Group #27137972 — subgroups

Subgroups — C₉₇×#2014 #27137972

Order 12416.

474 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₂²×SD₁₆×C₉₇	C₂	1	2	yes	maximal, complemented
2	C₂×#127×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#102×C₉₇	C₂	1	2	yes	maximal, complemented
4	#391×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₉₇	—	2	2	no	—
8	C₂²×D₄×C₉₇	C₂²	1	2	yes	—
9	C₂×SD₁₆×C₉₇	—	2	4	no	—
10	C₂×SD₁₆×C₉₇	—	2	4	no	—
11	C₂×SD₁₆×C₉₇	—	2	4	no	—
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	#127×C₉₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#102×C₉₇	—	2	4	no	—
17	#95×C₉₇	—	2	2	no	—
18	C₂⁴×C₉₇	—	2	2	no	—
19	C₂×Q₈×C₉₇	—	2	1	no	—
20	C₂×Q₈×C₉₇	—	2	2	no	—
21	C₂×Q₈×C₉₇	—	2	2	no	—
22	C₂×Q₈×C₉₇	D₄	1	2	yes	complemented
23	C₂×D₄×C₉₇	—	2	8	no	—
24	C₂×D₄×C₉₇	—	2	2	no	—
25	C₂×D₄×C₉₇	—	2	2	no	—
26	C₂×D₄×C₉₇	—	2	2	no	—
27	C₂×D₄×C₉₇	—	2	2	no	—
28	C₂×D₄×C₉₇	—	2	2	no	—
29	C₂×D₄×C₉₇	—	4	4	no	—
30	C₂×D₄×C₉₇	—	4	4	no	—
31	C₂×D₄×C₉₇	D₄	1	4	yes	—
32	C₂²×C₄×C₉₇	—	2	1	no	—
33	C₂²×C₄×C₉₇	—	2	1	no	—
34	C₂²×C₄×C₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
36	SD₁₆×C₉₇	—	2	16	no	—
37	C₂×C₈×C₉₇	—	2	2	no	—
38	C₂×C₈×C₉₇	—	2	2	no	—
39	C₂×C₈×C₉₇	D₄	1	4	yes	complemented
40	C₄⋊C₄×C₉₇	—	2	2	no	—
41	C₄²×C₉₇	—	2	2	no	—
42	C₂³×C₉₇	—	2	4	no	—
43	C₂³×C₉₇	—	4	2	no	—
44	C₂³×C₉₇	—	4	2	no	—
45	C₂³×C₉₇	—	4	4	no	—
46	C₂³×C₉₇	—	4	4	no	—
47	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₉₇	—	2	4	no	—
49	Q₈×C₉₇	—	4	2	no	—
50	D₄×C₉₇	—	2	4	no	—
51	D₄×C₉₇	—	2	4	no	—
52	D₄×C₉₇	—	4	16	no	—
53	D₄×C₉₇	—	4	2	no	—
54	D₄×C₉₇	—	4	2	no	—
55	C₂×C₄×C₉₇	—	2	2	no	—
56	C₂×C₄×C₉₇	—	2	2	no	—
57	C₂×C₄×C₉₇	—	2	2	no	—
58	C₂×C₄×C₉₇	—	4	1	no	—
59	C₂×C₄×C₉₇	—	4	1	no	—
60	C₂×C₄×C₉₇	—	4	1	no	—
61	C₂×C₄×C₉₇	—	4	2	no	—
62	C₂×C₄×C₉₇	SD₁₆	1	4	yes	complemented
63	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
67	C₈×C₉₇	—	2	4	no	—
68	C₂²×C₉₇	—	4	4	no	—
69	C₂²×C₉₇	—	4	8	no	—
70	C₂²×C₉₇	—	8	4	no	—
71	C₂²×C₉₇	—	8	2	no	—
72	C₂²×C₉₇	—	8	2	no	—
73	C₂²×C₉₇	#120	1	1	yes	characteristic
74	C₂²×C₉₇	#121	1	1	yes	characteristic
75	C₂²×C₉₇	#127	1	1	yes	characteristic
76	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₉₇	—	2	4	no	—
79	C₄×C₉₇	—	4	2	no	—
80	C₄×C₉₇	—	8	1	no	—
81	C₄×C₉₇	#120	1	2	yes	—
82	C₄×C₉₇	#121	1	2	yes	—
83	C₂×C₉₇	—	8	4	no	—
84	C₂×C₉₇	#392	1	1	yes	characteristic
85	C₂×C₉₇	#450	1	2	yes	—
86	C₂×C₉₇	#460	1	2	yes	—
87	C₂×C₉₇	#492	1	1	yes	characteristic
88	C₂×C₉₇	#496	1	1	yes	characteristic
89	#2014	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₉₇	#2014	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×SD₁₆	C₂×C₉₇	1	2	yes	complemented
92	C₂×#127	C₂×C₉₇	1	1	yes	characteristic
93	C₂×#102	C₂×C₉₇	1	2	yes	complemented
94	#391	C₂×C₉₇	1	1	yes	characteristic, complemented
95	#336	C₂×C₉₇	1	1	yes	characteristic, complemented
96	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
97	C₂²×D₄	—	2	2	no	—
98	C₂²×D₄	C₂²×C₉₇	1	2	yes	—
99	C₂×SD₁₆	—	2	4	no	—
100	C₂×SD₁₆	—	2	4	no	—
101	C₂×SD₁₆	—	2	4	no	—
102	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
103	#127	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
106	#102	—	2	4	no	—
107	#95	—	2	2	no	—
108	C₂⁴	—	2	2	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	D₄×C₉₇	1	2	yes	complemented
113	C₂×D₄	—	2	8	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	D₄×C₉₇	1	4	yes	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	D₄×C₉₇	1	2	yes	—
125	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
126	SD₁₆	—	2	16	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	D₄×C₉₇	1	4	yes	complemented
130	C₄⋊C₄	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	4	no	—
136	C₂³	—	4	4	no	—
137	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
138	Q₈	—	2	4	no	—
139	Q₈	—	4	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	SD₁₆×C₉₇	1	4	yes	complemented
153	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
156	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	8	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	4	no	—
163	C₂²	#120×C₉₇	1	1	yes	characteristic
164	C₂²	#121×C₉₇	1	1	yes	characteristic
165	C₂²	#127×C₉₇	1	1	yes	characteristic
166	C₂²	C₂×SD₁₆×C₉₇	1	2	yes	—
167	C₂²	AGL(1,ℤ/8)×C₉₇	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	1	no	—
171	C₄	#120×C₉₇	1	2	yes	—
172	C₄	#121×C₉₇	1	2	yes	—
173	C₂	—	8	4	no	—
174	C₂	#392×C₉₇	1	1	yes	characteristic
175	C₂	#450×C₉₇	1	2	yes	—
176	C₂	#460×C₉₇	1	2	yes	—
177	C₂	#492×C₉₇	1	1	yes	characteristic
178	C₂	#496×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×SD₁₆×C₉₇

C₂

yes

maximal, complemented

C₂×#127×C₉₇

C₂

yes

maximal, characteristic

C₂×#102×C₉₇

C₂

yes

maximal, complemented

#391×C₉₇

C₂

yes

maximal, characteristic, complemented

#336×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×D₄×C₉₇

—

C₂²×D₄×C₉₇

C₂²

yes

—

C₂×SD₁₆×C₉₇

—

C₂×SD₁₆×C₉₇

—

C₂×SD₁₆×C₉₇

—

C₂²×C₈×C₉₇

C₂²

yes

—

#127×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#102×C₉₇

—

#95×C₉₇

—

C₂⁴×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

complemented

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

SD₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

complemented

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

SD₁₆

yes

complemented

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

—

C₄×C₉₇

#121

yes

—

C₂×C₉₇

—

C₂×C₉₇

#392

yes

characteristic

C₂×C₉₇

#450

yes

—

C₂×C₉₇

#460

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2014

yes

characteristic, direct product factor, Sylow

C₂²×SD₁₆

C₂×C₉₇

yes

complemented

C₂×#127

C₂×C₉₇

yes

characteristic

C₂×#102

C₂×C₉₇

yes

complemented

#391

C₂×C₉₇

yes

characteristic, complemented

#336

C₂×C₉₇

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₉₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

C₂²×C₉₇

yes

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂×SD₁₆

—

102

C₂²×C₈

C₂²×C₉₇

yes

—

103

#127

—

104

C₂×C₄⋊C₄

C₂²×C₉₇

yes

characteristic

105

C₂×C₄²

C₂²×C₉₇

yes

characteristic

106

#102

—

107

#95

—

108

C₂⁴

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₄×C₉₇

yes

complemented

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

D₄×C₉₇

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

D₄×C₉₇

yes

—

125

C₂²×C₄

C₂³×C₉₇

yes

characteristic

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₄×C₉₇

yes

complemented

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×D₄×C₉₇

yes

characteristic

138

Q₈

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

SD₁₆×C₉₇

yes

complemented

153

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

154

C₂×C₄

C₂×D₄×C₉₇

yes

—

155

C₂×C₄

C₂×D₄×C₉₇

yes

—

156

C₂×C₄

#41×C₉₇

yes

characteristic

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#120×C₉₇

yes

characteristic

164

C₂²

#121×C₉₇

yes

characteristic

165

C₂²

#127×C₉₇

yes

characteristic

166

C₂²

C₂×SD₁₆×C₉₇

yes

—

167

C₂²

AGL(1,ℤ/8)×C₉₇

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#120×C₉₇

yes

—

172

C₄

#121×C₉₇

yes

—

173

C₂

—

174

C₂

#392×C₉₇

yes

characteristic

175

C₂

#450×C₉₇

yes

—

176

C₂

#460×C₉₇

yes

—

177

C₂

#492×C₉₇

yes

characteristic

178

C₂

#496×C₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂×#127×C₉₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₉₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

prime radical

C₂×#127×C₉₇

C₂

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
1. C₂²×SD₁₆×C₉₇	C₂	semidirect
3. C₂×#102×C₉₇	C₂	semidirect
4. #391×C₉₇	C₂	semidirect
5. #336×C₉₇	C₂	semidirect
22. C₂×Q₈×C₉₇	D₄	semidirect
39. C₂×C₈×C₉₇	D₄	semidirect
62. C₂×C₄×C₉₇	SD₁₆	semidirect
89. #2014	C₉₇	direct (N × H)
90. C₉₇	#2014	direct (N × H)
91. C₂²×SD₁₆	C₂×C₉₇	semidirect
93. C₂×#102	C₂×C₉₇	semidirect
94. #391	C₂×C₉₇	semidirect
95. #336	C₂×C₉₇	semidirect
112. C₂×Q₈	D₄×C₉₇	semidirect
129. C₂×C₈	D₄×C₉₇	semidirect
152. C₂×C₄	SD₁₆×C₉₇	semidirect

H ≅ G/N

type

1. C₂²×SD₁₆×C₉₇

C₂

semidirect

3. C₂×#102×C₉₇

C₂

semidirect

4. #391×C₉₇

C₂

semidirect

5. #336×C₉₇

C₂

semidirect

22. C₂×Q₈×C₉₇

D₄

semidirect

39. C₂×C₈×C₉₇

D₄

semidirect

62. C₂×C₄×C₉₇

SD₁₆

semidirect

89. #2014

C₉₇

direct (N × H)

90. C₉₇

#2014

direct (N × H)

91. C₂²×SD₁₆

C₂×C₉₇

semidirect

93. C₂×#102

C₂×C₉₇

semidirect

94. #391

C₂×C₉₇

semidirect

95. #336

C₂×C₉₇

semidirect

112. C₂×Q₈

D₄×C₉₇

semidirect

129. C₂×C₈

D₄×C₉₇

semidirect

152. C₂×C₄

SD₁₆×C₉₇

semidirect

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

subgroup	index
C₂²×SD₁₆×C₉₇	2
C₂×#127×C₉₇	2
C₂×#102×C₉₇	2
#391×C₉₇	2
#336×C₉₇	2
#2014	97

subgroup

index

C₂²×SD₁₆×C₉₇

C₂×#127×C₉₇

C₂×#102×C₉₇

#391×C₉₇

#336×C₉₇

#2014

Subgroups — C₉₇×#2014 #27137972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2014 #27137972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2014 #27137972