Subgroups — #27156867

Order 12416.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261640	C₂	1	2	yes	maximal, complemented
2	#20261111	C₂	1	1	yes	maximal, characteristic
3	#15690869	—	2	2	no	—
4	C₂×#13095496	—	2	2	no	—
5	#15690807	C₂²	1	1	yes	characteristic
6	C₄×#11715895	C₄	1	1	yes	characteristic, complemented
7	#127×C₉₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#87604	—	4	2	no	—
9	C₂×D₃₈₈	—	4	1	no	—
10	C₂×D₃₈₈	D₄	1	2	yes	—
11	#13095496	—	4	4	no	—
12	C₂×#11715895	—	2	1	no	—
13	C₂×D₄×C₉₇	—	2	1	no	—
14	C₂×D₄×C₉₇	—	4	1	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₉₇	—	4	1	no	—
17	C₂²×D₉₇	—	2	2	no	—
18	C₂×#87604	—	8	2	no	—
19	D₄×C₉₇	—	4	2	no	—
20	D₄×C₉₇	—	8	2	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	D₃₈₈	—	4	2	no	—
24	D₃₈₈	—	8	2	no	—
25	#11715895	—	2	2	no	—
26	C₂²×C₉₇	—	8	2	no	—
27	C₂²×C₉₇	—	8	1	no	—
28	C₂²×C₉₇	#99	1	1	yes	characteristic
29	C₂×D₉₇	—	4	2	no	—
30	C₂×D₉₇	—	8	2	no	—
31	#87604	—	16	2	no	—
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	—	2	1	no	—
34	C₂×C₉₇	—	2	1	no	—
35	C₂×C₉₇	—	16	1	no	—
36	C₂×C₉₇	#353	1	1	yes	characteristic
37	D₉₇	—	8	2	no	—
38	#1408	—	97	1	no	maximal, Sylow
39	C₉₇	#1408	1	1	yes	characteristic, complemented, Sylow
40	#493	—	97	1	no	—
41	#353	—	97	2	no	—
42	C₂×D₈	—	194	2	no	—
43	#127	—	97	1	no	—
44	#127	#87604	1	1	yes	characteristic, complemented
45	#99	—	194	2	no	—
46	C₄×C₈	—	97	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	97	2	no	—
50	C₂×D₄	—	388	1	no	—
51	D₈	—	388	4	no	—
52	C₂×C₈	—	194	1	no	—
53	C₂²⋊C₄	—	388	2	no	—
54	C₄²	C₂×#87604	1	1	yes	characteristic
55	C₂³	—	4	1	no	—
56	C₂³	—	194	2	no	—
57	D₄	—	4	2	no	—
58	D₄	—	8	2	no	—
59	D₄	—	776	2	no	—
60	D₄	—	388	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	776	2	no	—
63	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
64	C₈	—	194	2	no	—
65	C₂²	—	8	2	no	—
66	C₂²	—	8	1	no	—
67	C₂²	—	776	2	no	—
68	C₂²	—	388	2	no	—
69	C₂²	#15690868	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	1552	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	776	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#20261639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261639
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#127×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical	#20261111	C₂
cosocle	#15690807	C₂²
hypercenter	#127	#87604
hypocenter	C₉₇	#1408
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20261640	C₂	semidirect
6. C₄×#11715895	C₄	semidirect
7. #127×C₉₇	C₄	semidirect
39. C₉₇	#1408	semidirect
44. #127	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261640	2
#20261111	2
#1408	97