Group #27158914 — subgroups

Subgroups — #27158914

Order 12416.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20261854	C₂	1	2	yes	maximal, complemented
2	#20261834	C₂	1	4	yes	maximal, complemented
3	C₂×#15690898	C₂	1	1	yes	maximal, characteristic
4	#15690907	—	2	2	no	—
5	#15690906	C₂²	1	4	yes	—
6	#15690905	—	2	2	no	—
7	#15690898	—	2	2	no	—
8	#15690898	—	2	2	no	—
9	C₂×#13095521	C₄	1	1	yes	characteristic
10	C₂×#13095521	C₂²	1	2	yes	—
11	C₂×#13095519	C₄	1	2	yes	—
12	C₂×#13095519	C₂²	1	1	yes	characteristic
13	C₂×#13095512	C₄	1	1	yes	characteristic
14	#13095523	—	2	4	no	—
15	#13095521	—	2	2	no	—
16	#13095521	—	2	4	no	—
17	#13095519	—	2	2	no	—
18	#13095519	—	2	2	no	—
19	#13095519	D₄	1	2	yes	—
20	#13095519	D₄	1	2	yes	—
21	C₂×#11715897	C₂×C₄	1	2	yes	—
22	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
23	C₂×#11715897	C₂³	1	1	yes	characteristic
24	#13095512	—	2	2	no	—
25	#13095512	—	2	2	no	—
26	C₂×D₃₈₈	C₈	1	2	yes	—
27	C₂×D₃₈₈	C₈	1	1	yes	characteristic
28	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
31	C₂²×D₉₇	C₂×C₈	1	2	yes	—
32	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
37	D₃₈₈	—	2	4	no	—
38	D₃₈₈	—	2	2	no	—
39	C₄×D₉₇	—	2	2	no	—
40	C₄×D₉₇	—	2	2	no	—
41	C₄×D₉₇	C₂²⋊C₄	1	2	yes	—
42	C₄×D₉₇	M₁₆	1	2	yes	—
43	#11715897	—	2	2	no	—
44	#11715897	C₂²⋊C₄	1	1	yes	characteristic
45	#11715897	C₂²⋊C₄	1	1	yes	characteristic
46	#11715897	C₂×D₄	1	2	yes	—
47	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₉₇	—	2	2	no	—
49	C₂×D₉₇	—	4	1	no	—
50	C₂×D₉₇	—	4	2	no	—
51	C₂×D₉₇	C₂²⋊C₈	1	2	yes	—
52	C₄×C₉₇	—	2	2	no	—
53	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
54	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₉₇	—	2	1	no	—
57	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
58	D₉₇	—	2	2	no	—
59	D₉₇	—	4	2	no	—
60	#2115	—	97	1	no	maximal, Sylow
61	C₉₇	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	—	97	1	no	—
63	C₂×M₃₂	—	97	2	no	—
64	#349	—	97	4	no	—
65	C₂×#41	—	97	1	no	—
66	#131	—	194	2	no	—
67	#131	—	194	2	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂×M₁₆	—	97	1	no	—
70	C₂²×C₈	—	97	1	no	—
71	C₂²×C₈	—	97	2	no	—
72	M₃₂	—	97	4	no	—
73	M₃₂	—	194	2	no	—
74	C₂×C₁₆	—	194	2	no	—
75	#41	—	194	2	no	—
76	#41	—	194	2	no	—
77	C₂×Q₈	#11715906	1	1	yes	characteristic
78	C₂×D₄	—	97	2	no	—
79	C₂×D₄	—	97	1	no	—
80	C₂²×C₄	—	97	1	no	—
81	C₂²×C₄	—	97	2	no	—
82	M₁₆	—	194	4	no	—
83	M₁₆	—	194	2	no	—
84	C₂×C₈	—	97	1	no	—
85	C₂×C₈	—	97	1	no	—
86	C₂×C₈	—	97	2	no	—
87	C₂×C₈	—	97	2	no	—
88	C₂×C₈	—	97	2	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₁₆	—	194	4	no	—
92	C₂³	—	97	1	no	—
93	C₂³	—	97	2	no	—
94	Q₈	—	2	2	no	—
95	D₄	—	194	4	no	—
96	D₄	—	194	2	no	—
97	C₂×C₄	—	97	1	no	—
98	C₂×C₄	—	97	2	no	—
99	C₂×C₄	—	97	2	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	—	194	2	no	—
102	C₂×C₄	C₂×#11715906	1	2	yes	—
103	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
104	C₈	—	97	2	no	—
105	C₈	—	97	1	no	—
106	C₈	—	97	1	no	—
107	C₈	—	194	2	no	—
108	C₂²	—	97	2	no	—
109	C₂²	—	388	2	no	—
110	C₂²	—	388	1	no	—
111	C₂²	—	194	2	no	—
112	C₂²	C₂²×#11715906	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	97	1	no	—
115	C₄	—	97	1	no	—
116	C₄	C₂²⋊#11715906	1	2	yes	—
117	C₂	—	2	1	no	—
118	C₂	—	388	2	no	—
119	C₂	—	194	2	no	—
120	C₂	C₂×C₂²⋊#11715906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261854

C₂

yes

maximal, complemented

#20261834

C₂

yes

maximal, complemented

C₂×#15690898

C₂

yes

maximal, characteristic

#15690907

—

#15690906

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#13095521

C₂²

yes

—

C₂×#13095519

C₄

yes

—

C₂×#13095519

C₂²

yes

characteristic

C₂×#13095512

C₄

yes

—

—

—

—

—

D₄

yes

—

#13095519

D₄

yes

—

C₂×#11715897

C₂×C₄

yes

—

C₂×#11715897

C₂×C₄

yes

characteristic

C₂×#11715897

C₂³

yes

characteristic

#13095512

—

#13095512

—

C₂×D₃₈₈

C₈

yes

—

C₂×D₃₈₈

C₈

yes

characteristic

C₂×C₄×D₉₇

C₂×C₄

yes

—

C₂×C₄×D₉₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₉₇

C₈

yes

characteristic

C₂²×D₉₇

C₂×C₈

yes

—

C₂²×D₉₇

C₂×C₈

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

characteristic

C₂×Dic₉₇

C₂²×C₄

yes

characteristic

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

C₂²⋊C₄

yes

—

C₄×D₉₇

M₁₆

yes

—

#11715897

—

#11715897

C₂²⋊C₄

yes

characteristic

#11715897

C₂²⋊C₄

yes

characteristic

#11715897

C₂×D₄

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

Dic₉₇

C₂×C₂²⋊C₄

yes

characteristic

Dic₉₇

C₂×M₁₆

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₈

yes

characteristic

D₉₇

—

D₉₇

—

#2115

—

maximal, Sylow

C₉₇

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

#349

—

C₂×#41

—

#131

—

194

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

194

—

C₂×C₁₆

—

194

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

#11715906

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₁₆

—

194

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

194

—

D₄

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

C₂×#11715906

yes

—

103

C₂×C₄

C₂×#11715906

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

194

—

108

C₂²

—

109

C₂²

—

388

—

110

C₂²

—

388

—

111

C₂²

—

194

—

112

C₂²

C₂²×#11715906

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊#11715906

yes

—

117

C₂

—

118

C₂

—

388

—

119

C₂

—

194

—

120

C₂

C₂×C₂²⋊#11715906

yes

characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11715906
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₂×Q₈×C₉₇	C₈
socle	C₂×C₉₇	C₂×C₂²⋊C₈
prime radical	C₂×#13095512	C₄
cosocle	C₂×#11715897	C₂³
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#2115
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂

C₂×C₂²⋊#11715906

commutator

C₂²×C₉₇

C₂²×C₈

Frattini

C₂²

C₂²×#11715906

Fitting

C₂×Q₈×C₉₇

C₈

socle

C₂×C₉₇

C₂×C₂²⋊C₈

C₄

C₂³

C₂×Q₈

C₉₇

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
1. #20261854	C₂	semidirect
2. #20261834	C₂	semidirect
61. C₉₇	#2115	semidirect

H ≅ G/N

type

1. #20261854

C₂

semidirect

2. #20261834

C₂

semidirect

61. C₉₇

#2115

semidirect

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

subgroup	index
#20261854	2
#20261834	2
C₂×#15690898	2
#2115	97

subgroup

index

Subgroups — #27158914

Conjugacy classes of subgroups

Complements

Maximal subgroups