Group #27137985 — subgroups

Subgroups — C₉₇×#2027 #27137985

Order 12416.

282 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 84 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	#544×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#418×C₉₇	C₂	1	2	yes	maximal
3	#390×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#343×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#339×C₉₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₉₇	—	2	2	no	—
9	#125×C₉₇	—	2	2	no	—
10	C₂²⋊Q₈×C₉₇	—	2	2	no	—
11	#117×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₉₇	—	2	2	no	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#102×C₉₇	—	2	2	no	—
16	#102×C₉₇	—	2	2	no	—
17	C₂⁴×C₉₇	—	2	1	no	—
18	C₂×Q₈×C₉₇	—	2	2	no	—
19	C₂×D₄×C₉₇	—	2	2	no	—
20	C₂×D₄×C₉₇	—	2	2	no	—
21	C₂×D₄×C₉₇	—	2	1	no	—
22	C₂×D₄×C₉₇	Q₈	1	2	yes	complemented
23	C₂²×C₄×C₉₇	—	2	1	no	—
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₉₇	—	2	2	no	—
27	M₁₆×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	2	4	no	—
31	C₄⋊C₄×C₉₇	—	2	2	no	—
32	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
34	C₂²⋊C₄×C₉₇	—	2	2	no	—
35	C₂²⋊C₄×C₉₇	—	4	4	no	—
36	C₄²×C₉₇	—	2	2	no	—
37	C₄²×C₉₇	—	2	2	no	—
38	C₂³×C₉₇	—	2	2	no	—
39	C₂³×C₉₇	—	4	2	no	—
40	C₂³×C₉₇	—	4	2	no	—
41	C₂³×C₉₇	—	4	1	no	—
42	C₂³×C₉₇	—	4	1	no	—
43	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₉₇	—	8	2	no	—
45	D₄×C₉₇	—	2	4	no	—
46	D₄×C₉₇	—	4	2	no	—
47	C₂×C₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	—	2	2	no	—
49	C₂×C₄×C₉₇	—	2	2	no	—
50	C₂×C₄×C₉₇	—	2	2	no	—
51	C₂×C₄×C₉₇	—	4	1	no	—
52	C₂×C₄×C₉₇	—	4	1	no	—
53	C₂×C₄×C₉₇	—	4	2	no	—
54	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₉₇	#41	1	2	yes	—
57	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
58	C₈×C₉₇	—	4	2	no	—
59	C₂²×C₉₇	—	2	1	no	—
60	C₂²×C₉₇	—	2	1	no	—
61	C₂²×C₉₇	—	4	4	no	—
62	C₂²×C₉₇	—	4	2	no	—
63	C₂²×C₉₇	—	8	2	no	—
64	C₂²×C₉₇	—	8	1	no	—
65	C₂²×C₉₇	—	8	1	no	—
66	C₂²×C₉₇	#120	1	1	yes	characteristic
67	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₉₇	#124	1	1	yes	characteristic
69	C₄×C₉₇	—	4	2	no	—
70	C₄×C₉₇	—	4	2	no	—
71	C₄×C₉₇	—	4	2	no	—
72	C₄×C₉₇	—	4	2	no	—
73	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₉₇	#124	1	2	yes	—
75	C₂×C₉₇	—	2	1	no	—
76	C₂×C₉₇	—	2	1	no	—
77	C₂×C₉₇	—	8	2	no	—
78	C₂×C₉₇	#393	1	1	yes	characteristic
79	C₂×C₉₇	#453	1	1	yes	characteristic
80	C₂×C₉₇	#454	1	1	yes	characteristic
81	#2027	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₉₇	#2027	1	1	yes	characteristic, direct product factor, Sylow
83	#544	C₂×C₉₇	1	1	yes	characteristic, complemented
84	#418	C₂×C₉₇	1	2	yes	—
85	#390	C₂×C₉₇	1	1	yes	characteristic
86	#343	C₂×C₉₇	1	1	yes	characteristic, complemented
87	#339	C₂×C₉₇	1	2	yes	complemented
88	C₂²×D₄	C₂²×C₉₇	1	1	yes	characteristic
89	C₂×M₁₆	C₂²×C₉₇	1	2	yes	—
90	C₄⋊Q₈	—	2	2	no	—
91	#125	—	2	2	no	—
92	C₂²⋊Q₈	—	2	2	no	—
93	#117	C₂²×C₉₇	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	2	2	no	—
96	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
97	#102	—	2	2	no	—
98	#102	—	2	2	no	—
99	C₂⁴	—	2	1	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	Q₈×C₉₇	1	2	yes	complemented
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	D₄×C₉₇	1	2	yes	—
107	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
115	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	4	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	4	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
126	Q₈	—	8	2	no	—
127	D₄	—	2	4	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
137	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
138	C₂×C₄	#41×C₉₇	1	2	yes	—
139	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	#120×C₉₇	1	1	yes	characteristic
149	C₂²	C₂²⋊Q₈×C₉₇	1	1	yes	characteristic
150	C₂²	#124×C₉₇	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	C₂²⋊Q₈×C₉₇	1	2	yes	—
156	C₄	#124×C₉₇	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	8	2	no	—
160	C₂	#393×C₉₇	1	1	yes	characteristic
161	C₂	#453×C₉₇	1	1	yes	characteristic
162	C₂	#454×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#544×C₉₇

C₂

yes

maximal, characteristic, complemented

#418×C₉₇

C₂

yes

maximal

#390×C₉₇

C₂

yes

maximal, characteristic

#343×C₉₇

C₂

yes

maximal, characteristic, complemented

#339×C₉₇

C₂

yes

maximal, complemented

C₂²×D₄×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

—

#125×C₉₇

—

C₂²⋊Q₈×C₉₇

—

#117×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#102×C₉₇

—

#102×C₉₇

—

C₂⁴×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

Q₈

yes

complemented

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×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

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#124

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2027

yes

characteristic, direct product factor, Sylow

#544

C₂×C₉₇

yes

characteristic, complemented

#418

C₂×C₉₇

yes

—

#390

C₂×C₉₇

yes

characteristic

#343

C₂×C₉₇

yes

characteristic, complemented

#339

C₂×C₉₇

yes

complemented

C₂²×D₄

C₂²×C₉₇

yes

characteristic

C₂×M₁₆

C₂²×C₉₇

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

#102

—

#102

—

C₂⁴

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

Q₈×C₉₇

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

D₄×C₉₇

yes

—

107

C₂²×C₄

C₂³×C₉₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

D₄×C₉₇

yes

—

115

C₄⋊C₄

D₄×C₉₇

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×D₄×C₉₇

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×D₄×C₉₇

yes

—

137

C₂×C₄

C₂×Q₈×C₉₇

yes

characteristic

138

C₂×C₄

#41×C₉₇

yes

—

139

C₂×C₄

#41×C₉₇

yes

characteristic

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#120×C₉₇

yes

characteristic

149

C₂²

C₂²⋊Q₈×C₉₇

yes

characteristic

150

C₂²

#124×C₉₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₂²⋊Q₈×C₉₇

yes

—

156

C₄

#124×C₉₇

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#393×C₉₇

yes

characteristic

161

C₂

#453×C₉₇

yes

characteristic

162

C₂

#454×C₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂²×D₄×C₉₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₉₇

#120

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

prime radical

C₂²×D₄×C₉₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

N	H ≅ G/N	type
1. #544×C₉₇	C₂	semidirect
4. #343×C₉₇	C₂	semidirect
5. #339×C₉₇	C₂	semidirect
22. C₂×D₄×C₉₇	Q₈	semidirect
81. #2027	C₉₇	direct (N × H)
82. C₉₇	#2027	direct (N × H)
83. #544	C₂×C₉₇	semidirect
86. #343	C₂×C₉₇	semidirect
87. #339	C₂×C₉₇	semidirect
104. C₂×D₄	Q₈×C₉₇	semidirect

H ≅ G/N

type

1. #544×C₉₇

C₂

semidirect

4. #343×C₉₇

C₂

semidirect

5. #339×C₉₇

C₂

semidirect

22. C₂×D₄×C₉₇

Q₈

semidirect

81. #2027

C₉₇

direct (N × H)

82. C₉₇

#2027

direct (N × H)

83. #544

C₂×C₉₇

semidirect

86. #343

C₂×C₉₇

semidirect

87. #339

C₂×C₉₇

semidirect

104. C₂×D₄

Q₈×C₉₇

semidirect

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

subgroup	index
#544×C₉₇	2
#418×C₉₇	2
#390×C₉₇	2
#343×C₉₇	2
#339×C₉₇	2
#2027	97

subgroup

index

#544×C₉₇

#418×C₉₇

#390×C₉₇

#343×C₉₇

#339×C₉₇

#2027

Subgroups — C₉₇×#2027 #27137985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2027 #27137985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2027 #27137985