Group #27142878 — subgroups

Subgroups — #27142878

Order 12416.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690735	C₂	1	2	yes	maximal
3	#20260641	C₂	1	1	yes	maximal, characteristic
4	#20260498	C₂	1	2	yes	maximal
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	—	2	1	no	—
7	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
8	#15690810	—	2	4	no	—
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
12	#15690735	—	2	4	no	—
13	#15690717	—	2	1	no	—
14	#15690717	—	2	1	no	—
15	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	—	2	2	no	—
19	C₂×Dic₁₉₄	—	2	4	no	—
20	C₂×Dic₁₉₄	—	2	1	no	—
21	C₂×Dic₁₉₄	—	4	2	no	—
22	C₂×Dic₁₉₄	—	4	2	no	—
23	C₂×Dic₁₉₄	Q₈	1	2	yes	complemented
24	C₂²×Dic₉₇	—	2	2	no	—
25	C₂²×Dic₉₇	—	2	1	no	—
26	C₂²×Dic₉₇	—	2	1	no	—
27	C₄⋊Dic₉₇	—	2	4	no	—
28	C₄⋊Dic₉₇	—	2	2	no	—
29	C₄⋊Dic₉₇	—	2	2	no	—
30	C₄⋊Dic₉₇	D₄	1	4	yes	—
31	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₉₇	—	2	2	no	—
35	C₂×C₈×C₉₇	—	2	2	no	—
36	C₂×C₈×C₉₇	—	2	2	no	—
37	C₄²×C₉₇	—	2	2	no	—
38	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	—	2	1	no	—
42	C₂×C₄×C₉₇	—	2	1	no	—
43	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₉₇	#41	1	2	yes	—
49	C₂×Dic₉₇	—	2	4	no	—
50	C₂×Dic₉₇	—	2	2	no	—
51	C₂×Dic₉₇	—	4	1	no	—
52	C₂×Dic₉₇	—	4	1	no	—
53	C₂×Dic₉₇	—	4	2	no	—
54	C₂×Dic₉₇	—	4	2	no	—
55	C₂×Dic₉₇	—	4	1	no	—
56	C₂×Dic₉₇	—	4	1	no	—
57	C₂×Dic₉₇	—	4	1	no	—
58	Dic₁₉₄	—	2	4	no	—
59	Dic₁₉₄	—	4	8	no	—
60	Dic₁₉₄	—	4	2	no	—
61	C₈×C₉₇	—	4	2	no	—
62	C₂²×C₉₇	#120	1	1	yes	characteristic
63	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₉₇	#124	1	1	yes	characteristic
65	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₉₇	#137	1	1	yes	characteristic
69	C₄×C₉₇	—	2	2	no	—
70	C₄×C₉₇	—	2	2	no	—
71	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₉₇	#124	1	2	yes	—
73	Dic₉₇	—	4	2	no	—
74	Dic₉₇	—	4	4	no	—
75	Dic₉₇	—	8	1	no	—
76	C₂×C₉₇	#393	1	1	yes	characteristic
77	C₂×C₉₇	#450	1	1	yes	characteristic
78	C₂×C₉₇	#451	1	1	yes	characteristic
79	C₂×C₉₇	#475	1	1	yes	characteristic
80	C₂×C₉₇	#477	1	1	yes	characteristic
81	C₂×C₉₇	#487	1	1	yes	characteristic
82	C₂×C₉₇	#489	1	1	yes	characteristic
83	#2024	—	97	1	no	maximal, Sylow
84	C₉₇	#2024	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	—	97	1	no	—
86	C₂×#103	—	97	2	no	—
87	#391	—	97	1	no	—
88	#340	—	97	2	no	—
89	#336	D₉₇	1	1	yes	characteristic
90	C₂²×Q₈	—	97	1	no	—
91	C₂²×Q₈	—	194	1	no	—
92	C₂²×C₈	C₂×D₉₇	1	2	yes	—
93	C₄⋊Q₈	—	194	4	no	—
94	C₂×C₄⋊C₄	—	97	2	no	—
95	C₂×C₄⋊C₄	—	97	1	no	—
96	C₂×C₄⋊C₄	—	194	1	no	—
97	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
98	#103	—	194	4	no	—
99	#95	—	194	1	no	—
100	#95	—	194	1	no	—
101	C₂×Q₈	—	97	2	no	—
102	C₂×Q₈	—	388	2	no	—
103	C₂×Q₈	—	388	2	no	—
104	C₂×Q₈	—	194	4	no	—
105	C₂×Q₈	—	194	2	no	—
106	C₂×Q₈	—	194	2	no	—
107	C₂×Q₈	—	194	1	no	—
108	C₂²×C₄	—	194	1	no	—
109	C₂²×C₄	—	194	1	no	—
110	C₂²×C₄	—	194	2	no	—
111	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
112	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	97	4	no	—
118	C₄⋊C₄	—	194	4	no	—
119	C₄⋊C₄	—	194	2	no	—
120	C₄⋊C₄	—	194	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
123	Q₈	—	388	8	no	—
124	Q₈	—	388	2	no	—
125	Q₈	—	194	4	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	388	1	no	—
131	C₂×C₄	—	388	1	no	—
132	C₂×C₄	—	388	1	no	—
133	C₂×C₄	—	388	1	no	—
134	C₂×C₄	—	388	1	no	—
135	C₂×C₄	—	388	2	no	—
136	C₂×C₄	—	388	2	no	—
137	C₂×C₄	—	194	4	no	—
138	C₂×C₄	—	194	2	no	—
139	C₂×C₄	#13095500	1	2	yes	—
140	C₂×C₄	Dic₃₈₈	1	2	yes	—
141	C₂×C₄	D₄×D₉₇	1	2	yes	—
142	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
143	C₂×C₄	#13095510	1	1	yes	characteristic
144	C₂×C₄	#13095511	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#15690780	1	1	yes	characteristic
147	C₂²	#15690790	1	1	yes	characteristic
148	C₂²	#15690798	1	1	yes	characteristic
149	C₂²	C₂×#13095500	1	1	yes	characteristic
150	C₂²	C₂×Dic₃₈₈	1	1	yes	characteristic
151	C₂²	#15690838	1	1	yes	characteristic
152	C₂²	#15690843	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	776	1	no	—
156	C₄	—	388	4	no	—
157	C₄	—	388	2	no	—
158	C₄	#15690788	1	2	yes	—
159	C₄	#15690796	1	2	yes	—
160	C₂	#20260648	1	1	yes	characteristic
161	C₂	#20260888	1	1	yes	characteristic
162	C₂	#20260893	1	1	yes	characteristic
163	C₂	#20261021	1	1	yes	characteristic
164	C₂	#20261031	1	1	yes	characteristic
165	C₂	#20261086	1	1	yes	characteristic
166	C₂	#20261095	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690735

C₂

yes

maximal

#20260641

C₂

yes

maximal, characteristic

#20260498

C₂

yes

maximal

#336×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

—

C₂²×Dic₁₉₄

C₂²

yes

characteristic

#15690810

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690735

—

#15690717

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

Q₈

yes

complemented

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#124

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₉₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

194

—

C₂²×C₈

C₂×D₉₇

yes

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

#95

—

194

—

100

#95

—

194

—

101

C₂×Q₈

—

102

C₂×Q₈

—

388

—

103

C₂×Q₈

—

388

—

104

C₂×Q₈

—

194

—

105

C₂×Q₈

—

194

—

106

C₂×Q₈

—

194

—

107

C₂×Q₈

—

194

—

108

C₂²×C₄

—

194

—

109

C₂²×C₄

—

194

—

110

C₂²×C₄

—

194

—

111

C₂²×C₄

D₃₈₈

yes

characteristic

112

C₂²×C₄

D₃₈₈

yes

characteristic

113

C₂²×C₄

C₂²×D₉₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

194

—

119

C₄⋊C₄

—

194

—

120

C₄⋊C₄

—

194

—

121

C₄²

—

122

C₂³

C₂×D₃₈₈

yes

characteristic

123

Q₈

—

388

—

124

Q₈

—

388

—

125

Q₈

—

194

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

388

—

131

C₂×C₄

—

388

—

132

C₂×C₄

—

388

—

133

C₂×C₄

—

388

—

134

C₂×C₄

—

388

—

135

C₂×C₄

—

388

—

136

C₂×C₄

—

388

—

137

C₂×C₄

—

194

—

138

C₂×C₄

—

194

—

139

C₂×C₄

#13095500

yes

—

140

C₂×C₄

Dic₃₈₈

yes

—

141

C₂×C₄

D₄×D₉₇

yes

—

142

C₂×C₄

Q₈×D₉₇

yes

characteristic

143

C₂×C₄

#13095510

yes

characteristic

144

C₂×C₄

#13095511

yes

—

145

C₈

—

146

C₂²

#15690780

yes

characteristic

147

C₂²

#15690790

yes

characteristic

148

C₂²

#15690798

yes

characteristic

149

C₂²

C₂×#13095500

yes

characteristic

150

C₂²

C₂×Dic₃₈₈

yes

characteristic

151

C₂²

#15690838

yes

characteristic

152

C₂²

#15690843

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

776

—

156

C₄

—

388

—

157

C₄

—

388

—

158

C₄

#15690788

yes

—

159

C₄

#15690796

yes

—

160

C₂

#20260648

yes

characteristic

161

C₂

#20260888

yes

characteristic

162

C₂

#20260893

yes

characteristic

163

C₂

#20261021

yes

characteristic

164

C₂

#20261031

yes

characteristic

165

C₂

#20261086

yes

characteristic

166

C₂

#20261095

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#336×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#336	D₉₇
hypocenter	C₉₇	#2024
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#336×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₁₉₄	Q₈	semidirect
84. C₉₇	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₁₉₄

Q₈

semidirect

84. C₉₇

#2024

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690735	2
#20260641	2
#20260498	2
#336×C₉₇	2
#2024	97

subgroup

index

#336×C₉₇

Subgroups — #27142878

Conjugacy classes of subgroups

Complements

Maximal subgroups