Subgroups — #27149989

Order 12416.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260647	C₂	1	12	yes	maximal, complemented
2	#20260591	C₂	1	2	yes	maximal
3	#379×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₄	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊Dic₉₇	C₂²	1	12	yes	—
6	#15690717	C₂²	1	12	yes	complemented
7	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₉₇	C₂²	1	6	yes	—
9	C₂×Dic₁₉₄	—	4	12	no	—
10	C₂²×Dic₉₇	C₂³	1	8	yes	—
11	C₂²⋊Dic₉₇	—	4	12	no	—
12	C₂⁴×C₉₇	—	4	6	no	—
13	C₂⁴×C₉₇	Q₈	1	4	yes	complemented
14	C₂⁴×C₉₇	C₂³	1	3	yes	—
15	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
16	C₂²⋊C₄×C₉₇	—	4	6	no	—
17	C₂³×C₉₇	—	2	6	no	—
18	C₂³×C₉₇	—	2	12	no	—
19	C₂³×C₉₇	—	4	12	no	—
20	C₂³×C₉₇	—	4	16	no	—
21	C₂³×C₉₇	C₂×Q₈	1	6	yes	—
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	12	no	—
24	C₂×Dic₉₇	—	2	24	no	—
25	Dic₁₉₄	—	4	16	no	—
26	C₂²×C₉₇	—	2	6	no	—
27	C₂²×C₉₇	—	2	12	no	—
28	C₂²×C₉₇	—	4	16	no	—
29	C₂²×C₉₇	—	4	12	no	—
30	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
31	C₂²×C₉₇	H(5,𝔽₂)	1	6	yes	—
32	C₄×C₉₇	—	4	4	no	—
33	Dic₉₇	—	4	8	no	—
34	C₂×C₉₇	—	4	6	no	—
35	C₂×C₉₇	#543	1	3	yes	—
36	C₂×C₉₇	H(3,𝔽₄)	1	4	yes	—
37	#2840	—	97	1	no	maximal, Sylow
38	C₉₇	#2840	1	1	yes	characteristic, complemented, Sylow
39	#393	—	97	12	no	—
40	#379	—	97	2	no	—
41	#379	D₉₇	1	1	yes	characteristic
42	C₂⁵	C₂×D₉₇	1	1	yes	characteristic
43	C₂²×Q₈	—	97	4	no	—
44	C₂×C₂²⋊C₄	—	97	12	no	—
45	C₂×C₂²⋊C₄	C₂×D₉₇	1	6	yes	—
46	#95	—	97	12	no	—
47	C₂⁴	—	4	6	no	—
48	C₂⁴	Dic₁₉₄	1	4	yes	complemented
49	C₂⁴	C₂²×D₉₇	1	3	yes	—
50	C₂×Q₈	—	388	12	no	—
51	C₂²×C₄	—	97	8	no	—
52	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
53	C₂²⋊C₄	—	4	6	no	—
54	C₂²⋊C₄	—	388	12	no	—
55	C₂³	—	2	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	4	12	no	—
58	C₂³	—	4	16	no	—
59	C₂³	C₂×Dic₁₉₄	1	6	yes	—
60	C₂³	C₂³×D₉₇	1	1	yes	characteristic
61	Q₈	—	388	16	no	—
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	194	24	no	—
64	C₂²	—	2	6	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	4	16	no	—
67	C₂²	—	4	12	no	—
68	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
69	C₂²	#15690855	1	6	yes	—
70	C₄	—	4	4	no	—
71	C₄	—	388	8	no	—
72	C₂	—	4	6	no	—
73	C₂	#20261361	1	3	yes	—
74	C₂	#20261478	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#379×C₉₇	C₂
prime radical	C₂⁵×C₉₇	C₂²
hypercenter	#379	D₉₇
hypocenter	C₉₇	#2840
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #379

Complements

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

N	H ≅ G/N	type
1. #20260647	C₂	semidirect
4. C₂²×Dic₁₉₄	C₂²	semidirect
6. #15690717	C₂²	semidirect
13. C₂⁴×C₉₇	Q₈	semidirect
38. C₉₇	#2840	semidirect
48. C₂⁴	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20260647	2
#20260591	2
#379×C₉₇	2
#2840	97