Group #27142882 — subgroups

Subgroups — #27142882

Order 12416.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15690802	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690731	C₂	1	2	yes	maximal
3	#20260497	C₂	1	2	yes	maximal, complemented
4	#20260490	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690802	—	2	4	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×#13095490	—	2	2	no	—
10	#15690731	—	2	4	no	—
11	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₉₇	—	2	2	no	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	D₄	1	4	yes	—
21	#13095490	—	2	8	no	—
22	C₂⁴×C₉₇	—	2	1	no	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	—	2	2	no	—
25	C₂×D₄×C₉₇	—	2	1	no	—
26	C₂×D₄×C₉₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₉₇	—	2	1	no	—
28	C₂²×C₄×C₉₇	D₄	1	2	yes	—
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₂²⋊C₄×C₉₇	—	4	4	no	—
32	C₄²×C₉₇	—	2	2	no	—
33	C₂³×C₉₇	—	2	2	no	—
34	C₂³×C₉₇	—	4	1	no	—
35	C₂³×C₉₇	—	4	1	no	—
36	C₂³×C₉₇	—	4	2	no	—
37	C₂³×C₉₇	—	4	2	no	—
38	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₉₇	—	2	4	no	—
40	D₄×C₉₇	—	4	2	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	4	1	no	—
45	C₂×C₄×C₉₇	—	4	2	no	—
46	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₉₇	#41	1	2	yes	—
49	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
50	C₂×Dic₉₇	—	2	4	no	—
51	C₂×Dic₉₇	—	4	2	no	—
52	C₂×Dic₉₇	—	4	2	no	—
53	#11715895	—	4	2	no	—
54	C₂²×C₉₇	—	4	2	no	—
55	C₂²×C₉₇	—	4	4	no	—
56	C₂²×C₉₇	—	8	2	no	—
57	C₂²×C₉₇	—	8	1	no	—
58	C₂²×C₉₇	—	8	1	no	—
59	C₂²×C₉₇	#120	1	1	yes	characteristic
60	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₉₇	#124	1	1	yes	characteristic
62	C₂²×C₉₇	#135	1	2	yes	—
63	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₉₇	—	4	2	no	—
65	C₄×C₉₇	—	8	1	no	—
66	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₉₇	#124	1	2	yes	—
68	Dic₉₇	—	4	4	no	—
69	C₂×C₉₇	—	8	2	no	—
70	C₂×C₉₇	#393	1	1	yes	characteristic
71	C₂×C₉₇	#449	1	2	yes	—
72	C₂×C₉₇	#478	1	2	yes	—
73	C₂×C₉₇	#488	1	1	yes	characteristic
74	C₂×C₉₇	#490	1	1	yes	characteristic
75	#2025	—	97	1	no	maximal, Sylow
76	C₉₇	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	—	97	1	no	—
78	C₂×#102	—	97	2	no	—
79	#390	D₉₇	1	1	yes	characteristic
80	#340	—	97	2	no	—
81	#336	—	97	1	no	—
82	C₂²×D₄	C₂×D₉₇	1	1	yes	characteristic
83	C₂²×C₈	—	97	2	no	—
84	#125	—	194	4	no	—
85	C₂×C₄⋊C₄	—	97	2	no	—
86	C₂×C₄⋊C₄	—	194	2	no	—
87	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
90	#102	—	194	4	no	—
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	Dic₁₉₄	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	194	2	no	—
98	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
100	C₂×C₈	—	194	2	no	—
101	C₂×C₈	—	194	2	no	—
102	C₂×C₈	—	194	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	97	4	no	—
105	C₄⋊C₄	—	194	8	no	—
106	C₄⋊C₄	—	194	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	388	2	no	—
123	C₂×C₄	—	388	2	no	—
124	C₂×C₄	—	194	4	no	—
125	C₂×C₄	D₄×D₉₇	1	2	yes	—
126	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
127	C₂×C₄	#13095510	1	2	yes	—
128	C₂×C₄	#13095511	1	1	yes	characteristic
129	C₈	—	388	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#15690778	1	1	yes	characteristic
136	C₂²	#15690789	1	1	yes	characteristic
137	C₂²	#15690797	1	1	yes	characteristic
138	C₂²	#15690830	1	2	yes	—
139	C₂²	#15690836	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	388	4	no	—
143	C₄	#15690787	1	2	yes	—
144	C₄	#15690796	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#20260647	1	1	yes	characteristic
147	C₂	#20260878	1	2	yes	—
148	C₂	#20261035	1	2	yes	—
149	C₂	#20261089	1	1	yes	characteristic
150	C₂	#20261099	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic, complemented

C₂×#15690731

C₂

yes

maximal

#20260497

C₂

yes

maximal, complemented

#20260490

C₂

yes

maximal, characteristic, complemented

#390×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690802

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690731

—

C₂²×D₄×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

—

#13095490

—

C₂⁴×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#393

yes

characteristic

C₂×C₉₇

#449

yes

—

C₂×C₉₇

#478

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#102

—

#390

D₉₇

yes

characteristic

#340

—

#336

—

C₂²×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#102

—

194

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₉₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

100

C₂×C₈

—

194

—

101

C₂×C₈

—

194

—

102

C₂×C₈

—

194

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

194

—

106

C₄⋊C₄

—

194

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

388

—

123

C₂×C₄

—

388

—

124

C₂×C₄

—

194

—

125

C₂×C₄

D₄×D₉₇

yes

—

126

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

127

C₂×C₄

#13095510

yes

—

128

C₂×C₄

#13095511

yes

characteristic

129

C₈

—

388

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#15690778

yes

characteristic

136

C₂²

#15690789

yes

characteristic

137

C₂²

#15690797

yes

characteristic

138

C₂²

#15690830

yes

—

139

C₂²

#15690836

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

388

—

143

C₄

#15690787

yes

—

144

C₄

#15690796

yes

—

145

C₂

—

146

C₂

#20260647

yes

characteristic

147

C₂

#20260878

yes

—

148

C₂

#20261035

yes

—

149

C₂

#20261089

yes

characteristic

150

C₂

#20261099

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#390×C₉₇	C₂
socle	C₂³×C₉₇	C₂×D₄
prime radical	C₂²×D₄×C₉₇	C₂²
hypercenter	#390	D₉₇
hypocenter	C₉₇	#2025
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#390×C₉₇

C₂

socle

C₂³×C₉₇

C₂×D₄

prime radical

C₂²×D₄×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#15690802	C₂	semidirect
3. #20260497	C₂	semidirect
4. #20260490	C₂	semidirect
76. C₉₇	#2025	semidirect

H ≅ G/N

type

1. C₂×#15690802

C₂

semidirect

3. #20260497

C₂

semidirect

4. #20260490

C₂

semidirect

76. C₉₇

#2025

semidirect

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

subgroup	index
C₂×#15690802	2
C₂×#15690731	2
#20260497	2
#20260490	2
#390×C₉₇	2
#2025	97

subgroup

index

#390×C₉₇

Subgroups — #27142882

Conjugacy classes of subgroups

Complements

Maximal subgroups