Subgroups — #37572127

Order 24448.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133040	C₂	1	1	yes	maximal, characteristic
2	#27133011	C₂	1	4	yes	maximal, complemented
3	C₈×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₅₂₈	—	2	2	no	—
6	#20259319	C₂²	1	2	yes	—
7	#20259242	—	2	8	no	—
8	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₉₁	—	2	2	no	—
10	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
11	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×D₇₆₄	—	2	2	no	—
13	C₂×D₇₆₄	—	2	4	no	—
14	D₁₅₂₈	—	4	4	no	—
15	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
16	C₂×Q₈×C₁₉₁	—	2	2	no	—
17	Q₁₆×C₁₉₁	—	4	4	no	—
18	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₉₁	—	2	4	no	—
20	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
21	C₂²×D₁₉₁	—	4	2	no	—
22	D₇₆₄	—	4	4	no	—
23	D₇₆₄	—	4	8	no	—
24	C₁₉₁⋊C₈	—	2	4	no	—
25	Q₈×C₁₉₁	—	4	4	no	—
26	C₂×C₄×C₁₉₁	—	4	2	no	—
27	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₉₁	D₈	1	4	yes	—
30	C₂²×C₁₉₁	#127	1	1	yes	characteristic
31	C₂×D₁₉₁	—	8	2	no	—
32	C₂×D₁₉₁	—	8	4	no	—
33	C₄×C₁₉₁	—	8	2	no	—
34	C₄×C₁₉₁	C₂×D₈	1	2	yes	—
35	C₄×C₁₉₁	#135	1	4	yes	—
36	C₂×C₁₉₁	#493	1	1	yes	characteristic
37	C₂×C₁₉₁	#495	1	2	yes	—
38	D₁₉₁	—	16	2	no	—
39	C₁₉₁	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	191	1	no	maximal, Sylow
41	#494	D₁₉₁	1	1	yes	characteristic, complemented
42	#493	—	191	1	no	—
43	#486	—	191	4	no	—
44	C₈²	—	191	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	382	2	no	—
47	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
48	#127	—	191	2	no	—
49	#102	—	382	8	no	—
50	C₄×C₈	—	191	2	no	—
51	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	382	4	no	—
54	C₂×D₄	—	382	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	764	4	no	—
57	C₂×C₈	—	191	4	no	—
58	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
61	C₂³	—	764	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	764	4	no	—
64	D₄	—	764	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
68	C₈	—	382	4	no	—
69	C₈	#15690272	1	4	yes	—
70	C₂²	—	1528	2	no	—
71	C₂²	—	1528	4	no	—
72	C₂²	#20259318	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#15690272	1	2	yes	—
75	C₄	#20259345	1	4	yes	—
76	C₂	—	3056	2	no	—
77	C₂	#27133039	1	1	yes	characteristic
78	C₂	#27133048	1	2	yes	—

	subgroup	G/H
center	C₂²	#20259318
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#494×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#127
prime radical	#27133040	C₂
hypercenter	#494	D₁₉₁
hypocenter	C₁₉₁	#1709
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #27133011	C₂	semidirect
3. C₈×C₁₉₁⋊C₈	C₂	semidirect
4. #494×C₁₉₁	C₂	semidirect
39. C₁₉₁	#1709	semidirect
41. #494	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133040	2
#27133011	2
C₈×C₁₉₁⋊C₈	2
#494×C₁₉₁	2
#1709	191