Subgroups — #27139703

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260511	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260486	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	—	2	1	no	—
5	C₄⋊#11715895	—	2	1	no	—
6	C₂×D₈×C₉₇	C₄	1	1	yes	characteristic
7	C₄×D₄×C₉₇	—	2	1	no	—
8	#107×C₉₇	C₄	1	1	yes	characteristic
9	#102×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×#11715895	—	4	1	no	—
12	#13095486	—	2	2	no	—
13	C₂×D₄×C₉₇	—	2	1	no	—
14	C₂²×C₄×C₉₇	—	4	1	no	—
15	D₈×C₉₇	—	2	1	no	—
16	D₈×C₉₇	C₈	1	2	yes	complemented
17	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₉₇	—	2	1	no	—
20	C₂²⋊C₄×C₉₇	—	4	1	no	—
21	C₄²×C₉₇	D₄	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	—	4	1	no	—
26	C₂×C₄×C₉₇	—	8	2	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₉₇	—	2	1	no	—
31	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₉₇	M₁₆	1	1	yes	characteristic
33	#11715895	—	8	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	—	2	1	no	—
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	—	8	1	no	—
41	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₉₇	#104	1	1	yes	characteristic
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	D₉₇	1	1	yes	characteristic
50	#346	—	97	1	no	—
51	#334	—	97	1	no	—
52	C₂×D₈	Dic₉₇	1	1	yes	characteristic
53	C₄×D₄	—	2	1	no	—
54	C₂×C₁₆	—	194	1	no	—
55	#107	Dic₉₇	1	1	yes	characteristic
56	C₄⋊C₈	—	194	1	no	—
57	#102	—	2	1	no	—
58	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	D₈	—	2	1	no	—
62	D₈	#11715895	1	2	yes	complemented
63	C₂×C₈	—	388	1	no	—
64	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₂²⋊C₄	—	4	1	no	—
68	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
69	C₁₆	—	194	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
76	C₂×C₄	#13095496	1	1	yes	characteristic
77	C₂×C₄	#13095499	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	776	1	no	—
80	C₈	C₂×#11715895	1	1	yes	characteristic
81	C₈	#13095495	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#15690731	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	C₂²⋊#11715895	1	1	yes	characteristic
90	C₄	#15690737	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊#11715895	1	1	yes	characteristic
93	C₂	#20260545	1	1	yes	characteristic
94	C₂	#20260548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690731
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	C₄×D₈×C₉₇	C₂
socle	C₂²×C₉₇	#102
prime radical	C₂×D₈×C₉₇	C₄
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	C₄×D₈	D₉₇
hypocenter	C₉₇	#1333
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20260511	C₂	semidirect
2. #20260486	C₂	semidirect
16. D₈×C₉₇	C₈	semidirect
48. C₉₇	#1333	semidirect
62. D₈	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260511	2
#20260486	2
C₄×D₈×C₉₇	2
#1333	97