Group #27142659 — subgroups

Subgroups — #27142659

Order 12416.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
3	C₂×#15690733	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690810	—	2	2	no	—
8	#15690810	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	—	2	1	no	—
11	C₄⋊#11715895	—	2	2	no	—
12	#15690733	—	2	8	no	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	—	2	1	no	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#95×C₉₇	—	2	1	no	—
17	C₂×Dic₁₉₄	—	2	4	no	—
18	C₂×Dic₁₉₄	—	2	4	no	—
19	C₂×Dic₁₉₄	—	2	2	no	—
20	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
21	C₂²×Dic₉₇	—	2	2	no	—
22	C₂×#11715895	—	2	4	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₄⋊Dic₉₇	—	2	4	no	—
25	C₄⋊Dic₉₇	—	2	4	no	—
26	C₂²×C₄×C₉₇	—	2	2	no	—
27	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₄⋊C₄×C₉₇	D₄	1	4	yes	—
32	C₄²×C₉₇	—	4	2	no	—
33	C₄²×C₉₇	C₂×C₄	1	2	yes	—
34	C₄²×C₉₇	D₄	1	2	yes	—
35	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	4	no	—
38	C₂×C₄×C₉₇	—	4	2	no	—
39	C₂×C₄×C₉₇	—	4	2	no	—
40	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	C₂×Dic₉₇	—	2	4	no	—
49	C₂×Dic₉₇	—	4	4	no	—
50	Dic₁₉₄	—	2	8	no	—
51	Dic₁₉₄	—	4	2	no	—
52	Dic₁₉₄	—	4	2	no	—
53	#11715895	—	4	2	no	—
54	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₉₇	#121	1	2	yes	—
56	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₉₇	#137	1	2	yes	—
59	C₄×C₉₇	—	2	2	no	—
60	C₄×C₉₇	—	4	4	no	—
61	C₄×C₉₇	#103	1	4	yes	complemented
62	C₄×C₉₇	C₄×D₄	1	2	yes	—
63	C₄×C₉₇	#124	1	1	yes	characteristic
64	C₄×C₉₇	#127	1	1	yes	characteristic
65	Dic₉₇	—	4	4	no	—
66	C₂×C₉₇	#390	1	1	yes	characteristic
67	C₂×C₉₇	C₂×#103	1	1	yes	characteristic
68	C₂×C₉₇	#419	1	1	yes	characteristic
69	C₂×C₉₇	#461	1	2	yes	—
70	C₂×C₉₇	#462	1	2	yes	—
71	#1981	—	97	1	no	maximal, Sylow
72	C₉₇	#1981	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄⋊Q₈	—	97	1	no	—
74	C₂×C₄⋊C₈	—	97	1	no	—
75	C₂×#103	—	97	4	no	—
76	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
77	C₂²×Q₈	—	97	2	no	—
78	C₂²×C₈	—	97	2	no	—
79	C₄⋊Q₈	—	97	4	no	—
80	C₄⋊Q₈	—	194	2	no	—
81	C₂×C₄⋊C₄	—	194	1	no	—
82	C₂×C₄⋊C₄	—	194	1	no	—
83	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
86	C₄⋊C₈	—	194	2	no	—
87	#103	—	194	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	97	4	no	—
90	C₂×Q₈	—	194	4	no	—
91	C₂×Q₈	—	194	4	no	—
92	C₂×Q₈	—	194	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	194	2	no	—
95	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
98	C₂×C₈	—	194	4	no	—
99	C₂×C₈	—	194	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	194	4	no	—
102	C₄⋊C₄	—	194	4	no	—
103	C₄⋊C₄	D₃₈₈	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₉₇	1	2	yes	—
106	C₄²	D₃₈₈	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
108	Q₈	—	388	2	no	—
109	Q₈	—	388	2	no	—
110	Q₈	—	194	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	388	4	no	—
116	C₂×C₄	—	194	4	no	—
117	C₂×C₄	#13095488	1	2	yes	—
118	C₂×C₄	#13095488	1	2	yes	—
119	C₂×C₄	#13095499	1	2	yes	—
120	C₂×C₄	#13095501	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
123	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
124	C₂×C₄	#13095511	1	2	yes	—
125	C₈	—	388	2	no	—
126	C₂²	C₂×#13095488	1	1	yes	characteristic
127	C₂²	#15690784	1	2	yes	—
128	C₂²	C₂×#13095499	1	1	yes	characteristic
129	C₂²	C₂×#13095501	1	1	yes	characteristic
130	C₂²	#15690840	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	388	4	no	—
134	C₄	#15690733	1	4	yes	complemented
135	C₄	C₄×D₃₈₈	1	2	yes	—
136	C₄	#15690798	1	1	yes	characteristic
137	C₄	#15690807	1	1	yes	characteristic
138	C₂	#20260637	1	1	yes	characteristic
139	C₂	C₂×#15690733	1	1	yes	characteristic
140	C₂	#20260743	1	1	yes	characteristic
141	C₂	#20260940	1	2	yes	—
142	C₂	#20260945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₂×#15690733

C₂

yes

maximal

C₄×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂²×#11715895

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₄⋊#11715895

—

#15690733

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

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

D₄

yes

—

C₄²×C₉₇

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#121

yes

—

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

characteristic

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#103

yes

complemented

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#462

yes

—

#1981

—

maximal, Sylow

C₉₇

#1981

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₉₇

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#103

—

194

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

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

—

C₂×C₈

—

194

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

194

—

102

C₄⋊C₄

—

194

—

103

C₄⋊C₄

D₃₈₈

yes

—

104

C₄²

—

105

C₄²

C₄×D₉₇

yes

—

106

C₄²

D₃₈₈

yes

—

107

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

108

Q₈

—

388

—

109

Q₈

—

388

—

110

Q₈

—

194

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

388

—

116

C₂×C₄

—

194

—

117

C₂×C₄

#13095488

yes

—

118

C₂×C₄

#13095488

yes

—

119

C₂×C₄

#13095499

yes

—

120

C₂×C₄

#13095501

yes

—

121

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

122

C₂×C₄

C₂×D₃₈₈

yes

—

123

C₂×C₄

C₂×D₃₈₈

yes

characteristic

124

C₂×C₄

#13095511

yes

—

125

C₈

—

388

—

126

C₂²

C₂×#13095488

yes

characteristic

127

C₂²

#15690784

yes

—

128

C₂²

C₂×#13095499

yes

characteristic

129

C₂²

C₂×#13095501

yes

characteristic

130

C₂²

#15690840

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

388

—

134

C₄

#15690733

yes

complemented

135

C₄

C₄×D₃₈₈

yes

—

136

C₄

#15690798

yes

characteristic

137

C₄

#15690807

yes

characteristic

138

C₂

#20260637

yes

characteristic

139

C₂

C₂×#15690733

yes

characteristic

140

C₂

#20260743

yes

characteristic

141

C₂

#20260940

yes

—

142

C₂

#20260945

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#1981
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₄×C₄⋊C₄×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₉₇

hypocenter

C₉₇

#1981

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #15690810	C₄	semidirect
61. C₄×C₉₇	#103	semidirect
72. C₉₇	#1981	semidirect
134. C₄	#15690733	semidirect

H ≅ G/N

type

8. #15690810

C₄

semidirect

61. C₄×C₉₇

#103

semidirect

72. C₉₇

#1981

semidirect

134. C₄

#15690733

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×C₄⋊#11715895	2
C₂×#15690733	2
C₄×C₄⋊C₄×C₉₇	2
#1981	97

subgroup

index

C₄×C₄⋊C₄×C₉₇

Subgroups — #27142659

Conjugacy classes of subgroups

Complements

Maximal subgroups