Group #27137781 — subgroups

Subgroups — C₉₇×#1823 #27137781

Order 12416.

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

normal subgroups only

This group has 344 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	C₂²×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#380×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#340×C₉₇	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₉₇	—	2	1	no	—
6	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
9	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₉₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
14	C₂²⋊C₈×C₉₇	—	2	2	no	—
15	#95×C₉₇	—	2	1	no	—
16	C₂⁴×C₉₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	—	2	4	no	—
18	C₂²×C₄×C₉₇	—	2	4	no	—
19	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	—	2	2	no	—
21	C₂²×C₄×C₉₇	—	4	2	no	—
22	C₂²×C₄×C₉₇	—	4	2	no	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	—	2	4	no	—
29	C₄⋊C₄×C₉₇	—	2	4	no	—
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₄⋊C₄×C₉₇	—	2	4	no	—
32	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₉₇	—	4	2	no	—
34	C₂³×C₉₇	—	2	2	no	—
35	C₂³×C₉₇	—	2	2	no	—
36	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₉₇	D₈	1	2	yes	—
38	C₂³×C₉₇	SD₁₆	1	2	yes	—
39	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	—	2	4	no	—
43	C₂×C₄×C₉₇	—	2	4	no	—
44	C₂×C₄×C₉₇	—	4	8	no	—
45	C₂×C₄×C₉₇	—	4	2	no	—
46	C₂×C₄×C₉₇	—	4	2	no	—
47	C₂×C₄×C₉₇	—	4	2	no	—
48	C₂×C₄×C₉₇	—	4	2	no	—
49	C₂×C₄×C₉₇	—	4	2	no	—
50	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₉₇	#41	1	4	yes	—
54	C₈×C₉₇	—	4	2	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	—	2	2	no	—
57	C₂²×C₉₇	—	2	4	no	—
58	C₂²×C₉₇	—	2	4	no	—
59	C₂²×C₉₇	#102	1	4	yes	—
60	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₉₇	#123	1	2	yes	—
62	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₉₇	#137	1	2	yes	—
65	C₄×C₉₇	—	2	2	no	—
66	C₄×C₉₇	—	4	4	no	—
67	C₄×C₉₇	—	8	2	no	—
68	C₄×C₉₇	#117	1	2	yes	—
69	C₄×C₉₇	#123	1	2	yes	—
70	C₂×C₉₇	—	2	4	no	—
71	C₂×C₉₇	#381	1	1	yes	characteristic
72	C₂×C₉₇	C₂×#102	1	1	yes	characteristic
73	C₂×C₉₇	#419	1	1	yes	characteristic
74	C₂×C₉₇	#480	1	2	yes	—
75	C₂×C₉₇	#483	1	2	yes	—
76	#1823	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₉₇	#1823	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²×C₄⋊C₄	C₂×C₉₇	1	1	yes	characteristic
79	C₂×C₂²⋊C₈	C₂×C₉₇	1	1	yes	characteristic
80	#380	C₂×C₉₇	1	1	yes	characteristic
81	#340	C₂×C₉₇	1	4	yes	complemented
82	C₂³×C₄	—	2	1	no	—
83	C₂³×C₄	C₂²×C₉₇	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	C₄×C₉₇	1	4	yes	complemented
88	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	2	1	no	—
93	C₂⁴	D₄×C₉₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	C₂×C₄×C₉₇	1	2	yes	—
101	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
102	C₂²×C₄	D₄×C₉₇	1	2	yes	—
103	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	C₂×C₄×C₉₇	1	4	yes	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₂²⋊C₄×C₉₇	1	2	yes	—
114	C₂³	D₈×C₉₇	1	2	yes	—
115	C₂³	SD₁₆×C₉₇	1	2	yes	—
116	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₂²⋊C₄×C₉₇	1	2	yes	—
128	C₂×C₄	C₂²×C₄×C₉₇	1	1	yes	characteristic
129	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
130	C₂×C₄	#41×C₉₇	1	4	yes	—
131	C₈	—	4	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	4	no	—
136	C₂²	#102×C₉₇	1	4	yes	—
137	C₂²	C₂×C₂²⋊C₄×C₉₇	1	1	yes	characteristic
138	C₂²	#123×C₉₇	1	2	yes	—
139	C₂²	C₂×D₈×C₉₇	1	1	yes	characteristic
140	C₂²	C₂×SD₁₆×C₉₇	1	1	yes	characteristic
141	C₂²	#137×C₉₇	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	—	8	2	no	—
145	C₄	#117×C₉₇	1	2	yes	—
146	C₄	#123×C₉₇	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#381×C₉₇	1	1	yes	characteristic
149	C₂	C₂×#102×C₉₇	1	1	yes	characteristic
150	C₂	#419×C₉₇	1	1	yes	characteristic
151	C₂	#480×C₉₇	1	2	yes	—
152	C₂	#483×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₉₇

C₂

yes

maximal, characteristic

#380×C₉₇

C₂

yes

maximal, characteristic

#340×C₉₇

C₂

yes

maximal, complemented

C₂³×C₄×C₉₇

—

C₂³×C₄×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₉₇

—

C₂²⋊C₈×C₉₇

—

#95×C₉₇

—

C₂⁴×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂²⋊C₄

yes

—

C₂³×C₉₇

D₈

yes

—

C₂³×C₉₇

SD₁₆

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#102

yes

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#123

yes

—

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#117

yes

—

C₄×C₉₇

#123

yes

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#483

yes

—

#1823

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1823

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊C₄

C₂×C₉₇

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₉₇

yes

characteristic

#380

C₂×C₉₇

yes

characteristic

#340

C₂×C₉₇

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×C₉₇

yes

characteristic

C₂²×C₈

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₉₇

yes

complemented

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂⁴

D₄×C₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×C₄×C₉₇

yes

—

101

C₂²×C₄

D₄×C₉₇

yes

characteristic

102

C₂²×C₄

D₄×C₉₇

yes

—

103

C₂²×C₄

C₂³×C₉₇

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×C₄×C₉₇

yes

—

110

C₂²⋊C₄

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²⋊C₄×C₉₇

yes

—

114

C₂³

D₈×C₉₇

yes

—

115

C₂³

SD₁₆×C₉₇

yes

—

116

C₂³

C₂×D₄×C₉₇

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²⋊C₄×C₉₇

yes

—

128

C₂×C₄

C₂²×C₄×C₉₇

yes

characteristic

129

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

130

C₂×C₄

#41×C₉₇

yes

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#102×C₉₇

yes

—

137

C₂²

C₂×C₂²⋊C₄×C₉₇

yes

characteristic

138

C₂²

#123×C₉₇

yes

—

139

C₂²

C₂×D₈×C₉₇

yes

characteristic

140

C₂²

C₂×SD₁₆×C₉₇

yes

characteristic

141

C₂²

#137×C₉₇

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#117×C₉₇

yes

—

146

C₄

#123×C₉₇

yes

—

147

C₂

—

148

C₂

#381×C₉₇

yes

characteristic

149

C₂

C₂×#102×C₉₇

yes

characteristic

150

C₂

#419×C₉₇

yes

characteristic

151

C₂

#480×C₉₇

yes

—

152

C₂

#483×C₉₇

yes

—

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂⁴×C₉₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₉₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

prime radical

C₂⁴×C₉₇

D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

N	H ≅ G/N	type
4. #340×C₉₇	C₂	semidirect
10. C₂×C₄⋊C₄×C₉₇	C₄	semidirect
76. #1823	C₉₇	direct (N × H)
77. C₉₇	#1823	direct (N × H)
81. #340	C₂×C₉₇	semidirect
87. C₂×C₄⋊C₄	C₄×C₉₇	semidirect

H ≅ G/N

type

4. #340×C₉₇

C₂

semidirect

10. C₂×C₄⋊C₄×C₉₇

C₄

semidirect

76. #1823

C₉₇

direct (N × H)

77. C₉₇

#1823

direct (N × H)

81. #340

C₂×C₉₇

semidirect

87. C₂×C₄⋊C₄

C₄×C₉₇

semidirect

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

subgroup	index
C₂²×C₄⋊C₄×C₉₇	2
C₂×C₂²⋊C₈×C₉₇	2
#380×C₉₇	2
#340×C₉₇	2
#1823	97

subgroup

index

C₂²×C₄⋊C₄×C₉₇

C₂×C₂²⋊C₈×C₉₇

#380×C₉₇

#340×C₉₇

#1823

Subgroups — C₉₇×#1823 #27137781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1823 #27137781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1823 #27137781