Subgroups — #27156694

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261624	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261596	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₄⋊#11715897	—	2	1	no	—
6	D₄×Dic₉₇	—	2	1	no	—
7	#15690744	C₄	1	1	yes	characteristic
8	#15690731	—	2	1	no	—
9	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	4	1	no	—
12	#13095514	—	2	2	no	—
13	C₂²×Dic₉₇	—	4	1	no	—
14	C₂×#11715895	D₄	1	1	yes	characteristic
15	C₄⋊Dic₉₇	—	2	1	no	—
16	C₂²⋊Dic₉₇	—	4	1	no	—
17	C₄×Dic₉₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₉₇	—	2	1	no	—
19	D₈×C₉₇	—	2	1	no	—
20	D₈×C₉₇	C₈	1	2	yes	complemented
21	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₉₇	—	4	1	no	—
23	D₄×C₉₇	—	2	2	no	—
24	D₄×C₉₇	—	4	1	no	—
25	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₉₇	—	4	1	no	—
27	C₂×Dic₉₇	—	8	2	no	—
28	C₂×Dic₉₇	D₈	1	1	yes	characteristic
29	C₂×Dic₉₇	SD₁₆	1	1	yes	characteristic
30	#11715897	—	8	1	no	—
31	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₉₇	M₁₆	1	1	yes	characteristic
33	#11715895	—	2	1	no	—
34	C₂²×C₉₇	—	4	2	no	—
35	C₂²×C₉₇	—	8	1	no	—
36	C₂²×C₉₇	—	8	1	no	—
37	C₂²×C₉₇	#102	1	1	yes	characteristic
38	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₉₇	#104	1	1	yes	characteristic
40	Dic₉₇	—	2	1	no	—
41	Dic₉₇	—	2	1	no	—
42	Dic₉₇	—	8	1	no	—
43	C₂×C₉₇	—	8	2	no	—
44	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₉₇	#360	1	1	yes	characteristic
46	C₂×C₉₇	#361	1	1	yes	characteristic
47	#1333	—	97	1	no	maximal, Sylow
48	C₉₇	#1333	1	1	yes	characteristic, complemented, Sylow
49	C₄×D₈	—	97	1	no	—
50	#346	—	97	1	no	—
51	#334	—	97	1	no	—
52	C₂×D₈	#87604	1	1	yes	characteristic
53	C₄×D₄	—	194	1	no	—
54	C₂×C₁₆	—	194	1	no	—
55	#107	—	97	1	no	—
56	C₄⋊C₈	—	194	1	no	—
57	#102	—	194	1	no	—
58	C₄×C₈	—	97	1	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	388	1	no	—
61	D₈	—	2	1	no	—
62	D₈	#11715897	1	2	yes	complemented
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	—	388	1	no	—
65	C₂×C₈	C₂×#87604	1	1	yes	characteristic
66	C₄⋊C₄	—	194	1	no	—
67	C₂²⋊C₄	—	388	1	no	—
68	C₄²	—	97	1	no	—
69	C₁₆	—	194	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	776	2	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	—	97	1	no	—
76	C₂×C₄	—	388	1	no	—
77	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
78	C₈	—	776	1	no	—
79	C₈	—	194	1	no	—
80	C₈	C₂×#11715897	1	1	yes	characteristic
81	C₈	#13095520	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#15690872	1	1	yes	characteristic
86	C₄	—	776	1	no	—
87	C₄	—	194	1	no	—
88	C₄	—	194	1	no	—
89	C₄	C₂²⋊#11715897	1	1	yes	characteristic
90	C₄	#15690874	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊#11715897	1	1	yes	characteristic
93	C₂	#20261651	1	1	yes	characteristic
94	C₂	#20261653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690872
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting, prime radical	C₂×D₈×C₉₇	C₄
socle	C₂²×C₉₇	#102
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×D₈	#87604
hypocenter	C₉₇	#1333
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #20261624	C₂	semidirect
2. #20261596	C₂	semidirect
20. D₈×C₉₇	C₈	semidirect
48. C₉₇	#1333	semidirect
62. D₈	#11715897	semidirect

Maximal subgroups

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

subgroup	index
#20261624	2
#20261596	2
D₈×Dic₉₇	2
#1333	97