Group #27140955 — subgroups

Subgroups — #27140955

Order 12416.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	#20261087	C₂	1	1	yes	maximal, characteristic
3	#20261017	C₂	1	1	yes	maximal, characteristic
4	#20260483	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#11715895	C₂	1	1	yes	maximal, characteristic
6	#20260465	C₂	1	1	yes	maximal, characteristic
7	#462×C₉₇	C₂	1	1	yes	maximal, characteristic
8	#15690810	—	2	1	no	—
9	#15690810	C₂²	1	1	yes	characteristic
10	#15690809	—	2	2	no	—
11	#15690809	—	2	1	no	—
12	Q₈×Dic₉₇	—	2	1	no	—
13	C₄×Dic₁₉₄	—	2	1	no	—
14	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
15	#15690741	—	2	1	no	—
16	C₄⋊#11715895	C₂²	1	1	yes	characteristic
17	#15690733	—	2	1	no	—
18	#15690733	—	2	2	no	—
19	C₄×#11715895	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₉₇	—	2	1	no	—
21	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
24	#103×C₉₇	—	2	1	no	—
25	C₂×Dic₁₉₄	—	2	2	no	—
26	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
27	C₂×Dic₁₉₄	D₄	1	2	yes	—
28	C₂×#11715895	—	2	1	no	—
29	C₂×#11715895	—	2	1	no	—
30	C₄⋊Dic₉₇	—	2	2	no	—
31	C₄⋊Dic₉₇	—	2	1	no	—
32	C₄⋊Dic₉₇	—	2	1	no	—
33	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
34	#13095490	—	2	2	no	—
35	#13095490	—	2	1	no	—
36	C₄×Dic₉₇	—	2	2	no	—
37	C₄×Dic₉₇	—	2	1	no	—
38	C₂×Q₈×C₉₇	—	2	1	no	—
39	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
40	Q₁₆×C₉₇	—	4	2	no	—
41	C₂×C₈×C₉₇	—	2	1	no	—
42	C₄⋊C₄×C₉₇	—	2	1	no	—
43	C₄⋊C₄×C₉₇	—	2	1	no	—
44	C₄⋊C₄×C₉₇	—	2	1	no	—
45	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
46	C₄²×C₉₇	—	2	1	no	—
47	C₄²×C₉₇	C₂³	1	1	yes	characteristic
48	Q₈×C₉₇	—	2	1	no	—
49	Q₈×C₉₇	—	4	1	no	—
50	Q₈×C₉₇	—	4	1	no	—
51	Q₈×C₉₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₉₇	—	2	1	no	—
53	C₂×C₄×C₉₇	—	2	1	no	—
54	C₂×C₄×C₉₇	—	4	1	no	—
55	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₉₇	—	2	1	no	—
59	C₂×Dic₉₇	—	2	2	no	—
60	C₂×Dic₉₇	—	2	1	no	—
61	Dic₁₉₄	—	2	1	no	—
62	Dic₁₉₄	—	4	2	no	—
63	Dic₁₉₄	—	8	2	no	—
64	Dic₁₉₄	Q₁₆	1	2	yes	complemented
65	C₈×C₉₇	—	4	1	no	—
66	#11715895	—	4	1	no	—
67	#11715895	—	4	1	no	—
68	C₂²×C₉₇	#120	1	1	yes	characteristic
69	C₄×C₉₇	—	2	2	no	—
70	C₄×C₉₇	—	2	1	no	—
71	C₄×C₉₇	—	4	1	no	—
72	C₄×C₉₇	—	8	1	no	—
73	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₄×C₉₇	#137	1	1	yes	characteristic
76	C₄×C₉₇	#137	1	1	yes	characteristic
77	Dic₉₇	—	2	2	no	—
78	Dic₉₇	—	4	2	no	—
79	Dic₉₇	—	4	1	no	—
80	C₂×C₉₇	#448	1	1	yes	characteristic
81	C₂×C₉₇	#451	1	1	yes	characteristic
82	C₂×C₉₇	#455	1	1	yes	characteristic
83	#1645	—	97	1	no	maximal, Sylow
84	C₉₇	#1645	1	1	yes	characteristic, complemented, Sylow
85	Q₈²	—	97	1	no	—
86	#487	—	97	1	no	—
87	#475	—	97	1	no	—
88	#462	D₉₇	1	1	yes	characteristic
89	#333	—	97	1	no	—
90	Q₈⋊C₈	—	97	1	no	—
91	Q₈⋊C₈	—	97	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	—	97	1	no	—
94	C₄⋊Q₈	—	194	1	no	—
95	C₄⋊Q₈	—	194	2	no	—
96	C₄⋊Q₈	—	194	1	no	—
97	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
98	C₄×Q₈	—	97	1	no	—
99	C₄×Q₈	—	194	1	no	—
100	C₄×Q₈	—	194	1	no	—
101	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
102	#106	—	194	1	no	—
103	C₄⋊C₈	—	97	1	no	—
104	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	194	1	no	—
107	#103	—	194	2	no	—
108	C₄×C₈	—	97	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	97	1	no	—
111	C₂×Q₈	—	97	2	no	—
112	C₂×Q₈	—	194	2	no	—
113	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	194	1	no	—
117	C₂×C₈	—	194	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	97	1	no	—
122	C₄⋊C₄	—	194	2	no	—
123	C₄⋊C₄	—	194	2	no	—
124	C₄⋊C₄	—	194	1	no	—
125	C₄⋊C₄	—	194	1	no	—
126	C₄⋊C₄	—	194	1	no	—
127	C₄⋊C₄	C₂²⋊D₉₇	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	194	2	no	—
130	C₄²	—	194	1	no	—
131	C₄²	C₂²×D₉₇	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	776	2	no	—
136	Q₈	—	97	2	no	—
137	Q₈	—	388	2	no	—
138	Q₈	—	194	1	no	—
139	Q₈	#13095499	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	194	1	no	—
144	C₂×C₄	—	194	2	no	—
145	C₂×C₄	—	194	1	no	—
146	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
147	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	388	1	no	—
151	C₈	—	388	1	no	—
152	C₂²	#15690778	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	388	1	no	—
158	C₄	—	388	2	no	—
159	C₄	—	194	2	no	—
160	C₄	C₂×#13095499	1	1	yes	characteristic
161	C₄	Q₁₆×D₉₇	1	1	yes	characteristic
162	C₄	#15690840	1	1	yes	characteristic
163	C₄	#15690842	1	1	yes	characteristic
164	C₂	#20260873	1	1	yes	characteristic
165	C₂	#20260889	1	1	yes	characteristic
166	C₂	#20260911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₉₄

C₂

yes

maximal, characteristic

#20261087

C₂

yes

maximal, characteristic

#20261017

C₂

yes

maximal, characteristic

#20260483

C₂

yes

maximal, characteristic

Q₈⋊#11715895

C₂

yes

maximal, characteristic

#20260465

C₂

yes

maximal, characteristic

#462×C₉₇

C₂

yes

maximal, characteristic

#15690810

—

#15690810

C₂²

yes

characteristic

#15690809

—

#15690809

—

Q₈×Dic₉₇

—

C₄×Dic₁₉₄

—

C₄×Dic₁₉₄

C₂²

yes

characteristic

#15690741

—

C₄⋊#11715895

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×Q₁₆×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄×Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

characteristic

#103×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

D₄

yes

characteristic

C₂×Dic₁₉₄

D₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

characteristic

#13095490

—

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

characteristic

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

Q₁₆

yes

complemented

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

#120

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×SD₁₆

yes

characteristic

C₄×C₉₇

C₂×Q₁₆

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

Q₈²

—

#487

—

#475

—

#462

D₉₇

yes

characteristic

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

194

—

100

C₄×Q₈

—

194

—

101

C₄×Q₈

C₂×D₉₇

yes

characteristic

102

#106

—

194

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₉₇

yes

characteristic

105

#103

—

106

#103

—

194

—

107

#103

—

194

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

194

—

113

C₂×Q₈

C₂²⋊D₉₇

yes

characteristic

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

194

—

117

C₂×C₈

—

194

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

194

—

123

C₄⋊C₄

—

194

—

124

C₄⋊C₄

—

194

—

125

C₄⋊C₄

—

194

—

126

C₄⋊C₄

—

194

—

127

C₄⋊C₄

C₂²⋊D₉₇

yes

characteristic

128

C₄²

—

129

C₄²

—

194

—

130

C₄²

—

194

—

131

C₄²

C₂²×D₉₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

776

—

136

Q₈

—

137

Q₈

—

388

—

138

Q₈

—

194

—

139

Q₈

#13095499

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

194

—

144

C₂×C₄

—

194

—

145

C₂×C₄

—

194

—

146

C₂×C₄

D₄×D₉₇

yes

characteristic

147

C₂×C₄

D₄×D₉₇

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

149

C₈

—

150

C₈

—

388

—

151

C₈

—

388

—

152

C₂²

#15690778

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

388

—

158

C₄

—

388

—

159

C₄

—

194

—

160

C₄

C₂×#13095499

yes

characteristic

161

C₄

Q₁₆×D₉₇

yes

characteristic

162

C₄

#15690840

yes

characteristic

163

C₄

#15690842

yes

characteristic

164

C₂

#20260873

yes

characteristic

165

C₂

#20260889

yes

characteristic

166

C₂

#20260911

yes

characteristic

	subgroup	G/H
center	C₂²	#15690778
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#462×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#120
hypercenter	#462	D₉₇
hypocenter	C₉₇	#1645
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#15690778

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#462×C₉₇

C₂

C₂²×C₉₇

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
64. Dic₁₉₄	Q₁₆	semidirect
84. C₉₇	#1645	semidirect

H ≅ G/N

type

64. Dic₁₉₄

Q₁₆

semidirect

84. C₉₇

#1645

semidirect

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

subgroup	index
Q₈×Dic₁₉₄	2
#20261087	2
#20261017	2
#20260483	2
Q₈⋊#11715895	2
#20260465	2
#462×C₉₇	2
#1645	97

subgroup

index

Q₈×Dic₁₉₄

#462×C₉₇

Subgroups — #27140955

Conjugacy classes of subgroups

Complements

Maximal subgroups