Subgroups — #21462744

Order 7808.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840814	C₂	1	2	yes	maximal, complemented
2	#16840794	C₂	1	2	yes	maximal, complemented
3	#16840793	C₂	1	2	yes	maximal, complemented
4	#16840789	C₂	1	2	yes	maximal, complemented
5	#16840784	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840783	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840778	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840765	C₂	1	1	yes	maximal, characteristic, complemented
9	D₆₁⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#11758224	C₂	1	1	yes	maximal, characteristic
12	C₂×#11758217	C₂²	1	2	yes	—
13	C₂×D₆₁⋊C₈	C₂²	1	2	yes	—
14	C₂²×#92420	C₂²	1	2	yes	—
15	#14519844	C₂²	1	2	yes	—
16	#14519844	C₂²	1	2	yes	—
17	C₄⋊#92420	C₂²	1	1	yes	characteristic
18	C₄⋊#92420	C₂²	1	1	yes	characteristic
19	C₄⋊#92420	C₂²	1	2	yes	—
20	C₄⋊#92420	C₂²	1	1	yes	characteristic
21	C₄⋊#92420	C₂²	1	1	yes	characteristic
22	#14519834	C₂²	1	2	yes	—
23	C₂²⋊#92420	C₂²	1	2	yes	—
24	C₂²⋊#92420	C₂²	1	2	yes	—
25	#14519832	C₂²	1	2	yes	—
26	#14519832	C₂²	1	2	yes	—
27	C₄×#92420	C₂²	1	2	yes	—
28	C₂×#11758224	C₄	1	1	yes	characteristic
29	C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
30	#14519742	C₂²	1	1	yes	characteristic
31	D₄×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
34	#14519737	C₂²	1	1	yes	characteristic
35	#14519737	C₂²	1	1	yes	characteristic
36	#14519735	C₄	1	1	yes	characteristic
37	#14519733	C₄	1	1	yes	characteristic
38	#14519733	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
41	C₄²×D₆₁	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
43	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
52	#11758224	—	2	4	no	—
53	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
57	#11758217	—	4	2	no	—
58	C₂×#92420	—	4	2	no	—
59	C₂×#92420	C₂³	1	2	yes	—
60	C₂×#92420	C₂³	1	2	yes	—
61	C₂×#92420	C₂³	1	2	yes	—
62	C₂×#92420	C₂³	1	2	yes	—
63	D₆₁⋊C₈	—	4	2	no	—
64	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
70	#11758198	C₂×C₄	1	1	yes	characteristic
71	#11758198	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
73	#11758196	C₂×C₄	1	1	yes	characteristic
74	#11758196	C₂×C₄	1	1	yes	characteristic
75	#11758196	C₂×C₄	1	1	yes	characteristic
76	#11758196	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₆₁	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	C₂×C₄	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	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
86	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
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	C₂²⋊D₆₁	—	2	2	no	—
98	C₂²⋊D₆₁	M₁₆	1	4	yes	complemented
99	C₂×Dic₆₁	—	2	2	no	—
100	C₂×Dic₆₁	—	2	2	no	—
101	C₂×Dic₆₁	—	2	1	no	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	2	1	no	—
104	C₂×Dic₆₁	—	2	1	no	—
105	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
112	C₄×D₆₁	—	2	2	no	—
113	C₄×D₆₁	—	2	2	no	—
114	C₄×D₆₁	—	2	1	no	—
115	C₄×D₆₁	—	2	1	no	—
116	Dic₁₂₂	—	2	2	no	—
117	#92420	—	2	2	no	—
118	#92420	—	2	2	no	—
119	#92420	—	2	2	no	—
120	#92420	—	2	2	no	—
121	C₂²×C₆₁	—	2	1	no	—
122	C₂²×C₆₁	—	2	1	no	—
123	C₂²×C₆₁	—	2	1	no	—
124	C₂²×C₆₁	—	2	1	no	—
125	C₂²×C₆₁	—	2	1	no	—
126	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
127	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×D₆₁	—	2	1	no	—
129	C₂×D₆₁	—	2	1	no	—
130	C₂×D₆₁	C₂×M₁₆	1	2	yes	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	#131	1	2	yes	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	—	2	1	no	—
138	Dic₆₁	—	2	1	no	—
139	Dic₆₁	—	2	1	no	—
140	Dic₆₁	—	2	1	no	—
141	Dic₆₁	C₂×M₁₆	1	2	yes	—
142	Dic₆₁	H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₆₁	2₋¹⁺⁴	1	1	yes	characteristic
144	#2990	—	61	1	no	maximal, Sylow
145	C₂×C₆₁	—	2	2	no	—
146	C₂×C₆₁	—	4	1	no	—
147	C₂×C₆₁	#520	1	1	yes	characteristic
148	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
150	D₆₁	—	2	2	no	—
151	C₄×#41	—	61	1	no	—
152	#446	—	61	2	no	—
153	#435	—	61	2	no	—
154	#435	—	61	2	no	—
155	C₈×D₄	—	61	2	no	—
156	#432	—	61	1	no	—
157	#432	—	61	1	no	—
158	#431	—	61	1	no	—
159	#423	—	61	1	no	—
160	C₂×C₄⋊C₈	—	61	1	no	—
161	C₂×C₄⋊C₈	—	61	1	no	—
162	C₆₁	#2990	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	61	1	no	—
164	C₂×M₁₆	—	61	2	no	—
165	C₂²×C₈	—	61	2	no	—
166	C₂²×C₈	—	61	2	no	—
167	C₄×Q₈	—	61	1	no	—
168	C₄×Q₈	—	61	1	no	—
169	C₄×D₄	—	61	1	no	—
170	C₄×D₄	—	61	1	no	—
171	C₄×D₄	—	61	1	no	—
172	C₄×D₄	—	61	1	no	—
173	C₄×D₄	—	61	1	no	—
174	C₄×D₄	#6923	1	1	yes	characteristic
175	#117	—	61	1	no	—
176	#117	—	61	1	no	—
177	#117	—	61	1	no	—
178	C₂×C₄²	—	61	1	no	—
179	C₂×C₄²	—	61	1	no	—
180	C₂×C₄²	—	61	1	no	—
181	C₄⋊C₈	—	61	1	no	—
182	C₄⋊C₈	—	61	1	no	—
183	C₄⋊C₈	—	61	1	no	—
184	C₄⋊C₈	—	61	2	no	—
185	C₄⋊C₈	—	61	1	no	—
186	C₄⋊C₈	—	61	2	no	—
187	C₄⋊C₈	—	61	2	no	—
188	C₂²⋊C₈	—	61	2	no	—
189	C₂²⋊C₈	—	61	2	no	—
190	C₂²⋊C₈	—	61	2	no	—
191	#97	—	61	2	no	—
192	#97	—	61	2	no	—
193	C₄×C₈	—	61	2	no	—
194	#41	—	122	4	no	—
195	C₂×Q₈	—	61	1	no	—
196	C₂×D₄	—	61	1	no	—
197	C₂×D₄	—	61	1	no	—
198	C₂×D₄	C₂×#6923	1	1	yes	characteristic
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	61	1	no	—
205	C₂²×C₄	—	61	1	no	—
206	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
207	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
208	M₁₆	—	244	2	no	—
209	C₂×C₈	—	61	2	no	—
210	C₂×C₈	—	61	2	no	—
211	C₂×C₈	—	61	2	no	—
212	C₂×C₈	—	61	2	no	—
213	C₂×C₈	—	244	2	no	—
214	C₂×C₈	—	244	2	no	—
215	C₄⋊C₄	—	61	1	no	—
216	C₄⋊C₄	—	61	1	no	—
217	C₄⋊C₄	—	61	1	no	—
218	C₄⋊C₄	—	61	1	no	—
219	C₄⋊C₄	—	61	1	no	—
220	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
221	C₂²⋊C₄	—	61	1	no	—
222	C₂²⋊C₄	—	61	1	no	—
223	C₂²⋊C₄	—	61	1	no	—
224	C₂²⋊C₄	—	61	1	no	—
225	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#6923	1	1	yes	characteristic
227	C₄²	—	61	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	61	1	no	—
230	C₄²	—	61	1	no	—
231	C₄²	—	61	1	no	—
232	C₄²	—	61	1	no	—
233	C₄²	—	61	1	no	—
234	C₄²	—	61	1	no	—
235	C₄²	—	61	1	no	—
236	C₄²	C₂×#6923	1	1	yes	characteristic
237	C₂³	—	61	1	no	—
238	C₂³	C₂²×#6923	1	1	yes	characteristic
239	C₂³	C₂²×#6923	1	1	yes	characteristic
240	Q₈	—	122	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	61	4	no	—
243	D₄	—	122	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	61	1	no	—
250	C₂×C₄	—	61	1	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	61	1	no	—
253	C₂×C₄	—	122	1	no	—
254	C₂×C₄	—	122	1	no	—
255	C₂×C₄	—	122	2	no	—
256	C₂×C₄	—	122	1	no	—
257	C₂×C₄	—	122	1	no	—
258	C₂×C₄	—	122	1	no	—
259	C₂×C₄	—	122	1	no	—
260	C₂×C₄	—	122	2	no	—
261	C₂×C₄	—	122	2	no	—
262	C₂×C₄	—	122	2	no	—
263	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
264	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
265	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
266	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
267	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
268	C₈	—	122	2	no	—
269	C₈	—	122	2	no	—
270	C₈	—	122	2	no	—
271	C₈	—	122	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	61	2	no	—
278	C₂²	—	122	1	no	—
279	C₂²	—	122	1	no	—
280	C₂²	D₆₁⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#6923	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	61	1	no	—
287	C₄	—	61	1	no	—
288	C₄	—	61	2	no	—
289	C₄	—	122	1	no	—
290	C₄	—	122	1	no	—
291	C₄	—	122	1	no	—
292	C₄	—	122	1	no	—
293	C₄	—	122	1	no	—
294	C₄	#14519864	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	122	2	no	—
298	C₂	#16840831	1	1	yes	characteristic
299	C₂	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
300	C₂	C₂×#14519864	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×D₄×C₆₁	C₄
hypercenter	C₄×D₄	#6923
hypocenter	C₆₁	#2990
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #16840814	C₂	semidirect
2. #16840794	C₂	semidirect
3. #16840793	C₂	semidirect
4. #16840789	C₂	semidirect
5. #16840784	C₂	semidirect
6. #16840783	C₂	semidirect
7. #16840778	C₂	semidirect
8. #16840765	C₂	semidirect
9. D₆₁⋊(C₄⋊C₈)	C₂	semidirect
10. C₂×C₄⋊#92420	C₂	semidirect
98. C₂²⋊D₆₁	M₁₆	semidirect
162. C₆₁	#2990	semidirect