidirect
349. C₄⋊C₄	C₂²×D₂₆₉	semidirect
363. C₂²⋊C₄	C₂²×D₂₆₉	semidirect
365. C₄²	C₂²×D₂₆₉	semidirect
463. C₂	#31771188	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771188	2
C₂×#22575448	2
C₂×#22575447	2
C₂×#22575441	2
C₂×#22575439	2
C₂×#22575438	2
C₂×#22575436	2
C₂×#22575430	2
C₂×#22575427	2
C₂×#22575426	2
C₂×#22575425	2
C₂×#22575423	2
C₂×#22575421	2
C₂×C₄×C₂²⋊D₂₆₉	2
C₂²×#16864379	2
C₂×C₄×D₄×C₂₆₉	2
C₂×#538	269

Subgroups — C2×#31771188 #46824221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771188 #46824221