Group #27142694 — subgroups

Subgroups — #27142694

Order 12416.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	2	yes	maximal
4	#20260497	C₂	1	2	yes	maximal
5	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₉₇	—	2	1	no	—
9	C₄⋊#11715895	—	2	2	no	—
10	#15690717	—	2	1	no	—
11	#125×C₉₇	—	2	2	no	—
12	#125×C₉₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₇	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	4	no	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	Q₈	1	4	yes	—
23	C₄×Dic₉₇	—	4	2	no	—
24	C₂²×C₄×C₉₇	—	2	1	no	—
25	C₂²×C₄×C₉₇	—	2	1	no	—
26	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₉₇	—	2	4	no	—
30	C₄⋊C₄×C₉₇	—	2	4	no	—
31	C₄⋊C₄×C₉₇	—	2	1	no	—
32	C₄⋊C₄×C₉₇	—	2	1	no	—
33	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
35	C₄²×C₉₇	C₂×C₄	1	2	yes	—
36	C₄²×C₉₇	D₄	1	2	yes	—
37	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	4	2	no	—
44	C₂×C₄×C₉₇	—	4	1	no	—
45	C₂×C₄×C₉₇	—	4	1	no	—
46	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₉₇	#41	1	2	yes	—
52	C₂×Dic₉₇	—	2	4	no	—
53	C₂×Dic₉₇	—	4	2	no	—
54	C₂×Dic₉₇	—	4	2	no	—
55	#11715895	—	4	2	no	—
56	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₉₇	#135	1	2	yes	—
59	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₉₇	#137	1	1	yes	characteristic
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	2	2	no	—
63	C₄×C₉₇	—	4	2	no	—
64	C₄×C₉₇	—	4	2	no	—
65	C₄×C₉₇	C₄×Q₈	1	2	yes	—
66	C₄×C₉₇	#124	1	1	yes	characteristic
67	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₉₇	—	4	4	no	—
69	C₂×C₉₇	#391	1	1	yes	characteristic
70	C₂×C₉₇	#416	1	1	yes	characteristic
71	C₂×C₉₇	#417	1	1	yes	characteristic
72	C₂×C₉₇	#478	1	2	yes	—
73	C₂×C₉₇	#479	1	2	yes	—
74	#1989	—	97	1	no	maximal, Sylow
75	C₉₇	#1989	1	1	yes	characteristic, complemented, Sylow
76	C₂×#125	D₉₇	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	97	1	no	—
78	C₄×C₄⋊C₄	—	97	1	no	—
79	#340	—	97	2	no	—
80	#340	—	97	2	no	—
81	C₂²×C₈	—	97	2	no	—
82	#125	—	2	2	no	—
83	#125	Dic₉₇	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	97	2	no	—
87	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
89	C₂×C₄²	—	194	1	no	—
90	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
91	C₄⋊C₈	—	194	2	no	—
92	#95	—	194	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	194	2	no	—
96	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	C₂×C₈	—	194	2	no	—
100	C₂×C₈	—	194	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	97	4	no	—
106	C₄⋊C₄	—	194	2	no	—
107	C₄⋊C₄	C₂×Dic₉₇	1	2	yes	—
108	C₄⋊C₄	C₂×Dic₉₇	1	2	yes	—
109	C₄²	—	388	2	no	—
110	C₄²	C₂×Dic₉₇	1	2	yes	—
111	C₄²	C₂²⋊D₉₇	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	388	2	no	—
122	C₂×C₄	—	388	2	no	—
123	C₂×C₄	—	194	4	no	—
124	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
125	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
126	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₉₇	1	2	yes	—
129	C₂×C₄	#13095512	1	2	yes	—
130	C₈	—	388	2	no	—
131	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
132	C₂²	#15690786	1	2	yes	—
133	C₂²	#15690830	1	2	yes	—
134	C₂²	#15690835	1	1	yes	characteristic
135	C₂²	#15690840	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	388	4	no	—
141	C₄	Q₈×Dic₉₇	1	2	yes	—
142	C₄	#15690799	1	1	yes	characteristic
143	C₄	#15690808	1	1	yes	characteristic
144	C₂	#20260642	1	1	yes	characteristic
145	C₂	#20260731	1	1	yes	characteristic
146	C₂	#20260736	1	1	yes	characteristic
147	C₂	#20261032	1	2	yes	—
148	C₂	#20261039	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₉₇

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#20260497

C₂

yes

maximal

C₂×#125×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

—

C₄⋊#11715895

—

#15690717

—

#125×C₉₇

—

#125×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

Q₈

yes

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×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

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂²×C₉₇

#135

yes

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#479

yes

—

#1989

—

maximal, Sylow

C₉₇

#1989

yes

characteristic, complemented, Sylow

C₂×#125

D₉₇

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

#340

—

C₂²×C₈

—

#125

—

#125

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄²

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

100

C₂×C₈

—

194

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

C₂×Dic₉₇

yes

—

108

C₄⋊C₄

C₂×Dic₉₇

yes

—

109

C₄²

—

388

—

110

C₄²

C₂×Dic₉₇

yes

—

111

C₄²

C₂²⋊D₉₇

yes

—

112

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

388

—

122

C₂×C₄

—

388

—

123

C₂×C₄

—

194

—

124

C₂×C₄

C₂²⋊Dic₉₇

yes

—

125

C₂×C₄

C₂²⋊Dic₉₇

yes

—

126

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

128

C₂×C₄

Q₈×D₉₇

yes

—

129

C₂×C₄

#13095512

yes

—

130

C₈

—

388

—

131

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

132

C₂²

#15690786

yes

—

133

C₂²

#15690830

yes

—

134

C₂²

#15690835

yes

characteristic

135

C₂²

#15690840

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

388

—

141

C₄

Q₈×Dic₉₇

yes

—

142

C₄

#15690799

yes

characteristic

143

C₄

#15690808

yes

characteristic

144

C₂

#20260642

yes

characteristic

145

C₂

#20260731

yes

characteristic

146

C₂

#20260736

yes

characteristic

147

C₂

#20261032

yes

—

148

C₂

#20261039

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	#1989
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#125×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
12. #125×C₉₇	C₄	semidirect
75. C₉₇	#1989	semidirect
83. #125	Dic₉₇	semidirect

H ≅ G/N

type

12. #125×C₉₇

C₄

semidirect

75. C₉₇

#1989

semidirect

83. #125

Dic₉₇

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₄×C₄⋊Dic₉₇	2
#20260497	2
C₂×#125×C₉₇	2
#1989	97

subgroup

index

C₂×C₄⋊#11715895

C₄×C₄⋊Dic₉₇

#20260497

C₂×#125×C₉₇

#1989

Subgroups — #27142694

Conjugacy classes of subgroups

Complements

Maximal subgroups