Group #27137913 — subgroups

Subgroups — C₉₇×#1955 #27137913

Order 12416.

384 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 136 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×C₄×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₈×C₉₇	C₂	1	4	yes	maximal, complemented
4	C₂×#97×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₉₇	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₉₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
9	C₄×D₄×C₉₇	—	2	2	no	—
10	C₄×D₄×C₉₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₉₇	C₄	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₉₇	—	2	2	no	—
15	C₂²⋊C₈×C₉₇	—	2	8	no	—
16	#97×C₉₇	—	2	2	no	—
17	C₂⁴×C₉₇	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₉₇	—	2	4	no	—
19	C₂×D₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	—	2	4	no	—
21	C₂²×C₄×C₉₇	—	2	2	no	—
22	C₂²×C₄×C₉₇	—	2	2	no	—
23	C₂²×C₄×C₉₇	—	2	4	no	—
24	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	—	2	4	no	—
31	C₂×C₈×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	D₄	1	4	yes	complemented
33	C₄⋊C₄×C₉₇	—	2	2	no	—
34	C₂²⋊C₄×C₉₇	—	2	4	no	—
35	C₄²×C₉₇	D₄	1	2	yes	—
36	C₄²×C₉₇	D₄	1	2	yes	—
37	C₂³×C₉₇	—	2	2	no	—
38	C₂³×C₉₇	—	2	4	no	—
39	C₂³×C₉₇	—	2	2	no	—
40	C₂³×C₉₇	—	2	4	no	—
41	C₂³×C₉₇	M₁₆	1	4	yes	—
42	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	D₄×C₉₇	—	2	8	no	—
44	C₂×C₄×C₉₇	—	2	2	no	—
45	C₂×C₄×C₉₇	—	2	2	no	—
46	C₂×C₄×C₉₇	—	2	2	no	—
47	C₂×C₄×C₉₇	—	2	4	no	—
48	C₂×C₄×C₉₇	—	4	4	no	—
49	C₂×C₄×C₉₇	—	4	2	no	—
50	C₂×C₄×C₉₇	—	4	2	no	—
51	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	#41	1	2	yes	—
57	C₈×C₉₇	—	2	4	no	—
58	C₈×C₉₇	—	4	2	no	—
59	C₂²×C₉₇	—	2	4	no	—
60	C₂²×C₉₇	—	2	8	no	—
61	C₂²×C₉₇	—	4	2	no	—
62	C₂²×C₉₇	—	4	2	no	—
63	C₂²×C₉₇	—	4	4	no	—
64	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₉₇	C₄×D₄	1	2	yes	—
66	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
67	C₂²×C₉₇	#131	1	2	yes	—
68	C₄×C₉₇	—	2	2	no	—
69	C₄×C₉₇	—	2	2	no	—
70	C₄×C₉₇	—	4	2	no	—
71	C₄×C₉₇	#121	1	2	yes	—
72	C₄×C₉₇	#124	1	1	yes	characteristic
73	C₄×C₉₇	#127	1	1	yes	characteristic
74	C₂×C₉₇	—	4	4	no	—
75	C₂×C₉₇	#390	1	1	yes	characteristic
76	C₂×C₉₇	#407	1	1	yes	characteristic
77	C₂×C₉₇	#408	1	1	yes	characteristic
78	C₂×C₉₇	#435	1	4	yes	—
79	#1955	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₉₇	#1955	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₄×D₄	C₂×C₉₇	1	1	yes	characteristic
82	C₂×C₄⋊C₈	C₂×C₉₇	1	1	yes	characteristic, complemented
83	C₂×C₂²⋊C₈	C₂×C₉₇	1	4	yes	complemented
84	C₂×#97	C₂×C₉₇	1	1	yes	characteristic, complemented
85	C₂²×D₄	C₄×C₉₇	1	1	yes	characteristic
86	C₂³×C₄	C₂²×C₉₇	1	2	yes	—
87	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
88	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
89	C₄×D₄	—	2	2	no	—
90	C₄×D₄	—	2	2	no	—
91	C₂×C₄⋊C₄	C₄×C₉₇	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₄×C₉₇	1	2	yes	—
93	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₂²⋊C₈	—	2	8	no	—
96	#97	—	2	2	no	—
97	C₂⁴	C₂×C₄×C₉₇	1	2	yes	—
98	C₂×D₄	—	2	4	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	C₂×C₄×C₉₇	1	2	yes	—
105	C₂²×C₄	C₂×C₄×C₉₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×C₉₇	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	D₄×C₉₇	1	4	yes	complemented
113	C₄⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	4	no	—
115	C₄²	D₄×C₉₇	1	2	yes	—
116	C₄²	D₄×C₉₇	1	2	yes	—
117	C₂³	—	2	2	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	4	no	—
121	C₂³	M₁₆×C₉₇	1	4	yes	—
122	C₂³	C₂²×C₄×C₉₇	1	1	yes	characteristic
123	D₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊C₄×C₉₇	1	2	yes	—
132	C₂×C₄	C₂²⋊C₄×C₉₇	1	2	yes	—
133	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
136	C₂×C₄	#41×C₉₇	1	2	yes	—
137	C₈	—	2	4	no	—
138	C₈	—	4	2	no	—
139	C₂²	—	2	4	no	—
140	C₂²	—	2	8	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	4	4	no	—
144	C₂²	C₂×C₂²⋊C₄×C₉₇	1	1	yes	characteristic
145	C₂²	C₄×D₄×C₉₇	1	2	yes	—
146	C₂²	C₂×M₁₆×C₉₇	1	2	yes	—
147	C₂²	#131×C₉₇	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	2	no	—
151	C₄	#121×C₉₇	1	2	yes	—
152	C₄	#124×C₉₇	1	1	yes	characteristic
153	C₄	#127×C₉₇	1	1	yes	characteristic
154	C₂	—	4	4	no	—
155	C₂	#390×C₉₇	1	1	yes	characteristic
156	C₂	#407×C₉₇	1	1	yes	characteristic
157	C₂	#408×C₉₇	1	1	yes	characteristic
158	C₂	#435×C₉₇	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₈×C₉₇

C₂

yes

maximal, complemented

C₂×#97×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₉₇

C₄

yes

characteristic

C₂³×C₄×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₄×D₄×C₉₇

—

C₄×D₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²⋊C₈×C₉₇

—

#97×C₉₇

—

C₂⁴×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

complemented

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

M₁₆

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

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

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#121

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#1955

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1955

yes

characteristic, direct product factor, Sylow

C₂×C₄×D₄

C₂×C₉₇

yes

characteristic

C₂×C₄⋊C₈

C₂×C₉₇

yes

characteristic, complemented

C₂×C₂²⋊C₈

C₂×C₉₇

yes

complemented

C₂×#97

C₂×C₉₇

yes

characteristic, complemented

C₂²×D₄

C₄×C₉₇

yes

characteristic

C₂³×C₄

C₂²×C₉₇

yes

—

C₂²×C₈

C₂²×C₉₇

yes

—

C₂²×C₈

C₂²×C₉₇

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₄×C₉₇

yes

characteristic

C₂×C₂²⋊C₄

C₄×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂⁴

C₂×C₄×C₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×C₉₇

yes

—

105

C₂²×C₄

C₂×C₄×C₉₇

yes

characteristic

106

C₂²×C₄

C₂×C₄×C₉₇

yes

characteristic

107

C₂²×C₄

C₂³×C₉₇

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₄×C₉₇

yes

complemented

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

D₄×C₉₇

yes

—

116

C₄²

D₄×C₉₇

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

M₁₆×C₉₇

yes

—

122

C₂³

C₂²×C₄×C₉₇

yes

characteristic

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊C₄×C₉₇

yes

—

132

C₂×C₄

C₂²⋊C₄×C₉₇

yes

—

133

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

134

C₂×C₄

C₂×D₄×C₉₇

yes

—

135

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

136

C₂×C₄

#41×C₉₇

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊C₄×C₉₇

yes

characteristic

145

C₂²

C₄×D₄×C₉₇

yes

—

146

C₂²

C₂×M₁₆×C₉₇

yes

—

147

C₂²

#131×C₉₇

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#121×C₉₇

yes

—

152

C₄

#124×C₉₇

yes

characteristic

153

C₄

#127×C₉₇

yes

characteristic

154

C₂

—

155

C₂

#390×C₉₇

yes

characteristic

156

C₂

#407×C₉₇

yes

characteristic

157

C₂

#408×C₉₇

yes

characteristic

158

C₂

#435×C₉₇

yes

—

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

subgroup

G/H

center

C₂²×C₄×C₉₇

C₂³

commutator

C₂³

C₂²×C₄×C₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

socle

C₂³×C₉₇

C₂²×C₄

prime radical

C₂²×D₄×C₉₇

C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₉₇ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₉₇	C₂	semidirect
3. C₂×C₂²⋊C₈×C₉₇	C₂	semidirect
4. C₂×#97×C₉₇	C₂	semidirect
32. C₂×C₈×C₉₇	D₄	semidirect
79. #1955	C₉₇	direct (N × H)
80. C₉₇	#1955	direct (N × H)
82. C₂×C₄⋊C₈	C₂×C₉₇	semidirect
83. C₂×C₂²⋊C₈	C₂×C₉₇	semidirect
84. C₂×#97	C₂×C₉₇	semidirect
112. C₂×C₈	D₄×C₉₇	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₈×C₉₇

C₂

semidirect

3. C₂×C₂²⋊C₈×C₉₇

C₂

semidirect

4. C₂×#97×C₉₇

C₂

semidirect

32. C₂×C₈×C₉₇

D₄

semidirect

79. #1955

C₉₇

direct (N × H)

80. C₉₇

#1955

direct (N × H)

82. C₂×C₄⋊C₈

C₂×C₉₇

semidirect

83. C₂×C₂²⋊C₈

C₂×C₉₇

semidirect

84. C₂×#97

C₂×C₉₇

semidirect

112. C₂×C₈

D₄×C₉₇

semidirect

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

subgroup	index
C₂×C₄×D₄×C₉₇	2
C₂×C₄⋊C₈×C₉₇	2
C₂×C₂²⋊C₈×C₉₇	2
C₂×#97×C₉₇	2
#1955	97

subgroup

index

C₂×C₄×D₄×C₉₇

C₂×C₄⋊C₈×C₉₇

C₂×C₂²⋊C₈×C₉₇

C₂×#97×C₉₇

#1955

Subgroups — C₉₇×#1955 #27137913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1955 #27137913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1955 #27137913