Subgroups — #37574832

Order 24448.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₀₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#27132500	C₂	1	2	yes	maximal, complemented
3	#27132498	C₂	1	2	yes	maximal, complemented
4	#27132497	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₅₂₈	—	2	1	no	—
7	D₃₀₅₆	—	2	1	no	—
8	D₃₀₅₆	C₂²	1	2	yes	—
9	C₁₉₁⋊C₃₂	C₂²	1	2	yes	complemented
10	#135×C₁₉₁	—	2	1	no	—
11	Q₃₂×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
12	SD₃₂×C₁₉₁	—	2	1	no	—
13	D₁₆×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×D₇₆₄	—	4	1	no	—
16	D₁₅₂₈	—	2	2	no	—
17	D₁₅₂₈	—	4	1	no	—
18	#41×C₁₉₁	—	4	1	no	—
19	Q₁₆×C₁₉₁	—	2	1	no	—
20	SD₁₆×C₁₉₁	—	4	1	no	—
21	D₈×C₁₉₁	—	2	1	no	—
22	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
23	C₁₆×C₁₉₁	D₄	1	1	yes	characteristic
24	C₁₆×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₂²×D₁₉₁	—	8	1	no	—
26	D₇₆₄	—	4	2	no	—
27	D₇₆₄	—	8	1	no	—
28	Q₈×C₁₉₁	—	4	1	no	—
29	D₄×C₁₉₁	—	4	1	no	—
30	D₄×C₁₉₁	—	8	1	no	—
31	C₂×C₄×C₁₉₁	—	8	1	no	—
32	C₂×C₄×C₁₉₁	D₈	1	1	yes	characteristic
33	C₈×C₁₉₁	D₈	1	1	yes	characteristic
34	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₉₁	—	8	1	no	—
36	C₂²×C₁₉₁	D₁₆	1	1	yes	characteristic
37	C₂×D₁₉₁	—	8	2	no	—
38	C₂×D₁₉₁	—	16	2	no	—
39	C₄×C₁₉₁	—	8	1	no	—
40	C₄×C₁₉₁	D₁₆	1	1	yes	characteristic
41	C₄×C₁₉₁	C₂×D₈	1	1	yes	characteristic
42	C₂×C₁₉₁	—	2	1	no	—
43	C₂×C₁₉₁	—	16	1	no	—
44	C₂×C₁₉₁	C₂×D₁₆	1	1	yes	characteristic
45	D₁₉₁	—	16	2	no	—
46	C₁₉₁	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	191	1	no	maximal, Sylow
48	#508	D₁₉₁	1	1	yes	characteristic, complemented
49	C₂×D₁₆	—	191	1	no	—
50	SD₆₄	—	191	2	no	—
51	D₃₂	—	191	2	no	—
52	M₆₄	—	191	1	no	—
53	#135	—	2	1	no	—
54	C₂×D₈	—	382	1	no	—
55	Q₃₂	C₂×D₁₉₁	1	1	yes	characteristic, complemented
56	SD₃₂	—	2	1	no	—
57	D₁₆	—	382	1	no	—
58	D₁₆	—	191	2	no	—
59	D₁₆	C₂×D₁₉₁	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
61	C₃₂	—	191	2	no	—
62	#41	—	4	1	no	—
63	C₂×D₄	—	764	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	D₈	—	382	2	no	—
68	D₈	—	764	1	no	—
69	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
70	C₁₆	C₂²⋊D₁₉₁	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₉₁	1	1	yes	characteristic
72	C₂³	—	1528	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	8	1	no	—
76	D₄	—	1528	1	no	—
77	D₄	—	764	2	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#15690272	1	1	yes	characteristic
80	C₈	#15690272	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	3056	2	no	—
84	C₂²	—	1528	2	no	—
85	C₂²	#20259263	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	#20259263	1	1	yes	characteristic
88	C₄	C₂×#15690272	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	1	no	—
91	C₂	—	3056	2	no	—
92	C₂	C₂×#20259263	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#20259263
commutator, cosocle	C₁₆×C₁₉₁	C₂³
Frattini	C₁₆	C₂²×D₁₉₁
Fitting	#508×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂×D₁₆
hypercenter	#508	D₁₉₁
hypocenter	C₁₉₁	#2263
LCS	G ▷ C₁₆×C₁₉₁ ▷ C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₀₅₆	C₂	semidirect
2. #27132500	C₂	semidirect
3. #27132498	C₂	semidirect
4. #27132497	C₂	semidirect
5. #508×C₁₉₁	C₂	semidirect
9. C₁₉₁⋊C₃₂	C₂²	semidirect
11. Q₃₂×C₁₉₁	C₂²	semidirect
13. D₁₆×C₁₉₁	C₂²	semidirect
46. C₁₉₁	#2263	semidirect
48. #508	D₁₉₁	semidirect
55. Q₃₂	C₂×D₁₉₁	semidirect
59. D₁₆	C₂×D₁₉₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₃₀₅₆	2
#27132500	2
#27132498	2
#27132497	2
#508×C₁₉₁	2
#2263	191