Subgroups — #29834762

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456347	C₂	1	1	yes	maximal, characteristic, complemented
2	#21454735	C₂	1	1	yes	maximal, characteristic
3	#21447994	C₂	1	4	yes	maximal, complemented
4	#21447989	C₂	1	4	yes	maximal, complemented
5	#21447593	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447584	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447556	C₂	1	1	yes	maximal, characteristic
8	C₂×#16839571	C₂	1	1	yes	maximal, characteristic
9	C₂×C₆₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14519799	C₂²	1	1	yes	characteristic
11	#16840528	—	2	1	no	—
12	#16840393	—	2	1	no	—
13	C₂×#14519717	C₂²	1	1	yes	characteristic
14	#16840169	—	2	2	no	—
15	#16840162	C₂²	1	2	yes	complemented
16	#16840159	C₂²	1	2	yes	complemented
17	#16839883	—	2	2	no	—
18	#16839878	C₂²	1	4	yes	—
19	#16839802	—	2	1	no	—
20	#16839800	C₂²	1	1	yes	characteristic
21	#16839778	—	2	1	no	—
22	#16839717	C₂²	1	1	yes	characteristic
23	#16839586	C₂²	1	2	yes	complemented
24	#16839586	C₂²	1	2	yes	complemented
25	#16839582	C₂²	1	4	yes	complemented
26	#16839573	C₂²	1	2	yes	—
27	#16839573	C₂²	1	2	yes	—
28	#16839571	C₂²	1	4	yes	—
29	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#365	C₂²	1	4	yes	complemented
33	C₂×#11758222	—	2	1	no	—
34	#14519799	—	2	1	no	—
35	#14519799	—	2	1	no	—
36	#14519799	Q₈	1	2	yes	complemented
37	#14519799	C₂³	1	2	yes	—
38	C₂×Dic₂₄₄	Q₈	1	1	yes	characteristic, complemented
39	C₂×#11758190	—	2	1	no	—
40	C₂×D₄₈₈	Q₈	1	1	yes	characteristic, complemented
41	#14519785	—	2	4	no	—
42	#14519783	—	4	2	no	—
43	C₂×#11758194	D₄	1	1	yes	characteristic
44	M₁₆×D₆₁	—	4	2	no	—
45	#14519776	—	2	2	no	—
46	#14519776	—	4	1	no	—
47	#14519768	—	2	2	no	—
48	#14519768	—	4	1	no	—
49	#14519756	—	4	2	no	—
50	#14519736	—	2	1	no	—
51	#14519736	D₄	1	1	yes	characteristic
52	#14519717	—	2	1	no	—
53	#14519717	C₂³	1	2	yes	—
54	C₂×#11758185	—	2	1	no	—
55	#14519711	—	2	1	no	—
56	#14519710	—	2	2	no	—
57	#14519709	C₂³	1	1	yes	characteristic
58	#14519709	C₂³	1	1	yes	characteristic
59	#14519708	—	2	2	no	—
60	#14519703	—	2	4	no	—
61	#14519687	D₄	1	2	yes	—
62	C₈×Dic₆₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₆₁	—	2	1	no	—
64	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
66	M₃₂×C₆₁	—	2	1	no	—
67	M₃₂×C₆₁	C₂³	1	2	yes	—
68	C₂×C₁₆×C₆₁	—	2	1	no	—
69	#108×C₆₁	—	2	1	no	—
70	#108×C₆₁	C₂³	1	2	yes	—
71	#106×C₆₁	—	2	1	no	—
72	#106×C₆₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₆₁	—	4	1	no	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₆₁	—	4	1	no	—
76	#11758222	—	2	2	no	—
77	#11758222	—	2	2	no	—
78	#11758222	—	4	1	no	—
79	#11758222	—	4	1	no	—
80	C₂×D₂₄₄	—	2	1	no	—
81	C₂×C₄×D₆₁	—	4	1	no	—
82	C₂×Dic₁₂₂	—	2	1	no	—
83	C₂×Dic₁₂₂	—	4	2	no	—
84	M₁₆×C₆₁	—	2	2	no	—
85	M₁₆×C₆₁	—	4	1	no	—
86	C₂×C₈×C₆₁	—	4	1	no	—
87	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₂²⋊Dic₆₁	—	2	1	no	—
96	C₂²⋊Dic₆₁	—	4	1	no	—
97	C₄⋊Dic₆₁	—	2	1	no	—
98	C₄⋊Dic₆₁	—	2	1	no	—
99	C₄⋊Dic₆₁	—	2	1	no	—
100	C₄⋊Dic₆₁	—	4	2	no	—
101	#11758196	—	4	2	no	—
102	#11758196	—	4	2	no	—
103	C₄×Dic₆₁	—	4	1	no	—
104	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
105	#11758194	SD₁₆	1	2	yes	—
106	#11758194	SD₁₆	1	2	yes	—
107	C₂×#92418	C₂×D₄	1	2	yes	—
108	Dic₂₄₄	—	2	1	no	—
109	Dic₂₄₄	C₂×Q₈	1	2	yes	—
110	D₄₈₈	—	2	1	no	—
111	D₄₈₈	C₂×Q₈	1	2	yes	—
112	#11758190	—	2	2	no	—
113	#11758190	—	2	2	no	—
114	#11758189	—	4	4	no	—
115	C₈×D₆₁	—	4	4	no	—
116	C₁₆×C₆₁	—	2	2	no	—
117	#11758185	—	2	2	no	—
118	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	16	1	no	—
122	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₆₁	—	4	2	no	—
129	C₂²⋊D₆₁	—	4	2	no	—
130	C₂×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	4	1	no	—
132	C₂×Dic₆₁	—	4	1	no	—
133	C₂×Dic₆₁	—	4	1	no	—
134	D₂₄₄	—	2	2	no	—
135	D₂₄₄	—	4	1	no	—
136	C₄×D₆₁	—	4	2	no	—
137	C₄×D₆₁	—	4	2	no	—
138	Dic₁₂₂	—	2	2	no	—
139	Dic₁₂₂	—	4	1	no	—
140	Dic₁₂₂	—	16	2	no	—
141	C₈×C₆₁	—	4	2	no	—
142	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
146	#92418	C₂×SD₁₆	1	4	yes	—
147	#33638	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	—	4	1	no	—
149	C₂²×C₆₁	#476	1	1	yes	characteristic
150	C₂²×C₆₁	#476	1	1	yes	characteristic
151	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×D₆₁	—	4	2	no	—
153	C₂×D₆₁	—	16	1	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	#476	1	1	yes	characteristic
156	C₄×C₆₁	#476	1	1	yes	characteristic
157	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₆₁	—	4	2	no	—
160	Dic₆₁	—	4	2	no	—
161	Dic₆₁	—	8	1	no	—
162	Dic₆₁	—	8	1	no	—
163	#3163	—	61	1	no	—
164	#2957	—	61	1	no	—
165	#2231	—	61	4	no	—
166	#2230	—	61	4	no	—
167	#2155	—	61	1	no	—
168	C₂×#365	D₆₁	1	1	yes	characteristic, complemented
169	#2153	—	61	1	no	—
170	#2145	—	61	1	no	—
171	C₂×#360	—	61	1	no	—
172	C₂×C₆₁	—	2	1	no	—
173	C₂×C₆₁	—	2	1	no	—
174	C₂×C₆₁	—	2	1	no	—
175	C₂×C₆₁	C₂×#476	1	1	yes	characteristic
176	D₆₁	—	8	2	no	—
177	C₂×#135	—	61	1	no	—
178	#568	—	122	1	no	—
179	#544	—	122	1	no	—
180	C₂×M₃₂	—	61	1	no	—
181	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
182	#501	—	122	2	no	—
183	#499	—	61	2	no	—
184	#498	—	61	2	no	—
185	#444	—	122	2	no	—
186	#443	—	61	4	no	—
187	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
188	#428	—	122	1	no	—
189	#427	—	61	1	no	—
190	C₂×#106	C₂×D₆₁	1	1	yes	characteristic
191	#421	—	122	1	no	—
192	#405	—	61	1	no	—
193	#365	—	61	2	no	—
194	#365	—	61	2	no	—
195	#365	C₂×D₆₁	1	4	yes	complemented
196	#364	—	61	4	no	—
197	#360	—	61	2	no	—
198	#360	—	61	2	no	—
199	#360	—	61	4	no	—
200	C₆₁	#33638	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	122	1	no	—
202	#135	—	61	2	no	—
203	#135	—	61	2	no	—
204	#135	—	122	1	no	—
205	#135	—	122	1	no	—
206	C₂×Q₁₆	—	61	1	no	—
207	C₂×SD₁₆	—	122	1	no	—
208	C₂×D₈	—	61	1	no	—
209	#131	—	122	4	no	—
210	#131	—	244	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	61	1	no	—
213	C₂×M₁₆	—	244	2	no	—
214	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
215	C₄⋊Q₈	—	122	2	no	—
216	C₄⋊Q₈	—	244	1	no	—
217	#125	—	122	2	no	—
218	#125	—	244	1	no	—
219	C₂²⋊Q₈	—	244	2	no	—
220	#117	—	61	1	no	—
221	#117	—	122	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	61	2	no	—
225	M₃₂	—	122	1	no	—
226	M₃₂	C₂²×D₆₁	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	122	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂²×D₆₁	1	2	yes	—
231	#107	—	122	2	no	—
232	#107	—	122	1	no	—
233	#106	—	2	1	no	—
234	#106	—	61	1	no	—
235	#106	—	61	1	no	—
236	#106	—	122	2	no	—
237	#106	C₂²×D₆₁	1	2	yes	—
238	#104	—	122	4	no	—
239	#97	—	61	2	no	—
240	C₄×C₈	—	61	2	no	—
241	#41	—	122	2	no	—
242	#41	—	122	2	no	—
243	#41	—	244	1	no	—
244	#41	—	244	1	no	—
245	C₂×Q₈	—	122	1	no	—
246	C₂×Q₈	—	244	2	no	—
247	C₂×D₄	—	122	1	no	—
248	C₂×D₄	—	244	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	244	1	no	—
251	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
252	Q₁₆	—	61	2	no	—
253	Q₁₆	—	122	1	no	—
254	SD₁₆	—	122	2	no	—
255	SD₁₆	—	122	2	no	—
256	D₈	—	61	2	no	—
257	D₈	—	122	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	61	2	no	—
261	M₁₆	—	61	2	no	—
262	M₁₆	—	244	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	61	2	no	—
265	C₂×C₈	—	244	4	no	—
266	C₂×C₈	#11758222	1	1	yes	characteristic
267	C₂×C₈	#11758222	1	1	yes	characteristic
268	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	122	1	no	—
275	C₄⋊C₄	—	122	1	no	—
276	C₄⋊C₄	—	122	1	no	—
277	C₄⋊C₄	—	244	2	no	—
278	C₄⋊C₄	—	244	2	no	—
279	C₄⋊C₄	—	244	2	no	—
280	C₂²⋊C₄	—	122	1	no	—
281	C₂²⋊C₄	—	244	1	no	—
282	C₄²	—	61	2	no	—
283	C₄²	—	244	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	122	2	no	—
286	C₂³	—	244	1	no	—
287	C₂³	#14519804	1	1	yes	characteristic
288	Q₈	—	122	2	no	—
289	Q₈	—	976	2	no	—
290	Q₈	—	244	1	no	—
291	D₄	—	122	2	no	—
292	D₄	—	244	2	no	—
293	D₄	—	244	2	no	—
294	D₄	—	244	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	16	1	no	—
297	C₂×C₄	—	122	2	no	—
298	C₂×C₄	—	244	1	no	—
299	C₂×C₄	—	244	1	no	—
300	C₂×C₄	—	244	1	no	—
301	C₂×C₄	—	244	2	no	—
302	C₂×C₄	—	244	2	no	—
303	C₂×C₄	#14519755	1	1	yes	characteristic
304	C₂×C₄	#14519755	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
307	C₂×C₄	#14519804	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	61	4	no	—
311	C₈	#14519755	1	1	yes	characteristic
312	C₈	#14519755	1	1	yes	characteristic
313	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
314	C₈	C₂×#11758222	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	976	1	no	—
317	C₂²	—	244	2	no	—
318	C₂²	#16840051	1	1	yes	characteristic
319	C₂²	#16840051	1	1	yes	characteristic
320	C₂²	C₂×#14519804	1	1	yes	characteristic
321	C₄	—	8	2	no	—
322	C₄	—	488	1	no	—
323	C₄	—	488	1	no	—
324	C₄	—	244	2	no	—
325	C₄	—	244	2	no	—
326	C₄	#16840051	1	1	yes	characteristic
327	C₄	#16840051	1	1	yes	characteristic
328	C₄	C₂×#14519755	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₆₁	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	488	2	no	—
334	C₂	C₂×#16840051	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16840051
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#365	C₂
socle	C₂×C₆₁	C₂×#476
prime radical	C₂×#14519799	C₂²
hypercenter	C₂×#365	D₆₁
hypocenter	C₆₁	#33638
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
1. #21456347	C₂	semidirect
3. #21447994	C₂	semidirect
4. #21447989	C₂	semidirect
5. #21447593	C₂	semidirect
6. #21447584	C₂	semidirect
9. C₂×C₆₁×#365	C₂	semidirect
15. #16840162	C₂²	semidirect
16. #16840159	C₂²	semidirect
23. #16839586	C₂²	semidirect
24. #16839586	C₂²	semidirect
25. #16839582	C₂²	semidirect
32. C₆₁×#365	C₂²	semidirect
36. #14519799	Q₈	semidirect
38. C₂×Dic₂₄₄	Q₈	semidirect
40. C₂×D₄₈₈	Q₈	semidirect
168. C₂×#365	D₆₁	semidirect
195. #365	C₂×D₆₁	semidirect
200. C₆₁	#33638	semidirect

Maximal subgroups

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

subgroup	index
#21456347	2
#21454735	2
#21447994	2
#21447989	2
#21447593	2
#21447584	2
#21447556	2
C₂×#16839571	2
C₂×C₆₁×#365	2
#33638	61