Subgroups — #27150200

Order 12416.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261213	C₂	1	2	yes	maximal, complemented
2	#20260709	C₂	1	12	yes	maximal, complemented
3	C₂×2₋¹⁺⁴×C₉₇	C₂	1	1	yes	maximal, characteristic
4	Q₈×Dic₉₇	—	2	8	no	—
5	#15690769	C₂²	1	12	yes	—
6	#15690725	C₂²	1	16	yes	—
7	2₋¹⁺⁴×C₉₇	—	2	8	no	—
8	C₂×#41×C₉₇	C₄	1	4	yes	complemented
9	C₂×#41×C₉₇	C₂²	1	6	yes	—
10	C₂²×Q₈×C₉₇	C₄	1	4	yes	complemented
11	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₉₇	—	2	12	no	—
13	C₂²⋊Dic₉₇	C₂³	1	8	yes	—
14	C₄×Dic₉₇	—	2	12	no	—
15	#41×C₉₇	—	2	16	no	—
16	#41×C₉₇	—	2	24	no	—
17	C₂×Q₈×C₉₇	—	2	16	no	—
18	C₂×Q₈×C₉₇	C₂×C₄	1	6	yes	—
19	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
20	C₂×Q₈×C₉₇	D₄	1	8	yes	—
21	C₂×D₄×C₉₇	C₂×C₄	1	6	yes	—
22	C₂×D₄×C₉₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	12	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
25	C₂³×C₉₇	C₂²×C₄	1	4	yes	—
26	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
27	Q₈×C₉₇	—	2	12	no	—
28	Q₈×C₉₇	C₂²⋊C₄	1	16	yes	—
29	D₄×C₉₇	—	2	8	no	—
30	D₄×C₉₇	—	2	12	no	—
31	C₂×C₄×C₉₇	—	2	24	no	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
33	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
34	C₂×C₄×C₉₇	C₂×D₄	1	12	yes	—
35	C₂×Dic₉₇	—	2	8	no	—
36	C₂²×C₉₇	—	2	4	no	—
37	C₂²×C₉₇	—	4	1	no	—
38	C₂²×C₉₇	—	4	4	no	—
39	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
40	C₂²×C₉₇	C₂²×D₄	1	2	yes	—
41	C₄×C₉₇	—	2	4	no	—
42	C₄×C₉₇	C₂×C₂²⋊C₄	1	12	yes	—
43	Dic₉₇	—	4	8	no	—
44	C₂×C₉₇	—	2	2	no	—
45	C₂×C₉₇	—	2	1	no	—
46	C₂×C₉₇	—	4	4	no	—
47	C₂×C₉₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
48	#2884	—	97	1	no	maximal, Sylow
49	C₉₇	#2884	1	1	yes	characteristic, complemented, Sylow
50	C₂×2₋¹⁺⁴	D₉₇	1	1	yes	characteristic
51	#519	—	97	2	no	—
52	#410	—	97	12	no	—
53	2₋¹⁺⁴	—	2	8	no	—
54	C₂×#41	Dic₉₇	1	4	yes	complemented
55	C₂×#41	C₂×D₉₇	1	6	yes	—
56	C₂²×Q₈	Dic₉₇	1	4	yes	complemented
57	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
58	C₄×Q₈	—	194	8	no	—
59	#117	—	97	12	no	—
60	#99	—	97	16	no	—
61	#41	—	2	16	no	—
62	#41	—	2	24	no	—
63	C₂×Q₈	—	2	16	no	—
64	C₂×Q₈	C₂×Dic₉₇	1	6	yes	—
65	C₂×Q₈	C₂×Dic₉₇	1	4	yes	—
66	C₂×Q₈	C₂²⋊D₉₇	1	8	yes	—
67	C₂×D₄	C₂×Dic₉₇	1	6	yes	—
68	C₂×D₄	C₂²×D₉₇	1	4	yes	—
69	C₂²×C₄	C₂×Dic₉₇	1	12	yes	—
70	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
71	C₄⋊C₄	—	194	12	no	—
72	C₂²⋊C₄	—	97	8	no	—
73	C₄²	—	194	12	no	—
74	C₂³	C₂²×Dic₉₇	1	4	yes	—
75	C₂³	C₂³×D₉₇	1	1	yes	characteristic
76	Q₈	—	2	12	no	—
77	Q₈	C₂²⋊Dic₉₇	1	16	yes	—
78	D₄	—	2	8	no	—
79	D₄	—	2	12	no	—
80	C₂×C₄	—	2	24	no	—
81	C₂×C₄	—	194	8	no	—
82	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
83	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
84	C₂×C₄	C₂×C₂²⋊D₉₇	1	12	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
89	C₂²	C₂²×C₂²⋊D₉₇	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	388	8	no	—
92	C₄	C₂×C₂²⋊Dic₉₇	1	12	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	4	no	—
96	C₂	C₂²×C₂²⋊Dic₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₉₇
Fitting, prime radical	C₂×2₋¹⁺⁴×C₉₇	C₂
socle	C₂×C₉₇	C₂²×C₂²⋊C₄
cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×2₋¹⁺⁴	D₉₇
hypocenter	C₉₇	#2884
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #20261213	C₂	semidirect
2. #20260709	C₂	semidirect
8. C₂×#41×C₉₇	C₄	semidirect
10. C₂²×Q₈×C₉₇	C₄	semidirect
49. C₉₇	#2884	semidirect
54. C₂×#41	Dic₉₇	semidirect
56. C₂²×Q₈	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261213	2
#20260709	2
C₂×2₋¹⁺⁴×C₉₇	2
#2884	97