Group #27143269 — subgroups

Subgroups — #27143269

Order 12416.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15690741	C₂	1	2	yes	maximal, complemented
2	C₂×#15690731	C₂	1	2	yes	maximal
3	#20260605	C₂	1	1	yes	maximal, characteristic, 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	C₂×C₄⋊Dic₉₇	—	2	2	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	#15690741	—	2	4	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	D₄	1	4	yes	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	2	no	—
21	C₄⋊Dic₉₇	—	4	4	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	—	4	1	no	—
44	C₂×C₄×C₉₇	—	4	2	no	—
45	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
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	—	4	2	no	—
51	C₂×Dic₉₇	—	4	4	no	—
52	#11715895	—	2	4	no	—
53	C₂²×C₉₇	—	4	2	no	—
54	C₂²×C₉₇	—	4	4	no	—
55	C₂²×C₉₇	—	8	2	no	—
56	C₂²×C₉₇	—	8	1	no	—
57	C₂²×C₉₇	—	8	1	no	—
58	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₉₇	#123	1	1	yes	characteristic
60	C₂²×C₉₇	#127	1	1	yes	characteristic
61	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₉₇	—	2	4	no	—
64	C₄×C₉₇	—	8	1	no	—
65	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₉₇	#123	1	2	yes	—
67	Dic₉₇	—	8	2	no	—
68	C₂×C₉₇	—	8	2	no	—
69	C₂×C₉₇	#399	1	1	yes	characteristic
70	C₂×C₉₇	#476	1	2	yes	—
71	C₂×C₉₇	#481	1	2	yes	—
72	C₂×C₉₇	#492	1	1	yes	characteristic
73	C₂×C₉₇	#496	1	1	yes	characteristic
74	#2094	—	97	1	no	maximal, Sylow
75	C₉₇	#2094	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	97	2	no	—
77	C₂×#102	—	97	2	no	—
78	#390	D₉₇	1	1	yes	characteristic
79	#384	—	97	1	no	—
80	#336	—	97	1	no	—
81	C₂²×D₄	C₂×D₉₇	1	1	yes	characteristic
82	C₂²×C₈	—	97	2	no	—
83	C₂×C₄⋊C₄	—	97	2	no	—
84	C₂×C₄⋊C₄	—	194	2	no	—
85	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
88	#106	—	194	4	no	—
89	#102	—	194	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	Dic₁₉₄	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	194	2	no	—
97	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	C₂×C₈	—	97	4	no	—
100	C₂×C₈	—	194	2	no	—
101	C₂×C₈	—	194	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	388	4	no	—
104	C₄⋊C₄	—	194	2	no	—
105	C₄⋊C₄	—	194	2	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	388	2	no	—
121	C₂×C₄	—	388	4	no	—
122	C₂×C₄	#13095499	1	4	yes	—
123	C₂×C₄	D₄×D₉₇	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
125	C₂×C₄	#13095510	1	2	yes	—
126	C₂×C₄	#13095511	1	1	yes	characteristic
127	C₈	—	194	4	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	#15690789	1	1	yes	characteristic
134	C₂²	#15690791	1	1	yes	characteristic
135	C₂²	#15690806	1	1	yes	characteristic
136	C₂²	C₂×#13095499	1	2	yes	—
137	C₂²	#15690835	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	776	2	no	—
141	C₄	#15690787	1	2	yes	—
142	C₄	#15690792	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#20260671	1	1	yes	characteristic
145	C₂	#20261025	1	2	yes	—
146	C₂	#20261051	1	2	yes	—
147	C₂	#20261107	1	1	yes	characteristic
148	C₂	#20261125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690741

C₂

yes

maximal, complemented

C₂×#15690731

C₂

yes

maximal

#20260605

C₂

yes

maximal, characteristic, complemented

#20260490

C₂

yes

maximal, characteristic, complemented

#390×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690741

—

#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

D₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

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

SD₁₆

yes

—

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

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#123

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#399

yes

characteristic

C₂×C₉₇

#476

yes

—

C₂×C₉₇

#481

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#102

—

#390

D₉₇

yes

characteristic

#384

—

#336

—

C₂²×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#106

—

194

—

#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

C₂×C₈

—

100

C₂×C₈

—

194

—

101

C₂×C₈

—

194

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

388

—

104

C₄⋊C₄

—

194

—

105

C₄⋊C₄

—

194

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

388

—

121

C₂×C₄

—

388

—

122

C₂×C₄

#13095499

yes

—

123

C₂×C₄

D₄×D₉₇

yes

—

124

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

125

C₂×C₄

#13095510

yes

—

126

C₂×C₄

#13095511

yes

characteristic

127

C₈

—

194

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#15690789

yes

characteristic

134

C₂²

#15690791

yes

characteristic

135

C₂²

#15690806

yes

characteristic

136

C₂²

C₂×#13095499

yes

—

137

C₂²

#15690835

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

776

—

141

C₄

#15690787

yes

—

142

C₄

#15690792

yes

—

143

C₂

—

144

C₂

#20260671

yes

characteristic

145

C₂

#20261025

yes

—

146

C₂

#20261051

yes

—

147

C₂

#20261107

yes

characteristic

148

C₂

#20261125

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	#2094
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×#15690741	C₂	semidirect
3. #20260605	C₂	semidirect
4. #20260490	C₂	semidirect
75. C₉₇	#2094	semidirect

H ≅ G/N

type

1. C₂×#15690741

C₂

semidirect

3. #20260605

C₂

semidirect

4. #20260490

C₂

semidirect

75. C₉₇

#2094

semidirect

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

subgroup	index
C₂×#15690741	2
C₂×#15690731	2
#20260605	2
#20260490	2
#390×C₉₇	2
#2094	97

subgroup

index

#390×C₉₇

Subgroups — #27143269

Conjugacy classes of subgroups

Complements

Maximal subgroups