Group #27152047 — subgroups

Subgroups — C₂×#20261086 #27152047

Order 12416.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15690810	C₂	1	1	yes	maximal, characteristic
3	#20261086	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15690735	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	—	2	1	no	—
7	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
8	#15690810	—	2	2	no	—
9	#15690810	—	2	2	no	—
10	#15690810	C₂²	1	4	yes	—
11	#15690810	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₉₇	—	2	1	no	—
13	C₂×C₄⋊Dic₉₇	—	2	1	no	—
14	C₂×C₄⋊Dic₉₇	—	2	1	no	—
15	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
16	#15690735	C₂²	1	16	yes	—
17	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₉₇	C₂²	1	4	yes	—
20	C₂×Dic₁₉₄	—	2	8	no	—
21	C₂×Dic₁₉₄	—	2	2	no	—
22	C₂×Dic₁₉₄	—	2	4	no	—
23	C₂×Dic₁₉₄	—	4	4	no	—
24	C₂×Dic₁₉₄	C₂³	1	4	yes	—
25	C₂²×Dic₉₇	—	2	2	no	—
26	C₂²×Dic₉₇	—	2	2	no	—
27	C₄⋊Dic₉₇	—	2	4	no	—
28	C₄⋊Dic₉₇	—	2	4	no	—
29	C₄⋊Dic₉₇	—	2	4	no	—
30	C₄⋊Dic₉₇	—	2	2	no	—
31	C₄⋊Dic₉₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₉₇	—	2	4	no	—
36	C₂×C₈×C₉₇	C₂³	1	4	yes	—
37	C₄²×C₉₇	D₄	1	2	yes	—
38	C₄²×C₉₇	C₂³	1	2	yes	—
39	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₉₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₉₇	#41	1	4	yes	—
46	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₉₇	—	2	4	no	—
48	C₂×Dic₉₇	—	2	4	no	—
49	C₂×Dic₉₇	—	4	4	no	—
50	C₂×Dic₉₇	—	4	4	no	—
51	Dic₁₉₄	—	2	8	no	—
52	Dic₁₉₄	—	4	4	no	—
53	Dic₁₉₄	—	4	8	no	—
54	C₈×C₉₇	—	2	4	no	—
55	C₂²×C₉₇	#124	1	2	yes	—
56	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₉₇	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₉₇	#124	1	2	yes	—
60	C₄×C₉₇	C₂×SD₁₆	1	4	yes	—
61	C₄×C₉₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₉₇	C₂×#41	1	2	yes	—
63	Dic₉₇	—	4	4	no	—
64	Dic₉₇	—	4	4	no	—
65	C₂×C₉₇	#487	1	4	yes	complemented
66	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
67	C₂×C₉₇	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	97	1	no	maximal, Sylow
70	C₉₇	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	97	1	no	—
72	C₂×C₄⋊Q₈	—	97	1	no	—
73	#487	—	97	8	no	—
74	C₂×#103	—	97	4	no	—
75	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
76	C₂²×Q₈	—	97	2	no	—
77	C₂²×Q₈	—	194	1	no	—
78	C₂²×C₈	C₂×D₉₇	1	2	yes	—
79	C₄⋊Q₈	—	97	4	no	—
80	C₄⋊Q₈	—	97	4	no	—
81	C₄⋊Q₈	—	194	2	no	—
82	C₄⋊Q₈	—	194	2	no	—
83	C₂×C₄⋊C₄	—	97	2	no	—
84	C₂×C₄⋊C₄	—	194	1	no	—
85	C₂×C₄⋊C₄	—	194	1	no	—
86	C₂×C₄⋊C₄	—	194	1	no	—
87	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
88	#103	—	97	16	no	—
89	C₄×C₈	C₂×D₉₇	1	4	yes	—
90	C₂×Q₈	—	97	4	no	—
91	C₂×Q₈	—	388	4	no	—
92	C₂×Q₈	—	194	8	no	—
93	C₂×Q₈	—	194	4	no	—
94	C₂×Q₈	—	194	2	no	—
95	C₂²×C₄	—	194	2	no	—
96	C₂²×C₄	—	194	2	no	—
97	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
98	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂²×D₉₇	1	4	yes	—
102	C₄⋊C₄	—	97	4	no	—
103	C₄⋊C₄	—	194	2	no	—
104	C₄⋊C₄	—	194	4	no	—
105	C₄⋊C₄	—	194	4	no	—
106	C₄⋊C₄	—	194	4	no	—
107	C₄²	D₃₈₈	1	2	yes	—
108	C₄²	C₂²×D₉₇	1	2	yes	—
109	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
110	Q₈	—	388	4	no	—
111	Q₈	—	388	8	no	—
112	Q₈	—	194	8	no	—
113	C₂×C₄	—	388	4	no	—
114	C₂×C₄	—	388	4	no	—
115	C₂×C₄	—	194	4	no	—
116	C₂×C₄	—	194	4	no	—
117	C₂×C₄	#13095500	1	4	yes	—
118	C₂×C₄	Dic₃₈₈	1	4	yes	—
119	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
121	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
122	C₂×C₄	#13095511	1	4	yes	—
123	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	#15690798	1	2	yes	—
126	C₂²	C₂×#13095500	1	2	yes	—
127	C₂²	C₂×Dic₃₈₈	1	2	yes	—
128	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
129	C₄	—	388	4	no	—
130	C₄	—	388	4	no	—
131	C₄	#15690798	1	2	yes	—
132	C₄	C₂×#13095500	1	4	yes	—
133	C₄	C₂×Dic₃₈₈	1	4	yes	—
134	C₄	C₂×#13095511	1	2	yes	—
135	C₂	#20261086	1	4	yes	direct product factor
136	C₂	C₂×#15690798	1	1	yes	characteristic
137	C₂	C₂²×#13095500	1	1	yes	characteristic
138	C₂	C₂²×Dic₃₈₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690810

C₂

yes

maximal, characteristic

#20261086

C₂

yes

maximal, direct product factor

C₂×#15690735

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

—

C₂²×Dic₁₉₄

C₂²

yes

—

#15690810

—

#15690810

—

#15690810

C₂²

yes

—

#15690810

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690735

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂³

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂³

yes

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

C₂×Q₁₆

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

#124

yes

—

C₄×C₉₇

C₂×SD₁₆

yes

—

C₄×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

C₂²×SD₁₆

yes

characteristic

C₂×C₉₇

C₂²×Q₁₆

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#487

—

C₂×#103

—

C₂×C₄×C₈

D₉₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

194

—

C₂²×C₈

C₂×D₉₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

C₄×C₈

C₂×D₉₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₉₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

194

—

104

C₄⋊C₄

—

194

—

105

C₄⋊C₄

—

194

—

106

C₄⋊C₄

—

194

—

107

C₄²

D₃₈₈

yes

—

108

C₄²

C₂²×D₉₇

yes

—

109

C₂³

C₂×D₃₈₈

yes

characteristic

110

Q₈

—

388

—

111

Q₈

—

388

—

112

Q₈

—

194

—

113

C₂×C₄

—

388

—

114

C₂×C₄

—

388

—

115

C₂×C₄

—

194

—

116

C₂×C₄

—

194

—

117

C₂×C₄

#13095500

yes

—

118

C₂×C₄

Dic₃₈₈

yes

—

119

C₂×C₄

C₂×D₃₈₈

yes

characteristic

120

C₂×C₄

C₂×D₃₈₈

yes

—

121

C₂×C₄

C₂×D₃₈₈

yes

—

122

C₂×C₄

#13095511

yes

—

123

C₂×C₄

C₂³×D₉₇

yes

characteristic

124

C₈

—

125

C₂²

#15690798

yes

—

126

C₂²

C₂×#13095500

yes

—

127

C₂²

C₂×Dic₃₈₈

yes

—

128

C₂²

C₂²×D₃₈₈

yes

characteristic

129

C₄

—

388

—

130

C₄

—

388

—

131

C₄

#15690798

yes

—

132

C₄

C₂×#13095500

yes

—

133

C₄

C₂×Dic₃₈₈

yes

—

134

C₄

C₂×#13095511

yes

—

135

C₂

#20261086

yes

direct product factor

136

C₂

C₂×#15690798

yes

characteristic

137

C₂

C₂²×#13095500

yes

characteristic

138

C₂

C₂²×Dic₃₈₈

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	C₂×#487
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

C₂×#487

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #20261086	C₂	direct (N × H)
65. C₂×C₉₇	#487	semidirect
70. C₉₇	C₂×#487	semidirect
135. C₂	#20261086	direct (N × H)

H ≅ G/N

type

3. #20261086

C₂

direct (N × H)

65. C₂×C₉₇

#487

semidirect

70. C₉₇

C₂×#487

semidirect

135. C₂

#20261086

direct (N × H)

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

subgroup	index
C₂×#15690810	2
#20261086	2
C₂×#15690735	2
C₂×C₄×C₈×C₉₇	2
C₂×#487	97

subgroup

index

C₂×C₄×C₈×C₉₇

Subgroups — C₂×#20261086 #27152047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20261086 #27152047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261086 #27152047