Group #27150454 — subgroups

Subgroups — #27150454

Order 12416.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#15690817	C₂	1	2	yes	maximal, complemented
2	#20260807	C₂	1	8	yes	maximal, complemented
3	#20260759	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260702	C₂	1	2	yes	maximal, complemented
5	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₉₇	C₂	1	1	yes	maximal, characteristic
7	#15690817	—	2	8	no	—
8	C₂×#13095495	C₂²	1	2	yes	—
9	C₂×#13095495	C₂²	1	4	yes	—
10	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	C₂²×#11715895	C₂²	1	4	yes	—
12	C₂²×#11715895	C₂²	1	1	yes	characteristic
13	C₄⋊#11715895	C₂²	1	4	yes	—
14	C₂²⋊#11715895	C₂²	1	8	yes	—
15	#15690721	C₂²	1	4	yes	—
16	C₂×#41×C₉₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
18	#127×C₉₇	C₄	1	1	yes	characteristic
19	#124×C₉₇	C₄	1	2	yes	—
20	#121×C₉₇	C₄	1	4	yes	—
21	C₄×D₄×C₉₇	C₂²	1	4	yes	—
22	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
23	#13095495	—	2	4	no	—
24	#13095495	—	2	8	no	—
25	C₂×#11715895	—	2	2	no	—
26	C₂×#11715895	—	2	8	no	—
27	C₂×#11715895	D₄	1	4	yes	complemented
28	C₂×#11715895	C₂³	1	4	yes	—
29	C₂×#11715895	C₂³	1	2	yes	—
30	C₂×#11715895	C₂³	1	2	yes	—
31	#41×C₉₇	—	2	4	no	—
32	#41×C₉₇	—	2	4	no	—
33	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
39	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₉₇	C₂×C₄	1	8	yes	—
42	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₉₇	C₂³	1	2	yes	—
45	C₂³×C₉₇	C₂²×C₄	1	4	yes	—
46	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₉₇	—	2	4	no	—
48	D₄×C₉₇	—	2	8	no	—
49	D₄×C₉₇	—	2	4	no	—
50	D₄×C₉₇	—	4	4	no	—
51	C₂×C₄×C₉₇	—	2	4	no	—
52	C₂×C₄×C₉₇	—	2	4	no	—
53	C₂×C₄×C₉₇	—	4	2	no	—
54	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₉₇	#41	1	2	yes	—
60	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
61	#11715895	—	2	4	no	—
62	#11715895	—	2	2	no	—
63	#11715895	C₂×D₄	1	4	yes	—
64	C₂²×C₉₇	—	2	2	no	—
65	C₂²×C₉₇	—	2	4	no	—
66	C₂²×C₉₇	—	2	8	no	—
67	C₂²×C₉₇	C₄×D₄	1	2	yes	—
68	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₉₇	—	2	4	no	—
70	C₄×C₉₇	—	2	2	no	—
71	C₄×C₉₇	—	2	2	no	—
72	C₄×C₉₇	C₄×D₄	1	2	yes	—
73	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
75	C₂×C₉₇	—	2	2	no	—
76	C₂×C₉₇	—	4	4	no	—
77	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₉₇	#568	1	2	yes	—
79	#2932	—	97	1	no	maximal, Sylow
80	C₉₇	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	97	2	no	—
82	#532	D₉₇	1	1	yes	characteristic
83	#435	—	97	8	no	—
84	#423	—	97	1	no	—
85	#408	—	97	2	no	—
86	C₂×#97	—	97	1	no	—
87	C₂×#41	C₂×D₉₇	1	2	yes	—
88	#131	—	194	8	no	—
89	C₂×M₁₆	—	97	2	no	—
90	C₂×M₁₆	—	97	4	no	—
91	C₂²×C₈	—	97	1	no	—
92	C₂²×C₈	—	97	4	no	—
93	C₂²×C₈	—	97	1	no	—
94	C₄⋊Q₈	Dic₉₇	1	1	yes	characteristic
95	#127	Dic₉₇	1	1	yes	characteristic
96	#124	Dic₉₇	1	2	yes	—
97	#121	Dic₉₇	1	4	yes	—
98	C₄×D₄	C₂×D₉₇	1	4	yes	—
99	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
100	C₄⋊C₈	—	97	4	no	—
101	C₂²⋊C₈	—	97	8	no	—
102	#97	—	97	4	no	—
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×Dic₉₇	1	2	yes	—
106	C₂×D₄	C₂×Dic₉₇	1	4	yes	—
107	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
108	C₂×D₄	C₂×Dic₉₇	1	4	yes	—
109	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
110	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
111	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
112	M₁₆	—	194	8	no	—
113	M₁₆	—	194	4	no	—
114	C₂×C₈	—	97	4	no	—
115	C₂×C₈	—	97	2	no	—
116	C₂×C₈	—	97	2	no	—
117	C₂×C₈	—	97	4	no	—
118	C₂×C₈	—	194	8	no	—
119	C₂×C₈	—	194	2	no	—
120	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
121	C₂²⋊C₄	C₂×Dic₉₇	1	8	yes	—
122	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
123	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
124	C₄²	C₂²×D₉₇	1	2	yes	—
125	C₂³	C₂²×Dic₉₇	1	4	yes	—
126	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
135	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
136	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
137	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₉₇	1	2	yes	—
139	C₂×C₄	#13095510	1	2	yes	—
140	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
141	C₈	—	97	4	no	—
142	C₈	—	194	4	no	—
143	C₈	—	194	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₉₇	1	2	yes	—
148	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₉₇	1	2	yes	—
153	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
154	C₄	C₂×#13095510	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₉₇	1	1	yes	characteristic
158	C₂	#20261502	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690817

C₂

yes

maximal, complemented

#20260807

C₂

yes

maximal, complemented

#20260759

C₂

yes

maximal, characteristic, complemented

#20260702

C₂

yes

maximal, complemented

C₂×#15690721

C₂

yes

maximal, characteristic, complemented

#532×C₉₇

C₂

yes

maximal, characteristic

#15690817

—

C₂×#13095495

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₄⋊#11715895

C₂²

yes

—

C₂²⋊#11715895

C₂²

yes

—

#15690721

C₂²

yes

—

C₂×#41×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₄

yes

characteristic

#127×C₉₇

C₄

yes

characteristic

#124×C₉₇

C₄

yes

—

#121×C₉₇

C₄

yes

—

C₄×D₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

—

D₄

yes

complemented

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂²×C₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×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

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

—

—

C₂×D₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×D₄

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₄×D₄

yes

characteristic

C₂×C₉₇

#568

yes

—

#2932

—

maximal, Sylow

C₉₇

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

#532

D₉₇

yes

—

—

—

C₂×#97

—

C₂×#41

C₂×D₉₇

yes

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₉₇

yes

characteristic

#127

Dic₉₇

yes

characteristic

#124

Dic₉₇

yes

—

#121

Dic₉₇

yes

—

C₄×D₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×Dic₉₇

yes

—

106

C₂×D₄

C₂×Dic₉₇

yes

—

107

C₂×D₄

C₂×Dic₉₇

yes

—

108

C₂×D₄

C₂×Dic₉₇

yes

—

109

C₂²×C₄

C₂×Dic₉₇

yes

—

110

C₂²×C₄

C₂²×D₉₇

yes

—

111

C₂²×C₄

C₂²×D₉₇

yes

characteristic

112

M₁₆

—

194

—

113

M₁₆

—

194

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

194

—

119

C₂×C₈

—

194

—

120

C₄⋊C₄

C₂×Dic₉₇

yes

—

121

C₂²⋊C₄

C₂×Dic₉₇

yes

—

122

C₄²

C₂×Dic₉₇

yes

characteristic

123

C₄²

C₂×Dic₉₇

yes

characteristic

124

C₄²

C₂²×D₉₇

yes

—

125

C₂³

C₂²×Dic₉₇

yes

—

126

C₂³

C₂²×Dic₉₇

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂²×Dic₉₇

yes

—

135

C₂×C₄

C₂²×Dic₉₇

yes

—

136

C₂×C₄

C₂²×Dic₉₇

yes

—

137

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

138

C₂×C₄

D₄×D₉₇

yes

—

139

C₂×C₄

#13095510

yes

—

140

C₂×C₄

C₂³×D₉₇

yes

characteristic

141

C₈

—

142

C₈

—

194

—

143

C₈

—

194

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₉₇

yes

—

148

C₂²

C₂³×Dic₉₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₉₇

yes

—

153

C₄

C₂×D₄×D₉₇

yes

characteristic

154

C₄

C₂×#13095510

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₉₇

yes

characteristic

158

C₂

#20261502

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting, prime radical	#532×C₉₇	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#532	D₉₇
hypocenter	C₉₇	#2932
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂³×C₄

Fitting, prime radical

#532×C₉₇

C₂

cosocle

C₂×C₄×C₉₇

C₂⁴

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#15690817	C₂	semidirect
2. #20260807	C₂	semidirect
3. #20260759	C₂	semidirect
4. #20260702	C₂	semidirect
5. C₂×#15690721	C₂	semidirect
27. C₂×#11715895	D₄	semidirect
80. C₉₇	#2932	semidirect

H ≅ G/N

type

1. C₂×#15690817

C₂

semidirect

2. #20260807

C₂

semidirect

3. #20260759

C₂

semidirect

4. #20260702

C₂

semidirect

5. C₂×#15690721

C₂

semidirect

27. C₂×#11715895

D₄

semidirect

80. C₉₇

#2932

semidirect

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

subgroup	index
C₂×#15690817	2
#20260807	2
#20260759	2
#20260702	2
C₂×#15690721	2
#532×C₉₇	2
#2932	97

subgroup

index

#532×C₉₇

Subgroups — #27150454

Conjugacy classes of subgroups

Complements

Maximal subgroups