Subgroups — #21462649

Order 7808.

372 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840814	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840812	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840794	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840794	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840792	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840791	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840790	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840789	C₂	1	1	yes	maximal, characteristic, complemented
9	#16840787	C₂	1	1	yes	maximal, characteristic, complemented
10	#16840786	C₂	1	1	yes	maximal, characteristic, complemented
11	#16840781	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840727	C₂	1	1	yes	maximal, characteristic, complemented
13	#16840726	C₂	1	1	yes	maximal, characteristic, complemented
14	#16840721	C₂	1	1	yes	maximal, characteristic, complemented
15	#16840295	C₂	1	1	yes	maximal, characteristic
16	C₂×#11758213	C₂²	1	1	yes	characteristic
17	C₂×#11758217	C₂²	1	1	yes	characteristic
18	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×#92420	C₂²	1	1	yes	characteristic
22	#14519844	C₂²	1	1	yes	characteristic
23	#14519844	C₂²	1	1	yes	characteristic
24	C₄⋊#92420	C₂²	1	1	yes	characteristic
25	C₄⋊#92420	C₂²	1	1	yes	characteristic
26	C₄⋊#92420	C₂²	1	1	yes	characteristic
27	C₄⋊#92420	C₂²	1	1	yes	characteristic
28	#14519834	C₂²	1	1	yes	characteristic
29	#14519834	C₂²	1	1	yes	characteristic
30	#14519834	C₂²	1	1	yes	characteristic
31	#14519834	C₂²	1	1	yes	characteristic
32	C₂²⋊#92420	C₂²	1	1	yes	characteristic
33	C₂²⋊#92420	C₂²	1	1	yes	characteristic
34	#14519832	C₂²	1	1	yes	characteristic
35	#14519832	C₂²	1	1	yes	characteristic
36	#14519832	C₂²	1	1	yes	characteristic
37	#14519832	C₂²	1	1	yes	characteristic
38	#14519832	C₂²	1	1	yes	characteristic
39	#14519832	C₂²	1	1	yes	characteristic
40	C₄×#92420	C₂²	1	1	yes	characteristic
41	C₄×#92420	C₂²	1	1	yes	characteristic
42	C₄×#92420	C₂²	1	1	yes	characteristic
43	C₄×#92420	C₂²	1	1	yes	characteristic
44	#14519771	C₄	1	1	yes	characteristic
45	#14519767	C₄	1	1	yes	characteristic
46	#14519763	C₄	1	1	yes	characteristic
47	#14519762	C₄	1	1	yes	characteristic
48	#14519759	C₄	1	1	yes	characteristic
49	#14519755	C₄	1	1	yes	characteristic
50	#14519750	C₄	1	1	yes	characteristic
51	#14519742	C₂²	1	1	yes	characteristic
52	#14519739	C₂²	1	1	yes	characteristic
53	#14519737	C₂²	1	1	yes	characteristic
54	#14519737	C₂²	1	1	yes	characteristic
55	#14519735	C₂²	1	1	yes	characteristic
56	#14519733	C₂²	1	1	yes	characteristic
57	C₄²×D₆₁	C₂²	1	1	yes	characteristic
58	#126×C₆₁	C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
62	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
68	#11758217	—	2	2	no	—
69	C₂×#92420	—	2	2	no	—
70	C₂×#92420	C₂³	1	1	yes	characteristic
71	C₂×#92420	C₂³	1	1	yes	characteristic
72	C₂×#92420	C₂³	1	1	yes	characteristic
73	C₂×#92420	C₂³	1	1	yes	characteristic
74	C₂×#92420	C₂³	1	1	yes	characteristic
75	C₂×#92420	C₂³	1	1	yes	characteristic
76	C₂×#92420	C₂³	1	1	yes	characteristic
77	C₂×#92420	C₂³	1	1	yes	characteristic
78	#11758213	—	2	2	no	—
79	D₆₁⋊C₈	—	2	2	no	—
80	D₆₁⋊C₈	—	2	2	no	—
81	D₆₁⋊C₈	—	2	2	no	—
82	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
90	#11758198	C₂×C₄	1	1	yes	characteristic
91	#11758198	C₂×C₄	1	1	yes	characteristic
92	#11758198	C₂×C₄	1	1	yes	characteristic
93	#11758198	C₂×C₄	1	1	yes	characteristic
94	#11758198	C₂×C₄	1	1	yes	characteristic
95	#11758198	C₂×C₄	1	1	yes	characteristic
96	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
97	#11758196	C₂×C₄	1	1	yes	characteristic
98	#11758196	C₂×C₄	1	1	yes	characteristic
99	#11758196	C₂×C₄	1	1	yes	characteristic
100	#11758196	C₂×C₄	1	1	yes	characteristic
101	#11758196	C₂×C₄	1	1	yes	characteristic
102	#11758196	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
105	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
106	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
107	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
108	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
117	C₂²⋊D₆₁	—	2	2	no	—
118	C₂²⋊D₆₁	—	4	1	no	—
119	C₂×Dic₆₁	—	2	1	no	—
120	C₂×Dic₆₁	—	2	1	no	—
121	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
127	D₂₄₄	—	2	2	no	—
128	C₄×D₆₁	—	2	2	no	—
129	C₄×D₆₁	—	2	1	no	—
130	C₄×D₆₁	—	2	1	no	—
131	C₄×D₆₁	—	2	1	no	—
132	C₄×D₆₁	—	2	1	no	—
133	C₄×D₆₁	—	4	1	no	—
134	Dic₁₂₂	—	4	1	no	—
135	#92420	—	2	1	no	—
136	#92420	—	2	1	no	—
137	#92420	—	2	1	no	—
138	#92420	—	2	1	no	—
139	#92420	—	2	1	no	—
140	#92420	—	2	1	no	—
141	#92420	#41	1	2	yes	—
142	#92420	#41	1	2	yes	—
143	C₂²×C₆₁	—	2	1	no	—
144	C₂²×C₆₁	—	2	1	no	—
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₆₁	—	2	1	no	—
148	C₂×D₆₁	—	2	1	no	—
149	C₂×D₆₁	—	2	1	no	—
150	C₂×D₆₁	—	2	1	no	—
151	C₂×D₆₁	—	2	1	no	—
152	C₂×D₆₁	#131	1	2	yes	—
153	C₄×C₆₁	—	2	1	no	—
154	C₄×C₆₁	—	2	1	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	2	1	no	—
159	Dic₆₁	—	2	1	no	—
160	Dic₆₁	—	2	1	no	—
161	Dic₆₁	—	2	1	no	—
162	Dic₆₁	—	2	1	no	—
163	Dic₆₁	—	2	1	no	—
164	Dic₆₁	C₂×#41	1	1	yes	characteristic
165	Dic₆₁	C₂×#41	1	1	yes	characteristic
166	#2968	—	61	1	no	maximal, Sylow
167	C₂×C₆₁	—	4	1	no	—
168	C₂×C₆₁	C₄×#41	1	1	yes	characteristic
169	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
170	C₂×C₆₁	#568	1	1	yes	characteristic
171	D₆₁	—	2	2	no	—
172	D₆₁	—	4	1	no	—
173	#529	—	61	1	no	—
174	#446	—	61	1	no	—
175	#446	—	61	1	no	—
176	#435	—	61	1	no	—
177	#435	—	61	1	no	—
178	#435	—	61	1	no	—
179	#435	—	61	1	no	—
180	C₈×D₄	—	61	1	no	—
181	C₈×D₄	—	61	1	no	—
182	#433	—	61	1	no	—
183	#433	—	61	1	no	—
184	#431	—	61	1	no	—
185	#405	—	61	1	no	—
186	#405	—	61	1	no	—
187	C₂×#97	—	61	1	no	—
188	C₆₁	#2968	1	1	yes	characteristic, complemented, Sylow
189	C₂×M₁₆	—	61	1	no	—
190	C₂×M₁₆	—	61	1	no	—
191	C₂²×C₈	—	61	1	no	—
192	C₂²×C₈	—	61	1	no	—
193	C₂²×C₈	—	61	1	no	—
194	C₂²×C₈	—	61	1	no	—
195	#126	—	61	1	no	—
196	#126	#6923	1	1	yes	characteristic
197	#125	—	61	1	no	—
198	#124	—	61	1	no	—
199	#123	—	61	1	no	—
200	#123	—	61	1	no	—
201	C₂²⋊Q₈	—	61	1	no	—
202	#121	—	61	1	no	—
203	C₄×Q₈	—	61	1	no	—
204	C₄×D₄	—	61	1	no	—
205	C₄×D₄	—	61	1	no	—
206	C₄×D₄	—	61	1	no	—
207	#117	—	61	1	no	—
208	#117	—	61	1	no	—
209	C₂×C₄²	—	61	1	no	—
210	C₄⋊C₈	—	61	1	no	—
211	C₄⋊C₈	—	61	1	no	—
212	C₄⋊C₈	—	61	1	no	—
213	C₄⋊C₈	—	61	1	no	—
214	C₄⋊C₈	—	61	1	no	—
215	C₄⋊C₈	—	61	1	no	—
216	C₂²⋊C₈	—	61	1	no	—
217	C₂²⋊C₈	—	61	1	no	—
218	C₂²⋊C₈	—	61	1	no	—
219	C₂²⋊C₈	—	61	1	no	—
220	C₂²⋊C₈	—	61	1	no	—
221	C₂²⋊C₈	—	61	1	no	—
222	#97	—	61	1	no	—
223	#97	—	61	1	no	—
224	#97	—	61	1	no	—
225	#97	—	61	1	no	—
226	#97	—	61	1	no	—
227	#97	—	61	1	no	—
228	C₄×C₈	—	61	1	no	—
229	C₄×C₈	—	61	1	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	1	no	—
232	C₂×Q₈	—	61	1	no	—
233	C₂×D₄	—	61	1	no	—
234	C₂×D₄	—	61	1	no	—
235	C₂×D₄	—	61	1	no	—
236	C₂²×C₄	—	61	1	no	—
237	C₂²×C₄	—	61	1	no	—
238	C₂²×C₄	—	61	1	no	—
239	C₂²×C₄	—	61	1	no	—
240	C₂²×C₄	—	61	1	no	—
241	M₁₆	—	122	2	no	—
242	M₁₆	—	122	2	no	—
243	C₂×C₈	—	61	1	no	—
244	C₂×C₈	—	61	1	no	—
245	C₂×C₈	—	61	1	no	—
246	C₂×C₈	—	61	1	no	—
247	C₂×C₈	—	61	1	no	—
248	C₂×C₈	—	61	1	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	1	no	—
251	C₂×C₈	—	122	2	no	—
252	C₂×C₈	—	122	2	no	—
253	C₂×C₈	—	122	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₄⋊C₄	—	61	1	no	—
256	C₄⋊C₄	—	61	1	no	—
257	C₄⋊C₄	—	61	1	no	—
258	C₄⋊C₄	—	61	1	no	—
259	C₄⋊C₄	—	61	1	no	—
260	C₄⋊C₄	—	61	1	no	—
261	C₄⋊C₄	—	61	1	no	—
262	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
263	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
265	C₂²⋊C₄	—	61	1	no	—
266	C₂²⋊C₄	—	61	1	no	—
267	C₂²⋊C₄	—	61	1	no	—
268	C₂²⋊C₄	—	61	1	no	—
269	C₂²⋊C₄	—	61	1	no	—
270	C₂²⋊C₄	—	61	1	no	—
271	C₂²⋊C₄	—	61	1	no	—
272	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
275	C₄²	—	61	1	no	—
276	C₄²	—	61	1	no	—
277	C₄²	—	61	1	no	—
278	C₄²	—	61	1	no	—
279	C₄²	—	61	1	no	—
280	C₄²	C₂×#6923	1	1	yes	characteristic
281	C₂³	—	61	1	no	—
282	C₂³	—	61	1	no	—
283	C₂³	C₂²×#6923	1	1	yes	characteristic
284	Q₈	—	244	1	no	—
285	D₄	—	122	2	no	—
286	D₄	—	122	2	no	—
287	D₄	—	244	1	no	—
288	C₂×C₄	—	61	1	no	—
289	C₂×C₄	—	61	1	no	—
290	C₂×C₄	—	61	1	no	—
291	C₂×C₄	—	61	1	no	—
292	C₂×C₄	—	61	1	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	122	1	no	—
295	C₂×C₄	—	122	1	no	—
296	C₂×C₄	—	122	1	no	—
297	C₂×C₄	—	122	1	no	—
298	C₂×C₄	—	122	2	no	—
299	C₂×C₄	—	122	1	no	—
300	C₂×C₄	—	122	1	no	—
301	C₂×C₄	—	244	1	no	—
302	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
303	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
304	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
305	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
306	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
307	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
308	C₈	—	61	2	no	—
309	C₈	—	61	2	no	—
310	C₈	—	122	1	no	—
311	C₈	—	122	1	no	—
312	C₈	—	122	1	no	—
313	C₈	—	122	1	no	—
314	C₈	—	122	1	no	—
315	C₈	—	122	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	61	2	no	—
320	C₂²	—	122	1	no	—
321	C₂²	—	122	1	no	—
322	C₂²	—	122	1	no	—
323	C₂²	—	122	1	no	—
324	C₂²	—	122	1	no	—
325	C₂²	C₂³×#6923	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	61	1	no	—
333	C₄	—	61	1	no	—
334	C₄	—	122	1	no	—
335	C₄	—	122	1	no	—
336	C₄	—	122	1	no	—
337	C₄	—	122	1	no	—
338	C₄	—	122	1	no	—
339	C₂	—	4	1	no	—
340	C₂	—	122	2	no	—
341	C₂	—	244	1	no	—
342	C₂	#41×#6923	1	1	yes	characteristic
343	C₂	D₆₁⋊#131	1	1	yes	characteristic
344	C₂	#16840841	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	#126×C₆₁	C₄
hypercenter	#126	#6923
hypocenter	C₆₁	#2968
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. #16840814	C₂	semidirect
2. #16840812	C₂	semidirect
3. #16840794	C₂	semidirect
4. #16840794	C₂	semidirect
5. #16840792	C₂	semidirect
6. #16840791	C₂	semidirect
7. #16840790	C₂	semidirect
8. #16840789	C₂	semidirect
9. #16840787	C₂	semidirect
10. #16840786	C₂	semidirect
11. #16840781	C₂	semidirect
12. #16840727	C₂	semidirect
13. #16840726	C₂	semidirect
14. #16840721	C₂	semidirect
188. C₆₁	#2968	semidirect