Subgroups — #27150134

Order 12416.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690810	C₂	1	6	yes	maximal
2	#20260605	C₂	1	8	yes	maximal
3	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₄	C₂²	1	4	yes	—
5	#15690810	—	2	24	no	—
6	C₂×C₄⋊Dic₉₇	C₂²	1	24	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	4	yes	—
9	C₂×Dic₁₉₄	—	2	24	no	—
10	C₂²×Dic₉₇	C₂³	1	8	yes	—
11	C₄⋊Dic₉₇	—	2	48	no	—
12	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
14	C₄²×C₉₇	D₄	1	12	yes	—
15	C₄²×C₉₇	Q₈	1	16	yes	complemented
16	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
17	C₂×C₄×C₉₇	C₂×D₄	1	6	yes	—
18	C₂×C₄×C₉₇	C₂×D₄	1	12	yes	—
19	C₂×C₄×C₉₇	C₂×Q₈	1	24	yes	—
20	C₂×Dic₉₇	—	2	24	no	—
21	Dic₁₉₄	—	4	16	no	—
22	C₂²×C₉₇	C₂²×D₄	1	3	yes	—
23	C₂²×C₉₇	C₂²×Q₈	1	4	yes	—
24	C₄×C₉₇	#127	1	4	yes	—
25	C₄×C₉₇	C₄⋊Q₈	1	24	yes	complemented
26	Dic₉₇	—	4	8	no	—
27	C₂×C₉₇	C₂×#127	1	1	yes	characteristic
28	C₂×C₉₇	C₂×C₄⋊Q₈	1	6	yes	—
29	#2868	—	97	1	no	maximal, Sylow
30	C₉₇	#2868	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₄⋊Q₈	—	97	6	no	—
32	#384	—	97	8	no	—
33	C₄³	D₉₇	1	1	yes	characteristic
34	C₂²×Q₈	—	97	4	no	—
35	C₄⋊Q₈	—	194	24	no	—
36	C₂×C₄⋊C₄	—	97	24	no	—
37	C₂×C₄²	C₂×D₉₇	1	4	yes	—
38	C₂×C₄²	C₂×D₉₇	1	3	yes	—
39	C₂×Q₈	—	194	24	no	—
40	C₂²×C₄	—	97	8	no	—
41	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
42	C₂²×C₄	C₂²×D₉₇	1	6	yes	—
43	C₄⋊C₄	—	194	48	no	—
44	C₄²	Dic₁₉₄	1	16	yes	complemented
45	C₄²	D₃₈₈	1	12	yes	—
46	C₂³	C₂³×D₉₇	1	1	yes	characteristic
47	Q₈	—	388	16	no	—
48	C₂×C₄	—	194	24	no	—
49	C₂×C₄	C₂×D₃₈₈	1	6	yes	—
50	C₂×C₄	C₂×D₃₈₈	1	12	yes	—
51	C₂×C₄	C₂×Dic₁₉₄	1	24	yes	—
52	C₂²	C₂²×D₃₈₈	1	3	yes	—
53	C₂²	C₂²×Dic₁₉₄	1	4	yes	—
54	C₄	—	388	8	no	—
55	C₄	#15690807	1	4	yes	—
56	C₄	#15690810	1	24	yes	complemented
57	C₂	C₂×#15690807	1	1	yes	characteristic
58	C₂	C₂×#15690810	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₄³×C₉₇	C₂
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#2868
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
15. C₄²×C₉₇	Q₈	semidirect
25. C₄×C₉₇	C₄⋊Q₈	semidirect
30. C₉₇	#2868	semidirect
44. C₄²	Dic₁₉₄	semidirect
56. C₄	#15690810	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690810	2
#20260605	2
C₄³×C₉₇	2
#2868	97