Group #28400884 — subgroups

Subgroups — C₂×#20298948 #28400884

Order 14464.

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×#16813622	C₂	1	1	yes	maximal, characteristic
2	#20298948	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16813546	C₂	1	4	yes	maximal
4	C₂×C₄×#11744751	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₆	—	2	1	no	—
7	C₂²×#11744751	C₂²	1	2	yes	—
8	#16813622	—	2	2	no	—
9	#16813622	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₁₃	—	2	1	no	—
11	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
12	#16813546	C₂²	1	16	yes	—
13	C₄×#11744751	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₁₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
16	C₄⋊Q₈×C₁₁₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
19	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₂₆	—	2	4	no	—
21	C₂×Dic₂₂₆	—	4	4	no	—
22	C₂²×Dic₁₁₃	—	2	2	no	—
23	C₂×#11744751	—	2	4	no	—
24	C₂×#11744751	C₂³	1	4	yes	—
25	C₄⋊Dic₁₁₃	—	2	4	no	—
26	C₄⋊Dic₁₁₃	—	2	2	no	—
27	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₁₃	—	2	2	no	—
29	C₂×Q₈×C₁₁₃	—	2	8	no	—
30	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₁₃	—	2	4	no	—
36	C₄⋊C₄×C₁₁₃	—	2	4	no	—
37	C₄²×C₁₁₃	D₄	1	2	yes	—
38	C₄²×C₁₁₃	C₂³	1	2	yes	—
39	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₁₃	—	2	8	no	—
41	Q₈×C₁₁₃	—	4	4	no	—
42	C₂×C₄×C₁₁₃	—	2	4	no	—
43	C₂×C₄×C₁₁₃	—	4	4	no	—
44	C₂×C₄×C₁₁₃	SD₁₆	1	4	yes	—
45	C₂×C₄×C₁₁₃	Q₁₆	1	4	yes	—
46	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₁₃	#41	1	4	yes	—
50	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₁₃	—	2	4	no	—
52	C₂×Dic₁₁₃	—	4	4	no	—
53	Dic₂₂₆	—	4	8	no	—
54	#11744751	—	2	4	no	—
55	C₂²×C₁₁₃	#124	1	2	yes	—
56	C₂²×C₁₁₃	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₁₃	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₁₃	—	4	4	no	—
60	C₄×C₁₁₃	#124	1	2	yes	—
61	C₄×C₁₁₃	C₂×SD₁₆	1	4	yes	—
62	C₄×C₁₁₃	C₂×Q₁₆	1	4	yes	—
63	C₄×C₁₁₃	C₂×#41	1	2	yes	—
64	Dic₁₁₃	—	4	4	no	—
65	C₂×C₁₁₃	#487	1	4	yes	complemented
66	C₂×C₁₁₃	C₂×#124	1	1	yes	characteristic
67	C₂×C₁₁₃	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₁₁₃	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	113	1	no	maximal, Sylow
70	C₁₁₃	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	113	1	no	—
72	C₂×C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic
73	#487	—	113	8	no	—
74	C₂×#103	—	113	4	no	—
75	C₂×C₄×C₈	—	113	1	no	—
76	C₂²×Q₈	—	226	1	no	—
77	C₂²×Q₈	C₂×D₁₁₃	1	2	yes	—
78	C₂²×C₈	—	113	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	113	4	no	—
81	C₄⋊Q₈	—	226	2	no	—
82	C₄⋊Q₈	C₂×D₁₁₃	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	113	2	no	—
86	C₂×C₄⋊C₄	—	226	1	no	—
87	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
88	#103	—	113	16	no	—
89	C₄×C₈	—	113	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	452	4	no	—
93	C₂×Q₈	—	226	4	no	—
94	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	226	2	no	—
97	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
100	C₂×C₈	—	113	4	no	—
101	C₂×C₈	—	226	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	113	4	no	—
105	C₄⋊C₄	—	226	2	no	—
106	C₄⋊C₄	—	226	4	no	—
107	C₄²	C₂²⋊D₁₁₃	1	2	yes	—
108	C₄²	C₂²×D₁₁₃	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	452	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	452	4	no	—
116	C₂×C₄	—	226	4	no	—
117	C₂×C₄	#14268634	1	4	yes	—
118	C₂×C₄	#14268637	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
122	C₂×C₄	#14268646	1	4	yes	—
123	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
124	C₈	—	226	4	no	—
125	C₂²	#16813609	1	2	yes	—
126	C₂²	C₂×#14268634	1	2	yes	—
127	C₂²	C₂×#14268637	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	452	4	no	—
131	C₄	#16813609	1	2	yes	—
132	C₄	C₂×#14268634	1	4	yes	—
133	C₄	C₂×#14268637	1	4	yes	—
134	C₄	C₂×#14268646	1	2	yes	—
135	C₂	#20298948	1	4	yes	direct product factor
136	C₂	C₂×#16813609	1	1	yes	characteristic
137	C₂	C₂²×#14268634	1	1	yes	characteristic
138	C₂	C₂²×#14268637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813622

C₂

yes

maximal, characteristic

#20298948

C₂

yes

maximal, direct product factor

C₂×#16813546

C₂

yes

maximal

C₂×C₄×#11744751

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₂₆

—

C₂²×#11744751

C₂²

yes

—

#16813622

—

#16813622

C₂²

yes

—

C₂×C₄⋊Dic₁₁₃

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

#16813546

C₂²

yes

—

C₄×#11744751

C₂²

yes

—

C₂²×Q₈×C₁₁₃

C₂²

yes

—

C₄⋊Q₈×C₁₁₃

—

C₄⋊Q₈×C₁₁₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₁₃

—

C₂×C₄⋊C₄×C₁₁₃

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂²×Dic₁₁₃

—

C₂×#11744751

—

C₂×#11744751

C₂³

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

C₂³

yes

—

C₂×Q₈×C₁₁₃

—

C₂×Q₈×C₁₁₃

—

C₂×Q₈×C₁₁₃

C₂³

yes

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

D₄

yes

—

C₄²×C₁₁₃

C₂³

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₁₃

—

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

SD₁₆

yes

—

C₂×C₄×C₁₁₃

Q₁₆

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

Dic₂₂₆

—

#11744751

—

C₂²×C₁₁₃

#124

yes

—

C₂²×C₁₁₃

C₂×SD₁₆

yes

—

C₂²×C₁₁₃

C₂×Q₁₆

yes

—

C₂²×C₁₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

#124

yes

—

C₄×C₁₁₃

C₂×SD₁₆

yes

—

C₄×C₁₁₃

C₂×Q₁₆

yes

—

C₄×C₁₁₃

C₂×#41

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

C₂²×SD₁₆

yes

characteristic

C₂×C₁₁₃

C₂²×Q₁₆

yes

characteristic

C₂×#487

—

113

maximal, Sylow

C₁₁₃

C₂×#487

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

113

—

C₂×C₄⋊Q₈

D₁₁₃

yes

characteristic

#487

—

113

—

C₂×#103

—

113

—

C₂×C₄×C₈

—

113

—

C₂²×Q₈

—

226

—

C₂²×Q₈

C₂×D₁₁₃

yes

—

C₂²×C₈

—

113

—

C₄⋊Q₈

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

226

—

C₄⋊Q₈

C₂×D₁₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#103

—

113

—

C₄×C₈

—

113

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

452

—

C₂×Q₈

—

226

—

C₂×Q₈

C₂²×D₁₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

100

C₂×C₈

—

113

—

101

C₂×C₈

—

226

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

113

—

105

C₄⋊C₄

—

226

—

106

C₄⋊C₄

—

226

—

107

C₄²

C₂²⋊D₁₁₃

yes

—

108

C₄²

C₂²×D₁₁₃

yes

—

109

C₂³

C₂×C₂²⋊D₁₁₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

452

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

452

—

116

C₂×C₄

—

226

—

117

C₂×C₄

#14268634

yes

—

118

C₂×C₄

#14268637

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

122

C₂×C₄

#14268646

yes

—

123

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

124

C₈

—

226

—

125

C₂²

#16813609

yes

—

126

C₂²

C₂×#14268634

yes

—

127

C₂²

C₂×#14268637

yes

—

128

C₂²

C₂²×C₂²⋊D₁₁₃

yes

characteristic

129

C₄

—

130

C₄

—

452

—

131

C₄

#16813609

yes

—

132

C₄

C₂×#14268634

yes

—

133

C₄

C₂×#14268637

yes

—

134

C₄

C₂×#14268646

yes

—

135

C₂

#20298948

yes

direct product factor

136

C₂

C₂×#16813609

yes

characteristic

137

C₂

C₂²×#14268634

yes

characteristic

138

C₂

C₂²×#14268637

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator, cosocle	C₂×C₄×C₁₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₂×C₄⋊Q₈×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂×#487
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator, cosocle

C₂×C₄×C₁₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁₃

Fitting

C₂×C₄⋊Q₈×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₁₃

hypocenter

C₁₁₃

C₂×#487

LCS

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20298948	C₂	direct (N × H)
65. C₂×C₁₁₃	#487	semidirect
70. C₁₁₃	C₂×#487	semidirect
135. C₂	#20298948	direct (N × H)

H ≅ G/N

type

2. #20298948

C₂

direct (N × H)

65. C₂×C₁₁₃

#487

semidirect

70. C₁₁₃

C₂×#487

semidirect

135. C₂

#20298948

direct (N × H)

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

subgroup	index
C₂×#16813622	2
#20298948	2
C₂×#16813546	2
C₂×C₄×#11744751	2
C₂×C₄⋊Q₈×C₁₁₃	2
C₂×#487	113

subgroup

index

C₂×C₄⋊Q₈×C₁₁₃

113

Subgroups — C₂×#20298948 #28400884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20298948 #28400884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20298948 #28400884