Group #27140793 — subgroups

Subgroups — #27140793

Order 12416.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20261471	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260932	C₂	1	2	yes	maximal, complemented
3	#20260479	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260467	C₂	1	2	yes	maximal, complemented
5	#493×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095499	—	2	2	no	—
7	#15690797	—	2	2	no	—
8	#15690791	—	2	2	no	—
9	C₄×Dic₁₉₄	C₂²	1	2	yes	—
10	D₄×Dic₉₇	—	2	2	no	—
11	C₄⋊#11715895	C₂²	1	2	yes	—
12	#15690731	—	2	2	no	—
13	C₂×D₈×C₉₇	—	2	2	no	—
14	#127×C₉₇	C₂²	1	1	yes	characteristic
15	#127×C₉₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉₄	D₄	1	2	yes	—
18	C₂²×Dic₉₇	—	2	2	no	—
19	#13095499	—	4	4	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₄⋊Dic₉₇	D₄	1	2	yes	—
22	#13095490	—	2	2	no	—
23	C₂²⋊Dic₉₇	—	2	4	no	—
24	C₂²⋊Dic₉₇	—	2	2	no	—
25	C₄×Dic₉₇	—	2	2	no	—
26	C₂×D₄×C₉₇	—	2	2	no	—
27	C₂×D₄×C₉₇	—	2	2	no	—
28	C₂×D₄×C₉₇	—	2	1	no	—
29	C₂×D₄×C₉₇	D₄	1	2	yes	—
30	D₈×C₉₇	—	4	4	no	—
31	C₂×C₈×C₉₇	—	2	1	no	—
32	C₄²×C₉₇	C₂³	1	1	yes	characteristic
33	C₂³×C₉₇	—	2	2	no	—
34	C₂³×C₉₇	—	4	1	no	—
35	D₄×C₉₇	—	4	2	no	—
36	D₄×C₉₇	—	4	2	no	—
37	D₄×C₉₇	—	4	2	no	—
38	D₄×C₉₇	—	4	2	no	—
39	D₄×C₉₇	—	4	2	no	—
40	D₄×C₉₇	—	4	2	no	—
41	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₉₇	—	2	2	no	—
44	C₂×Dic₉₇	—	2	2	no	—
45	C₂×Dic₉₇	—	8	2	no	—
46	Dic₁₉₄	—	2	2	no	—
47	Dic₁₉₄	—	2	2	no	—
48	C₈×C₉₇	—	2	2	no	—
49	#11715895	—	4	2	no	—
50	C₂²×C₉₇	—	4	4	no	—
51	C₂²×C₉₇	—	4	2	no	—
52	C₂²×C₉₇	—	8	2	no	—
53	C₂²×C₉₇	—	8	1	no	—
54	C₂²×C₉₇	#120	1	1	yes	characteristic
55	C₄×C₉₇	—	2	1	no	—
56	C₄×C₉₇	#135	1	2	yes	—
57	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₉₇	—	4	2	no	—
59	Dic₉₇	—	4	2	no	—
60	C₂×C₉₇	—	8	2	no	—
61	C₂×C₉₇	—	16	1	no	—
62	C₂×C₉₇	#449	1	2	yes	—
63	C₂×C₉₇	#453	1	1	yes	characteristic
64	#1620	—	97	1	no	maximal, Sylow
65	C₉₇	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	—	97	1	no	—
67	#493	D₉₇	1	1	yes	characteristic
68	#460	—	97	2	no	—
69	#331	—	97	1	no	—
70	Q₈⋊C₈	—	97	2	no	—
71	C₂×SD₁₆	—	194	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂×D₉₇	1	1	yes	characteristic
74	#127	C₂×D₉₇	1	1	yes	characteristic
75	#124	—	194	2	no	—
76	#123	—	194	2	no	—
77	C₄×Q₈	—	97	2	no	—
78	C₄×D₄	—	194	2	no	—
79	C₄⋊C₈	—	97	2	no	—
80	#102	—	194	2	no	—
81	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
82	C₂×Q₈	—	97	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	C₂²⋊D₉₇	1	2	yes	—
87	C₂²×C₄	—	194	2	no	—
88	SD₁₆	—	388	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	194	2	no	—
92	C₄⋊C₄	—	97	2	no	—
93	C₄⋊C₄	—	194	2	no	—
94	C₂²⋊C₄	—	194	4	no	—
95	C₂²⋊C₄	—	194	2	no	—
96	C₄²	—	194	2	no	—
97	C₄²	C₂²×D₉₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	194	2	no	—
101	Q₈	—	194	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	776	2	no	—
109	C₂×C₄	—	194	2	no	—
110	C₂×C₄	—	194	2	no	—
111	C₂×C₄	D₄×D₉₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	388	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	#15690778	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	388	2	no	—
122	C₄	—	388	2	no	—
123	C₄	#15690829	1	2	yes	—
124	C₄	#15690834	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	#20260877	1	2	yes	—
128	C₂	#20260902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261471

C₂

yes

maximal, characteristic, complemented

#20260932

C₂

yes

maximal, complemented

#20260479

C₂

yes

maximal, characteristic, complemented

#20260467

C₂

yes

maximal, complemented

#493×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095499

—

#15690797

—

#15690791

—

C₄×Dic₁₉₄

C₂²

yes

—

D₄×Dic₉₇

—

C₄⋊#11715895

C₂²

yes

—

#15690731

—

C₂×D₈×C₉₇

—

#127×C₉₇

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

D₄

yes

—

C₂²×Dic₉₇

—

#13095499

—

C₂×#11715895

—

C₄⋊Dic₉₇

D₄

yes

—

#13095490

—

C₂²⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

D₄

yes

—

D₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#135

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#449

yes

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#559

—

#493

D₉₇

yes

characteristic

#460

—

#331

—

Q₈⋊C₈

—

C₂×SD₁₆

—

194

—

C₂×D₈

—

#127

C₂×D₉₇

yes

characteristic

#127

C₂×D₉₇

yes

characteristic

#124

—

194

—

#123

—

194

—

C₄×Q₈

—

C₄×D₄

—

194

—

C₄⋊C₈

—

#102

—

194

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

—

194

—

SD₁₆

—

388

—

D₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

194

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

194

—

101

Q₈

—

194

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

776

—

109

C₂×C₄

—

194

—

110

C₂×C₄

—

194

—

111

C₂×C₄

D₄×D₉₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₈

—

114

C₈

—

388

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#15690778

yes

characteristic

120

C₄

—

121

C₄

—

388

—

122

C₄

—

388

—

123

C₄

#15690829

yes

—

124

C₄

#15690834

yes

—

125

C₂

—

126

C₂

—

127

C₂

#20260877

yes

—

128

C₂

#20260902

yes

characteristic

	subgroup	G/H
center	C₂²	#15690778
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting, prime radical	#493×C₉₇	C₂
socle	C₂²×C₉₇	#120
hypercenter	#493	D₉₇
hypocenter	C₉₇	#1620
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15690778

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting, prime radical

#493×C₉₇

C₂

C₂²×C₉₇

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20261471	C₂	semidirect
2. #20260932	C₂	semidirect
3. #20260479	C₂	semidirect
4. #20260467	C₂	semidirect
65. C₉₇	#1620	semidirect

H ≅ G/N

type

1. #20261471

C₂

semidirect

2. #20260932

C₂

semidirect

3. #20260479

C₂

semidirect

4. #20260467

C₂

semidirect

65. C₉₇

#1620

semidirect

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

subgroup	index
#20261471	2
#20260932	2
#20260479	2
#20260467	2
#493×C₉₇	2
#1620	97

subgroup

index

#493×C₉₇

Subgroups — #27140793

Conjugacy classes of subgroups

Complements

Maximal subgroups