Subgroups — #21462511

Order 7808.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840787	C₂	1	2	yes	maximal, complemented
2	#16840787	C₂	1	2	yes	maximal, complemented
3	#16840785	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840784	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840781	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840780	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840768	C₂	1	2	yes	maximal, complemented
8	#16840734	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519834	C₂	1	1	yes	maximal, characteristic
10	#16840726	C₂	1	2	yes	maximal, complemented
11	#117×D₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#11758217	C₂²	1	2	yes	—
13	C₂²×#92420	C₂²	1	2	yes	—
14	#14519844	C₂²	1	2	yes	complemented
15	#14519844	C₂²	1	2	yes	complemented
16	C₄⋊#92420	C₂²	1	2	yes	complemented
17	C₄⋊#92420	C₂²	1	2	yes	complemented
18	#14519834	C₂²	1	1	yes	characteristic
19	#14519834	C₂²	1	1	yes	characteristic
20	#14519834	C₂²	1	2	yes	—
21	#14519834	C₂²	1	1	yes	characteristic
22	#14519834	C₂²	1	1	yes	characteristic
23	#14519834	C₂²	1	2	yes	—
24	#14519832	C₂²	1	2	yes	complemented
25	#14519832	C₂²	1	2	yes	complemented
26	C₄×#92420	C₂²	1	2	yes	complemented
27	C₄×#92420	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₆₁	C₂²	1	1	yes	characteristic
29	#14519736	C₄	1	1	yes	characteristic
30	#14519735	C₂²	1	1	yes	characteristic
31	#14519735	C₂²	1	1	yes	characteristic
32	#14519734	C₄	1	1	yes	characteristic
33	#14519734	C₄	1	1	yes	characteristic
34	#14519733	C₂²	1	1	yes	characteristic
35	#14519733	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₆₁	C₄	1	1	yes	characteristic
37	C₄⋊C₄×D₆₁	C₄	1	1	yes	characteristic
38	C₂²⋊C₄×D₆₁	C₄	1	1	yes	characteristic
39	C₂²⋊C₄×D₆₁	C₄	1	1	yes	characteristic
40	C₄²×D₆₁	C₂²	1	1	yes	characteristic
41	C₄²×D₆₁	C₂²	1	1	yes	characteristic
42	#117×C₆₁	C₄	1	1	yes	characteristic
43	C₂³×D₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₆₁	—	2	4	no	—
47	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
55	#11758217	—	2	4	no	—
56	C₂×#92420	—	2	4	no	—
57	C₂×#92420	C₂³	1	2	yes	—
58	C₂×#92420	C₂³	1	2	yes	—
59	C₂×#92420	C₂³	1	2	yes	—
60	C₂×#92420	C₂³	1	2	yes	—
61	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
69	#11758198	C₂×C₄	1	1	yes	characteristic
70	#11758198	C₂×C₄	1	1	yes	characteristic
71	#11758198	C₂×C₄	1	1	yes	characteristic
72	#11758198	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
75	#11758196	C₂×C₄	1	1	yes	characteristic
76	#11758196	C₂×C₄	1	1	yes	characteristic
77	#11758196	C₂×C₄	1	1	yes	characteristic
78	#11758196	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
82	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
84	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
85	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₆₁	—	2	2	no	—
87	C₂²×D₆₁	—	2	2	no	—
88	C₂²×D₆₁	—	4	1	no	—
89	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₆₁	#41	1	4	yes	—
104	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
105	C₄×D₆₁	—	2	2	no	—
106	C₄×D₆₁	—	2	2	no	—
107	C₄×D₆₁	—	2	2	no	—
108	C₄×D₆₁	—	2	2	no	—
109	C₄×D₆₁	—	2	2	no	—
110	C₄×D₆₁	—	2	2	no	—
111	C₄×D₆₁	—	4	1	no	—
112	C₄×D₆₁	—	4	1	no	—
113	#92420	—	2	2	no	—
114	#92420	—	2	2	no	—
115	#92420	—	2	2	no	—
116	#92420	—	2	2	no	—
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	#117	1	2	yes	—
120	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₆₁	—	2	1	no	—
122	C₂×D₆₁	—	2	1	no	—
123	C₂×D₆₁	—	2	1	no	—
124	C₂×D₆₁	—	2	1	no	—
125	C₂×D₆₁	—	2	1	no	—
126	C₂×D₆₁	—	4	4	no	—
127	C₂×D₆₁	#131	1	2	yes	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	2	1	no	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	—	2	1	no	—
138	Dic₆₁	#117	1	2	yes	—
139	Dic₆₁	C₂×#41	1	2	yes	—
140	#2924	—	61	1	no	maximal, Sylow
141	C₂×C₆₁	—	2	2	no	—
142	C₂×C₆₁	C₂×#117	1	1	yes	characteristic
143	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
144	C₂×C₆₁	#568	1	1	yes	characteristic
145	D₆₁	—	2	2	no	—
146	D₆₁	—	4	1	no	—
147	C₂×#117	—	61	1	no	—
148	#433	—	61	2	no	—
149	#433	—	61	2	no	—
150	#432	—	61	1	no	—
151	#432	—	61	1	no	—
152	#431	—	61	1	no	—
153	#431	—	61	1	no	—
154	#424	—	61	2	no	—
155	#407	—	61	1	no	—
156	C₂×C₂²⋊C₈	—	61	1	no	—
157	#405	—	61	2	no	—
158	C₆₁	#2924	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	61	1	no	—
160	C₂×M₁₆	—	61	2	no	—
161	C₂²×C₈	—	61	2	no	—
162	#117	—	61	1	no	—
163	#117	—	61	1	no	—
164	#117	—	61	1	no	—
165	#117	—	61	1	no	—
166	#117	—	61	1	no	—
167	#117	—	61	1	no	—
168	#117	—	61	1	no	—
169	#117	#6923	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	61	1	no	—
171	C₂×C₄⋊C₄	—	61	1	no	—
172	C₂×C₂²⋊C₄	—	61	1	no	—
173	C₂×C₂²⋊C₄	—	61	1	no	—
174	C₂×C₄²	—	61	1	no	—
175	C₂×C₄²	—	61	1	no	—
176	C₄⋊C₈	—	61	2	no	—
177	C₄⋊C₈	—	61	2	no	—
178	C₄⋊C₈	—	61	2	no	—
179	C₄⋊C₈	—	61	2	no	—
180	C₂²⋊C₈	—	61	1	no	—
181	C₂²⋊C₈	—	61	1	no	—
182	C₂²⋊C₈	—	61	2	no	—
183	C₂²⋊C₈	—	61	1	no	—
184	C₂²⋊C₈	—	61	1	no	—
185	C₂²⋊C₈	—	61	2	no	—
186	#97	—	61	2	no	—
187	#97	—	61	2	no	—
188	C₄×C₈	—	61	2	no	—
189	C₄×C₈	—	61	2	no	—
190	C₂⁴	—	61	1	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	61	1	no	—
193	C₂²×C₄	—	61	1	no	—
194	C₂²×C₄	—	61	1	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	61	1	no	—
197	C₂²×C₄	—	61	1	no	—
198	C₂²×C₄	—	61	1	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	122	4	no	—
201	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
202	M₁₆	—	122	4	no	—
203	C₂×C₈	—	61	2	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	—	61	2	no	—
206	C₂×C₈	—	61	2	no	—
207	C₂×C₈	—	122	4	no	—
208	C₄⋊C₄	—	61	1	no	—
209	C₄⋊C₄	—	61	1	no	—
210	C₄⋊C₄	—	61	1	no	—
211	C₄⋊C₄	—	61	1	no	—
212	C₄⋊C₄	—	61	1	no	—
213	C₄⋊C₄	—	61	1	no	—
214	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
216	C₂²⋊C₄	—	61	1	no	—
217	C₂²⋊C₄	—	61	1	no	—
218	C₂²⋊C₄	—	61	1	no	—
219	C₂²⋊C₄	—	61	1	no	—
220	C₂²⋊C₄	—	61	1	no	—
221	C₂²⋊C₄	—	61	1	no	—
222	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
223	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
224	C₄²	—	61	1	no	—
225	C₄²	—	61	1	no	—
226	C₄²	—	61	1	no	—
227	C₄²	—	61	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	61	1	no	—
230	C₄²	C₂×#6923	1	1	yes	characteristic
231	C₄²	C₂×#6923	1	1	yes	characteristic
232	C₂³	—	61	1	no	—
233	C₂³	—	61	1	no	—
234	C₂³	—	122	2	no	—
235	C₂³	—	122	2	no	—
236	C₂³	—	244	1	no	—
237	C₂³	C₂²×#6923	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	61	4	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	122	2	no	—
247	C₂×C₄	—	122	2	no	—
248	C₂×C₄	—	122	2	no	—
249	C₂×C₄	—	122	2	no	—
250	C₂×C₄	—	122	2	no	—
251	C₂×C₄	—	122	2	no	—
252	C₂×C₄	—	244	1	no	—
253	C₂×C₄	—	244	1	no	—
254	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
255	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
256	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
257	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
258	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
259	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
260	C₈	—	122	2	no	—
261	C₈	—	122	2	no	—
262	C₈	—	122	2	no	—
263	C₈	—	122	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	61	2	no	—
267	C₂²	—	122	1	no	—
268	C₂²	—	122	1	no	—
269	C₂²	—	122	1	no	—
270	C₂²	—	122	1	no	—
271	C₂²	—	122	1	no	—
272	C₂²	—	244	4	no	—
273	C₂²	#14519856	1	2	yes	—
274	C₂²	C₂³×#6923	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	61	2	no	—
282	C₄	—	61	2	no	—
283	C₄	—	122	1	no	—
284	C₄	—	122	1	no	—
285	C₄	—	122	1	no	—
286	C₄	—	122	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	122	2	no	—
289	C₂	—	244	1	no	—
290	C₂	C₂×#14519856	1	1	yes	characteristic
291	C₂	D₆₁⋊#131	1	1	yes	characteristic
292	C₂	#16840841	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	#117×C₆₁	C₄
hypercenter	#117	#6923
hypocenter	C₆₁	#2924
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #16840787	C₂	semidirect
2. #16840787	C₂	semidirect
3. #16840785	C₂	semidirect
4. #16840784	C₂	semidirect
5. #16840781	C₂	semidirect
6. #16840780	C₂	semidirect
7. #16840768	C₂	semidirect
10. #16840726	C₂	semidirect
14. #14519844	C₂²	semidirect
15. #14519844	C₂²	semidirect
16. C₄⋊#92420	C₂²	semidirect
17. C₄⋊#92420	C₂²	semidirect
24. #14519832	C₂²	semidirect
25. #14519832	C₂²	semidirect
26. C₄×#92420	C₂²	semidirect
27. C₄×#92420	C₂²	semidirect
158. C₆₁	#2924	semidirect