Group #27139203 — subgroups

Subgroups — C₉₇×#3245 #27139203

Order 12416.

436 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 188 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	#532×C₉₇	C₂	1	2	yes	maximal, complemented
2	#474×C₉₇	C₂	1	4	yes	maximal, complemented
3	#466×C₉₇	C₂	1	4	yes	maximal, complemented
4	#459×C₉₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₉₇	—	2	2	no	—
7	C₂×D₈×C₉₇	C₂²	1	4	yes	—
8	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
10	#127×C₉₇	C₂²	1	2	yes	—
11	#124×C₉₇	—	2	2	no	—
12	#121×C₉₇	—	2	2	no	—
13	#121×C₉₇	C₂²	1	4	yes	—
14	C₄×D₄×C₉₇	—	2	2	no	—
15	C₄×D₄×C₉₇	C₂²	1	4	yes	—
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	#107×C₉₇	C₂²	1	4	yes	—
18	C₄⋊C₈×C₉₇	C₂²	1	4	yes	—
19	#102×C₉₇	C₂²	1	8	yes	—
20	#41×C₉₇	—	4	8	no	—
21	C₂×Q₈×C₉₇	—	2	2	no	—
22	C₂×D₄×C₉₇	—	2	4	no	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	—	2	2	no	—
25	C₂×D₄×C₉₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₉₇	—	2	4	no	—
27	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	D₈×C₉₇	—	4	4	no	—
31	C₂×C₈×C₉₇	—	4	2	no	—
32	C₂×C₈×C₉₇	C₂³	1	4	yes	—
33	C₄⋊C₄×C₉₇	—	2	2	no	—
34	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₉₇	—	2	4	no	—
36	C₂²⋊C₄×C₉₇	—	2	4	no	—
37	C₄²×C₉₇	D₄	1	2	yes	—
38	C₄²×C₉₇	C₂³	1	2	yes	—
39	C₂³×C₉₇	—	2	4	no	—
40	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₉₇	—	4	4	no	—
42	D₄×C₉₇	—	2	4	no	—
43	D₄×C₉₇	—	2	4	no	—
44	D₄×C₉₇	—	4	4	no	—
45	D₄×C₉₇	—	4	8	no	—
46	D₄×C₉₇	—	4	4	no	—
47	C₂×C₄×C₉₇	—	2	1	no	—
48	C₂×C₄×C₉₇	—	2	1	no	—
49	C₂×C₄×C₉₇	—	2	4	no	—
50	C₂×C₄×C₉₇	—	4	1	no	—
51	C₂×C₄×C₉₇	—	4	8	no	—
52	C₂×C₄×C₉₇	D₈	1	4	yes	complemented
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	2	yes	—
56	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
57	C₈×C₉₇	—	2	4	no	—
58	C₂²×C₉₇	—	2	1	no	—
59	C₂²×C₉₇	—	2	1	no	—
60	C₂²×C₉₇	—	4	4	no	—
61	C₂²×C₉₇	—	4	4	no	—
62	C₂²×C₉₇	—	4	4	no	—
63	C₂²×C₉₇	C₂×D₈	1	2	yes	—
64	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₉₇	—	2	1	no	—
66	C₄×C₉₇	—	2	2	no	—
67	C₄×C₉₇	—	4	4	no	—
68	C₄×C₉₇	C₂×D₈	1	4	yes	—
69	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
70	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
71	C₂×C₉₇	—	2	2	no	—
72	C₂×C₉₇	—	8	4	no	—
73	C₂×C₉₇	#537	1	1	yes	characteristic
74	C₂×C₉₇	C₂²×D₈	1	1	yes	characteristic
75	C₂×C₉₇	#575	1	1	yes	characteristic
76	#3245	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₉₇	#3245	1	1	yes	characteristic, direct product factor, Sylow
78	#532	C₂×C₉₇	1	2	yes	complemented
79	#474	C₂×C₉₇	1	4	yes	complemented
80	#466	C₂×C₉₇	1	4	yes	complemented
81	#459	C₂×C₉₇	1	4	yes	complemented
82	C₂×C₄⋊C₈	C₂×C₉₇	1	1	yes	characteristic, complemented
83	C₂×#41	—	2	2	no	—
84	C₂×D₈	C₂²×C₉₇	1	4	yes	—
85	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
86	C₄⋊Q₈	C₂²×C₉₇	1	2	yes	—
87	#127	C₂²×C₉₇	1	2	yes	—
88	#124	—	2	2	no	—
89	#121	—	2	2	no	—
90	#121	C₂²×C₉₇	1	4	yes	—
91	C₄×D₄	—	2	2	no	—
92	C₄×D₄	C₂²×C₉₇	1	4	yes	—
93	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
94	#107	C₂²×C₉₇	1	4	yes	—
95	C₄⋊C₈	C₂²×C₉₇	1	4	yes	—
96	#102	C₂²×C₉₇	1	8	yes	—
97	#41	—	4	8	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	2	4	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	C₂³×C₉₇	1	4	yes	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
105	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
107	D₈	—	4	4	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	C₂³×C₉₇	1	4	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂³×C₉₇	1	4	yes	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₂²⋊C₄	—	2	4	no	—
114	C₄²	D₄×C₉₇	1	2	yes	—
115	C₄²	C₂³×C₉₇	1	2	yes	—
116	C₂³	—	2	4	no	—
117	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
118	Q₈	—	4	4	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	4	no	—
121	D₄	—	4	4	no	—
122	D₄	—	4	8	no	—
123	D₄	—	4	4	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	8	no	—
129	C₂×C₄	D₈×C₉₇	1	4	yes	complemented
130	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
132	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
133	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	4	no	—
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	C₂×D₈×C₉₇	1	2	yes	—
141	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	C₂×D₈×C₉₇	1	4	yes	—
146	C₄	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
147	C₄	2₋¹⁺⁴×C₉₇	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	—	8	4	no	—
150	C₂	#537×C₉₇	1	1	yes	characteristic
151	C₂	C₂²×D₈×C₉₇	1	1	yes	characteristic
152	C₂	#575×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#532×C₉₇

C₂

yes

maximal, complemented

#474×C₉₇

C₂

yes

maximal, complemented

#466×C₉₇

C₂

yes

maximal, complemented

#459×C₉₇

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₉₇

—

C₂×D₈×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

#127×C₉₇

C₂²

yes

—

#124×C₉₇

—

#121×C₉₇

—

#121×C₉₇

C₂²

yes

—

C₄×D₄×C₉₇

—

C₄×D₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#107×C₉₇

C₂²

yes

—

C₄⋊C₈×C₉₇

C₂²

yes

—

#102×C₉₇

C₂²

yes

—

#41×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

D₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂³

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

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

D₈

yes

complemented

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₈

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₈

yes

—

C₄×C₉₇

H(5,𝔽₂)

yes

characteristic

C₄×C₉₇

2₋¹⁺⁴

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#537

yes

characteristic

C₂×C₉₇

C₂²×D₈

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#3245

yes

characteristic, direct product factor, Sylow

#532

C₂×C₉₇

yes

complemented

#474

C₂×C₉₇

yes

complemented

#466

C₂×C₉₇

yes

complemented

#459

C₂×C₉₇

yes

complemented

C₂×C₄⋊C₈

C₂×C₉₇

yes

characteristic, complemented

C₂×#41

—

C₂×D₈

C₂²×C₉₇

yes

—

C₂²×C₈

C₂²×C₉₇

yes

—

C₄⋊Q₈

C₂²×C₉₇

yes

—

#127

C₂²×C₉₇

yes

—

#124

—

#121

—

#121

C₂²×C₉₇

yes

—

C₄×D₄

—

C₄×D₄

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

#107

C₂²×C₉₇

yes

—

C₄⋊C₈

C₂²×C₉₇

yes

—

#102

C₂²×C₉₇

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

C₂³×C₉₇

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

D₄×C₉₇

yes

characteristic

105

C₂²×C₄

D₄×C₉₇

yes

characteristic

106

C₂²×C₄

C₂³×C₉₇

yes

characteristic

107

D₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂³×C₉₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂³×C₉₇

yes

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

D₄×C₉₇

yes

—

115

C₄²

C₂³×C₉₇

yes

—

116

C₂³

—

117

C₂³

C₂×D₄×C₉₇

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₈×C₉₇

yes

complemented

130

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

131

C₂×C₄

C₂×D₄×C₉₇

yes

—

132

C₂×C₄

C₂×D₄×C₉₇

yes

—

133

C₂×C₄

C₂⁴×C₉₇

yes

characteristic

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×D₈×C₉₇

yes

—

141

C₂²

C₂²×D₄×C₉₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

C₂×D₈×C₉₇

yes

—

146

C₄

H(5,𝔽₂)×C₉₇

yes

characteristic

147

C₄

2₋¹⁺⁴×C₉₇

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

#537×C₉₇

yes

characteristic

151

C₂

C₂²×D₈×C₉₇

yes

characteristic

152

C₂

#575×C₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₉₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₉₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
1. #532×C₉₇	C₂	semidirect
2. #474×C₉₇	C₂	semidirect
3. #466×C₉₇	C₂	semidirect
4. #459×C₉₇	C₂	semidirect
5. C₂×C₄⋊C₈×C₉₇	C₂	semidirect
52. C₂×C₄×C₉₇	D₈	semidirect
76. #3245	C₉₇	direct (N × H)
77. C₉₇	#3245	direct (N × H)
78. #532	C₂×C₉₇	semidirect
79. #474	C₂×C₉₇	semidirect
80. #466	C₂×C₉₇	semidirect
81. #459	C₂×C₉₇	semidirect
82. C₂×C₄⋊C₈	C₂×C₉₇	semidirect
129. C₂×C₄	D₈×C₉₇	semidirect

H ≅ G/N

type

1. #532×C₉₇

C₂

semidirect

2. #474×C₉₇

C₂

semidirect

3. #466×C₉₇

C₂

semidirect

4. #459×C₉₇

C₂

semidirect

5. C₂×C₄⋊C₈×C₉₇

C₂

semidirect

52. C₂×C₄×C₉₇

D₈

semidirect

76. #3245

C₉₇

direct (N × H)

77. C₉₇

#3245

direct (N × H)

78. #532

C₂×C₉₇

semidirect

79. #474

C₂×C₉₇

semidirect

80. #466

C₂×C₉₇

semidirect

81. #459

C₂×C₉₇

semidirect

82. C₂×C₄⋊C₈

C₂×C₉₇

semidirect

129. C₂×C₄

D₈×C₉₇

semidirect

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

subgroup	index
#532×C₉₇	2
#474×C₉₇	2
#466×C₉₇	2
#459×C₉₇	2
C₂×C₄⋊C₈×C₉₇	2
#3245	97

subgroup

index

#532×C₉₇

#474×C₉₇

#466×C₉₇

#459×C₉₇

C₂×C₄⋊C₈×C₉₇

#3245

Subgroups — C₉₇×#3245 #27139203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#3245 #27139203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3245 #27139203