Group #27157929 — subgroups

Subgroups — #27157929

Order 12416.

260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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	#20261738	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095520	C₂	1	1	yes	maximal, characteristic
3	#20261615	C₂	1	4	yes	maximal
4	C₂×#15690809	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095520	C₂²	1	2	yes	—
6	C₂×#13095520	C₂²	1	2	yes	—
7	#15690877	—	2	1	no	—
8	C₄⋊#11715897	—	2	1	no	—
9	#15690865	—	2	1	no	—
10	C₄×#11715897	—	2	1	no	—
11	C₂²×Dic₁₉₄	C₄	1	2	yes	—
12	#15690809	—	2	1	no	—
13	#15690809	—	2	1	no	—
14	#15690809	—	2	1	no	—
15	#15690809	—	2	1	no	—
16	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
17	C₂×#13095490	C₂²	1	2	yes	—
18	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
20	#13095520	—	2	2	no	—
21	#13095520	—	2	4	no	—
22	#13095520	Q₈	1	4	yes	complemented
23	C₂×#11715897	—	2	2	no	—
24	C₂×#11715897	—	2	2	no	—
25	C₂×Dic₁₉₄	—	2	2	no	—
26	C₂×Dic₁₉₄	—	2	2	no	—
27	C₂×Dic₁₉₄	—	4	4	no	—
28	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
30	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₉₇	—	2	1	no	—
32	C₄⋊Dic₉₇	—	2	1	no	—
33	#13095490	—	2	2	no	—
34	#13095490	—	2	2	no	—
35	C₄×Dic₉₇	D₄	1	1	yes	characteristic
36	C₄×Dic₉₇	D₄	1	1	yes	characteristic
37	C₄×Dic₉₇	D₄	1	1	yes	characteristic
38	C₄×Dic₉₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₉₇	—	2	1	no	—
42	C₄⋊C₄×C₉₇	—	2	1	no	—
43	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	—	2	2	no	—
45	C₂×C₄×C₉₇	—	2	2	no	—
46	C₂×C₄×C₉₇	—	4	2	no	—
47	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₉₇	—	2	2	no	—
50	C₂×Dic₉₇	—	2	2	no	—
51	C₂×Dic₉₇	—	4	2	no	—
52	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₉₇	C₂×Q₈	1	2	yes	—
57	C₂×Dic₉₇	#41	1	2	yes	—
58	Dic₁₉₄	—	4	4	no	—
59	Dic₁₉₄	—	4	4	no	—
60	#11715897	—	2	4	no	—
61	#11715897	—	4	2	no	—
62	C₂²×C₉₇	—	2	1	no	—
63	C₂²×C₉₇	—	2	1	no	—
64	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
66	C₄×C₉₇	—	2	1	no	—
67	C₄×C₉₇	—	4	2	no	—
68	C₄×C₉₇	#101	1	2	yes	—
69	Dic₉₇	—	2	1	no	—
70	Dic₉₇	—	4	2	no	—
71	Dic₉₇	#100	1	2	yes	—
72	Dic₉₇	C₂²⋊Q₈	1	2	yes	—
73	Dic₉₇	#124	1	1	yes	characteristic
74	Dic₉₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₉₇	—	2	2	no	—
76	C₂×C₉₇	#391	1	1	yes	characteristic
77	C₂×C₉₇	C₂×#100	1	1	yes	characteristic
78	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
79	#1997	—	97	1	no	maximal, Sylow
80	C₉₇	#1997	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄⋊Q₈	—	97	1	no	—
82	#423	—	97	1	no	—
83	C₄×M₁₆	—	97	1	no	—
84	#343	—	97	4	no	—
85	C₂²×Q₈	—	97	2	no	—
86	C₂×M₁₆	—	97	2	no	—
87	C₂×M₁₆	—	97	2	no	—
88	C₄⋊Q₈	—	194	1	no	—
89	C₄⋊Q₈	—	194	1	no	—
90	C₄⋊Q₈	—	194	1	no	—
91	C₄⋊Q₈	—	194	1	no	—
92	C₂×C₄⋊C₄	—	97	2	no	—
93	C₂×C₄⋊C₄	—	97	1	no	—
94	C₂×C₄⋊C₄	#87604	1	1	yes	characteristic
95	C₂×C₄²	—	97	1	no	—
96	C₄⋊C₈	—	194	1	no	—
97	C₄⋊C₈	—	194	1	no	—
98	#97	—	194	1	no	—
99	C₄×C₈	—	194	1	no	—
100	C₂×Q₈	—	388	4	no	—
101	C₂×Q₈	—	194	2	no	—
102	C₂×Q₈	—	194	2	no	—
103	C₂²×C₄	—	97	1	no	—
104	C₂²×C₄	—	97	1	no	—
105	C₂²×C₄	—	97	2	no	—
106	C₂²×C₄	C₂×#87604	1	2	yes	—
107	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
108	M₁₆	—	97	4	no	—
109	M₁₆	—	194	4	no	—
110	M₁₆	—	194	2	no	—
111	C₂×C₈	—	194	2	no	—
112	C₂×C₈	—	194	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	194	2	no	—
116	C₄⋊C₄	—	194	2	no	—
117	C₄⋊C₄	—	194	1	no	—
118	C₄⋊C₄	—	194	1	no	—
119	C₄²	—	97	1	no	—
120	C₄²	—	97	1	no	—
121	C₄²	—	97	1	no	—
122	C₄²	—	97	1	no	—
123	C₂³	C₂²×#87604	1	1	yes	characteristic
124	Q₈	—	388	4	no	—
125	Q₈	—	388	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	97	1	no	—
130	C₂×C₄	—	97	1	no	—
131	C₂×C₄	—	97	1	no	—
132	C₂×C₄	—	97	2	no	—
133	C₂×C₄	—	97	2	no	—
134	C₂×C₄	—	97	1	no	—
135	C₂×C₄	—	388	2	no	—
136	C₂×C₄	—	194	2	no	—
137	C₂×C₄	—	194	2	no	—
138	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
140	C₈	—	388	2	no	—
141	C₈	—	194	4	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	C₂×C₂²⋊#87604	1	1	yes	characteristic
145	C₂²	Q₈×#87604	1	2	yes	—
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	97	1	no	—
149	C₄	—	97	2	no	—
150	C₄	—	97	1	no	—
151	C₄	—	97	2	no	—
152	C₄	—	388	2	no	—
153	C₄	—	194	1	no	—
154	C₄	#15690871	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	(C₂×#87604)⋊Q₈	1	1	yes	characteristic
157	C₂	D₉₇⋊#100	1	1	yes	characteristic
158	C₂	C₂×#15690871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261738

C₂

yes

maximal, characteristic

C₄×#13095520

C₂

yes

maximal, characteristic

#20261615

C₂

yes

maximal

C₂×#15690809

C₂

yes

maximal, characteristic

C₂×#13095520

C₂²

yes

—

C₂×#13095520

C₂²

yes

—

—

—

—

—

C₂²×Dic₁₉₄

C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₉₇

C₄

yes

characteristic

C₂×#13095490

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

—

Q₈

yes

complemented

C₂×#11715897

—

C₂×#11715897

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂³

yes

characteristic

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

#13095490

—

#13095490

—

C₄×Dic₉₇

D₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

#41

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715897

—

#11715897

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#101

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#100

yes

—

Dic₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

#124

yes

characteristic

Dic₉₇

C₄⋊Q₈

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

—

#343

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#87604

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

#97

—

194

—

C₄×C₈

—

194

—

100

C₂×Q₈

—

388

—

101

C₂×Q₈

—

194

—

102

C₂×Q₈

—

194

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×#87604

yes

—

107

C₂²×C₄

C₂×#87604

yes

characteristic

108

M₁₆

—

109

M₁₆

—

194

—

110

M₁₆

—

194

—

111

C₂×C₈

—

194

—

112

C₂×C₈

—

194

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

194

—

116

C₄⋊C₄

—

194

—

117

C₄⋊C₄

—

194

—

118

C₄⋊C₄

—

194

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

C₂²×#87604

yes

characteristic

124

Q₈

—

388

—

125

Q₈

—

388

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

388

—

136

C₂×C₄

—

194

—

137

C₂×C₄

—

194

—

138

C₂×C₄

C₂²⋊#87604

yes

characteristic

139

C₂×C₄

C₂²⋊#87604

yes

characteristic

140

C₈

—

388

—

141

C₈

—

194

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊#87604

yes

characteristic

145

C₂²

Q₈×#87604

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

388

—

153

C₄

—

194

—

154

C₄

#15690871

yes

—

155

C₂

—

156

C₂

(C₂×#87604)⋊Q₈

yes

characteristic

157

C₂

D₉₇⋊#100

yes

characteristic

158

C₂

C₂×#15690871

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87604
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂³	C₂²×#87604
Fitting	C₂×C₄⋊C₄×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	C₂×C₄⋊C₄	#87604
hypocenter	C₉₇	#1997
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87604

commutator, prime radical

C₂³×C₉₇

C₂²×C₄

Frattini

C₂³

C₂²×#87604

Fitting

C₂×C₄⋊C₄×C₉₇

C₄

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₉₇

C₂³

C₂×C₄⋊C₄

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
22. #13095520	Q₈	semidirect
80. C₉₇	#1997	semidirect

H ≅ G/N

type

22. #13095520

Q₈

semidirect

80. C₉₇

#1997

semidirect

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

subgroup	index
#20261738	2
C₄×#13095520	2
#20261615	2
C₂×#15690809	2
#1997	97

subgroup

index

Subgroups — #27157929

Conjugacy classes of subgroups

Complements

Maximal subgroups