Subgroups — #27143520

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260773	C₂	1	2	yes	maximal
2	#20260563	C₂	1	2	yes	maximal, complemented
3	#20260561	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690769	—	2	2	no	—
6	#15690744	C₂²	1	4	yes	—
7	#15690742	—	2	2	no	—
8	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₉₇	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₉₇	—	2	4	no	—
13	C₂²⋊Dic₉₇	—	4	2	no	—
14	C₄×Dic₉₇	—	4	2	no	—
15	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	Q₈	1	2	yes	—
19	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
20	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
21	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
22	C₂³×C₉₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₉₇	—	4	4	no	—
28	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
29	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₉₇	—	2	2	no	—
32	C₂²×C₉₇	#107	1	2	yes	—
33	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
34	C₄×C₉₇	#107	1	2	yes	—
35	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₉₇	—	8	4	no	—
38	C₂×C₉₇	—	2	2	no	—
39	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
40	C₂×C₉₇	#508	1	2	yes	—
41	#2158	—	97	1	no	maximal, Sylow
42	C₉₇	#2158	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₁₆	D₉₇	1	1	yes	characteristic
44	#427	—	97	2	no	—
45	#367	—	97	2	no	—
46	#366	—	97	2	no	—
47	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
48	#117	—	194	2	no	—
49	C₂×C₁₆	Dic₉₇	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
52	#107	—	97	4	no	—
53	#106	—	194	2	no	—
54	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
55	C₂×C₈	Dic₁₉₄	1	2	yes	—
56	C₂×C₈	D₃₈₈	1	1	yes	characteristic
57	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
58	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
59	C₄⋊C₄	—	194	4	no	—
60	C₂²⋊C₄	—	388	2	no	—
61	C₄²	—	388	2	no	—
62	C₁₆	C₂×Dic₉₇	1	2	yes	—
63	C₁₆	C₂×Dic₉₇	1	2	yes	—
64	C₂³	D₇₇₆	1	1	yes	characteristic
65	C₂×C₄	—	388	4	no	—
66	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
67	C₂×C₄	D₇₇₆	1	1	yes	characteristic
68	C₂×C₄	Dic₃₈₈	1	2	yes	—
69	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
70	C₈	C₄⋊Dic₉₇	1	2	yes	—
71	C₈	C₂²×Dic₉₇	1	1	yes	characteristic
72	C₈	C₂×Dic₁₉₄	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#15690744	1	2	yes	—
75	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
76	C₄	—	776	4	no	—
77	C₄	#15690744	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
79	C₄	C₂×Dic₃₈₈	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#15690744	1	1	yes	characteristic
82	C₂	#20261167	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₇₇₆
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	C₂²×C₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
prime radical	C₂³×C₉₇	D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	C₂²×C₁₆	D₉₇
hypocenter	C₉₇	#2158
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20260563	C₂	semidirect
3. #20260561	C₂	semidirect
9. C₂×C₁₆×C₉₇	C₄	semidirect
42. C₉₇	#2158	semidirect
49. C₂×C₁₆	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260773	2
#20260563	2
#20260561	2
C₂²×C₁₆×C₉₇	2
#2158	97