Subgroups — #21454970

Order 7808.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840296	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839841	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839837	C₂	1	2	yes	maximal, complemented
6	#16839836	C₂	1	2	yes	maximal, complemented
7	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839822	C₂	1	2	yes	maximal, complemented
9	#16839787	C₂	1	2	yes	maximal, complemented
10	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11758194	C₂²	1	2	yes	—
13	C₂×#11758189	C₂²	1	2	yes	—
14	C₂×C₈×D₆₁	C₂²	1	2	yes	—
15	#14519771	C₄	1	2	yes	—
16	#14519769	C₄	1	1	yes	characteristic
17	#14519765	C₄	1	1	yes	characteristic
18	#14519760	C₄	1	2	yes	—
19	#14519756	C₄	1	1	yes	characteristic
20	#14519751	C₄	1	1	yes	characteristic
21	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
22	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₆₁	C₂²	1	2	yes	—
24	#14519734	C₂²	1	2	yes	—
25	C₄⋊#92418	C₂²	1	2	yes	—
26	C₄⋊#92418	C₂²	1	1	yes	characteristic
27	C₄⋊#92418	C₂²	1	1	yes	characteristic
28	#14519706	C₂²	1	2	yes	—
29	#14519706	C₂²	1	2	yes	—
30	#14519689	C₂²	1	2	yes	—
31	#14519689	C₂²	1	2	yes	—
32	#14519687	C₂²	1	2	yes	—
33	C₈×Dic₆₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
45	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₆₁	C₂³	1	2	yes	—
47	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
57	#11758198	C₂×C₄	1	2	yes	—
58	#11758198	C₂×C₄	1	2	yes	—
59	#11758198	C₂×C₄	1	2	yes	—
60	#11758198	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
63	#11758196	C₂×C₄	1	2	yes	—
64	#11758196	C₂×C₄	1	2	yes	—
65	#11758196	C₂×C₄	1	2	yes	—
66	#11758196	C₂×C₄	1	2	yes	—
67	C₄×Dic₆₁	C₂³	1	2	yes	—
68	#11758194	—	4	2	no	—
69	C₂×#92418	C₂³	1	2	yes	—
70	C₂×#92418	C₂³	1	2	yes	—
71	#11758189	—	4	2	no	—
72	C₈×D₆₁	—	4	2	no	—
73	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	4	1	no	—
79	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₆₁	—	4	2	no	—
86	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
89	D₂₄₄	—	2	2	no	—
90	C₄×D₆₁	—	2	2	no	—
91	C₄×D₆₁	—	2	2	no	—
92	Dic₁₂₂	—	2	2	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	—	2	1	no	—
95	C₈×C₆₁	—	2	1	no	—
96	C₈×C₆₁	—	2	1	no	—
97	#92418	—	2	2	no	—
98	#92418	—	2	2	no	—
99	C₂²×C₆₁	—	2	1	no	—
100	C₂²×C₆₁	—	2	1	no	—
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₆₁	—	2	2	no	—
104	C₂×D₆₁	—	2	2	no	—
105	C₂×D₆₁	—	2	2	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	#131	1	2	yes	—
111	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₆₁	—	2	2	no	—
114	Dic₆₁	—	2	2	no	—
115	Dic₆₁	—	2	2	no	—
116	#2993	—	61	1	no	maximal, Sylow
117	C₂×C₆₁	—	4	1	no	—
118	C₂×C₆₁	#520	1	1	yes	characteristic
119	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
120	C₂×C₆₁	#568	1	1	yes	characteristic
121	D₆₁	—	4	2	no	—
122	#529	—	61	1	no	—
123	#446	—	61	1	no	—
124	C₈×Q₈	—	61	1	no	—
125	#436	—	61	2	no	—
126	#436	—	61	1	no	—
127	#435	—	61	2	no	—
128	C₈×D₄	—	61	1	no	—
129	#433	—	61	2	no	—
130	#431	D₆₁	1	1	yes	characteristic, complemented
131	#424	—	61	2	no	—
132	#423	—	61	1	no	—
133	C₆₁	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	61	2	no	—
135	C₂×M₁₆	—	61	2	no	—
136	C₂²×C₈	—	61	2	no	—
137	#126	—	61	2	no	—
138	#125	—	61	1	no	—
139	#124	—	61	1	no	—
140	#123	—	61	2	no	—
141	C₂²⋊Q₈	—	61	1	no	—
142	#121	—	61	1	no	—
143	C₄×Q₈	—	61	1	no	—
144	C₄×D₄	—	61	2	no	—
145	C₄×D₄	—	61	1	no	—
146	#117	—	61	2	no	—
147	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
148	C₄⋊C₈	—	61	2	no	—
149	C₄⋊C₈	—	61	2	no	—
150	C₄⋊C₈	—	61	2	no	—
151	C₄⋊C₈	—	61	1	no	—
152	C₄⋊C₈	—	61	1	no	—
153	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
155	C₂²⋊C₈	—	61	2	no	—
156	C₂²⋊C₈	—	61	2	no	—
157	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
159	#97	—	61	2	no	—
160	C₄×C₈	—	61	2	no	—
161	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
162	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
163	C₂×Q₈	—	61	1	no	—
164	C₂×D₄	—	61	2	no	—
165	C₂×D₄	—	61	1	no	—
166	C₂²×C₄	—	61	2	no	—
167	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
170	M₁₆	—	244	2	no	—
171	M₁₆	—	244	2	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	244	2	no	—
175	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
179	C₄⋊C₄	—	61	2	no	—
180	C₄⋊C₄	—	61	2	no	—
181	C₄⋊C₄	—	61	2	no	—
182	C₄⋊C₄	—	61	2	no	—
183	C₄⋊C₄	—	61	1	no	—
184	C₄⋊C₄	—	61	1	no	—
185	C₂²⋊C₄	—	61	2	no	—
186	C₂²⋊C₄	—	61	2	no	—
187	C₂²⋊C₄	—	61	2	no	—
188	C₂²⋊C₄	—	61	2	no	—
189	C₂²⋊C₄	—	61	2	no	—
190	C₄²	—	61	2	no	—
191	C₄²	C₄×D₆₁	1	1	yes	characteristic
192	C₄²	C₄×D₆₁	1	1	yes	characteristic
193	C₄²	C₂²×D₆₁	1	1	yes	characteristic
194	C₄²	C₂²×D₆₁	1	1	yes	characteristic
195	C₂³	—	61	2	no	—
196	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
197	Q₈	—	122	2	no	—
198	D₄	—	122	2	no	—
199	D₄	—	244	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	61	2	no	—
205	C₂×C₄	—	61	2	no	—
206	C₂×C₄	—	61	2	no	—
207	C₂×C₄	—	122	2	no	—
208	C₂×C₄	—	122	2	no	—
209	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	122	2	no	—
220	C₈	—	122	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	122	2	no	—
225	C₂²	—	122	2	no	—
226	C₂²	—	122	2	no	—
227	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	122	2	no	—
233	C₄	—	122	2	no	—
234	C₄	—	122	2	no	—
235	C₄	#14519783	1	2	yes	—
236	C₄	#14519823	1	1	yes	characteristic
237	C₄	#14519824	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	244	2	no	—
240	C₂	#16840242	1	1	yes	characteristic
241	C₂	C₂×#14519783	1	1	yes	characteristic
242	C₂	#16840528	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#431	C₂
hypercenter	#431	D₆₁
hypocenter	C₆₁	#2993
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. C₈×Dic₁₂₂	C₂	semidirect
4. #16839841	C₂	semidirect
5. #16839837	C₂	semidirect
6. #16839836	C₂	semidirect
7. C₈×D₂₄₄	C₂	semidirect
8. #16839822	C₂	semidirect
9. #16839787	C₂	semidirect
10. #16839786	C₂	semidirect
11. C₆₁×#431	C₂	semidirect
130. #431	D₆₁	semidirect
133. C₆₁	#2993	semidirect