Group #27140192 — subgroups

Subgroups — #27140192

Order 12416.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20260759	C₂	1	1	yes	maximal, characteristic
3	#20260467	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	1	no	—
6	#15690810	C₄	1	1	yes	characteristic
7	#15690810	C₄	1	1	yes	characteristic
8	#15690802	—	2	1	no	—
9	#15690789	—	2	1	no	—
10	#15690789	C₄	1	2	yes	—
11	C₄×Dic₁₉₄	—	2	1	no	—
12	C₄×Dic₁₉₄	C₂²	1	2	yes	—
13	#15690769	—	2	1	no	—
14	C₄⋊#11715895	—	2	1	no	—
15	C₄⋊#11715895	C₂²	1	2	yes	—
16	C₂²×C₈×C₉₇	—	2	1	no	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₉₇	—	2	1	no	—
19	C₄×C₈×C₉₇	C₂²	1	2	yes	—
20	C₂×Dic₁₉₄	—	2	1	no	—
21	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
22	#13095495	—	4	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₄⋊Dic₉₇	—	2	1	no	—
25	C₄⋊Dic₉₇	—	2	1	no	—
26	C₄⋊Dic₉₇	—	2	1	no	—
27	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
28	#13095490	—	2	2	no	—
29	#13095490	—	2	2	no	—
30	C₂²⋊Dic₉₇	—	2	2	no	—
31	C₄×Dic₉₇	—	2	2	no	—
32	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₉₇	—	2	4	no	—
36	C₂×C₈×C₉₇	—	2	2	no	—
37	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₉₇	D₄	1	1	yes	characteristic
39	C₄²×C₉₇	D₄	1	1	yes	characteristic
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₉₇	—	2	2	no	—
53	C₂×Dic₉₇	—	2	2	no	—
54	Dic₁₉₄	—	2	4	no	—
55	Dic₁₉₄	—	4	2	no	—
56	C₈×C₉₇	—	2	4	no	—
57	#11715895	—	4	2	no	—
58	C₂²×C₉₇	—	2	1	no	—
59	C₂²×C₉₇	—	2	1	no	—
60	C₂²×C₉₇	#103	1	2	yes	—
61	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₉₇	—	2	1	no	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	#103	1	2	yes	—
65	C₄×C₉₇	#104	1	2	yes	—
66	C₄×C₉₇	#131	1	2	yes	—
67	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₉₇	—	4	2	no	—
70	Dic₉₇	—	4	2	no	—
71	C₂×C₉₇	—	2	2	no	—
72	C₂×C₉₇	#408	1	1	yes	characteristic
73	C₂×C₉₇	C₂×#103	1	1	yes	characteristic
74	C₂×C₉₇	C₂×#104	1	1	yes	characteristic
75	#1493	—	97	1	no	maximal, Sylow
76	C₉₇	#1493	1	1	yes	characteristic, complemented, Sylow
77	#533	—	97	1	no	—
78	#423	—	97	1	no	—
79	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
80	Q₈⋊C₈	—	97	4	no	—
81	C₂×M₁₆	—	194	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₄⋊Q₈	—	97	1	no	—
84	C₄⋊Q₈	—	97	1	no	—
85	#125	—	194	1	no	—
86	C₂²⋊Q₈	—	97	2	no	—
87	C₂²⋊Q₈	—	194	1	no	—
88	C₄×Q₈	—	97	2	no	—
89	C₄×Q₈	—	194	1	no	—
90	#117	—	194	1	no	—
91	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
92	C₄⋊C₈	—	97	2	no	—
93	C₄⋊C₈	—	194	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	C₂×D₉₇	1	2	yes	—
96	C₂×Q₈	—	97	2	no	—
97	C₂×Q₈	—	194	1	no	—
98	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
99	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
101	M₁₆	—	388	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	194	2	no	—
105	C₄⋊C₄	—	97	2	no	—
106	C₄⋊C₄	—	194	2	no	—
107	C₄⋊C₄	—	194	1	no	—
108	C₄⋊C₄	—	194	1	no	—
109	C₄⋊C₄	—	194	1	no	—
110	C₄⋊C₄	—	194	2	no	—
111	C₂²⋊C₄	—	194	2	no	—
112	C₄²	—	194	2	no	—
113	C₄²	C₄×D₉₇	1	1	yes	characteristic
114	C₄²	D₃₈₈	1	1	yes	characteristic
115	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
116	C₄²	C₂²×D₉₇	1	1	yes	characteristic
117	C₂³	#13095488	1	1	yes	characteristic
118	Q₈	—	388	2	no	—
119	Q₈	—	194	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	194	2	no	—
123	C₂×C₄	—	194	2	no	—
124	C₂×C₄	#13095488	1	1	yes	characteristic
125	C₂×C₄	#13095488	1	1	yes	characteristic
126	C₂×C₄	#13095488	1	1	yes	characteristic
127	C₂×C₄	#13095500	1	2	yes	—
128	C₂×C₄	Dic₃₈₈	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
131	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	388	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#15690735	1	2	yes	—
137	C₂²	C₂×#13095488	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	388	2	no	—
141	C₄	—	388	2	no	—
142	C₄	#15690735	1	2	yes	—
143	C₄	#15690738	1	2	yes	—
144	C₄	#15690816	1	2	yes	—
145	C₄	C₂×#13095500	1	1	yes	characteristic
146	C₄	C₂×Dic₃₈₈	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#20260701	1	1	yes	characteristic
149	C₂	C₂×#15690735	1	1	yes	characteristic
150	C₂	C₂×#15690738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

#20260759

C₂

yes

maximal, characteristic

#20260467

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

—

#15690810

C₄

yes

characteristic

#15690810

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₉₄

—

C₄×Dic₁₉₄

C₂²

yes

—

#15690769

—

C₄⋊#11715895

—

C₄⋊#11715895

C₂²

yes

—

C₂²×C₈×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

C₄×C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

#13095490

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#103

yes

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#103

yes

—

C₄×C₉₇

#104

yes

—

C₄×C₉₇

#131

yes

—

C₄×C₉₇

C₂×SD₁₆

yes

characteristic

C₄×C₉₇

C₂×Q₁₆

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄×C₈

D₉₇

yes

characteristic

Q₈⋊C₈

—

C₂×M₁₆

—

194

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

194

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

C₄×Q₈

—

194

—

#117

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

D₃₈₈

yes

characteristic

100

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

101

M₁₆

—

388

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

194

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

—

194

—

108

C₄⋊C₄

—

194

—

109

C₄⋊C₄

—

194

—

110

C₄⋊C₄

—

194

—

111

C₂²⋊C₄

—

194

—

112

C₄²

—

194

—

113

C₄²

C₄×D₉₇

yes

characteristic

114

C₄²

D₃₈₈

yes

characteristic

115

C₄²

C₂²⋊D₉₇

yes

characteristic

116

C₄²

C₂²×D₉₇

yes

characteristic

117

C₂³

#13095488

yes

characteristic

118

Q₈

—

388

—

119

Q₈

—

194

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

194

—

123

C₂×C₄

—

194

—

124

C₂×C₄

#13095488

yes

characteristic

125

C₂×C₄

#13095488

yes

characteristic

126

C₂×C₄

#13095488

yes

characteristic

127

C₂×C₄

#13095500

yes

—

128

C₂×C₄

Dic₃₈₈

yes

—

129

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

131

C₂×C₄

C₂×D₃₈₈

yes

characteristic

132

C₈

—

133

C₈

—

388

—

134

C₂²

—

135

C₂²

—

136

C₂²

#15690735

yes

—

137

C₂²

C₂×#13095488

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

388

—

141

C₄

—

388

—

142

C₄

#15690735

yes

—

143

C₄

#15690738

yes

—

144

C₄

#15690816

yes

—

145

C₄

C₂×#13095500

yes

characteristic

146

C₄

C₂×Dic₃₈₈

yes

characteristic

147

C₂

—

148

C₂

#20260701

yes

characteristic

149

C₂

C₂×#15690735

yes

characteristic

150

C₂

C₂×#15690738

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1493
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂²⋊C₄

cosocle

C₄²×C₉₇

C₂³

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1493

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #20260467	C₂	semidirect
76. C₉₇	#1493	semidirect

H ≅ G/N

type

3. #20260467

C₂

semidirect

76. C₉₇

#1493

semidirect

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

subgroup	index
#20261292	2
#20260759	2
#20260467	2
C₂×C₄×C₈×C₉₇	2
#1493	97

subgroup

index

C₂×C₄×C₈×C₉₇

Subgroups — #27140192

Conjugacy classes of subgroups

Complements

Maximal subgroups