Subgroups — #15690780

Order 3104.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₈₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#13095488	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×D₉₇	—	2	2	no	—
8	C₂²×D₉₇	—	2	4	no	—
9	C₂²×D₉₇	C₂²	1	2	yes	—
10	C₂²×D₉₇	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₄×C₉₇	C₂²	1	2	yes	complemented
12	C₂²⋊D₉₇	—	2	2	no	—
13	C₂×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
14	D₃₈₈	—	2	4	no	—
15	C₂²×C₉₇	—	2	2	no	—
16	C₂²×C₉₇	D₄	1	2	yes	complemented
17	C₂²×C₉₇	C₂³	1	1	yes	characteristic
18	C₂×D₉₇	—	2	8	no	—
19	C₂×D₉₇	—	2	2	no	—
20	C₂×D₉₇	—	2	2	no	—
21	C₂×D₉₇	—	2	2	no	—
22	C₂×D₉₇	—	2	1	no	—
23	C₂×D₉₇	D₄	1	4	yes	complemented
24	C₄×C₉₇	—	2	2	no	—
25	Dic₉₇	—	2	1	no	—
26	C₂×C₉₇	—	2	2	no	—
27	C₂×C₉₇	C₂×D₄	1	2	yes	—
28	C₂×C₉₇	C₂×D₄	1	1	yes	characteristic
29	D₉₇	—	2	4	no	—
30	D₉₇	—	4	1	no	—
31	C₉₇	#120	1	1	yes	characteristic, complemented, Sylow
32	#120	—	97	1	no	maximal, Sylow
33	C₂⁴	—	97	1	no	—
34	C₂×D₄	—	97	1	no	—
35	C₂×D₄	—	97	2	no	—
36	C₂²⋊C₄	—	97	2	no	—
37	C₂²⋊C₄	D₉₇	1	1	yes	characteristic, complemented
38	C₂³	—	97	2	no	—
39	C₂³	—	97	1	no	—
40	C₂³	—	194	4	no	—
41	C₂³	—	194	2	no	—
42	C₂³	C₂×D₉₇	1	1	yes	characteristic
43	D₄	—	194	4	no	—
44	D₄	—	194	2	no	—
45	C₂×C₄	—	97	1	no	—
46	C₂×C₄	C₂×D₉₇	1	2	yes	complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	97	4	no	—
49	C₂²	—	194	8	no	—
50	C₂²	—	194	2	no	—
51	C₂²	—	194	1	no	—
52	C₂²	—	194	2	no	—
53	C₂²	—	194	2	no	—
54	C₂²	D₃₈₈	1	2	yes	complemented
55	C₂²	C₂²×D₉₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	194	1	no	—
58	C₂	—	2	2	no	—
59	C₂	—	388	1	no	—
60	C₂	—	194	4	no	—
61	C₂	D₄×D₉₇	1	2	yes	—
62	C₂	C₂×D₃₈₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂³
Fitting	C₂²⋊C₄×C₉₇	C₂
hypercenter	C₂²⋊C₄	D₉₇
hypocenter	C₉₇	#120
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₉₇	C₂	semidirect
2. C₂×D₃₈₈	C₂	semidirect
3. C₂×C₂²⋊D₉₇	C₂	semidirect
4. #13095488	C₂	semidirect
5. C₂²⋊C₄×C₉₇	C₂	semidirect
10. C₂²×D₉₇	C₂²	semidirect
11. C₂×C₄×C₉₇	C₂²	semidirect
13. C₂×Dic₉₇	C₂²	semidirect
16. C₂²×C₉₇	D₄	semidirect
23. C₂×D₉₇	D₄	semidirect
31. C₉₇	#120	semidirect
37. C₂²⋊C₄	D₉₇	semidirect
46. C₂×C₄	C₂×D₉₇	semidirect
54. C₂²	D₃₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₉₇	2
C₂×D₃₈₈	2
C₂×C₂²⋊D₉₇	2
#13095488	2
C₂²⋊C₄×C₉₇	2
#120	97