Subgroups — #27144913

Order 12416.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260615	C₂	1	12	yes	maximal, complemented
2	#20260603	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₉₇	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₉₇	C₂²	1	12	yes	—
6	#15690717	C₂²	1	12	yes	—
7	C₂²×D₄×C₉₇	C₂²	1	6	yes	—
8	#127×C₉₇	C₄	1	8	yes	complemented
9	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₉₇	—	4	12	no	—
11	C₂²×Dic₉₇	C₂³	1	8	yes	—
12	C₂²⋊Dic₉₇	—	4	12	no	—
13	C₂⁴×C₉₇	C₂³	1	4	yes	—
14	C₂×D₄×C₉₇	—	4	6	no	—
15	C₂×D₄×C₉₇	—	4	6	no	—
16	C₂×D₄×C₉₇	C₂×C₄	1	24	yes	complemented
17	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
18	C₄²×C₉₇	C₂×C₄	1	4	yes	complemented
19	C₂³×C₉₇	—	2	12	no	—
20	C₂³×C₉₇	—	2	12	no	—
21	C₂³×C₉₇	C₂²×C₄	1	8	yes	—
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	D₄×C₉₇	—	4	24	no	—
24	C₂×C₄×C₉₇	—	2	6	no	—
25	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
26	C₂×Dic₉₇	—	2	24	no	—
27	C₂×Dic₉₇	—	4	16	no	—
28	C₂²×C₉₇	—	2	24	no	—
29	C₂²×C₉₇	—	4	4	no	—
30	C₂²×C₉₇	—	4	12	no	—
31	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₉₇	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₉₇	—	2	6	no	—
34	Dic₉₇	—	4	8	no	—
35	C₂×C₉₇	—	4	8	no	—
36	C₂×C₉₇	#518	1	3	yes	—
37	C₂×C₉₇	#560	1	4	yes	—
38	#2380	—	97	1	no	maximal, Sylow
39	C₉₇	#2380	1	1	yes	characteristic, complemented, Sylow
40	C₂×#127	D₉₇	1	1	yes	characteristic
41	#386	—	97	12	no	—
42	#383	—	97	2	no	—
43	C₂²×D₄	C₂×D₉₇	1	6	yes	—
44	C₂³×C₄	—	97	4	no	—
45	#127	Dic₉₇	1	8	yes	complemented
46	C₂×C₂²⋊C₄	—	97	12	no	—
47	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
48	#95	—	97	12	no	—
49	C₂⁴	C₂²×D₉₇	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₉₇	1	24	yes	complemented
53	C₂²×C₄	—	97	8	no	—
54	C₂²×C₄	—	388	12	no	—
55	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
56	C₂²⋊C₄	—	388	12	no	—
57	C₄²	C₂×Dic₉₇	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₉₇	1	8	yes	—
61	C₂³	C₂³×D₉₇	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	388	16	no	—
65	C₂×C₄	—	194	24	no	—
66	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
71	C₂²	#15690855	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	388	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#20261210	1	3	yes	—
76	C₂	#20261472	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting, prime radical	C₂×#127×C₉₇	C₂
socle, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×#127	D₉₇
hypocenter	C₉₇	#2380
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #20260615	C₂	semidirect
2. #20260603	C₂	semidirect
8. #127×C₉₇	C₄	semidirect
16. C₂×D₄×C₉₇	C₂×C₄	semidirect
18. C₄²×C₉₇	C₂×C₄	semidirect
39. C₉₇	#2380	semidirect
45. #127	Dic₉₇	semidirect
52. C₂×D₄	C₂×Dic₉₇	semidirect
57. C₄²	C₂×Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260615	2
#20260603	2
C₂×#127×C₉₇	2
#2380	97