Subgroups — #27141896

Order 12416.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690744	C₂	1	4	yes	maximal
2	#20260605	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₉₇	—	2	4	no	—
5	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
6	#15690744	—	2	8	no	—
7	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₉₇	C₄	1	4	yes	complemented
10	C₂²×Dic₉₇	—	2	4	no	—
11	C₄⋊Dic₉₇	—	2	8	no	—
12	C₄⋊Dic₉₇	—	4	8	no	—
13	C₂²×C₄×C₉₇	D₄	1	2	yes	—
14	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₉₇	D₄	1	4	yes	—
17	C₂×C₈×C₉₇	Q₈	1	4	yes	—
18	C₄²×C₉₇	C₂×C₄	1	2	yes	—
19	C₄²×C₉₇	Q₈	1	2	yes	—
20	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₉₇	D₈	1	4	yes	—
23	C₂×C₄×C₉₇	Q₁₆	1	4	yes	—
24	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
27	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	4	no	—
29	C₂×Dic₉₇	—	4	8	no	—
30	C₈×C₉₇	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₉₇	#127	1	1	yes	characteristic
33	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₉₇	C₂×D₈	1	2	yes	—
35	C₂²×C₉₇	C₂×Q₁₆	1	2	yes	—
36	C₄×C₉₇	#107	1	8	yes	complemented
37	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
39	Dic₉₇	—	8	4	no	—
40	C₂×C₉₇	#384	1	1	yes	characteristic
41	C₂×C₉₇	C₂×#107	1	2	yes	—
42	C₂×C₉₇	#493	1	1	yes	characteristic
43	C₂×C₉₇	#494	1	1	yes	characteristic
44	C₂×C₉₇	#500	1	2	yes	—
45	#1845	—	97	1	no	maximal, Sylow
46	C₉₇	#1845	1	1	yes	characteristic, complemented, Sylow
47	C₂×#107	—	97	4	no	—
48	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
49	#384	—	97	2	no	—
50	C₂²×C₈	C₂×D₉₇	1	2	yes	—
51	C₂×C₄⋊C₄	—	97	4	no	—
52	C₂×C₄⋊C₄	—	194	4	no	—
53	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
54	#107	—	194	8	no	—
55	C₄×C₈	Dic₉₇	1	4	yes	complemented
56	C₂²×C₄	—	194	4	no	—
57	C₂²×C₄	D₃₈₈	1	2	yes	—
58	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
59	C₂×C₈	Dic₁₉₄	1	4	yes	—
60	C₂×C₈	D₃₈₈	1	4	yes	—
61	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
62	C₄⋊C₄	—	388	8	no	—
63	C₄⋊C₄	—	194	8	no	—
64	C₄²	Dic₁₉₄	1	2	yes	—
65	C₄²	C₂×Dic₉₇	1	2	yes	—
66	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
67	C₂×C₄	—	388	4	no	—
68	C₂×C₄	—	388	8	no	—
69	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
70	C₂×C₄	D₇₇₆	1	4	yes	—
71	C₂×C₄	Dic₃₈₈	1	4	yes	—
72	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
73	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
74	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
75	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
76	C₈	C₄⋊Dic₉₇	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
78	C₂²	#15690807	1	1	yes	characteristic
79	C₂²	#15690810	1	1	yes	characteristic
80	C₂²	C₂×D₇₇₆	1	2	yes	—
81	C₂²	C₂×Dic₃₈₈	1	2	yes	—
82	C₄	—	776	4	no	—
83	C₄	#15690744	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₉₇	1	2	yes	—
85	C₄	#15690810	1	2	yes	—
86	C₂	#20260605	1	1	yes	characteristic
87	C₂	C₂×#15690744	1	2	yes	—
88	C₂	#20261112	1	1	yes	characteristic
89	C₂	#20261115	1	1	yes	characteristic
90	C₂	#20261140	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1845
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₉₇	C₄	semidirect
30. C₈×C₉₇	C₄⋊C₄	semidirect
36. C₄×C₉₇	#107	semidirect
46. C₉₇	#1845	semidirect
55. C₄×C₈	Dic₉₇	semidirect
76. C₈	C₄⋊Dic₉₇	semidirect
83. C₄	#15690744	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690744	2
#20260605	2
C₂×C₄×C₈×C₉₇	2
#1845	97