Group #27143452 — subgroups

Subgroups — #27143452

Order 12416.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690750	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260773	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260536	C₂	1	4	yes	maximal, complemented
4	C₂²×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690769	—	2	1	no	—
6	#15690750	—	2	2	no	—
7	C₂×#13095486	C₂²	1	2	yes	complemented
8	#15690744	C₂²	1	2	yes	complemented
9	#15690742	—	2	1	no	—
10	C₂²×D₄×C₉₇	—	2	1	no	—
11	C₂×D₈×C₉₇	—	2	1	no	—
12	C₂×D₈×C₉₇	—	2	2	no	—
13	C₂×D₈×C₉₇	C₄	1	4	yes	complemented
14	C₂×D₈×C₉₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₉₇	—	2	2	no	—
17	C₂²⋊Dic₉₇	—	4	1	no	—
18	C₄×Dic₉₇	—	4	1	no	—
19	#13095486	—	2	2	no	—
20	C₂⁴×C₉₇	—	4	1	no	—
21	C₂×D₄×C₉₇	—	2	4	no	—
22	C₂×D₄×C₉₇	—	2	2	no	—
23	C₂×D₄×C₉₇	—	4	1	no	—
24	C₂×D₄×C₉₇	—	4	2	no	—
25	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
26	D₈×C₉₇	—	2	4	no	—
27	D₈×C₉₇	—	2	2	no	—
28	D₈×C₉₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₉₇	D₄	1	2	yes	—
32	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
33	C₂³×C₉₇	—	4	4	no	—
34	C₂³×C₉₇	—	4	2	no	—
35	C₂³×C₉₇	—	8	4	no	—
36	C₂³×C₉₇	D₈	1	1	yes	characteristic
37	D₄×C₉₇	—	2	4	no	—
38	D₄×C₉₇	—	4	4	no	—
39	D₄×C₉₇	—	4	2	no	—
40	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
42	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₉₇	—	4	2	no	—
45	C₈×C₉₇	C₂²⋊C₄	1	2	yes	—
46	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₉₇	—	2	2	no	—
49	C₂²×C₉₇	—	4	4	no	—
50	C₂²×C₉₇	—	8	8	no	—
51	C₂²×C₉₇	—	8	4	no	—
52	C₂²×C₉₇	#102	1	2	yes	—
53	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₉₇	#102	1	2	yes	—
55	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₉₇	—	8	2	no	—
58	C₂×C₉₇	—	2	2	no	—
59	C₂×C₉₇	—	8	4	no	—
60	C₂×C₉₇	C₂×#102	1	1	yes	characteristic
61	C₂×C₉₇	#509	1	2	yes	—
62	#2140	—	97	1	no	maximal, Sylow
63	C₉₇	#2140	1	1	yes	characteristic, complemented, Sylow
64	C₂²×D₈	D₉₇	1	1	yes	characteristic
65	C₂×M₃₂	—	97	1	no	—
66	#427	—	97	1	no	—
67	#357	—	97	4	no	—
68	C₂²×D₄	—	2	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	Dic₉₇	1	4	yes	complemented
72	C₂×D₈	C₂×D₉₇	1	2	yes	—
73	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
74	#117	—	194	1	no	—
75	M₃₂	—	194	2	no	—
76	C₂×C₁₆	—	97	2	no	—
77	#107	—	97	2	no	—
78	#106	—	194	1	no	—
79	C₂⁴	—	4	1	no	—
80	C₂×D₄	—	2	4	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
85	D₈	—	2	4	no	—
86	D₈	—	2	2	no	—
87	D₈	C₂×Dic₉₇	1	4	yes	—
88	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
91	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
92	C₄⋊C₄	—	194	2	no	—
93	C₂²⋊C₄	—	388	1	no	—
94	C₄²	—	388	1	no	—
95	C₁₆	—	194	2	no	—
96	C₂³	—	4	2	no	—
97	C₂³	—	4	4	no	—
98	C₂³	—	8	4	no	—
99	C₂³	#13095496	1	1	yes	characteristic
100	D₄	—	2	4	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	4	no	—
103	C₂×C₄	—	388	2	no	—
104	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
105	C₂×C₄	#13095496	1	1	yes	characteristic
106	C₂×C₄	#13095499	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
108	C₈	C₂²⋊Dic₉₇	1	2	yes	—
109	C₈	C₂²×Dic₉₇	1	1	yes	characteristic
110	C₈	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	8	no	—
114	C₂²	—	8	4	no	—
115	C₂²	#15690731	1	2	yes	—
116	C₂²	C₂×#13095496	1	1	yes	characteristic
117	C₄	—	776	2	no	—
118	C₄	#15690731	1	2	yes	—
119	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
120	C₄	C₂×#13095499	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	8	4	no	—
123	C₂	C₂×#15690731	1	1	yes	characteristic
124	C₂	#20261171	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690750

C₂

yes

maximal, characteristic, complemented

#20260773

C₂

yes

maximal, characteristic, complemented

#20260536

C₂

yes

maximal, complemented

C₂²×D₈×C₉₇

C₂

yes

maximal, characteristic

#15690769

—

#15690750

—

C₂×#13095486

C₂²

yes

complemented

#15690744

C₂²

yes

complemented

#15690742

—

C₂²×D₄×C₉₇

—

C₂×D₈×C₉₇

—

C₂×D₈×C₉₇

—

C₂×D₈×C₉₇

C₄

yes

complemented

C₂×D₈×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

#13095486

—

C₂⁴×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

D₈×C₉₇

—

D₈×C₉₇

—

D₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

D₈

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

—

C₈×C₉₇

C₂²×C₄

yes

characteristic

C₈×C₉₇

C₂×D₄

yes

characteristic

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#102

yes

—

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₄×C₉₇

#102

yes

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

C₂×SD₁₆

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#102

yes

characteristic

C₂×C₉₇

#509

yes

—

#2140

—

maximal, Sylow

C₉₇

#2140

yes

characteristic, complemented, Sylow

C₂²×D₈

D₉₇

yes

characteristic

C₂×M₃₂

—

#427

—

#357

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₉₇

yes

complemented

C₂×D₈

C₂×D₉₇

yes

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

194

—

M₃₂

—

194

—

C₂×C₁₆

—

#107

—

#106

—

194

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

D₈

—

D₈

—

D₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂²⋊D₉₇

yes

characteristic

C₂×C₈

C₂²⋊D₉₇

yes

—

C₂×C₈

C₂²×D₉₇

yes

characteristic

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

388

—

C₄²

—

388

—

C₁₆

—

194

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#13095496

yes

characteristic

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

388

—

104

C₂×C₄

C₂²⋊Dic₉₇

yes

—

105

C₂×C₄

#13095496

yes

characteristic

106

C₂×C₄

#13095499

yes

—

107

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

108

C₈

C₂²⋊Dic₉₇

yes

—

109

C₈

C₂²×Dic₉₇

yes

characteristic

110

C₈

C₂×C₂²⋊D₉₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#15690731

yes

—

116

C₂²

C₂×#13095496

yes

characteristic

117

C₄

—

776

—

118

C₄

#15690731

yes

—

119

C₄

C₂×C₂²⋊Dic₉₇

yes

characteristic

120

C₄

C₂×#13095499

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₂×#15690731

yes

characteristic

124

C₂

#20261171

yes

—

	subgroup	G/H
center	C₂²	C₂×#13095496
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting, prime radical	C₂²×D₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	C₂²×D₈	D₉₇
hypocenter	C₉₇	#2140
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

C₂×#13095496

commutator

C₈×C₉₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₉₇

Fitting, prime radical

C₂²×D₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×D₈

cosocle

C₂×C₈×C₉₇

C₂³

hypercenter

C₂²×D₈

D₉₇

hypocenter

C₉₇

#2140

LCS

G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#15690750	C₂	semidirect
2. #20260773	C₂	semidirect
3. #20260536	C₂	semidirect
7. C₂×#13095486	C₂²	semidirect
8. #15690744	C₂²	semidirect
13. C₂×D₈×C₉₇	C₄	semidirect
63. C₉₇	#2140	semidirect
71. C₂×D₈	Dic₉₇	semidirect

H ≅ G/N

type

1. C₂×#15690750

C₂

semidirect

2. #20260773

C₂

semidirect

3. #20260536

C₂

semidirect

7. C₂×#13095486

C₂²

semidirect

8. #15690744

C₂²

semidirect

13. C₂×D₈×C₉₇

C₄

semidirect

63. C₉₇

#2140

semidirect

71. C₂×D₈

Dic₉₇

semidirect

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

subgroup	index
C₂×#15690750	2
#20260773	2
#20260536	2
C₂²×D₈×C₉₇	2
#2140	97

subgroup

index

C₂²×D₈×C₉₇

Subgroups — #27143452

Conjugacy classes of subgroups

Complements

Maximal subgroups