Subgroups — #46820422

Order 34432.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575478	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771169	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771088	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770829	C₂	1	2	yes	maximal, complemented
5	#31770822	C₂	1	2	yes	maximal, complemented
6	#31770803	C₂	1	2	yes	maximal, complemented
7	#31770798	C₂	1	2	yes	maximal, complemented
8	#31770631	C₂	1	1	yes	maximal, characteristic, complemented
9	#31770606	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22575375	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#135×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
13	#22575478	—	2	4	no	—
14	C₂×#16864389	C₂²	1	2	yes	—
15	C₂×#16864387	C₂²	1	2	yes	—
16	C₂×#16864386	C₂²	1	1	yes	characteristic
17	#22575449	C₂²	1	2	yes	—
18	#22575442	C₂²	1	2	yes	—
19	#22575427	—	2	1	no	—
20	#22575427	—	2	1	no	—
21	#22575421	—	2	2	no	—
22	Q₈×Dic₂₆₉	C₂²	1	2	yes	complemented
23	D₄×Dic₂₆₉	—	2	2	no	—
24	D₄×Dic₂₆₉	C₂²	1	2	yes	complemented
25	#22575412	—	2	1	no	—
26	#22575412	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
28	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
29	#22575382	C₂²	1	2	yes	complemented
30	#22575382	C₂²	1	2	yes	complemented
31	#22575377	C₂²	1	2	yes	complemented
32	#22575375	C₂²	1	2	yes	—
33	#22575375	C₂²	1	2	yes	—
34	#22575374	C₂²	1	2	yes	complemented
35	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
36	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
37	#135×C₂₆₉	—	2	4	no	—
38	C₂×Q₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
40	C₂×SD₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
41	C₂×D₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
42	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
43	C₂³×D₂₆₉	—	2	1	no	—
44	C₂×D₁₀₇₆	—	2	2	no	—
45	C₂×D₁₀₇₆	—	2	1	no	—
46	C₂×D₁₀₇₆	—	2	2	no	—
47	C₂×D₁₀₇₆	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₂₆₉	—	2	2	no	—
49	C₂×C₂²⋊D₂₆₉	—	2	2	no	—
50	C₂²×Dic₂₆₉	—	2	2	no	—
51	C₂²×Dic₂₆₉	—	2	1	no	—
52	#16864389	—	2	4	no	—
53	#16864387	—	2	4	no	—
54	#16864386	—	2	2	no	—
55	C₂×#14524130	C₂³	1	2	yes	—
56	C₄⋊Dic₂₆₉	—	2	2	no	—
57	C₄⋊Dic₂₆₉	—	2	1	no	—
58	C₄⋊Dic₂₆₉	D₄	1	1	yes	characteristic
59	C₄⋊Dic₂₆₉	D₄	1	1	yes	characteristic
60	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
61	C₂²⋊Dic₂₆₉	—	2	2	no	—
62	C₂²⋊Dic₂₆₉	D₄	1	2	yes	—
63	#16864379	—	2	2	no	—
64	#16864379	—	2	2	no	—
65	C₄×Dic₂₆₉	—	2	2	no	—
66	C₄×Dic₂₆₉	C₂³	1	2	yes	—
67	#41×C₂₆₉	—	2	2	no	—
68	#41×C₂₆₉	—	2	2	no	—
69	#41×C₂₆₉	—	2	2	no	—
70	#41×C₂₆₉	D₄	1	4	yes	complemented
71	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₂₆₉	—	2	1	no	—
74	C₂×D₄×C₂₆₉	—	2	1	no	—
75	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₆₉	—	2	1	no	—
78	C₂²×C₄×C₂₆₉	—	2	1	no	—
79	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
80	Q₁₆×C₂₆₉	—	2	2	no	—
81	SD₁₆×C₂₆₉	—	2	2	no	—
82	SD₁₆×C₂₆₉	—	2	2	no	—
83	D₈×C₂₆₉	—	2	2	no	—
84	C₂×C₈×C₂₆₉	—	2	2	no	—
85	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
87	C₂²×D₂₆₉	—	2	2	no	—
88	C₂²×D₂₆₉	—	4	2	no	—
89	C₂²×D₂₆₉	—	8	1	no	—
90	C₂²×D₂₆₉	—	8	1	no	—
91	D₁₀₇₆	—	2	2	no	—
92	D₁₀₇₆	—	2	2	no	—
93	D₁₀₇₆	—	4	2	no	—
94	C₂²⋊D₂₆₉	—	8	2	no	—
95	C₂²⋊D₂₆₉	—	8	2	no	—
96	C₂×Dic₂₆₉	—	2	2	no	—
97	C₂×Dic₂₆₉	—	8	2	no	—
98	C₂×Dic₂₆₉	—	8	1	no	—
99	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
100	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
101	#14524130	—	2	2	no	—
102	C₂³×C₂₆₉	—	2	1	no	—
103	C₂³×C₂₆₉	—	2	1	no	—
104	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₂₆₉	—	2	1	no	—
106	Q₈×C₂₆₉	—	2	1	no	—
107	Q₈×C₂₆₉	—	2	1	no	—
108	Q₈×C₂₆₉	C₂×D₄	1	2	yes	—
109	D₄×C₂₆₉	—	2	4	no	—
110	D₄×C₂₆₉	—	2	1	no	—
111	D₄×C₂₆₉	—	2	1	no	—
112	D₄×C₂₆₉	—	2	1	no	—
113	D₄×C₂₆₉	—	4	2	no	—
114	D₄×C₂₆₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₂₆₉	—	2	1	no	—
116	C₂×C₄×C₂₆₉	—	2	4	no	—
117	C₂×C₄×C₂₆₉	—	2	1	no	—
118	C₂×C₄×C₂₆₉	—	4	2	no	—
119	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₂₆₉	#41	1	2	yes	—
122	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
123	C₈×C₂₆₉	—	2	1	no	—
124	C₈×C₂₆₉	—	2	1	no	—
125	C₂²×C₂₆₉	—	2	1	no	—
126	C₂²×C₂₆₉	—	2	1	no	—
127	C₂²×C₂₆₉	—	2	2	no	—
128	C₂²×C₂₆₉	—	4	1	no	—
129	C₂²×C₂₆₉	—	4	1	no	—
130	C₂²×C₂₆₉	—	4	1	no	—
131	C₂²×C₂₆₉	—	4	1	no	—
132	C₂²×C₂₆₉	—	4	1	no	—
133	C₂²×C₂₆₉	#121	1	2	yes	—
134	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₂₆₉	—	4	2	no	—
136	C₂×D₂₆₉	—	4	2	no	—
137	C₂×D₂₆₉	—	4	2	no	—
138	C₂×D₂₆₉	—	8	4	no	—
139	C₄×C₂₆₉	—	2	2	no	—
140	C₄×C₂₆₉	—	4	1	no	—
141	C₄×C₂₆₉	#121	1	2	yes	—
142	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
144	Dic₂₆₉	—	2	2	no	—
145	Dic₂₆₉	—	2	2	no	—
146	Dic₂₆₉	—	4	2	no	—
147	C₂×C₂₆₉	—	2	2	no	—
148	C₂×C₂₆₉	—	4	2	no	—
149	C₂×C₂₆₉	—	8	1	no	—
150	C₂×C₂₆₉	C₂×#121	1	1	yes	characteristic
151	C₂×C₂₆₉	#576	1	1	yes	characteristic
152	C₂×C₂₆₉	#577	1	1	yes	characteristic
153	D₂₆₉	—	8	2	no	—
154	C₂₆₉	#3045	1	1	yes	characteristic, complemented, Sylow
155	#3045	—	269	1	no	maximal, Sylow
156	C₂×AGL(1,ℤ/8)	—	269	1	no	—
157	C₂×#135	D₂₆₉	1	1	yes	characteristic, complemented
158	#535	—	269	1	no	—
159	#520	—	269	1	no	—
160	#464	—	269	2	no	—
161	#463	—	269	2	no	—
162	#460	—	269	2	no	—
163	#459	—	269	2	no	—
164	#424	—	269	1	no	—
165	#417	—	269	1	no	—
166	C₂×#102	—	269	1	no	—
167	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
168	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
169	C₂²×D₄	—	269	1	no	—
170	AGL(1,ℤ/8)	—	538	4	no	—
171	#135	—	2	4	no	—
172	C₂×Q₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	—	269	2	no	—
174	C₂×SD₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
175	C₂×SD₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
176	C₂×D₈	—	269	2	no	—
177	C₂×D₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
178	C₂×M₁₆	—	269	1	no	—
179	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
180	#127	—	269	2	no	—
181	#124	—	269	2	no	—
182	#121	—	538	1	no	—
183	#121	—	538	1	no	—
184	#120	—	538	2	no	—
185	C₄×Q₈	—	269	2	no	—
186	C₄×D₄	—	269	2	no	—
187	C₄×D₄	—	538	2	no	—
188	#117	—	269	1	no	—
189	#117	—	538	1	no	—
190	C₂×C₄⋊C₄	—	269	1	no	—
191	C₂×C₄⋊C₄	—	538	1	no	—
192	C₄⋊C₈	—	269	2	no	—
193	C₄⋊C₈	—	269	2	no	—
194	#103	—	269	2	no	—
195	#102	—	269	2	no	—
196	#102	—	269	2	no	—
197	#102	—	269	2	no	—
198	C₂⁴	—	538	1	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	C₂²⋊D₂₆₉	1	4	yes	complemented
203	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
204	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	269	2	no	—
208	C₂×D₄	—	538	2	no	—
209	C₂×D₄	—	538	2	no	—
210	C₂×D₄	—	538	1	no	—
211	C₂×D₄	—	538	2	no	—
212	C₂×D₄	—	538	2	no	—
213	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
214	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	538	2	no	—
218	C₂²×C₄	—	538	1	no	—
219	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	SD₁₆	—	538	4	no	—
224	D₈	—	2	2	no	—
225	D₈	—	538	4	no	—
226	M₁₆	—	538	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	269	2	no	—
229	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
231	C₄⋊C₄	—	269	1	no	—
232	C₄⋊C₄	—	269	1	no	—
233	C₄⋊C₄	—	269	2	no	—
234	C₄⋊C₄	—	538	2	no	—
235	C₄⋊C₄	—	538	1	no	—
236	C₂²⋊C₄	—	269	2	no	—
237	C₂²⋊C₄	—	538	2	no	—
238	C₂²⋊C₄	—	538	2	no	—
239	C₂²⋊C₄	—	538	2	no	—
240	C₄²	—	269	2	no	—
241	C₄²	—	538	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	538	2	no	—
245	C₂³	—	1076	2	no	—
246	C₂³	—	2152	1	no	—
247	C₂³	—	2152	1	no	—
248	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	Q₈	C₂×C₂²⋊D₂₆₉	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	1	no	—
257	D₄	—	4	2	no	—
258	D₄	—	538	2	no	—
259	D₄	—	538	2	no	—
260	D₄	—	1076	2	no	—
261	D₄	—	2152	2	no	—
262	D₄	—	2152	2	no	—
263	D₄	C₂×C₂²⋊D₂₆₉	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	269	2	no	—
269	C₂×C₄	—	269	2	no	—
270	C₂×C₄	—	538	2	no	—
271	C₂×C₄	—	2152	2	no	—
272	C₂×C₄	—	2152	1	no	—
273	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	2	yes	—
275	C₂×C₄	#16864401	1	2	yes	—
276	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	538	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	1076	2	no	—
289	C₂²	—	1076	2	no	—
290	C₂²	—	1076	2	no	—
291	C₂²	—	2152	4	no	—
292	C₂²	#22575425	1	2	yes	—
293	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	538	2	no	—
297	C₄	—	538	2	no	—
298	C₄	—	1076	2	no	—
299	C₄	#22575425	1	2	yes	—
300	C₄	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
301	C₄	C₂×#16864401	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	1	no	—
305	C₂	—	2152	2	no	—
306	C₂	C₂×#22575425	1	1	yes	characteristic
307	C₂	#31771407	1	1	yes	characteristic
308	C₂	#31771412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×#135×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	C₂×#135	D₂₆₉
hypocenter	C₂₆₉	#3045
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#22575478	C₂	semidirect
2. #31771169	C₂	semidirect
3. #31771088	C₂	semidirect
4. #31770829	C₂	semidirect
5. #31770822	C₂	semidirect
6. #31770803	C₂	semidirect
7. #31770798	C₂	semidirect
8. #31770631	C₂	semidirect
9. #31770606	C₂	semidirect
10. C₂×#22575375	C₂	semidirect
11. C₂×#135×C₂₆₉	C₂	semidirect
22. Q₈×Dic₂₆₉	C₂²	semidirect
24. D₄×Dic₂₆₉	C₂²	semidirect
29. #22575382	C₂²	semidirect
30. #22575382	C₂²	semidirect
31. #22575377	C₂²	semidirect
34. #22575374	C₂²	semidirect
38. C₂×Q₁₆×C₂₆₉	C₂²	semidirect
39. C₂×SD₁₆×C₂₆₉	C₂²	semidirect
40. C₂×SD₁₆×C₂₆₉	C₂²	semidirect
41. C₂×D₈×C₂₆₉	C₂²	semidirect
70. #41×C₂₆₉	D₄	semidirect
154. C₂₆₉	#3045	semidirect
157. C₂×#135	D₂₆₉	semidirect
172. C₂×Q₁₆	C₂×D₂₆₉	semidirect
174. C₂×SD₁₆	C₂×D₂₆₉	semidirect
175. C₂×SD₁₆	C₂×D₂₆₉	semidirect
177. C₂×D₈	C₂×D₂₆₉	semidirect
202. #41	C₂²⋊D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575478	2
#31771169	2
#31771088	2
#31770829	2
#31770822	2
#31770803	2
#31770798	2
#31770631	2
#31770606	2
C₂×#22575375	2
C₂×#135×C₂₆₉	2
#3045	269