Group #27137916 — subgroups

Subgroups — C₉₇×#1958 #27137916

Order 12416.

336 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 128 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×#124×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₉₇	C₂	1	2	yes	maximal, complemented
3	C₂×#102×C₉₇	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#382×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
9	#124×C₉₇	—	2	2	no	—
10	#124×C₉₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₉₇	—	2	2	no	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#103×C₉₇	—	2	4	no	—
15	#102×C₉₇	—	2	4	no	—
16	C₄×C₈×C₉₇	—	2	2	no	—
17	#95×C₉₇	—	2	2	no	—
18	C₂⁴×C₉₇	—	2	1	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	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	—	2	1	no	—
25	C₂×D₄×C₉₇	—	2	2	no	—
26	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉₇	—	2	1	no	—
28	C₂²×C₄×C₉₇	—	2	2	no	—
29	C₂²×C₄×C₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	—	2	2	no	—
33	C₂×C₈×C₉₇	D₄	1	4	yes	—
34	C₄⋊C₄×C₉₇	—	2	2	no	—
35	C₄⋊C₄×C₉₇	—	2	2	no	—
36	C₂²⋊C₄×C₉₇	—	2	8	no	—
37	C₄²×C₉₇	C₂×C₄	1	2	yes	—
38	C₄²×C₉₇	D₄	1	2	yes	—
39	C₂³×C₉₇	—	2	2	no	—
40	C₂³×C₉₇	—	4	1	no	—
41	C₂³×C₉₇	—	4	1	no	—
42	C₂³×C₉₇	—	4	4	no	—
43	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₉₇	—	2	4	no	—
45	Q₈×C₉₇	—	4	2	no	—
46	D₄×C₉₇	—	2	4	no	—
47	D₄×C₉₇	—	4	2	no	—
48	C₂×C₄×C₉₇	—	2	4	no	—
49	C₂×C₄×C₉₇	—	2	2	no	—
50	C₂×C₄×C₉₇	—	4	1	no	—
51	C₂×C₄×C₉₇	—	4	1	no	—
52	C₂×C₄×C₉₇	—	4	2	no	—
53	C₂×C₄×C₉₇	—	4	4	no	—
54	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	#41	1	2	yes	—
59	C₈×C₉₇	—	2	4	no	—
60	C₂²×C₉₇	—	4	2	no	—
61	C₂²×C₉₇	—	4	4	no	—
62	C₂²×C₉₇	—	8	2	no	—
63	C₂²×C₉₇	—	8	2	no	—
64	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₉₇	#124	1	1	yes	characteristic
66	C₂²×C₉₇	#127	1	1	yes	characteristic
67	C₂²×C₉₇	#135	1	4	yes	—
68	C₄×C₉₇	—	2	4	no	—
69	C₄×C₉₇	—	4	2	no	—
70	C₄×C₉₇	—	8	2	no	—
71	C₄×C₉₇	C₄×D₄	1	2	yes	—
72	C₄×C₉₇	#121	1	2	yes	—
73	C₂×C₉₇	—	8	2	no	—
74	C₂×C₉₇	#390	1	1	yes	characteristic
75	C₂×C₉₇	#416	1	2	yes	—
76	C₂×C₉₇	#488	1	2	yes	—
77	C₂×C₉₇	#495	1	2	yes	—
78	#1958	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₉₇	#1958	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×#124	C₂×C₉₇	1	1	yes	characteristic
81	C₂×#103	C₂×C₉₇	1	2	yes	complemented
82	C₂×#102	C₂×C₉₇	1	2	yes	—
83	C₂×C₄×C₈	C₂×C₉₇	1	1	yes	characteristic, complemented
84	#382	C₂×C₉₇	1	1	yes	characteristic, complemented
85	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
86	C₂²×D₄	C₂²×C₉₇	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
88	#124	—	2	2	no	—
89	#124	C₄×C₉₇	1	4	yes	complemented
90	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
93	#103	—	2	4	no	—
94	#102	—	2	4	no	—
95	C₄×C₈	—	2	2	no	—
96	#95	—	2	2	no	—
97	C₂⁴	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×C₄×C₉₇	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	C₂×C₄×C₉₇	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	D₄×C₉₇	1	2	yes	—
109	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	D₄×C₉₇	1	4	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₄²	C₂×C₄×C₉₇	1	2	yes	—
117	C₄²	D₄×C₉₇	1	2	yes	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	4	no	—
122	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	C₂²⋊C₄×C₉₇	1	4	yes	—
134	C₂×C₄	C₂²×C₄×C₉₇	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
136	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
137	C₂×C₄	#41×C₉₇	1	2	yes	—
138	C₈	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂×C₂²⋊C₄×C₉₇	1	1	yes	characteristic
144	C₂²	#124×C₉₇	1	1	yes	characteristic
145	C₂²	#127×C₉₇	1	1	yes	characteristic
146	C₂²	#135×C₉₇	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	2	no	—
150	C₄	C₄×D₄×C₉₇	1	2	yes	—
151	C₄	#121×C₉₇	1	2	yes	—
152	C₂	—	8	2	no	—
153	C₂	#390×C₉₇	1	1	yes	characteristic
154	C₂	#416×C₉₇	1	2	yes	—
155	C₂	#488×C₉₇	1	2	yes	—
156	C₂	#495×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₉₇

C₂

yes

maximal, characteristic

C₂×#103×C₉₇

C₂

yes

maximal, complemented

C₂×#102×C₉₇

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic, complemented

#382×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂²×D₄×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

—

#124×C₉₇

—

#124×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#103×C₉₇

—

#102×C₉₇

—

C₄×C₈×C₉₇

—

#95×C₉₇

—

C₂⁴×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

#121

yes

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#416

yes

—

C₂×C₉₇

#488

yes

—

C₂×C₉₇

#495

yes

—

#1958

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1958

yes

characteristic, direct product factor, Sylow

C₂×#124

C₂×C₉₇

yes

characteristic

C₂×#103

C₂×C₉₇

yes

complemented

C₂×#102

C₂×C₉₇

yes

—

C₂×C₄×C₈

C₂×C₉₇

yes

characteristic, complemented

#382

C₂×C₉₇

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₉₇

yes

characteristic

C₂²×D₄

C₂²×C₉₇

yes

characteristic

C₂²×C₈

C₂²×C₉₇

yes

—

#124

—

#124

C₄×C₉₇

yes

complemented

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

#103

—

#102

—

C₄×C₈

—

#95

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×C₄×C₉₇

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×C₄×C₉₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×C₉₇

yes

—

109

C₂²×C₄

C₂³×C₉₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₄×C₉₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₂×C₄×C₉₇

yes

—

117

C₄²

D₄×C₉₇

yes

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×D₄×C₉₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂²⋊C₄×C₉₇

yes

—

134

C₂×C₄

C₂²×C₄×C₉₇

yes

characteristic

135

C₂×C₄

C₂×D₄×C₉₇

yes

—

136

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

137

C₂×C₄

#41×C₉₇

yes

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊C₄×C₉₇

yes

characteristic

144

C₂²

#124×C₉₇

yes

characteristic

145

C₂²

#127×C₉₇

yes

characteristic

146

C₂²

#135×C₉₇

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₄×D₄×C₉₇

yes

—

151

C₄

#121×C₉₇

yes

—

152

C₂

—

153

C₂

#390×C₉₇

yes

characteristic

154

C₂

#416×C₉₇

yes

—

155

C₂

#488×C₉₇

yes

—

156

C₂

#495×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	C₂²×D₄×C₉₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×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

C₂²×D₄×C₉₇

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
2. C₂×#103×C₉₇	C₂	semidirect
4. C₂×C₄×C₈×C₉₇	C₂	semidirect
5. #382×C₉₇	C₂	semidirect
10. #124×C₉₇	C₄	semidirect
78. #1958	C₉₇	direct (N × H)
79. C₉₇	#1958	direct (N × H)
81. C₂×#103	C₂×C₉₇	semidirect
83. C₂×C₄×C₈	C₂×C₉₇	semidirect
84. #382	C₂×C₉₇	semidirect
89. #124	C₄×C₉₇	semidirect

H ≅ G/N

type

2. C₂×#103×C₉₇

C₂

semidirect

4. C₂×C₄×C₈×C₉₇

C₂

semidirect

5. #382×C₉₇

C₂

semidirect

10. #124×C₉₇

C₄

semidirect

78. #1958

C₉₇

direct (N × H)

79. C₉₇

#1958

direct (N × H)

81. C₂×#103

C₂×C₉₇

semidirect

83. C₂×C₄×C₈

C₂×C₉₇

semidirect

84. #382

C₂×C₉₇

semidirect

89. #124

C₄×C₉₇

semidirect

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

subgroup	index
C₂×#124×C₉₇	2
C₂×#103×C₉₇	2
C₂×#102×C₉₇	2
C₂×C₄×C₈×C₉₇	2
#382×C₉₇	2
#1958	97

subgroup

index

C₂×#124×C₉₇

C₂×#103×C₉₇

C₂×#102×C₉₇

C₂×C₄×C₈×C₉₇

#382×C₉₇

#1958

Subgroups — C₉₇×#1958 #27137916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1958 #27137916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1958 #27137916