Subgroups — #27156744

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261658	C₂	1	2	yes	maximal, complemented
2	#20261111	C₂	1	1	yes	maximal, characteristic
3	#15690884	—	2	2	no	—
4	C₂×#13095496	—	2	2	no	—
5	#15690807	C₄	1	1	yes	characteristic
6	C₄×#11715895	C₂²	1	1	yes	characteristic
7	#127×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	C₂×D₃₈₈	—	2	1	no	—
10	C₂×D₃₈₈	—	4	1	no	—
11	#13095496	—	4	2	no	—
12	#13095496	—	4	2	no	—
13	C₂×#11715895	D₄	1	2	yes	—
14	C₂×D₄×C₉₇	—	2	1	no	—
15	C₂×D₄×C₉₇	—	4	1	no	—
16	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₉₇	—	4	1	no	—
18	C₂²×D₉₇	—	4	1	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	D₈	1	2	yes	—
26	#11715895	SD₁₆	1	2	yes	—
27	C₂²×C₉₇	—	8	2	no	—
28	C₂²×C₉₇	—	8	1	no	—
29	C₂²×C₉₇	#100	1	1	yes	characteristic
30	C₂×D₉₇	—	8	2	no	—
31	C₂×D₉₇	—	8	1	no	—
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	#102	1	2	yes	—
34	C₂×C₉₇	—	2	1	no	—
35	C₂×C₉₇	—	16	1	no	—
36	C₂×C₉₇	#331	1	1	yes	characteristic
37	D₉₇	—	16	1	no	—
38	#1357	—	97	1	no	maximal, Sylow
39	C₉₇	#1357	1	1	yes	characteristic, complemented, Sylow
40	#493	—	97	1	no	—
41	#364	—	97	2	no	—
42	C₂×D₈	—	194	2	no	—
43	#127	—	97	1	no	—
44	#127	#87604	1	1	yes	characteristic
45	M₃₂	—	194	2	no	—
46	C₄×C₈	—	97	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	388	1	no	—
50	C₂×D₄	—	194	1	no	—
51	D₈	—	388	2	no	—
52	D₈	—	388	2	no	—
53	C₂×C₈	—	97	2	no	—
54	C₄²	C₂×#87604	1	1	yes	characteristic
55	C₁₆	—	388	2	no	—
56	C₂³	—	4	1	no	—
57	C₂³	—	388	1	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	776	2	no	—
61	D₄	—	388	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
64	C₈	—	97	2	no	—
65	C₈	—	97	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	1	no	—
68	C₂²	—	776	2	no	—
69	C₂²	—	776	1	no	—
70	C₂²	#15690870	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	#15690872	1	2	yes	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	1552	1	no	—
76	C₂	#20261593	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261593
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#127×C₉₇	C₄
socle	C₂×C₉₇	#331
prime radical	#20261111	C₂
cosocle	C₄×#11715895	C₂²
hypercenter	#127	#87604
hypocenter	C₉₇	#1357
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20261658	C₂	semidirect
39. C₉₇	#1357	semidirect

Maximal subgroups

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

subgroup	index
#20261658	2
#20261111	2
#1357	97