Group #27157652 — subgroups

Subgroups — #27157652

Order 12416.

260 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 106 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×#15690877	C₂	1	2	yes	maximal
2	#20261601	C₂	1	2	yes	maximal
3	#20261601	C₂	1	2	yes	maximal
4	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715897	C₂²	1	4	yes	—
6	#15690877	—	2	4	no	—
7	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
14	C₂×#11715897	—	2	4	no	—
15	C₂×#11715897	—	2	4	no	—
16	C₂×#11715897	—	2	4	no	—
17	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
21	C₄×Dic₉₇	—	2	2	no	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₄×Dic₉₇	—	2	2	no	—
24	C₄×Dic₉₇	—	2	2	no	—
25	C₄×Dic₉₇	C₈	1	4	yes	complemented
26	C₄×Dic₉₇	D₄	1	2	yes	—
27	C₄×Dic₉₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	1	no	—
34	C₂×C₄×C₉₇	—	2	1	no	—
35	C₂×C₄×C₉₇	—	2	1	no	—
36	C₂×C₄×C₉₇	—	2	1	no	—
37	C₂×C₄×C₉₇	—	2	1	no	—
38	C₂×C₄×C₉₇	—	2	1	no	—
39	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
42	C₂×Dic₉₇	—	2	1	no	—
43	C₂×Dic₉₇	—	2	1	no	—
44	C₂×Dic₉₇	—	2	1	no	—
45	C₂×Dic₉₇	—	2	1	no	—
46	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
47	C₂×Dic₉₇	C₂×C₈	1	2	yes	—
48	C₂×Dic₉₇	C₂×C₈	1	2	yes	—
49	C₂×Dic₉₇	M₁₆	1	2	yes	—
50	C₂×Dic₉₇	M₁₆	1	2	yes	—
51	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×Dic₉₇	#41	1	2	yes	—
54	C₂×Dic₉₇	#41	1	2	yes	—
55	#11715897	—	4	4	no	—
56	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	#117	1	1	yes	characteristic
58	C₂²×C₉₇	#117	1	1	yes	characteristic
59	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
60	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
63	C₄×C₉₇	—	2	2	no	—
64	C₄×C₉₇	—	2	2	no	—
65	C₄×C₉₇	—	2	2	no	—
66	Dic₉₇	—	2	2	no	—
67	Dic₉₇	—	2	2	no	—
68	Dic₉₇	C₄⋊C₈	1	4	yes	—
69	Dic₉₇	#124	1	2	yes	—
70	Dic₉₇	#125	1	2	yes	—
71	C₂×C₉₇	#382	1	1	yes	characteristic
72	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₉₇	#423	1	1	yes	characteristic
74	C₂×C₉₇	#431	1	1	yes	characteristic
75	C₂×C₉₇	#431	1	1	yes	characteristic
76	C₂×C₉₇	#432	1	1	yes	characteristic
77	C₂×C₉₇	#432	1	1	yes	characteristic
78	#1831	—	97	1	no	maximal, Sylow
79	C₉₇	#1831	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄⋊C₈	—	97	2	no	—
81	C₄³	—	97	1	no	—
82	#336	—	97	2	no	—
83	#336	—	97	2	no	—
84	C₂²×C₈	—	97	4	no	—
85	C₂×C₄²	—	97	1	no	—
86	C₂×C₄²	—	97	1	no	—
87	C₂×C₄²	—	97	1	no	—
88	C₂×C₄²	—	97	1	no	—
89	C₂×C₄²	—	97	1	no	—
90	C₂×C₄²	—	97	1	no	—
91	C₂×C₄²	#87604	1	1	yes	characteristic
92	C₄⋊C₈	—	194	4	no	—
93	C₂²×C₄	—	97	1	no	—
94	C₂²×C₄	—	97	1	no	—
95	C₂²×C₄	—	97	1	no	—
96	C₂²×C₄	—	97	1	no	—
97	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
98	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
99	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
100	C₂×C₈	—	194	4	no	—
101	C₂×C₈	—	194	4	no	—
102	C₂×C₈	—	194	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	97	2	no	—
105	C₄²	—	97	2	no	—
106	C₄²	—	97	4	no	—
107	C₄²	—	194	2	no	—
108	C₄²	—	194	2	no	—
109	C₄²	—	194	2	no	—
110	C₄²	—	194	2	no	—
111	C₂³	C₂²×#87604	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	97	1	no	—
119	C₂×C₄	—	97	2	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	97	2	no	—
122	C₂×C₄	—	97	2	no	—
123	C₂×C₄	—	97	2	no	—
124	C₂×C₄	—	97	2	no	—
125	C₂×C₄	—	97	2	no	—
126	C₂×C₄	—	97	2	no	—
127	C₂×C₄	—	194	1	no	—
128	C₂×C₄	—	194	1	no	—
129	C₂×C₄	—	194	1	no	—
130	C₂×C₄	—	194	1	no	—
131	C₂×C₄	C₄⋊#87604	1	2	yes	—
132	C₂×C₄	#13095519	1	2	yes	—
133	C₂×C₄	#13095520	1	2	yes	—
134	C₈	—	388	4	no	—
135	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
136	C₂²	#15690889	1	1	yes	characteristic
137	C₂²	#15690889	1	1	yes	characteristic
138	C₂²	C₂×#13095519	1	1	yes	characteristic
139	C₂²	C₂×#13095520	1	1	yes	characteristic
140	C₂²	D₉₇⋊M₁₆	1	1	yes	characteristic
141	C₂²	D₉₇⋊M₁₆	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	97	2	no	—
146	C₄	—	97	2	no	—
147	C₄	—	97	4	no	—
148	C₄	—	194	2	no	—
149	C₄	—	194	2	no	—
150	C₂	#20261679	1	1	yes	characteristic
151	C₂	#20261736	1	1	yes	characteristic
152	C₂	#20261738	1	1	yes	characteristic
153	C₂	#20261752	1	1	yes	characteristic
154	C₂	#20261753	1	1	yes	characteristic
155	C₂	#20261755	1	1	yes	characteristic
156	C₂	#20261758	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690877

C₂

yes

maximal

#20261601

C₂

yes

maximal

#20261601

C₂

yes

maximal

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

C₂²×#11715897

C₂²

yes

—

#15690877

—

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₄

yes

—

—

—

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂³

yes

characteristic

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₈

yes

complemented

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

M₁₆

yes

—

C₂×Dic₉₇

M₁₆

yes

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

#41

yes

—

C₂×Dic₉₇

#41

yes

—

#11715897

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₄⋊C₈

yes

—

Dic₉₇

#124

yes

—

Dic₉₇

#125

yes

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₄³

—

#336

—

#336

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#87604

yes

characteristic

C₄⋊C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂×C₈

—

194

—

101

C₂×C₈

—

194

—

102

C₂×C₈

—

194

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

194

—

108

C₄²

—

194

—

109

C₄²

—

194

—

110

C₄²

—

194

—

111

C₂³

C₂²×#87604

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

194

—

128

C₂×C₄

—

194

—

129

C₂×C₄

—

194

—

130

C₂×C₄

—

194

—

131

C₂×C₄

C₄⋊#87604

yes

—

132

C₂×C₄

#13095519

yes

—

133

C₂×C₄

#13095520

yes

—

134

C₈

—

388

—

135

C₂²

C₂×C₄⋊#87604

yes

characteristic

136

C₂²

#15690889

yes

characteristic

137

C₂²

#15690889

yes

characteristic

138

C₂²

C₂×#13095519

yes

characteristic

139

C₂²

C₂×#13095520

yes

characteristic

140

C₂²

D₉₇⋊M₁₆

yes

characteristic

141

C₂²

D₉₇⋊M₁₆

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

194

—

149

C₄

—

194

—

150

C₂

#20261679

yes

characteristic

151

C₂

#20261736

yes

characteristic

152

C₂

#20261738

yes

characteristic

153

C₂

#20261752

yes

characteristic

154

C₂

#20261753

yes

characteristic

155

C₂

#20261755

yes

characteristic

156

C₂

#20261758

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×C₄²×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#1831
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#87604

commutator

C₂²×C₉₇

C₂²×C₈

Fitting

C₂×C₄²×C₉₇

C₄

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×Dic₉₇

C₂³

C₂×C₄²

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
25. C₄×Dic₉₇	C₈	semidirect
79. C₉₇	#1831	semidirect

H ≅ G/N

type

25. C₄×Dic₉₇

C₈

semidirect

79. C₉₇

#1831

semidirect

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

subgroup	index
C₂×#15690877	2
#20261601	2
C₄²×Dic₉₇	2
#1831	97

subgroup

index

C₂×#15690877

#20261601

C₄²×Dic₉₇

#1831

Subgroups — #27157652

Conjugacy classes of subgroups

Complements

Maximal subgroups