Group #27152084 — subgroups

Subgroups — #27152084

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	#20261097	C₂	1	4	yes	maximal, complemented
5	#20260729	C₂	1	4	yes	maximal, complemented
6	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13095511	C₂²	1	2	yes	—
9	C₂²×#11715895	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	#15690721	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
20	#125×C₉₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₉₇	—	2	2	no	—
22	C₄×Q₈×C₉₇	—	2	2	no	—
23	#117×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	C₂×#11715895	—	2	4	no	—
32	C₂×#11715895	C₂³	1	4	yes	—
33	C₄⋊Dic₉₇	—	2	2	no	—
34	#13095488	—	2	4	no	—
35	C₂×Q₈×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	2	no	—
40	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₉₇	—	2	2	no	—
42	C₄²×C₉₇	—	2	2	no	—
43	C₄²×C₉₇	D₄	1	2	yes	—
44	C₄²×C₉₇	C₂³	1	1	yes	characteristic
45	C₄²×C₉₇	C₂³	1	1	yes	characteristic
46	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₉₇	—	2	2	no	—
48	Q₈×C₉₇	—	4	4	no	—
49	C₂×C₄×C₉₇	—	2	4	no	—
50	C₂×C₄×C₉₇	—	4	2	no	—
51	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	#41	1	4	yes	—
55	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₉₇	—	4	4	no	—
57	C₂×Dic₉₇	—	2	2	no	—
58	D₃₈₈	—	2	4	no	—
59	D₃₈₈	—	4	4	no	—
60	C₄×D₉₇	—	4	4	no	—
61	Dic₁₉₄	—	2	4	no	—
62	#11715895	—	2	4	no	—
63	C₂²×C₉₇	—	2	2	no	—
64	C₂²×C₉₇	#124	1	2	yes	—
65	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₉₇	—	4	2	no	—
67	C₂×D₉₇	—	4	4	no	—
68	C₄×C₉₇	—	2	2	no	—
69	C₄×C₉₇	—	2	2	no	—
70	C₄×C₉₇	—	4	4	no	—
71	C₄×C₉₇	#124	1	2	yes	—
72	C₄×C₉₇	C₂×#41	1	2	yes	—
73	Dic₉₇	—	4	2	no	—
74	C₂×C₉₇	—	2	2	no	—
75	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₉₇	#575	1	2	yes	—
77	D₉₇	—	8	2	no	—
78	#3135	—	97	1	no	maximal, Sylow
79	C₉₇	#3135	1	1	yes	characteristic, complemented, Sylow
80	#533	D₉₇	1	1	yes	characteristic, complemented
81	#532	—	97	1	no	—
82	#491	—	97	2	no	—
83	#490	—	97	2	no	—
84	#489	—	97	4	no	—
85	#416	—	97	4	no	—
86	C₂×#97	—	97	1	no	—
87	C₂×#41	—	97	2	no	—
88	C₂²×C₈	—	97	2	no	—
89	C₄⋊Q₈	—	97	1	no	—
90	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
91	#127	—	97	1	no	—
92	#125	C₂×D₉₇	1	2	yes	—
93	#124	—	97	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	194	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	194	2	no	—
98	#117	C₂×D₉₇	1	2	yes	—
99	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
100	#103	—	97	8	no	—
101	#102	—	97	8	no	—
102	#97	—	97	2	no	—
103	#97	—	97	2	no	—
104	#41	—	194	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	97	2	no	—
107	C₂×D₄	—	97	2	no	—
108	C₂×D₄	—	194	2	no	—
109	C₂×D₄	—	194	2	no	—
110	C₂²×C₄	—	194	2	no	—
111	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
113	C₂×C₈	—	97	4	no	—
114	C₂×C₈	—	194	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	—	2	2	no	—
120	C₂²⋊C₄	—	194	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₉₇	1	2	yes	—
123	C₄²	C₂²×D₉₇	1	1	yes	characteristic
124	C₄²	C₂²×D₉₇	1	1	yes	characteristic
125	C₂³	—	194	2	no	—
126	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	194	4	no	—
129	D₄	—	388	4	no	—
130	D₄	—	388	4	no	—
131	D₄	—	194	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	388	4	no	—
135	C₂×C₄	—	194	2	no	—
136	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
139	C₂×C₄	#13095512	1	4	yes	—
140	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
141	C₈	—	194	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	388	2	no	—
144	C₂²	—	388	4	no	—
145	C₂²	#15690799	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	388	2	no	—
151	C₄	#15690799	1	2	yes	—
152	C₄	C₂×#13095512	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	776	2	no	—
155	C₂	C₂×#15690799	1	1	yes	characteristic
156	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

#20261097

C₂

yes

maximal, complemented

#20260729

C₂

yes

maximal, complemented

C₂×#15690721

C₂

yes

maximal, characteristic, complemented

#533×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

C₂²×#11715895

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

—

#15690721

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

#125×C₉₇

C₂²

yes

—

C₂²⋊Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

#117×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

—

C₂×#11715895

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095488

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂²×D₉₇

—

Q₈×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

—

#3135

—

maximal, Sylow

C₉₇

#3135

yes

characteristic, complemented, Sylow

#533

D₉₇

yes

characteristic, complemented

—

—

—

—

—

C₂×#97

—

C₂×#41

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#127

—

#125

C₂×D₉₇

yes

—

#124

—

C₂²⋊Q₈

—

#121

—

194

—

C₄×Q₈

—

C₄×D₄

—

194

—

#117

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

100

#103

—

101

#102

—

102

#97

—

103

#97

—

104

#41

—

194

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

194

—

109

C₂×D₄

—

194

—

110

C₂²×C₄

—

194

—

111

C₂²×C₄

C₂²⋊D₉₇

yes

—

112

C₂²×C₄

C₂²×D₉₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

194

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

194

—

118

C₄⋊C₄

C₂²×D₉₇

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

194

—

121

C₄²

—

122

C₄²

C₂²⋊D₉₇

yes

—

123

C₄²

C₂²×D₉₇

yes

characteristic

124

C₄²

C₂²×D₉₇

yes

characteristic

125

C₂³

—

194

—

126

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

127

Q₈

—

128

Q₈

—

194

—

129

D₄

—

388

—

130

D₄

—

388

—

131

D₄

—

194

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

388

—

135

C₂×C₄

—

194

—

136

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

139

C₂×C₄

#13095512

yes

—

140

C₂×C₄

C₂³×D₉₇

yes

characteristic

141

C₈

—

194

—

142

C₂²

—

143

C₂²

—

388

—

144

C₂²

—

388

—

145

C₂²

#15690799

yes

—

146

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

388

—

151

C₄

#15690799

yes

—

152

C₄

C₂×#13095512

yes

—

153

C₂

—

154

C₂

—

776

—

155

C₂

C₂×#15690799

yes

characteristic

156

C₂

#20261533

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#533×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	#20261285	C₂
hypercenter	#533	D₉₇
hypocenter	C₉₇	#3135
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

#533×C₉₇

C₂

socle

C₂²×C₉₇

C₂²×D₄

C₂

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20261105	C₂	semidirect
3. #20261101	C₂	semidirect
4. #20261097	C₂	semidirect
5. #20260729	C₂	semidirect
6. C₂×#15690721	C₂	semidirect
7. #533×C₉₇	C₂	semidirect
79. C₉₇	#3135	semidirect
80. #533	D₉₇	semidirect

H ≅ G/N

type

2. #20261105

C₂

semidirect

3. #20261101

C₂

semidirect

4. #20261097

C₂

semidirect

5. #20260729

C₂

semidirect

6. C₂×#15690721

C₂

semidirect

7. #533×C₉₇

C₂

semidirect

79. C₉₇

#3135

semidirect

80. #533

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
#20261097	2
#20260729	2
C₂×#15690721	2
#533×C₉₇	2
#3135	97

subgroup

index

#533×C₉₇

Subgroups — #27152084

Conjugacy classes of subgroups

Complements

Maximal subgroups