Subgroups — #27141354

Order 12416.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261140	C₂	1	4	yes	maximal
2	#20261115	C₂	1	2	yes	maximal
3	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₈	—	2	4	no	—
5	#15690810	C₂²	1	4	yes	—
6	#15690744	—	2	8	no	—
7	C₄×C₈×C₉₇	C₂²	1	2	yes	—
8	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₉₄	—	2	4	no	—
10	Dic₃₈₈	—	4	8	no	—
11	C₄⋊Dic₉₇	—	2	8	no	—
12	C₂×C₈×C₉₇	D₄	1	2	yes	—
13	C₂×C₈×C₉₇	D₄	1	4	yes	—
14	C₄²×C₉₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
17	C₂×Dic₉₇	—	4	4	no	—
18	Dic₁₉₄	—	4	8	no	—
19	C₈×C₉₇	D₈	1	4	yes	—
20	C₈×C₉₇	Q₁₆	1	8	yes	complemented
21	C₂²×C₉₇	#127	1	1	yes	characteristic
22	C₄×C₉₇	C₂×D₈	1	2	yes	—
23	C₄×C₉₇	C₂×Q₁₆	1	4	yes	—
24	Dic₉₇	—	8	4	no	—
25	C₂×C₉₇	#493	1	1	yes	characteristic
26	C₂×C₉₇	#494	1	2	yes	—
27	#1713	—	97	1	no	maximal, Sylow
28	C₉₇	#1713	1	1	yes	characteristic, complemented, Sylow
29	#500	—	97	4	no	—
30	#494	—	97	2	no	—
31	C₈²	D₉₇	1	1	yes	characteristic
32	C₂×Q₁₆	—	194	4	no	—
33	C₄⋊Q₈	—	97	4	no	—
34	#107	—	194	8	no	—
35	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
36	C₄×C₈	C₂×D₉₇	1	2	yes	—
37	C₂×Q₈	—	194	4	no	—
38	Q₁₆	—	388	8	no	—
39	C₂×C₈	D₃₈₈	1	2	yes	—
40	C₂×C₈	D₃₈₈	1	4	yes	—
41	C₄⋊C₄	—	194	8	no	—
42	C₄²	C₂²×D₉₇	1	1	yes	characteristic
43	Q₈	—	388	8	no	—
44	C₂×C₄	—	388	4	no	—
45	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
47	C₈	D₇₇₆	1	4	yes	—
48	C₈	Dic₃₈₈	1	8	yes	complemented
49	C₂²	#15690807	1	1	yes	characteristic
50	C₄	—	776	4	no	—
51	C₄	C₂×D₇₇₆	1	2	yes	—
52	C₄	C₂×Dic₃₈₈	1	4	yes	—
53	C₂	#20261112	1	1	yes	characteristic
54	C₂	#20261115	1	2	yes	—

	subgroup	G/H
center	C₂²	#15690807
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	C₈²×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#127
hypercenter	C₈²	D₉₇
hypocenter	C₉₇	#1713
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₉₇	Q₁₆	semidirect
28. C₉₇	#1713	semidirect
48. C₈	Dic₃₈₈	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20261115	2
C₈²×C₉₇	2
#1713	97