Group #27152139 — subgroups

Subgroups — #27152139

Order 12416.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20261285	C₂	1	1	yes	maximal, characteristic
2	#20261105	C₂	1	2	yes	maximal, complemented
3	#20261101	C₂	1	2	yes	maximal, complemented
4	#20261092	C₂	1	4	yes	maximal, complemented
5	#20260734	C₂	1	4	yes	maximal, complemented
6	C₄×#13095495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13095511	C₂²	1	2	yes	—
9	C₂×#13095495	C₂²	1	2	yes	—
10	#15690810	C₂²	1	1	yes	characteristic
11	#15690807	C₂²	1	1	yes	characteristic
12	#15690798	C₂²	1	2	yes	—
13	#15690784	—	2	2	no	—
14	C₄×D₃₈₈	—	2	2	no	—
15	#15690733	C₂²	1	8	yes	—
16	#15690730	C₂²	1	8	yes	—
17	#15690721	C₂²	1	2	yes	—
18	C₄×#11715895	C₂²	1	2	yes	—
19	#125×C₉₇	—	2	2	no	—
20	#125×C₉₇	C₂²	1	2	yes	—
21	#125×C₉₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
25	#13095511	—	2	8	no	—
26	C₂×Dic₁₉₄	C₂³	1	2	yes	—
27	C₂×D₃₈₈	—	2	2	no	—
28	C₂×D₃₈₈	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₉₇	—	2	2	no	—
30	C₂×C₄×D₉₇	—	2	2	no	—
31	#13095495	—	2	4	no	—
32	C₂×#11715895	C₂³	1	4	yes	—
33	C₄⋊Dic₉₇	—	2	2	no	—
34	#13095488	—	2	4	no	—
35	C₂²×C₄×C₉₇	—	2	2	no	—
36	C₂²×C₄×C₉₇	D₄	1	2	yes	—
37	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₉₇	—	2	4	no	—
39	C₄⋊C₄×C₉₇	—	2	4	no	—
40	C₄⋊C₄×C₉₇	—	2	2	no	—
41	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
42	C₄²×C₉₇	D₄	1	2	yes	—
43	C₄²×C₉₇	C₂³	1	1	yes	characteristic
44	C₄²×C₉₇	C₂³	1	1	yes	characteristic
45	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₉₇	—	2	2	no	—
47	C₂×C₄×C₉₇	—	2	4	no	—
48	C₂×C₄×C₉₇	—	4	2	no	—
49	C₂×C₄×C₉₇	—	4	4	no	—
50	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₉₇	#41	1	4	yes	—
54	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₉₇	—	4	4	no	—
56	C₂×Dic₉₇	—	2	2	no	—
57	D₃₈₈	—	2	4	no	—
58	D₃₈₈	—	4	4	no	—
59	C₄×D₉₇	—	4	4	no	—
60	Dic₁₉₄	—	2	4	no	—
61	#11715895	—	2	4	no	—
62	C₂²×C₉₇	—	2	2	no	—
63	C₂²×C₉₇	#124	1	2	yes	—
64	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₉₇	—	4	2	no	—
66	C₂×D₉₇	—	4	4	no	—
67	C₄×C₉₇	—	2	2	no	—
68	C₄×C₉₇	—	2	2	no	—
69	C₄×C₉₇	—	4	4	no	—
70	C₄×C₉₇	#124	1	2	yes	—
71	C₄×C₉₇	C₂×#41	1	2	yes	—
72	Dic₉₇	—	4	2	no	—
73	C₂×C₉₇	—	2	2	no	—
74	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
75	C₂×C₉₇	#575	1	2	yes	—
76	D₉₇	—	8	2	no	—
77	#3142	—	97	1	no	maximal, Sylow
78	C₉₇	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	—	97	1	no	—
80	C₂×#125	D₉₇	1	1	yes	characteristic, complemented
81	#491	—	97	2	no	—
82	#490	—	97	2	no	—
83	#488	—	97	4	no	—
84	#417	—	97	4	no	—
85	C₄×M₁₆	—	97	1	no	—
86	C₂×#41	—	97	2	no	—
87	C₂×M₁₆	—	97	2	no	—
88	C₄⋊Q₈	—	97	1	no	—
89	#127	—	97	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₉₇	1	2	yes	—
92	#125	C₂×D₉₇	1	2	yes	—
93	#124	—	97	2	no	—
94	#121	—	194	2	no	—
95	C₄×D₄	—	194	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
98	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
99	#103	—	97	8	no	—
100	#102	—	97	8	no	—
101	#97	—	97	2	no	—
102	C₄×C₈	—	97	2	no	—
103	#41	—	194	8	no	—
104	C₂×Q₈	—	97	2	no	—
105	C₂×D₄	—	97	2	no	—
106	C₂×D₄	—	194	2	no	—
107	C₂×D₄	—	194	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	194	2	no	—
110	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
112	M₁₆	—	194	4	no	—
113	C₂×C₈	—	97	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	194	2	no	—
118	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
119	C₂²⋊C₄	—	194	4	no	—
120	C₄²	C₂²⋊D₉₇	1	2	yes	—
121	C₄²	C₂²×D₉₇	1	1	yes	characteristic
122	C₄²	C₂²×D₉₇	1	1	yes	characteristic
123	C₂³	—	194	2	no	—
124	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
125	Q₈	—	194	4	no	—
126	D₄	—	388	4	no	—
127	D₄	—	388	4	no	—
128	D₄	—	194	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	388	4	no	—
133	C₂×C₄	—	194	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
137	C₂×C₄	#13095512	1	4	yes	—
138	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
139	C₈	—	194	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	388	2	no	—
142	C₂²	—	388	4	no	—
143	C₂²	#15690799	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	388	2	no	—
149	C₄	#15690799	1	2	yes	—
150	C₄	C₂×#13095512	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	776	2	no	—
153	C₂	C₂×#15690799	1	1	yes	characteristic
154	C₂	#20261533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261285

C₂

yes

maximal, characteristic

#20261105

C₂

yes

maximal, complemented

#20261101

C₂

yes

maximal, complemented

#20261092

C₂

yes

maximal, complemented

#20260734

C₂

yes

maximal, complemented

C₄×#13095495

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

#15690810

C₂²

yes

characteristic

#15690807

C₂²

yes

characteristic

#15690798

C₂²

yes

—

#15690784

—

C₄×D₃₈₈

—

#15690733

C₂²

yes

—

#15690730

C₂²

yes

—

#15690721

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

#125×C₉₇

—

#125×C₉₇

C₂²

yes

—

#125×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#13095511

—

C₂×Dic₁₉₄

C₂³

yes

—

C₂×D₃₈₈

—

C₂×D₃₈₈

C₂³

yes

—

C₂×C₂²⋊D₉₇

—

C₂×C₄×D₉₇

—

#13095495

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095488

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#124

yes

characteristic

C₂×C₉₇

#575

yes

—

D₉₇

—

#3142

—

maximal, Sylow

C₉₇

#3142

yes

characteristic, complemented, Sylow

#532

—

C₂×#125

D₉₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₉₇

yes

—

#125

C₂×D₉₇

yes

—

#124

—

#121

—

194

—

C₄×D₄

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

194

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

194

—

107

C₂×D₄

—

194

—

108

C₂²×C₄

—

109

C₂²×C₄

—

194

—

110

C₂²×C₄

C₂²⋊D₉₇

yes

—

111

C₂²×C₄

C₂²×D₉₇

yes

characteristic

112

M₁₆

—

194

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

194

—

118

C₄⋊C₄

C₂²×D₉₇

yes

—

119

C₂²⋊C₄

—

194

—

120

C₄²

C₂²⋊D₉₇

yes

—

121

C₄²

C₂²×D₉₇

yes

characteristic

122

C₄²

C₂²×D₉₇

yes

characteristic

123

C₂³

—

194

—

124

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

125

Q₈

—

194

—

126

D₄

—

388

—

127

D₄

—

388

—

128

D₄

—

194

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

388

—

133

C₂×C₄

—

194

—

134

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

135

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

137

C₂×C₄

#13095512

yes

—

138

C₂×C₄

C₂³×D₉₇

yes

characteristic

139

C₈

—

194

—

140

C₂²

—

141

C₂²

—

388

—

142

C₂²

—

388

—

143

C₂²

#15690799

yes

—

144

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

388

—

149

C₄

#15690799

yes

—

150

C₄

C₂×#13095512

yes

—

151

C₂

—

152

C₂

—

776

—

153

C₂

C₂×#15690799

yes

characteristic

154

C₂

#20261533

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	#20261285	C₂
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	#3142
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×#125×C₉₇

C₂

socle

C₂²×C₉₇

C₂²×D₄

C₂

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20261105	C₂	semidirect
3. #20261101	C₂	semidirect
4. #20261092	C₂	semidirect
5. #20260734	C₂	semidirect
6. C₄×#13095495	C₂	semidirect
7. C₂×#125×C₉₇	C₂	semidirect
78. C₉₇	#3142	semidirect
80. C₂×#125	D₉₇	semidirect

H ≅ G/N

type

2. #20261105

C₂

semidirect

3. #20261101

C₂

semidirect

4. #20261092

C₂

semidirect

5. #20260734

C₂

semidirect

6. C₄×#13095495

C₂

semidirect

7. C₂×#125×C₉₇

C₂

semidirect

78. C₉₇

#3142

semidirect

80. C₂×#125

D₉₇

semidirect

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

subgroup	index
#20261285	2
#20261105	2
#20261101	2
#20261092	2
#20260734	2
C₄×#13095495	2
C₂×#125×C₉₇	2
#3142	97

subgroup

index

C₂×#125×C₉₇

Subgroups — #27152139

Conjugacy classes of subgroups

Complements

Maximal subgroups