Group #27158908 — subgroups

Subgroups — #27158908

Order 12416.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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×#15690907	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690905	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261832	C₂	1	4	yes	maximal, complemented
4	C₂×#15690898	C₂	1	1	yes	maximal, characteristic
5	#15690907	—	2	2	no	—
6	#15690905	—	2	2	no	—
7	C₂×#13095523	C₂²	1	2	yes	—
8	C₂×#13095523	C₂²	1	2	yes	—
9	#15690898	—	2	2	no	—
10	#15690898	—	2	2	no	—
11	C₂×#13095521	C₄	1	2	yes	—
12	C₂×#13095521	C₄	1	1	yes	characteristic
13	C₂×#13095519	C₂²	1	2	yes	—
14	C₂×#13095519	C₂²	1	1	yes	characteristic
15	C₂×#13095512	C₄	1	1	yes	characteristic
16	#13095523	—	2	2	no	—
17	#13095523	—	2	2	no	—
18	#13095521	—	2	2	no	—
19	#13095521	—	2	4	no	—
20	#13095519	—	2	2	no	—
21	#13095519	—	2	2	no	—
22	#13095519	D₄	1	2	yes	—
23	#13095519	D₄	1	2	yes	—
24	C₂×#11715897	C₂×C₄	1	2	yes	—
25	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
26	C₂×#11715897	C₂³	1	1	yes	characteristic
27	#13095512	—	2	2	no	—
28	#13095512	—	2	2	no	—
29	C₂×D₃₈₈	C₈	1	2	yes	—
30	C₂×D₃₈₈	C₈	1	1	yes	characteristic
31	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
34	C₂²×D₉₇	C₂×C₈	1	2	yes	—
35	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
36	Q₈×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
40	D₃₈₈	—	2	4	no	—
41	D₃₈₈	—	2	2	no	—
42	C₄×D₉₇	—	2	2	no	—
43	C₄×D₉₇	—	2	2	no	—
44	C₄×D₉₇	C₂²⋊C₄	1	2	yes	—
45	C₄×D₉₇	M₁₆	1	2	yes	—
46	#11715897	—	2	2	no	—
47	#11715897	C₂²⋊C₄	1	2	yes	—
48	#11715897	C₂×D₄	1	1	yes	characteristic
49	#11715897	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×D₉₇	—	2	2	no	—
52	C₂×D₉₇	—	2	2	no	—
53	C₂×D₉₇	—	2	4	no	—
54	C₂×D₉₇	C₂²⋊C₈	1	2	yes	—
55	C₄×C₉₇	—	2	2	no	—
56	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
57	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₉₇	#504	1	2	yes	—
61	D₉₇	—	2	2	no	—
62	D₉₇	—	4	2	no	—
63	#2113	—	97	1	no	maximal, Sylow
64	C₉₇	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	97	1	no	—
66	C₂×M₃₂	—	97	1	no	—
67	C₂²×C₁₆	—	97	1	no	—
68	C₂²⋊C₁₆	—	97	4	no	—
69	C₂×#41	—	97	1	no	—
70	#131	—	194	2	no	—
71	#131	—	194	2	no	—
72	C₂×M₁₆	—	97	2	no	—
73	C₂×M₁₆	—	97	1	no	—
74	C₂²×C₈	—	97	1	no	—
75	C₂²×C₈	—	97	2	no	—
76	M₃₂	—	194	2	no	—
77	C₂×C₁₆	—	97	2	no	—
78	C₂×C₁₆	—	97	2	no	—
79	C₂×C₁₆	—	194	2	no	—
80	#41	—	194	2	no	—
81	#41	—	194	2	no	—
82	C₂×Q₈	#11715906	1	1	yes	characteristic
83	C₂×D₄	—	97	2	no	—
84	C₂×D₄	—	97	1	no	—
85	C₂²×C₄	—	97	1	no	—
86	C₂²×C₄	—	97	2	no	—
87	M₁₆	—	194	4	no	—
88	M₁₆	—	194	2	no	—
89	C₂×C₈	—	97	1	no	—
90	C₂×C₈	—	97	1	no	—
91	C₂×C₈	—	97	2	no	—
92	C₂×C₈	—	97	2	no	—
93	C₂×C₈	—	97	2	no	—
94	C₂×C₈	—	194	2	no	—
95	C₂×C₈	—	194	2	no	—
96	C₁₆	—	194	2	no	—
97	C₁₆	—	194	2	no	—
98	C₂³	—	97	1	no	—
99	C₂³	—	97	2	no	—
100	Q₈	—	2	2	no	—
101	D₄	—	194	4	no	—
102	D₄	—	194	2	no	—
103	C₂×C₄	—	97	1	no	—
104	C₂×C₄	—	97	2	no	—
105	C₂×C₄	—	97	2	no	—
106	C₂×C₄	—	194	2	no	—
107	C₂×C₄	—	194	2	no	—
108	C₂×C₄	C₂×#11715906	1	2	yes	—
109	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
110	C₈	—	97	1	no	—
111	C₈	—	97	1	no	—
112	C₈	—	97	2	no	—
113	C₈	—	194	2	no	—
114	C₂²	—	97	2	no	—
115	C₂²	—	194	2	no	—
116	C₂²	—	194	4	no	—
117	C₂²	—	194	2	no	—
118	C₂²	C₂²×#11715906	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	97	1	no	—
121	C₄	—	97	1	no	—
122	C₄	C₂²⋊#11715906	1	2	yes	—
123	C₂	—	388	2	no	—
124	C₂	—	194	2	no	—
125	C₂	C₂×C₂²⋊#11715906	1	1	yes	characteristic
126	C₂	#20261857	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690907

C₂

yes

maximal, characteristic, complemented

C₂×#15690905

C₂

yes

maximal, characteristic, complemented

#20261832

C₂

yes

maximal, complemented

C₂×#15690898

C₂

yes

maximal, characteristic

#15690907

—

#15690905

—

C₂×#13095523

C₂²

yes

—

C₂×#13095523

C₂²

yes

—

#15690898

—

#15690898

—

C₂×#13095521

C₄

yes

—

C₂×#13095521

C₄

yes

characteristic

C₂×#13095519

C₂²

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

—

#11715897

C₂×D₄

yes

characteristic

#11715897

C₂×D₄

yes

characteristic

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×C₂²⋊C₈

yes

characteristic

C₂×C₉₇

#504

yes

—

D₉₇

—

D₉₇

—

#2113

—

maximal, Sylow

C₉₇

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂×#41

—

#131

—

194

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

194

—

C₂×C₁₆

—

C₂×C₁₆

—

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

—

C₁₆

—

194

—

C₂³

—

C₂³

—

100

Q₈

—

101

D₄

—

194

—

102

D₄

—

194

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

194

—

107

C₂×C₄

—

194

—

108

C₂×C₄

C₂×#11715906

yes

—

109

C₂×C₄

C₂×#11715906

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

194

—

114

C₂²

—

115

C₂²

—

194

—

116

C₂²

—

194

—

117

C₂²

—

194

—

118

C₂²

C₂²×#11715906

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂²⋊#11715906

yes

—

123

C₂

—

388

—

124

C₂

—

194

—

125

C₂

C₂×C₂²⋊#11715906

yes

characteristic

126

C₂

#20261857

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×Q₈×C₉₇	C₈
prime radical	C₂×#13095512	C₄
cosocle	C₂×#11715897	C₂³
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#2113
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×#11715906

commutator, socle

C₂²×C₉₇

C₂²×C₈

Fitting

C₂×Q₈×C₉₇

C₈

C₄

C₂³

C₂×Q₈

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
1. C₂×#15690907	C₂	semidirect
2. C₂×#15690905	C₂	semidirect
3. #20261832	C₂	semidirect
64. C₉₇	#2113	semidirect

H ≅ G/N

type

1. C₂×#15690907

C₂

semidirect

2. C₂×#15690905

C₂

semidirect

3. #20261832

C₂

semidirect

64. C₉₇

#2113

semidirect

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

subgroup	index
C₂×#15690907	2
C₂×#15690905	2
#20261832	2
C₂×#15690898	2
#2113	97

subgroup

index

Subgroups — #27158908

Conjugacy classes of subgroups

Complements

Maximal subgroups