Subgroups — #21462591

Order 7808.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14519864	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840809	C₂	1	2	yes	maximal, complemented
4	#16840807	C₂	1	2	yes	maximal, complemented
5	#16840805	C₂	1	2	yes	maximal, complemented
6	#16840804	C₂	1	2	yes	maximal, complemented
7	#16840777	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840760	C₂	1	1	yes	maximal, characteristic, complemented
9	#16840759	C₂	1	1	yes	maximal, characteristic, complemented
10	#16840728	C₂	1	1	yes	maximal, characteristic, complemented
11	#16840558	C₂	1	1	yes	maximal, characteristic
12	#14519865	C₂²	1	1	yes	characteristic, complemented
13	#14519865	C₂²	1	1	yes	characteristic, complemented
14	#14519864	—	2	1	no	—
15	#14519864	—	2	1	no	—
16	#14519864	—	2	1	no	—
17	#14519864	—	2	1	no	—
18	#14519864	C₂²	1	1	yes	characteristic, complemented
19	#14519864	C₂²	1	2	yes	complemented
20	#14519864	C₂²	1	2	yes	complemented
21	#14519864	C₂²	1	1	yes	characteristic, complemented
22	C₂×#11758213	C₂²	1	1	yes	characteristic
23	D₆₁⋊M₁₆	—	2	1	no	—
24	D₆₁⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#11758217	—	2	1	no	—
26	C₂×D₆₁⋊C₈	—	2	1	no	—
27	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×#92420	—	2	1	no	—
29	#14519856	C₂²	1	1	yes	characteristic
30	#14519848	C₂²	1	2	yes	complemented
31	#14519847	C₂²	1	2	yes	complemented
32	#14519842	C₂²	1	2	yes	complemented
33	#14519841	C₂²	1	2	yes	complemented
34	#14519841	C₂²	1	2	yes	complemented
35	#14519841	C₂²	1	2	yes	complemented
36	#14519831	C₂²	1	2	yes	complemented
37	C₈×#6923	C₂²	1	2	yes	complemented
38	C₂×#11758224	C₂²	1	1	yes	characteristic
39	#41×D₆₁	C₂²	1	1	yes	characteristic
40	#14519810	C₄	1	1	yes	characteristic, complemented
41	#14519807	C₄	1	1	yes	characteristic, complemented
42	#14519806	C₄	1	2	yes	complemented
43	#14519803	C₄	1	1	yes	characteristic, complemented
44	#14519801	C₄	1	2	yes	complemented
45	#14519798	C₂²	1	2	yes	—
46	#14519796	C₂²	1	2	yes	—
47	M₁₆×D₆₁	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₆₁	C₄	1	1	yes	characteristic, complemented
49	#41×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₆₁	—	2	1	no	—
52	C₂²×Dic₆₁	—	2	1	no	—
53	#11758226	C₂³	1	1	yes	characteristic
54	Q₈×D₆₁	C₂×C₄	1	1	yes	characteristic
55	#11758224	—	2	1	no	—
56	#11758224	—	2	1	no	—
57	#11758224	—	2	1	no	—
58	#11758224	C₂×C₄	1	2	yes	—
59	#11758224	C₂³	1	1	yes	characteristic
60	#11758224	C₂³	1	2	yes	—
61	D₄×D₆₁	—	2	1	no	—
62	D₄×D₆₁	C₂×C₄	1	1	yes	characteristic
63	#11758222	—	2	1	no	—
64	#11758222	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₆₁	—	2	1	no	—
66	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊#6923	—	2	1	no	—
69	#11758217	—	2	1	no	—
70	#11758217	—	2	2	no	—
71	#11758217	—	2	2	no	—
72	#11758217	—	2	1	no	—
73	#11758217	—	2	1	no	—
74	C₂×#92420	—	2	1	no	—
75	C₂×#92420	—	2	2	no	—
76	C₂×#92420	—	2	2	no	—
77	C₂×#92420	—	2	1	no	—
78	C₂×#92420	—	2	1	no	—
79	C₂×#92420	D₄	1	1	yes	characteristic
80	C₂×#92420	D₄	1	1	yes	characteristic
81	C₄⋊#6923	—	2	1	no	—
82	C₄×#6923	C₂³	1	2	yes	—
83	#11758213	—	2	1	no	—
84	#11758213	—	2	1	no	—
85	#11758213	—	2	1	no	—
86	#11758213	C₂³	1	1	yes	characteristic
87	#11758213	C₂³	1	1	yes	characteristic
88	#11758213	C₂³	1	2	yes	—
89	D₆₁⋊C₈	—	2	1	no	—
90	D₆₁⋊C₈	—	2	1	no	—
91	D₆₁⋊C₈	—	2	1	no	—
92	D₆₁⋊C₈	D₄	1	1	yes	characteristic
93	D₆₁⋊C₈	D₄	1	1	yes	characteristic
94	D₆₁⋊C₈	C₂³	1	1	yes	characteristic
95	D₆₁⋊C₈	C₂³	1	1	yes	characteristic
96	SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
97	D₈×C₆₁	C₂×C₄	1	2	yes	—
98	M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
99	#11758202	C₂×C₄	1	2	yes	—
100	#11758200	C₂×C₄	1	2	yes	—
101	#11758200	C₂×C₄	1	2	yes	—
102	#11758199	C₂×C₄	1	2	yes	—
103	#11758194	C₂×C₄	1	1	yes	characteristic
104	Dic₂₄₄	C₂×C₄	1	2	yes	—
105	#11758190	C₂×C₄	1	2	yes	—
106	#11758189	C₂×C₄	1	2	yes	—
107	C₈×D₆₁	C₂³	1	2	yes	—
108	C₂³×C₆₁	—	2	1	no	—
109	C₂²×D₆₁	—	2	1	no	—
110	C₂²×D₆₁	#41	1	1	yes	characteristic
111	C₂×#6923	—	2	2	no	—
112	Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₆₁	—	2	1	no	—
114	D₄×C₆₁	—	2	1	no	—
115	D₄×C₆₁	C₂²×C₄	1	2	yes	—
116	D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₆₁	—	2	1	no	—
120	C₂²⋊D₆₁	—	2	2	no	—
121	C₂²⋊D₆₁	—	2	1	no	—
122	C₂²⋊D₆₁	—	2	2	no	—
123	C₂²⋊D₆₁	—	2	1	no	—
124	C₂×Dic₆₁	—	2	1	no	—
125	C₂×Dic₆₁	—	2	2	no	—
126	C₂×Dic₆₁	—	2	1	no	—
127	C₂×Dic₆₁	—	2	2	no	—
128	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
129	D₂₄₄	—	2	1	no	—
130	D₂₄₄	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₆₁	—	2	1	no	—
132	C₄×D₆₁	—	2	1	no	—
133	C₄×D₆₁	—	2	1	no	—
134	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
136	C₄×D₆₁	#41	1	1	yes	characteristic
137	C₄×D₆₁	C₂⁴	1	1	yes	characteristic
138	Dic₁₂₂	—	2	1	no	—
139	Dic₁₂₂	—	2	1	no	—
140	Dic₁₂₂	C₂²×C₄	1	2	yes	—
141	Dic₁₂₂	C₂²×C₄	1	1	yes	characteristic
142	#92420	—	2	1	no	—
143	#92420	—	2	1	no	—
144	#92420	—	2	2	no	—
145	#92420	C₂×D₄	1	1	yes	characteristic
146	#92420	C₂×D₄	1	1	yes	characteristic
147	#92420	C₂×D₄	1	1	yes	characteristic
148	#92420	C₂×D₄	1	1	yes	characteristic
149	C₈×C₆₁	C₂²×C₄	1	2	yes	—
150	#92418	C₂²×C₄	1	2	yes	—
151	C₂²×C₆₁	—	2	2	no	—
152	C₂²×C₆₁	—	2	1	no	—
153	C₂²×C₆₁	—	4	2	no	—
154	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
155	C₂×D₆₁	—	2	1	no	—
156	C₂×D₆₁	—	4	1	no	—
157	C₂×D₆₁	—	4	2	no	—
158	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
159	C₂×D₆₁	C₂×#41	1	1	yes	characteristic
160	#6923	—	4	2	no	—
161	C₄×C₆₁	—	2	1	no	—
162	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
163	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
164	Dic₆₁	—	2	1	no	—
165	Dic₆₁	—	2	2	no	—
166	Dic₆₁	C₄×D₄	1	1	yes	characteristic
167	Dic₆₁	C₂²×D₄	1	1	yes	characteristic
168	#2955	—	61	1	no	maximal, Sylow
169	C₂×C₆₁	—	2	1	no	—
170	C₂×C₆₁	—	4	1	no	—
171	C₂×C₆₁	—	4	2	no	—
172	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
173	D₆₁	—	2	1	no	—
174	D₆₁	—	2	1	no	—
175	D₆₁	—	4	1	no	—
176	#575	—	61	1	no	—
177	C₂×#131	—	61	1	no	—
178	C₂×#131	—	61	1	no	—
179	#444	—	61	2	no	—
180	#444	—	61	2	no	—
181	#443	—	61	2	no	—
182	#443	—	61	2	no	—
183	#430	—	61	1	no	—
184	#421	—	61	1	no	—
185	#421	—	61	1	no	—
186	#405	—	61	1	no	—
187	C₆₁	#2955	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	61	1	no	—
189	C₂×#41	—	61	1	no	—
190	#137	—	61	1	no	—
191	#137	—	61	2	no	—
192	#137	—	61	1	no	—
193	AGL(1,ℤ/8)	—	61	1	no	—
194	AGL(1,ℤ/8)	—	61	2	no	—
195	AGL(1,ℤ/8)	#6923	1	1	yes	characteristic, complemented
196	#135	—	61	2	no	—
197	#135	—	61	2	no	—
198	#131	—	61	2	no	—
199	#131	—	61	1	no	—
200	#131	—	61	2	no	—
201	#131	—	61	1	no	—
202	#131	—	61	1	no	—
203	#131	—	61	1	no	—
204	#131	—	122	1	no	—
205	#131	—	122	1	no	—
206	#131	—	122	1	no	—
207	#131	—	122	1	no	—
208	C₂×M₁₆	—	61	1	no	—
209	C₂×M₁₆	—	61	1	no	—
210	C₂×M₁₆	—	61	1	no	—
211	C₂×M₁₆	—	122	1	no	—
212	C₂×M₁₆	—	122	1	no	—
213	C₂²×C₈	—	61	1	no	—
214	C₂²×C₈	—	122	1	no	—
215	C₂²×C₈	—	122	1	no	—
216	#117	—	61	1	no	—
217	#108	—	61	2	no	—
218	#108	—	61	2	no	—
219	#104	—	61	2	no	—
220	#104	—	61	2	no	—
221	#104	—	61	2	no	—
222	#104	—	61	2	no	—
223	#97	—	61	2	no	—
224	C₄×C₈	—	61	2	no	—
225	#41	—	61	1	no	—
226	#41	—	61	2	no	—
227	#41	—	61	1	no	—
228	#41	—	61	2	no	—
229	#41	—	61	1	no	—
230	#41	—	122	1	no	—
231	#41	—	122	1	no	—
232	#41	—	122	1	no	—
233	#41	—	122	1	no	—
234	#41	C₂×#6923	1	1	yes	characteristic
235	C₂×Q₈	—	61	1	no	—
236	C₂×Q₈	—	61	1	no	—
237	C₂×D₄	—	61	1	no	—
238	C₂×D₄	—	122	1	no	—
239	C₂×D₄	—	122	1	no	—
240	C₂×D₄	C₂×#6923	1	1	yes	characteristic
241	C₂²×C₄	—	61	1	no	—
242	C₂²×C₄	—	122	1	no	—
243	C₂²×C₄	—	122	1	no	—
244	Q₁₆	—	61	2	no	—
245	Q₁₆	—	61	2	no	—
246	SD₁₆	—	61	2	no	—
247	SD₁₆	—	61	2	no	—
248	SD₁₆	—	61	2	no	—
249	SD₁₆	C₂×#6923	1	2	yes	—
250	D₈	—	61	2	no	—
251	D₈	C₂×#6923	1	2	yes	—
252	M₁₆	—	61	1	no	—
253	M₁₆	—	61	2	no	—
254	M₁₆	—	61	1	no	—
255	M₁₆	—	61	1	no	—
256	M₁₆	—	61	2	no	—
257	M₁₆	—	122	1	no	—
258	M₁₆	—	122	1	no	—
259	M₁₆	—	122	2	no	—
260	M₁₆	—	122	1	no	—
261	M₁₆	—	122	2	no	—
262	M₁₆	—	122	1	no	—
263	M₁₆	—	122	1	no	—
264	M₁₆	—	122	1	no	—
265	M₁₆	C₂×#6923	1	1	yes	characteristic
266	C₂×C₈	—	61	1	no	—
267	C₂×C₈	—	61	1	no	—
268	C₂×C₈	—	61	1	no	—
269	C₂×C₈	—	61	1	no	—
270	C₂×C₈	—	61	1	no	—
271	C₂×C₈	—	61	1	no	—
272	C₂×C₈	—	61	2	no	—
273	C₂×C₈	—	122	1	no	—
274	C₂×C₈	—	122	1	no	—
275	C₂×C₈	—	122	2	no	—
276	C₂×C₈	—	122	1	no	—
277	C₂×C₈	—	122	2	no	—
278	C₂×C₈	—	122	1	no	—
279	C₂×C₈	—	122	1	no	—
280	C₂×C₈	—	122	1	no	—
281	C₄⋊C₄	—	122	1	no	—
282	C₂²⋊C₄	—	122	1	no	—
283	C₄²	—	61	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	61	1	no	—
286	C₂³	—	122	1	no	—
287	Q₈	—	61	2	no	—
288	Q₈	—	61	1	no	—
289	Q₈	—	122	1	no	—
290	Q₈	—	122	1	no	—
291	Q₈	C₂²×#6923	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	61	1	no	—
295	D₄	—	122	1	no	—
296	D₄	—	122	1	no	—
297	D₄	—	122	1	no	—
298	D₄	—	122	2	no	—
299	D₄	—	122	2	no	—
300	D₄	—	122	1	no	—
301	D₄	C₂²×#6923	1	2	yes	—
302	D₄	C₂²×#6923	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	61	1	no	—
306	C₂×C₄	—	61	1	no	—
307	C₂×C₄	—	61	1	no	—
308	C₂×C₄	—	61	1	no	—
309	C₂×C₄	—	122	1	no	—
310	C₂×C₄	—	122	1	no	—
311	C₂×C₄	—	122	1	no	—
312	C₂×C₄	—	122	2	no	—
313	C₂×C₄	—	122	2	no	—
314	C₂×C₄	—	122	1	no	—
315	C₂×C₄	—	122	1	no	—
316	C₂×C₄	—	122	2	no	—
317	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
318	C₈	—	61	1	no	—
319	C₈	—	61	1	no	—
320	C₈	—	61	1	no	—
321	C₈	—	61	1	no	—
322	C₈	—	61	2	no	—
323	C₈	—	122	1	no	—
324	C₈	—	122	1	no	—
325	C₈	—	122	2	no	—
326	C₈	C₂²×#6923	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	61	1	no	—
331	C₂²	—	61	1	no	—
332	C₂²	—	122	1	no	—
333	C₂²	—	244	1	no	—
334	C₂²	—	244	2	no	—
335	C₂²	D₄×#6923	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	61	1	no	—
338	C₄	—	61	1	no	—
339	C₄	—	122	1	no	—
340	C₄	—	122	2	no	—
341	C₄	—	244	2	no	—
342	C₄	D₄×#6923	1	1	yes	characteristic
343	C₄	C₂³×#6923	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	122	1	no	—
348	C₂	—	122	1	no	—
349	C₂	—	244	1	no	—
350	C₂	C₂×D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6923
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₄	C₂³×#6923
Fitting	AGL(1,ℤ/8)×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	AGL(1,ℤ/8)	#6923
hypocenter	C₆₁	#2955
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₆₁⋊#131	C₂	semidirect
2. C₂×#14519864	C₂	semidirect
3. #16840809	C₂	semidirect
4. #16840807	C₂	semidirect
5. #16840805	C₂	semidirect
6. #16840804	C₂	semidirect
7. #16840777	C₂	semidirect
8. #16840760	C₂	semidirect
9. #16840759	C₂	semidirect
10. #16840728	C₂	semidirect
12. #14519865	C₂²	semidirect
13. #14519865	C₂²	semidirect
18. #14519864	C₂²	semidirect
19. #14519864	C₂²	semidirect
20. #14519864	C₂²	semidirect
21. #14519864	C₂²	semidirect
30. #14519848	C₂²	semidirect
31. #14519847	C₂²	semidirect
32. #14519842	C₂²	semidirect
33. #14519841	C₂²	semidirect
34. #14519841	C₂²	semidirect
35. #14519841	C₂²	semidirect
36. #14519831	C₂²	semidirect
37. C₈×#6923	C₂²	semidirect
40. #14519810	C₄	semidirect
41. #14519807	C₄	semidirect
42. #14519806	C₄	semidirect
43. #14519803	C₄	semidirect
44. #14519801	C₄	semidirect
48. AGL(1,ℤ/8)×C₆₁	C₄	semidirect
187. C₆₁	#2955	semidirect
195. AGL(1,ℤ/8)	#6923	semidirect