Subgroups — #27150013

Order 12416.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690779	C₂	1	7	yes	maximal, complemented
2	#20260591	C₂	1	7	yes	maximal, complemented
3	C₂⁶×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₉₇	C₂²	1	7	yes	complemented
5	#15690779	—	2	28	no	—
6	C₂×C₂²⋊Dic₉₇	C₂²	1	21	yes	complemented
7	C₂⁵×C₉₇	—	2	28	no	—
8	C₂⁵×C₉₇	C₂²	1	7	yes	—
9	C₂³×D₉₇	C₂³	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₉₇	—	2	42	no	—
11	C₂²×Dic₉₇	C₂³	1	7	yes	complemented
12	C₂²⋊Dic₉₇	—	2	42	no	—
13	C₂⁴×C₉₇	—	2	224	no	—
14	C₂⁴×C₉₇	—	2	84	no	—
15	C₂⁴×C₉₇	D₄	1	28	yes	complemented
16	C₂⁴×C₉₇	C₂³	1	7	yes	—
17	C₂³×C₉₇	—	2	256	no	—
18	C₂³×C₉₇	—	2	224	no	—
19	C₂³×C₉₇	—	2	168	no	—
20	C₂³×C₉₇	—	2	28	no	—
21	C₂³×C₉₇	C₂×D₄	1	42	yes	—
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	C₂²×D₉₇	—	2	7	no	—
24	C₂²⋊D₉₇	—	4	28	no	—
25	C₂×Dic₉₇	—	2	21	no	—
26	C₂²×C₉₇	—	2	224	no	—
27	C₂²×C₉₇	—	2	84	no	—
28	C₂²×C₉₇	#120	1	28	yes	complemented
29	C₂²×C₉₇	C₂²×D₄	1	7	yes	—
30	C₂×D₉₇	—	4	7	no	—
31	Dic₉₇	—	4	7	no	—
32	C₂×C₉₇	—	2	28	no	—
33	C₂×C₉₇	C₂×#120	1	7	yes	—
34	D₉₇	—	8	1	no	—
35	#2846	—	97	1	no	maximal, Sylow
36	C₉₇	#2846	1	1	yes	characteristic, complemented, Sylow
37	C₂⁶	D₉₇	1	1	yes	characteristic, complemented
38	C₂×#120	—	97	7	no	—
39	#379	—	97	7	no	—
40	C₂⁵	—	2	28	no	—
41	C₂⁵	C₂×D₉₇	1	7	yes	—
42	C₂²×D₄	—	97	7	no	—
43	#120	—	194	28	no	—
44	C₂×C₂²⋊C₄	—	97	21	no	—
45	C₂⁴	—	2	224	no	—
46	C₂⁴	—	2	84	no	—
47	C₂⁴	—	97	1	no	—
48	C₂⁴	C₂²⋊D₉₇	1	28	yes	complemented
49	C₂⁴	C₂²×D₉₇	1	7	yes	—
50	C₂×D₄	—	194	42	no	—
51	C₂²×C₄	—	97	7	no	—
52	C₂²⋊C₄	—	194	42	no	—
53	C₂³	—	2	256	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	2	168	no	—
56	C₂³	—	2	28	no	—
57	C₂³	—	194	7	no	—
58	C₂³	C₂×C₂²⋊D₉₇	1	42	yes	—
59	C₂³	C₂³×D₉₇	1	1	yes	characteristic
60	D₄	—	388	28	no	—
61	C₂×C₄	—	194	21	no	—
62	C₂²	—	2	224	no	—
63	C₂²	—	2	84	no	—
64	C₂²	—	388	7	no	—
65	C₂²	#15690779	1	28	yes	complemented
66	C₂²	C₂²×C₂²⋊D₉₇	1	7	yes	—
67	C₄	—	388	7	no	—
68	C₂	—	2	28	no	—
69	C₂	—	776	1	no	—
70	C₂	C₂×#15690779	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₂⁶×C₉₇	C₂
hypercenter	C₂⁶	D₉₇
hypocenter	C₉₇	#2846
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂×#15690779	C₂	semidirect
2. #20260591	C₂	semidirect
3. C₂⁶×C₉₇	C₂	semidirect
4. C₂²×C₂²⋊D₉₇	C₂²	semidirect
6. C₂×C₂²⋊Dic₉₇	C₂²	semidirect
9. C₂³×D₉₇	C₂³	semidirect
11. C₂²×Dic₉₇	C₂³	semidirect
15. C₂⁴×C₉₇	D₄	semidirect
28. C₂²×C₉₇	#120	semidirect
36. C₉₇	#2846	semidirect
37. C₂⁶	D₉₇	semidirect
48. C₂⁴	C₂²⋊D₉₇	semidirect
65. C₂²	#15690779	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690779	2
#20260591	2
C₂⁶×C₉₇	2
#2846	97