Group #27142289 — subgroups

Subgroups — #27142289

Order 12416.

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

normal subgroups only

This group has 308 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	#20261502	C₂	1	1	yes	maximal, characteristic
2	#20260719	C₂	1	1	yes	maximal, characteristic
3	C₂×#15690729	C₂	1	2	yes	maximal, complemented
4	#20260505	C₂	1	2	yes	maximal, complemented
5	#536×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690817	—	2	4	no	—
7	C₂×#13095495	—	2	2	no	—
8	C₂×#13095495	C₂²	1	2	yes	—
9	C₂×#13095495	C₂²	1	2	yes	—
10	#15690729	—	2	4	no	—
11	#15690729	—	2	1	no	—
12	#15690727	—	2	1	no	—
13	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₉₇	—	2	1	no	—
16	#124×C₉₇	—	2	1	no	—
17	#123×C₉₇	C₄	1	4	yes	—
18	C₂²⋊Q₈×C₉₇	—	2	2	no	—
19	#117×C₉₇	C₂²	1	1	yes	characteristic
20	#13095495	—	2	4	no	—
21	#13095495	—	2	2	no	—
22	#13095495	—	2	2	no	—
23	#13095495	—	2	2	no	—
24	#13095495	D₄	1	4	yes	—
25	C₂×#11715895	—	2	4	no	—
26	C₂×#11715895	—	2	2	no	—
27	C₂×#11715895	—	2	2	no	—
28	#41×C₉₇	—	2	2	no	—
29	#41×C₉₇	—	2	2	no	—
30	C₂×Q₈×C₉₇	—	2	1	no	—
31	C₂×Q₈×C₉₇	—	2	1	no	—
32	C₂×Q₈×C₉₇	—	2	2	no	—
33	C₂×Q₈×C₉₇	—	2	2	no	—
34	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉₇	D₄	1	2	yes	—
39	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	—	2	4	no	—
41	C₄⋊C₄×C₉₇	—	2	1	no	—
42	C₂²⋊C₄×C₉₇	—	2	4	no	—
43	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
44	C₄²×C₉₇	—	2	1	no	—
45	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₉₇	—	2	2	no	—
48	Q₈×C₉₇	—	4	1	no	—
49	Q₈×C₉₇	—	4	2	no	—
50	Q₈×C₉₇	—	4	1	no	—
51	D₄×C₉₇	—	2	4	no	—
52	D₄×C₉₇	—	2	2	no	—
53	C₂×C₄×C₉₇	—	2	2	no	—
54	C₂×C₄×C₉₇	—	2	2	no	—
55	C₂×C₄×C₉₇	—	2	2	no	—
56	C₂×C₄×C₉₇	—	2	2	no	—
57	C₂×C₄×C₉₇	—	2	4	no	—
58	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₉₇	#41	1	2	yes	—
63	#11715895	—	2	4	no	—
64	#11715895	—	4	2	no	—
65	C₂²×C₉₇	—	2	2	no	—
66	C₂²×C₉₇	—	4	1	no	—
67	C₂²×C₉₇	—	4	2	no	—
68	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₉₇	C₄×D₄	1	2	yes	—
70	C₄×C₉₇	—	2	2	no	—
71	C₄×C₉₇	—	4	2	no	—
72	C₄×C₉₇	—	4	2	no	—
73	C₄×C₉₇	#120	1	1	yes	characteristic
74	C₄×C₉₇	#121	1	2	yes	—
75	C₄×C₉₇	#123	1	1	yes	characteristic
76	C₂×C₉₇	—	2	1	no	—
77	C₂×C₉₇	—	2	2	no	—
78	C₂×C₉₇	—	4	2	no	—
79	C₂×C₉₇	#386	1	1	yes	characteristic
80	#1915	—	97	1	no	maximal, Sylow
81	C₉₇	#1915	1	1	yes	characteristic, complemented, Sylow
82	#568	—	97	1	no	—
83	#536	D₉₇	1	1	yes	characteristic
84	#413	—	97	1	no	—
85	C₂×#101	—	97	2	no	—
86	#344	—	97	2	no	—
87	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
89	#131	—	194	4	no	—
90	C₂×M₁₆	—	97	2	no	—
91	C₂×M₁₆	—	97	2	no	—
92	C₂×M₁₆	—	194	2	no	—
93	C₄⋊Q₈	—	2	1	no	—
94	#124	—	2	1	no	—
95	#123	Dic₉₇	1	4	yes	—
96	C₂²⋊Q₈	—	2	2	no	—
97	#117	C₂×D₉₇	1	1	yes	characteristic
98	#101	—	194	4	no	—
99	#101	—	194	1	no	—
100	#100	—	194	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
109	C₂×D₄	C₂²⋊D₉₇	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
111	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
113	M₁₆	—	97	4	no	—
114	M₁₆	—	194	2	no	—
115	M₁₆	—	194	2	no	—
116	M₁₆	—	194	4	no	—
117	M₁₆	—	194	2	no	—
118	C₂×C₈	—	194	2	no	—
119	C₂×C₈	—	194	4	no	—
120	C₂×C₈	—	194	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊Dic₉₇	1	2	yes	—
127	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
140	C₂×C₄	D₄×D₉₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
143	C₂×C₄	#13095510	1	2	yes	—
144	C₈	—	388	2	no	—
145	C₈	—	194	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
150	C₂²	D₄×Dic₉₇	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#15690778	1	1	yes	characteristic
155	C₄	#15690782	1	2	yes	—
156	C₄	#15690791	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#20260615	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261502

C₂

yes

maximal, characteristic

#20260719

C₂

yes

maximal, characteristic

C₂×#15690729

C₂

yes

maximal, complemented

#20260505

C₂

yes

maximal, complemented

#536×C₉₇

C₂

yes

maximal, characteristic

#15690817

—

C₂×#13095495

—

C₂×#13095495

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

#15690729

—

#15690729

—

#15690727

—

C₂×#41×C₉₇

C₂²

yes

characteristic

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

—

#124×C₉₇

—

#123×C₉₇

C₄

yes

—

C₂²⋊Q₈×C₉₇

—

#117×C₉₇

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

characteristic

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂²⋊C₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

#121

yes

—

C₄×C₉₇

#123

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#568

—

#536

D₉₇

yes

—

—

—

C₂×#41

C₂×D₉₇

yes

characteristic

C₂²×Q₈

C₂×D₉₇

yes

characteristic

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

194

—

C₄⋊Q₈

—

#124

—

#123

Dic₉₇

yes

—

C₂²⋊Q₈

—

#117

C₂×D₉₇

yes

characteristic

#101

—

194

—

#101

—

194

—

100

#100

—

194

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₉₇

yes

characteristic

108

C₂×D₄

C₂×Dic₉₇

yes

—

109

C₂×D₄

C₂²⋊D₉₇

yes

characteristic

110

C₂²×C₄

C₂×Dic₉₇

yes

—

111

C₂²×C₄

C₂²⋊D₉₇

yes

—

112

C₂²×C₄

C₂²×D₉₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

194

—

115

M₁₆

—

194

—

116

M₁₆

—

194

—

117

M₁₆

—

194

—

118

C₂×C₈

—

194

—

119

C₂×C₈

—

194

—

120

C₂×C₈

—

194

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×Dic₉₇

yes

—

125

C₄²

—

126

C₂³

C₂²⋊Dic₉₇

yes

—

127

C₂³

C₂²×Dic₉₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₉₇

yes

—

140

C₂×C₄

D₄×D₉₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

143

C₂×C₄

#13095510

yes

—

144

C₈

—

388

—

145

C₈

—

194

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

150

C₂²

D₄×Dic₉₇

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#15690778

yes

characteristic

155

C₄

#15690782

yes

—

156

C₄

#15690791

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#20260615

yes

characteristic

	subgroup	G/H
center	C₂	#20260615
commutator	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#536×C₉₇	C₂
socle	C₂×C₉₇	#386
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#536	D₉₇
hypocenter	C₉₇	#1915
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#20260615

commutator

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#536×C₉₇

C₂

socle

C₂×C₉₇

#386

prime radical

C₂×#41×C₉₇

C₂²

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
3. C₂×#15690729	C₂	semidirect
4. #20260505	C₂	semidirect
81. C₉₇	#1915	semidirect

H ≅ G/N

type

3. C₂×#15690729

C₂

semidirect

4. #20260505

C₂

semidirect

81. C₉₇

#1915

semidirect

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

subgroup	index
#20261502	2
#20260719	2
C₂×#15690729	2
#20260505	2
#536×C₉₇	2
#1915	97

subgroup

index

#536×C₉₇

Subgroups — #27142289

Conjugacy classes of subgroups

Complements

Maximal subgroups