Subgroups — #27139904

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261114	C₂	1	1	yes	maximal, characteristic
2	#20260534	C₂	1	1	yes	maximal, characteristic
3	#356×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095501	—	2	1	no	—
5	C₂×#13095501	—	2	1	no	—
6	#15690810	C₄	1	1	yes	characteristic
7	#15690729	—	2	1	no	—
8	C₄×#11715895	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	#101×C₉₇	—	2	1	no	—
11	C₂×Dic₁₉₄	—	2	1	no	—
12	#13095501	—	4	2	no	—
13	#13095501	—	4	2	no	—
14	#13095495	—	4	1	no	—
15	C₂×#11715895	—	2	1	no	—
16	C₄⋊Dic₉₇	—	4	1	no	—
17	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
19	M₁₆×C₉₇	—	4	1	no	—
20	C₄⋊C₄×C₉₇	—	4	1	no	—
21	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
22	Q₈×C₉₇	—	4	1	no	—
23	Q₈×C₉₇	—	4	1	no	—
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	1	no	—
29	Dic₁₉₄	—	4	1	no	—
30	Dic₁₉₄	—	4	1	no	—
31	C₈×C₉₇	—	8	1	no	—
32	#11715895	—	2	2	no	—
33	#11715895	—	8	1	no	—
34	C₂²×C₉₇	#99	1	1	yes	characteristic
35	C₄×C₉₇	—	2	2	no	—
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	—	4	1	no	—
38	C₄×C₉₇	—	4	1	no	—
39	Dic₉₇	—	8	1	no	—
40	C₂×C₉₇	—	2	1	no	—
41	C₂×C₉₇	#353	1	1	yes	characteristic
42	#1411	—	97	1	no	maximal, Sylow
43	C₉₇	#1411	1	1	yes	characteristic, complemented, Sylow
44	#494	—	97	1	no	—
45	#356	—	97	1	no	—
46	#356	D₉₇	1	1	yes	characteristic
47	C₂×Q₁₆	—	194	1	no	—
48	C₂×Q₁₆	—	194	1	no	—
49	C₄⋊Q₈	—	97	1	no	—
50	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
51	#101	—	2	1	no	—
52	#101	—	194	1	no	—
53	C₄×C₈	—	97	1	no	—
54	C₂×Q₈	—	194	1	no	—
55	C₂×Q₈	D₃₈₈	1	1	yes	characteristic
56	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
57	Q₁₆	—	388	2	no	—
58	Q₁₆	—	388	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	388	1	no	—
61	C₂×C₈	—	194	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	388	1	no	—
64	C₄²	C₄×D₉₇	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	388	1	no	—
68	Q₈	—	388	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	388	1	no	—
73	C₂×C₄	#13095488	1	1	yes	characteristic
74	C₈	—	8	1	no	—
75	C₈	—	776	1	no	—
76	C₈	—	194	2	no	—
77	C₂²	#15690724	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	776	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#20260527	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260527
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₉₇
Fitting	#356×C₉₇	C₂
socle	C₂×C₉₇	#353
prime radical	C₂²×C₉₇	#99
cosocle	C₄⋊Q₈×C₉₇	C₂²
hypercenter	#356	D₉₇
hypocenter	C₉₇	#1411
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
43. C₉₇	#1411	semidirect

Maximal subgroups

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

subgroup	index
#20261114	2
#20260534	2
#356×C₉₇	2
#1411	97