Group #28390690 — subgroups

Subgroups — #28390690

Order 14464.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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⋊#11744751	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	#20298352	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11744751	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₁₃	C₄	1	2	yes	—
7	C₂×C₄×Dic₁₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
10	C₄⋊#11744751	—	2	2	no	—
11	C₂²×C₈×C₁₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₁₃	—	2	2	no	—
14	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
16	C₂×#11744751	—	2	2	no	—
17	C₂×#11744751	—	2	4	no	—
18	C₄×Dic₁₁₃	—	2	2	no	—
19	C₄×Dic₁₁₃	—	2	2	no	—
20	C₄×Dic₁₁₃	—	2	2	no	—
21	C₄×Dic₁₁₃	—	2	4	no	—
22	C₄×Dic₁₁₃	C₈	1	4	yes	complemented
23	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₁₃	—	2	2	no	—
27	C₂×C₈×C₁₁₃	—	2	4	no	—
28	C₄²×C₁₁₃	D₄	1	2	yes	—
29	C₄²×C₁₁₃	Q₈	1	2	yes	—
30	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₁₃	—	2	2	no	—
32	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₁₁₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	#41	1	2	yes	—
39	C₂×C₄×C₁₁₃	#41	1	2	yes	—
40	C₂×Dic₁₁₃	—	2	2	no	—
41	C₂×Dic₁₁₃	—	2	2	no	—
42	C₂×Dic₁₁₃	—	2	2	no	—
43	C₂×Dic₁₁₃	—	2	2	no	—
44	C₂×Dic₁₁₃	C₂×C₈	1	4	yes	—
45	C₂×Dic₁₁₃	M₁₆	1	4	yes	—
46	C₈×C₁₁₃	—	4	2	no	—
47	#11744751	—	4	2	no	—
48	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₁₃	#117	1	2	yes	—
50	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
51	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
52	C₂²×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₁₁₃	—	2	2	no	—
54	C₄×C₁₁₃	C₄⋊C₈	1	4	yes	—
55	C₄×C₁₁₃	#124	1	1	yes	characteristic
56	C₄×C₁₁₃	#124	1	1	yes	characteristic
57	C₄×C₁₁₃	#125	1	2	yes	—
58	Dic₁₁₃	—	2	4	no	—
59	Dic₁₁₃	—	2	4	no	—
60	C₂×C₁₁₃	#382	1	1	yes	characteristic
61	C₂×C₁₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₁₁₃	#423	1	1	yes	characteristic
63	C₂×C₁₁₃	#431	1	2	yes	—
64	C₂×C₁₁₃	#432	1	2	yes	—
65	#1831	—	113	1	no	maximal, Sylow
66	C₁₁₃	#1831	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	113	1	no	—
68	C₂×C₄⋊C₈	D₁₁₃	1	1	yes	characteristic
69	C₄³	—	113	1	no	—
70	#336	—	113	4	no	—
71	C₂²×C₈	—	113	2	no	—
72	C₂²×C₈	C₂×D₁₁₃	1	2	yes	—
73	C₂×C₄²	—	113	1	no	—
74	C₂×C₄²	—	113	2	no	—
75	C₂×C₄²	—	113	1	no	—
76	C₂×C₄²	—	113	2	no	—
77	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	226	2	no	—
80	C₂²×C₄	—	113	2	no	—
81	C₂²×C₄	—	113	2	no	—
82	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	226	2	no	—
88	C₂×C₈	—	226	4	no	—
89	C₄²	—	113	4	no	—
90	C₄²	—	226	4	no	—
91	C₄²	—	226	2	no	—
92	C₄²	—	226	2	no	—
93	C₄²	—	226	2	no	—
94	C₄²	Dic₂₂₆	1	2	yes	—
95	C₄²	C₂²⋊D₁₁₃	1	2	yes	—
96	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	113	4	no	—
99	C₂×C₄	—	113	4	no	—
100	C₂×C₄	—	226	2	no	—
101	C₂×C₄	—	226	2	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	—	226	2	no	—
104	C₂×C₄	#14268626	1	2	yes	—
105	C₂×C₄	#14268626	1	2	yes	—
106	C₂×C₄	C₈×D₁₁₃	1	2	yes	—
107	C₂×C₄	#14268630	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
110	C₂×C₄	#14268646	1	2	yes	—
111	C₂×C₄	#14268648	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	452	2	no	—
114	C₂²	C₂×#14268626	1	1	yes	characteristic
115	C₂²	#16813580	1	2	yes	—
116	C₂²	C₂×C₈×D₁₁₃	1	1	yes	characteristic
117	C₂²	M₁₆×D₁₁₃	1	2	yes	—
118	C₂²	C₂×#14268630	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	226	4	no	—
121	C₄	—	226	4	no	—
122	C₄	#16813551	1	4	yes	—
123	C₄	#16813609	1	1	yes	characteristic
124	C₄	#16813611	1	1	yes	characteristic
125	C₄	#16813613	1	2	yes	—
126	C₂	#20298463	1	1	yes	characteristic
127	C₂	C₂×#16813551	1	1	yes	characteristic
128	C₂	#20298621	1	1	yes	characteristic
129	C₂	#20298650	1	2	yes	—
130	C₂	#20298654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11744751

C₂

yes

maximal, characteristic

C₄²×Dic₁₁₃

C₂

yes

maximal, characteristic

#20298352

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄×Dic₁₁₃

C₄

yes

—

C₂×C₄×Dic₁₁₃

C₄

yes

characteristic

C₂×C₄×Dic₁₁₃

C₄

yes

characteristic

C₂×C₄×Dic₁₁₃

C₂²

yes

—

C₄⋊#11744751

—

C₂²×C₈×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₁₃

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

C₂×#11744751

—

C₂×#11744751

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

C₈

yes

complemented

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₄²×C₁₁₃

D₄

yes

—

C₄²×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂×C₈

yes

—

C₂×C₄×C₁₁₃

M₁₆

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂×C₈

yes

—

C₂×Dic₁₁₃

M₁₆

yes

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

#117

yes

—

C₂²×C₁₁₃

C₂²×C₈

yes

characteristic

C₂²×C₁₁₃

C₂×M₁₆

yes

—

C₂²×C₁₁₃

C₂×M₁₆

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄⋊C₈

yes

—

C₄×C₁₁₃

yes

C₄×C₁₁₃

yes

C₄×C₁₁₃

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

#382

yes

characteristic

C₂×C₁₁₃

C₂×C₄⋊C₈

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

C₂×C₁₁₃

#432

yes

—

#1831

—

113

maximal, Sylow

C₁₁₃

#1831

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

113

—

C₂×C₄⋊C₈

D₁₁₃

yes

characteristic

C₄³

—

113

—

#336

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

—

C₂×C₄²

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄²

—

113

—

C₄²

—

226

—

C₄²

—

226

—

C₄²

—

226

—

C₄²

—

226

—

C₄²

Dic₂₂₆

yes

—

C₄²

C₂²⋊D₁₁₃

yes

—

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

100

C₂×C₄

—

226

—

101

C₂×C₄

—

226

—

102

C₂×C₄

—

226

—

103

C₂×C₄

—

226

—

104

C₂×C₄

#14268626

yes

—

105

C₂×C₄

#14268626

yes

—

106

C₂×C₄

C₈×D₁₁₃

yes

—

107

C₂×C₄

#14268630

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

109

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

110

C₂×C₄

#14268646

yes

—

111

C₂×C₄

#14268648

yes

—

112

C₈

—

113

C₈

—

452

—

114

C₂²

C₂×#14268626

yes

characteristic

115

C₂²

#16813580

yes

—

116

C₂²

C₂×C₈×D₁₁₃

yes

characteristic

117

C₂²

M₁₆×D₁₁₃

yes

—

118

C₂²

C₂×#14268630

yes

characteristic

119

C₄

—

120

C₄

—

226

—

121

C₄

—

226

—

122

C₄

#16813551

yes

—

123

C₄

#16813609

yes

characteristic

124

C₄

#16813611

yes

characteristic

125

C₄

#16813613

yes

—

126

C₂

#20298463

yes

characteristic

127

C₂

C₂×#16813551

yes

characteristic

128

C₂

#20298621

yes

characteristic

129

C₂

#20298650

yes

—

130

C₂

#20298654

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₁₃
hypocenter	C₁₁₃	#1831
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁₃

commutator

C₂²×C₁₁₃

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₄×Dic₁₁₃	C₈	semidirect
66. C₁₁₃	#1831	semidirect

H ≅ G/N

type

22. C₄×Dic₁₁₃

C₈

semidirect

66. C₁₁₃

#1831

semidirect

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

subgroup	index
C₂×C₄⋊#11744751	2
C₄²×Dic₁₁₃	2
#20298352	2
C₂×C₄⋊C₈×C₁₁₃	2
#1831	113

subgroup

index

C₂×C₄⋊#11744751

C₄²×Dic₁₁₃

#20298352

C₂×C₄⋊C₈×C₁₁₃

#1831

113

Subgroups — #28390690

Conjugacy classes of subgroups

Complements

Maximal subgroups