Subgroups — #46824554

Order 34432.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575499	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771274	C₂	1	2	yes	maximal, complemented
3	#31771266	C₂	1	2	yes	maximal, complemented
4	#31771243	C₂	1	6	yes	maximal, complemented
5	D₄×C₂²⋊D₂₆₉	C₂	1	6	yes	maximal, complemented
6	#31771183	C₂	1	3	yes	maximal, complemented
7	#31771169	C₂	1	3	yes	maximal, complemented
8	#31771116	C₂	1	3	yes	maximal, complemented
9	C₂×#22575427	C₂	1	3	yes	maximal, complemented
10	#31771088	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#22575499	—	2	8	no	—
13	C₂×#16864403	C₂²	1	2	yes	complemented
14	C₂×#16864402	C₂²	1	3	yes	complemented
15	C₂²×D₁₀₇₆	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊D₂₆₉	C₂²	1	6	yes	complemented
17	C₂×D₄×D₂₆₉	C₂²	1	6	yes	complemented
18	#22575449	C₂²	1	6	yes	complemented
19	#22575442	C₂²	1	6	yes	complemented
20	#22575436	C₂²	1	6	yes	complemented
21	#22575432	C₂²	1	3	yes	complemented
22	#22575432	C₂²	1	1	yes	characteristic, complemented
23	#22575429	C₂²	1	6	yes	complemented
24	#22575427	C₂²	1	3	yes	complemented
25	#22575427	C₂²	1	6	yes	complemented
26	#22575427	C₂²	1	6	yes	complemented
27	#22575427	C₂²	1	6	yes	complemented
28	#22575427	C₂²	1	3	yes	complemented
29	#22575425	C₂²	1	12	yes	complemented
30	#22575424	C₂²	1	6	yes	complemented
31	#22575422	C₂²	1	6	yes	complemented
32	#22575421	C₂²	1	12	yes	complemented
33	Q₈×Dic₂₆₉	C₂²	1	2	yes	complemented
34	D₄×Dic₂₆₉	C₂²	1	6	yes	complemented
35	C₄×C₂²⋊D₂₆₉	C₂²	1	6	yes	complemented
36	C₄×C₂²⋊D₂₆₉	C₂²	1	6	yes	complemented
37	#22575412	C₂²	1	3	yes	complemented
38	C₂×C₄⋊Dic₂₆₉	C₂²	1	3	yes	complemented
39	C₂×#16864379	C₂²	1	6	yes	complemented
40	C₂×#41×C₂₆₉	—	2	8	no	—
41	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
42	C₂×#41×C₂₆₉	C₂²	1	3	yes	complemented
43	C₂×#41×C₂₆₉	C₂²	1	3	yes	complemented
44	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
45	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
46	C₂²×D₄×C₂₆₉	C₂²	1	3	yes	complemented
47	C₂³×C₄×C₂₆₉	C₂²	1	3	yes	complemented
48	C₂³×D₂₆₉	C₂³	1	6	yes	—
49	#16864403	—	2	8	no	—
50	#16864402	—	2	12	no	—
51	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
52	C₂×D₁₀₇₆	—	2	12	no	—
53	C₂×D₁₀₇₆	C₂³	1	3	yes	—
54	C₂×D₁₀₇₆	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₂₆₉	—	4	12	no	—
56	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₂₆₉	C₂³	1	12	yes	—
60	D₄×D₂₆₉	—	2	24	no	—
61	C₂²×Dic₂₆₉	C₂³	1	6	yes	—
62	C₂×C₄×D₂₆₉	C₂³	1	6	yes	—
63	C₄⋊Dic₂₆₉	C₂³	1	3	yes	—
64	C₄⋊Dic₂₆₉	C₂³	1	6	yes	—
65	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
66	#16864381	C₂³	1	2	yes	—
67	#16864381	C₂³	1	6	yes	—
68	C₂²⋊Dic₂₆₉	C₂³	1	6	yes	—
69	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
70	C₂²⋊Dic₂₆₉	C₂³	1	6	yes	—
71	#16864379	C₂³	1	6	yes	—
72	#16864379	C₂³	1	12	yes	—
73	#16864379	C₂³	1	6	yes	—
74	C₄×Dic₂₆₉	C₂³	1	6	yes	—
75	C₂⁴×C₂₆₉	C₂³	1	3	yes	—
76	#41×C₂₆₉	—	2	4	no	—
77	#41×C₂₆₉	—	2	12	no	—
78	#41×C₂₆₉	—	2	12	no	—
79	#41×C₂₆₉	D₄	1	8	yes	complemented
80	C₂×Q₈×C₂₆₉	—	2	4	no	—
81	C₂×Q₈×C₂₆₉	C₂³	1	3	yes	—
82	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
83	C₂×D₄×C₂₆₉	—	2	12	no	—
84	C₂×D₄×C₂₆₉	C₂³	1	6	yes	—
85	C₂×D₄×C₂₆₉	C₂³	1	3	yes	—
86	C₂×D₄×C₂₆₉	C₂³	1	3	yes	—
87	C₂²×C₄×C₂₆₉	—	2	12	no	—
88	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
89	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
90	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
91	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
93	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
94	C₂²×D₂₆₉	—	2	12	no	—
95	C₂²×D₂₆₉	—	4	6	no	—
96	C₂²×D₂₆₉	—	4	6	no	—
97	C₂²×D₂₆₉	C₂⁴	1	6	yes	—
98	Dic₅₃₈	—	2	2	no	—
99	D₁₀₇₆	—	2	12	no	—
100	D₁₀₇₆	—	2	6	no	—
101	C₂²⋊D₂₆₉	—	2	12	no	—
102	C₂²⋊D₂₆₉	—	2	12	no	—
103	C₂²⋊D₂₆₉	—	4	12	no	—
104	C₂²⋊D₂₆₉	—	4	6	no	—
105	C₂×Dic₂₆₉	—	4	6	no	—
106	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
107	C₂×Dic₂₆₉	C₂⁴	1	6	yes	—
108	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
109	C₄×D₂₆₉	—	2	12	no	—
110	C₂³×C₂₆₉	—	2	6	no	—
111	C₂³×C₂₆₉	—	2	6	no	—
112	C₂³×C₂₆₉	—	4	3	no	—
113	C₂³×C₂₆₉	C₂⁴	1	3	yes	—
114	C₂³×C₂₆₉	C₂⁴	1	3	yes	—
115	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
116	Q₈×C₂₆₉	—	2	6	no	—
117	Q₈×C₂₆₉	C₂×D₄	1	4	yes	complemented
118	D₄×C₂₆₉	—	2	12	no	—
119	D₄×C₂₆₉	—	2	6	no	—
120	D₄×C₂₆₉	C₂×D₄	1	12	yes	complemented
121	C₂×C₄×C₂₆₉	—	2	6	no	—
122	C₂×C₄×C₂₆₉	—	2	2	no	—
123	C₂×C₄×C₂₆₉	—	2	6	no	—
124	C₂×C₄×C₂₆₉	—	2	6	no	—
125	C₂×C₄×C₂₆₉	—	2	6	no	—
126	C₂×C₄×C₂₆₉	C₂×D₄	1	12	yes	complemented
127	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
129	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
131	C₂²×C₂₆₉	—	2	1	no	—
132	C₂²×C₂₆₉	—	2	3	no	—
133	C₂²×C₂₆₉	—	2	1	no	—
134	C₂²×C₂₆₉	—	2	3	no	—
135	C₂²×C₂₆₉	—	2	3	no	—
136	C₂²×C₂₆₉	—	2	3	no	—
137	C₂²×C₂₆₉	—	2	3	no	—
138	C₂²×C₂₆₉	—	4	12	no	—
139	C₂²×C₂₆₉	C₂²×D₄	1	6	yes	—
140	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	complemented
141	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
142	C₂×D₂₆₉	—	2	6	no	—
143	C₂×D₂₆₉	—	2	6	no	—
144	C₂×D₂₆₉	—	2	6	no	—
145	C₂×D₂₆₉	—	8	12	no	—
146	C₄×C₂₆₉	—	2	1	no	—
147	C₄×C₂₆₉	—	2	3	no	—
148	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
149	C₄×C₂₆₉	C₂²×D₄	1	6	yes	—
150	Dic₂₆₉	—	2	2	no	—
151	Dic₂₆₉	—	2	6	no	—
152	Dic₂₆₉	—	2	2	no	—
153	C₂×C₂₆₉	—	2	6	no	—
154	C₂×C₂₆₉	—	2	2	no	—
155	C₂×C₂₆₉	—	4	3	no	—
156	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
157	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₂₆₉	#585	1	1	yes	characteristic
159	D₂₆₉	—	4	6	no	—
160	C₂₆₉	#3488	1	1	yes	characteristic, complemented, Sylow
161	#3488	—	269	1	no	maximal, Sylow
162	C₂×H(5,𝔽₂)	—	269	1	no	—
163	C₂²×#41	D₂₆₉	1	1	yes	characteristic, complemented
164	#550	—	269	2	no	—
165	#548	—	269	2	no	—
166	#546	—	269	6	no	—
167	D₄²	—	269	6	no	—
168	#537	—	269	3	no	—
169	#535	—	269	3	no	—
170	#525	—	269	3	no	—
171	C₂×#121	—	269	3	no	—
172	#520	—	269	1	no	—
173	H(5,𝔽₂)	—	538	8	no	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	269	2	no	—
176	C₂×#41	—	269	3	no	—
177	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
178	C₂×#41	C₂×D₂₆₉	1	3	yes	complemented
179	C₂×#41	C₂×D₂₆₉	1	3	yes	complemented
180	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic, complemented
181	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
182	C₂²×D₄	—	269	6	no	—
183	C₂²×D₄	—	269	6	no	—
184	C₂²×D₄	—	269	3	no	—
185	C₂²×D₄	C₂×D₂₆₉	1	3	yes	complemented
186	C₂³×C₄	C₂×D₂₆₉	1	3	yes	complemented
187	#127	—	269	6	no	—
188	#124	—	269	6	no	—
189	#123	—	269	6	no	—
190	C₂²⋊Q₈	—	269	6	no	—
191	C₂²⋊Q₈	—	269	1	no	—
192	C₂²⋊Q₈	—	269	3	no	—
193	#121	—	269	12	no	—
194	#121	—	269	6	no	—
195	#121	—	269	3	no	—
196	#121	—	269	6	no	—
197	#121	—	269	6	no	—
198	#121	—	269	6	no	—
199	#121	—	269	3	no	—
200	#120	—	269	6	no	—
201	#120	—	269	12	no	—
202	C₄×Q₈	—	269	2	no	—
203	C₄×D₄	—	269	6	no	—
204	C₄×D₄	—	269	6	no	—
205	C₄×D₄	—	269	6	no	—
206	#117	—	269	3	no	—
207	C₂×C₄⋊C₄	—	269	3	no	—
208	C₂×C₂²⋊C₄	—	269	6	no	—
209	C₂⁴	—	269	6	no	—
210	C₂⁴	C₂²×D₂₆₉	1	3	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	12	no	—
213	#41	—	2	12	no	—
214	#41	—	538	8	no	—
215	#41	—	538	12	no	—
216	#41	C₂²⋊D₂₆₉	1	8	yes	complemented
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	269	1	no	—
219	C₂×Q₈	C₂²×D₂₆₉	1	3	yes	—
220	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	—	269	6	no	—
223	C₂×D₄	—	269	6	no	—
224	C₂×D₄	—	269	6	no	—
225	C₂×D₄	—	269	12	no	—
226	C₂×D₄	—	269	3	no	—
227	C₂×D₄	—	269	6	no	—
228	C₂×D₄	—	538	24	no	—
229	C₂×D₄	—	538	12	no	—
230	C₂×D₄	—	1076	12	no	—
231	C₂×D₄	C₂²×D₂₆₉	1	6	yes	—
232	C₂×D₄	C₂²×D₂₆₉	1	3	yes	—
233	C₂×D₄	C₂²×D₂₆₉	1	3	yes	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	269	6	no	—
236	C₂²×C₄	—	269	6	no	—
237	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
238	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
239	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
240	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
242	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
243	C₄⋊C₄	—	269	3	no	—
244	C₄⋊C₄	—	269	6	no	—
245	C₄⋊C₄	—	269	1	no	—
246	C₄⋊C₄	—	269	2	no	—
247	C₄⋊C₄	—	269	6	no	—
248	C₂²⋊C₄	—	269	6	no	—
249	C₂²⋊C₄	—	269	2	no	—
250	C₂²⋊C₄	—	269	6	no	—
251	C₂²⋊C₄	—	269	6	no	—
252	C₂²⋊C₄	—	269	12	no	—
253	C₂²⋊C₄	—	269	6	no	—
254	C₄²	—	269	6	no	—
255	C₂³	—	2	6	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	4	3	no	—
258	C₂³	—	269	6	no	—
259	C₂³	—	538	12	no	—
260	C₂³	—	1076	6	no	—
261	C₂³	—	1076	6	no	—
262	C₂³	C₂³×D₂₆₉	1	3	yes	—
263	C₂³	C₂³×D₂₆₉	1	3	yes	—
264	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
265	Q₈	—	2	6	no	—
266	Q₈	—	538	2	no	—
267	Q₈	C₂×C₂²⋊D₂₆₉	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	6	no	—
270	D₄	—	538	12	no	—
271	D₄	—	538	12	no	—
272	D₄	—	538	6	no	—
273	D₄	—	538	12	no	—
274	D₄	—	1076	12	no	—
275	D₄	—	1076	6	no	—
276	D₄	C₂×C₂²⋊D₂₆₉	1	12	yes	complemented
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	269	2	no	—
283	C₂×C₄	—	269	6	no	—
284	C₂×C₄	—	269	2	no	—
285	C₂×C₄	—	538	12	no	—
286	C₂×C₄	—	1076	6	no	—
287	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	12	yes	complemented
288	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
290	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
291	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	3	no	—
296	C₂²	—	2	3	no	—
297	C₂²	—	2	3	no	—
298	C₂²	—	2	3	no	—
299	C₂²	—	4	12	no	—
300	C₂²	—	538	6	no	—
301	C₂²	—	538	6	no	—
302	C₂²	—	538	6	no	—
303	C₂²	—	2152	12	no	—
304	C₂²	C₂²×C₂²⋊D₂₆₉	1	6	yes	—
305	C₂²	#22575499	1	2	yes	complemented
306	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	3	no	—
309	C₄	—	538	2	no	—
310	C₄	—	538	6	no	—
311	C₄	—	538	2	no	—
312	C₄	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
313	C₄	C₂²×C₂²⋊D₂₆₉	1	6	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	2	6	no	—
316	C₂	—	4	3	no	—
317	C₂	—	1076	6	no	—
318	C₂	C₂³×C₂²⋊D₂₆₉	1	1	yes	characteristic
319	C₂	C₂×#22575499	1	1	yes	characteristic
320	C₂	#31771441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁵
Fitting	C₂²×#41×C₂₆₉	C₂
hypercenter	C₂²×#41	D₂₆₉
hypocenter	C₂₆₉	#3488
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#22575499	C₂	semidirect
2. #31771274	C₂	semidirect
3. #31771266	C₂	semidirect
4. #31771243	C₂	semidirect
5. D₄×C₂²⋊D₂₆₉	C₂	semidirect
6. #31771183	C₂	semidirect
7. #31771169	C₂	semidirect
8. #31771116	C₂	semidirect
9. C₂×#22575427	C₂	semidirect
10. #31771088	C₂	semidirect
11. C₂²×#41×C₂₆₉	C₂	semidirect
13. C₂×#16864403	C₂²	semidirect
14. C₂×#16864402	C₂²	semidirect
15. C₂²×D₁₀₇₆	C₂²	semidirect
16. C₂²×C₂²⋊D₂₆₉	C₂²	semidirect
17. C₂×D₄×D₂₆₉	C₂²	semidirect
18. #22575449	C₂²	semidirect
19. #22575442	C₂²	semidirect
20. #22575436	C₂²	semidirect
21. #22575432	C₂²	semidirect
22. #22575432	C₂²	semidirect
23. #22575429	C₂²	semidirect
24. #22575427	C₂²	semidirect
25. #22575427	C₂²	semidirect
26. #22575427	C₂²	semidirect
27. #22575427	C₂²	semidirect
28. #22575427	C₂²	semidirect
29. #22575425	C₂²	semidirect
30. #22575424	C₂²	semidirect
31. #22575422	C₂²	semidirect
32. #22575421	C₂²	semidirect
33. Q₈×Dic₂₆₉	C₂²	semidirect
34. D₄×Dic₂₆₉	C₂²	semidirect
35. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
36. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
37. #22575412	C₂²	semidirect
38. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
39. C₂×#16864379	C₂²	semidirect
42. C₂×#41×C₂₆₉	C₂²	semidirect
43. C₂×#41×C₂₆₉	C₂²	semidirect
44. C₂×#41×C₂₆₉	C₂²	semidirect
45. C₂²×Q₈×C₂₆₉	C₂²	semidirect
46. C₂²×D₄×C₂₆₉	C₂²	semidirect
47. C₂³×C₄×C₂₆₉	C₂²	semidirect
79. #41×C₂₆₉	D₄	semidirect
117. Q₈×C₂₆₉	C₂×D₄	semidirect
120. D₄×C₂₆₉	C₂×D₄	semidirect
126. C₂×C₄×C₂₆₉	C₂×D₄	semidirect
140. C₂²×C₂₆₉	H(5,𝔽₂)	semidirect
160. C₂₆₉	#3488	semidirect
163. C₂²×#41	D₂₆₉	semidirect
178. C₂×#41	C₂×D₂₆₉	semidirect
179. C₂×#41	C₂×D₂₆₉	semidirect
180. C₂×#41	C₂×D₂₆₉	semidirect
181. C₂²×Q₈	C₂×D₂₆₉	semidirect
185. C₂²×D₄	C₂×D₂₆₉	semidirect
186. C₂³×C₄	C₂×D₂₆₉	semidirect
216. #41	C₂²⋊D₂₆₉	semidirect
267. Q₈	C₂×C₂²⋊D₂₆₉	semidirect
276. D₄	C₂×C₂²⋊D₂₆₉	semidirect
287. C₂×C₄	C₂×C₂²⋊D₂₆₉	semidirect
305. C₂²	#22575499	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575499	2
#31771274	2
#31771266	2
#31771243	2
D₄×C₂²⋊D₂₆₉	2
#31771183	2
#31771169	2
#31771116	2
C₂×#22575427	2
#31771088	2
C₂²×#41×C₂₆₉	2
#3488	269