Group #27152136 — subgroups

Subgroups — #27152136

Order 12416.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690802	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261103	C₂	1	2	yes	maximal, complemented
3	#20261099	C₂	1	2	yes	maximal, complemented
4	#20261089	C₂	1	4	yes	maximal, complemented
5	#20260736	C₂	1	4	yes	maximal, complemented
6	C₄×#13095495	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095495	C₂²	1	2	yes	—
9	#15690802	—	2	2	no	—
10	#15690802	C₂²	1	2	yes	—
11	#15690802	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
13	C₂×#13095490	—	2	2	no	—
14	#15690734	C₂²	1	8	yes	—
15	#15690731	C₂²	1	8	yes	—
16	#15690721	C₂²	1	2	yes	—
17	C₄×#11715895	C₂²	1	2	yes	—
18	C₂×#41×C₉₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
20	#127×C₉₇	C₂²	1	1	yes	characteristic
21	#124×C₉₇	C₂²	1	2	yes	—
22	#121×C₉₇	—	2	2	no	—
23	C₄×D₄×C₉₇	—	2	2	no	—
24	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₇	—	2	2	no	—
26	#13095495	—	2	4	no	—
27	C₂×#11715895	C₂³	1	4	yes	—
28	C₄⋊Dic₉₇	—	2	2	no	—
29	C₄⋊Dic₉₇	C₂³	1	4	yes	—
30	#13095490	—	2	4	no	—
31	#13095490	—	2	4	no	—
32	#41×C₉₇	—	2	8	no	—
33	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
34	C₂×D₄×C₉₇	—	2	2	no	—
35	C₂×D₄×C₉₇	—	2	2	no	—
36	C₂×D₄×C₉₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₉₇	—	2	2	no	—
38	C₂²×C₄×C₉₇	D₄	1	2	yes	—
39	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	—	2	2	no	—
41	C₂²⋊C₄×C₉₇	—	2	4	no	—
42	C₄²×C₉₇	D₄	1	2	yes	—
43	C₄²×C₉₇	C₂³	1	1	yes	characteristic
44	C₄²×C₉₇	C₂³	1	1	yes	characteristic
45	C₂³×C₉₇	—	2	2	no	—
46	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₉₇	—	2	4	no	—
48	D₄×C₉₇	—	2	4	no	—
49	D₄×C₉₇	—	4	4	no	—
50	D₄×C₉₇	—	4	4	no	—
51	C₂×C₄×C₉₇	—	2	2	no	—
52	C₂×C₄×C₉₇	—	4	2	no	—
53	C₂×C₄×C₉₇	—	4	4	no	—
54	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	#41	1	4	yes	—
58	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₉₇	—	2	4	no	—
60	C₂×Dic₉₇	—	4	4	no	—
61	#11715895	—	2	4	no	—
62	C₂²×C₉₇	—	2	2	no	—
63	C₂²×C₉₇	—	4	2	no	—
64	C₂²×C₉₇	—	4	4	no	—
65	C₂²×C₉₇	#124	1	2	yes	—
66	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₉₇	—	2	2	no	—
68	C₄×C₉₇	—	2	2	no	—
69	C₄×C₉₇	—	4	2	no	—
70	C₄×C₉₇	#124	1	2	yes	—
71	C₄×C₉₇	C₂×#41	1	2	yes	—
72	Dic₉₇	—	4	4	no	—
73	C₂×C₉₇	—	2	2	no	—
74	C₂×C₉₇	—	8	2	no	—
75	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₉₇	#575	1	2	yes	—
77	#3142	—	97	1	no	maximal, Sylow
78	C₉₇	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	D₉₇	1	1	yes	characteristic
80	C₂×#125	—	97	1	no	—
81	#491	—	97	2	no	—
82	#490	—	97	2	no	—
83	#488	—	97	4	no	—
84	#417	—	97	4	no	—
85	C₄×M₁₆	—	97	1	no	—
86	C₂×#41	C₂×D₉₇	1	2	yes	—
87	C₂×M₁₆	—	97	2	no	—
88	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
89	#127	C₂×D₉₇	1	1	yes	characteristic
90	#125	—	97	2	no	—
91	#125	—	97	2	no	—
92	#125	—	194	2	no	—
93	#124	C₂×D₉₇	1	2	yes	—
94	#121	—	2	2	no	—
95	C₄×D₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	97	2	no	—
97	C₂×C₄⋊C₄	—	194	2	no	—
98	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
99	#103	—	97	8	no	—
100	#102	—	97	8	no	—
101	#97	—	97	2	no	—
102	C₄×C₈	—	97	2	no	—
103	#41	—	2	8	no	—
104	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂²×D₉₇	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	194	2	no	—
110	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
112	M₁₆	—	194	4	no	—
113	C₂×C₈	—	97	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	97	4	no	—
116	C₄⋊C₄	—	194	4	no	—
117	C₄⋊C₄	—	194	4	no	—
118	C₄⋊C₄	—	194	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₉₇	1	2	yes	—
121	C₄²	C₂²×D₉₇	1	1	yes	characteristic
122	C₄²	C₂²×D₉₇	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	388	4	no	—
133	C₂×C₄	—	194	4	no	—
134	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
137	C₂×C₄	#13095510	1	4	yes	—
138	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
139	C₈	—	194	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#15690797	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	388	4	no	—
149	C₄	#15690797	1	2	yes	—
150	C₄	C₂×#13095510	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	8	2	no	—
153	C₂	C₂×#15690797	1	1	yes	characteristic
154	C₂	#20261533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic, complemented

#20261103

C₂

yes

maximal, complemented

#20261099

C₂

yes

maximal, complemented

#20261089

C₂

yes

maximal, complemented

#20260736

C₂

yes

maximal, complemented

C₄×#13095495

C₂

yes

maximal, characteristic, complemented

#532×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690802

—

#15690802

C₂²

yes

—

#15690802

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690734

C₂²

yes

—

#15690731

C₂²

yes

—

#15690721

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

C₂×#41×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

#124×C₉₇

C₂²

yes

—

#121×C₉₇

—

C₄×D₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

#13095495

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

#13095490

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂³

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#124

yes

characteristic

C₂×C₉₇

#575

yes

—

#3142

—

maximal, Sylow

C₉₇

#3142

yes

characteristic, complemented, Sylow

#532

D₉₇

yes

—

—

—

—

—

C₄×M₁₆

—

C₂×#41

C₂×D₉₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

#127

C₂×D₉₇

yes

—

—

—

194

—

#124

C₂×D₉₇

yes

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

104

C₂×Q₈

C₂²×D₉₇

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₉₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

194

—

110

C₂²×C₄

C₂²⋊D₉₇

yes

—

111

C₂²×C₄

C₂²×D₉₇

yes

characteristic

112

M₁₆

—

194

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

194

—

117

C₄⋊C₄

—

194

—

118

C₄⋊C₄

—

194

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₉₇

yes

—

121

C₄²

C₂²×D₉₇

yes

characteristic

122

C₄²

C₂²×D₉₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

388

—

133

C₂×C₄

—

194

—

134

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

135

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

137

C₂×C₄

#13095510

yes

—

138

C₂×C₄

C₂³×D₉₇

yes

characteristic

139

C₈

—

194

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#15690797

yes

—

144

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

388

—

149

C₄

#15690797

yes

—

150

C₄

C₂×#13095510

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#15690797

yes

characteristic

154

C₂

#20261533

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting, prime radical	#532×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
hypercenter	#532	D₉₇
hypocenter	C₉₇	#3142
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting, prime radical

#532×C₉₇

C₂

socle

C₂²×C₉₇

C₂²×D₄

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#15690802	C₂	semidirect
2. #20261103	C₂	semidirect
3. #20261099	C₂	semidirect
4. #20261089	C₂	semidirect
5. #20260736	C₂	semidirect
6. C₄×#13095495	C₂	semidirect
78. C₉₇	#3142	semidirect

H ≅ G/N

type

1. C₂×#15690802

C₂

semidirect

2. #20261103

C₂

semidirect

3. #20261099

C₂

semidirect

4. #20261089

C₂

semidirect

5. #20260736

C₂

semidirect

6. C₄×#13095495

C₂

semidirect

78. C₉₇

#3142

semidirect

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

subgroup	index
C₂×#15690802	2
#20261103	2
#20261099	2
#20261089	2
#20260736	2
C₄×#13095495	2
#532×C₉₇	2
#3142	97

subgroup

index

#532×C₉₇

Subgroups — #27152136

Conjugacy classes of subgroups

Complements

Maximal subgroups