Group #28390426 — subgroups

Subgroups — #28390426

Order 14464.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16813551	C₂	1	2	yes	maximal
2	C₂×#16813532	C₂	1	2	yes	maximal
3	C₄²×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#20298353	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11744751	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₁₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₁₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
11	#16813551	C₄	1	8	yes	complemented
12	#16813532	—	2	4	no	—
13	C₂²×C₈×C₁₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#11744751	—	2	2	no	—
19	C₂×#11744751	—	2	2	no	—
20	C₂×#11744751	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₁₃	—	2	4	no	—
22	C₄×Dic₁₁₃	—	2	4	no	—
23	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₁₃	D₄	1	2	yes	—
25	C₄×Dic₁₁₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₁₃	—	2	2	no	—
29	C₂×C₈×C₁₁₃	—	2	2	no	—
30	C₂×C₈×C₁₁₃	C₂×C₄	1	4	yes	—
31	C₄²×C₁₁₃	—	2	2	no	—
32	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄×C₁₁₃	M₁₆	1	4	yes	—
36	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
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	C₄²	1	2	yes	—
44	C₂×Dic₁₁₃	C₄²	1	2	yes	—
45	C₂×Dic₁₁₃	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₁₁₃	M₁₆	1	4	yes	—
47	C₈×C₁₁₃	—	2	4	no	—
48	#11744751	—	2	4	no	—
49	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₁₃	#117	1	1	yes	characteristic
52	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₁₃	—	2	4	no	—
55	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
56	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
57	Dic₁₁₃	—	2	2	no	—
58	Dic₁₁₃	—	2	4	no	—
59	Dic₁₁₃	#97	1	4	yes	complemented
60	C₂×C₁₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₁₃	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₁₃	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₁₃	#423	1	2	yes	—
64	C₂×C₁₁₃	#432	1	2	yes	—
65	#1767	—	113	1	no	maximal, Sylow
66	C₁₁₃	#1767	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	113	2	no	—
68	C₂×#97	—	113	2	no	—
69	C₄³	—	113	1	no	—
70	#336	—	113	1	no	—
71	#336	D₁₁₃	1	1	yes	characteristic
72	C₂²×C₈	—	113	2	no	—
73	C₂²×C₈	C₂×D₁₁₃	1	2	yes	—
74	C₂×C₄²	—	113	1	no	—
75	C₂×C₄²	—	113	1	no	—
76	C₂×C₄²	—	113	2	no	—
77	C₂×C₄²	—	113	2	no	—
78	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
79	C₄⋊C₈	—	113	8	no	—
80	#97	—	226	4	no	—
81	C₂²×C₄	—	113	1	no	—
82	C₂²×C₄	—	113	1	no	—
83	C₂²×C₄	—	113	2	no	—
84	C₂²×C₄	C₄×D₁₁₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	113	4	no	—
89	C₂×C₈	—	226	2	no	—
90	C₂×C₈	—	226	2	no	—
91	C₂×C₈	C₄×D₁₁₃	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	113	4	no	—
94	C₄²	—	113	2	no	—
95	C₄²	—	113	2	no	—
96	C₄²	—	226	4	no	—
97	C₄²	—	226	4	no	—
98	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	113	2	no	—
102	C₂×C₄	—	113	4	no	—
103	C₂×C₄	—	113	2	no	—
104	C₂×C₄	—	113	4	no	—
105	C₂×C₄	—	226	2	no	—
106	C₂×C₄	—	226	2	no	—
107	C₂×C₄	—	226	2	no	—
108	C₂×C₄	#14268630	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
110	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₁₃	1	1	yes	characteristic
112	C₂×C₄	#14268646	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	226	4	no	—
115	C₂²	C₄²×D₁₁₃	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₁₃	1	1	yes	characteristic
117	C₂²	#16813578	1	1	yes	characteristic
118	C₂²	M₁₆×D₁₁₃	1	2	yes	—
119	C₂²	C₂×#14268630	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	113	4	no	—
122	C₄	—	226	2	no	—
123	C₄	—	226	4	no	—
124	C₄	#16813582	1	2	yes	—
125	C₄	#16813587	1	2	yes	—
126	C₂	#20298449	1	1	yes	characteristic
127	C₂	#97×D₁₁₃	1	1	yes	characteristic
128	C₂	C₄×#14268630	1	1	yes	characteristic
129	C₂	#20298620	1	2	yes	—
130	C₂	#20298652	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813551

C₂

yes

maximal

C₂×#16813532

C₂

yes

maximal

C₄²×Dic₁₁₃

C₂

yes

maximal, characteristic

#20298353

C₂

yes

maximal, characteristic

#336×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄×Dic₁₁₃

C₄

yes

—

C₂×C₄×Dic₁₁₃

C₄

yes

—

C₂×C₄×Dic₁₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₁₃

C₂²

yes

characteristic

#16813551

C₄

yes

complemented

#16813532

—

C₂²×C₈×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₁₃

C₂×C₄

yes

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

D₄

yes

—

C₄×Dic₁₁₃

Q₈

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

M₁₆

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₄²

yes

—

C₂×Dic₁₁₃

C₄²

yes

—

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₁₃

M₁₆

yes

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄²

yes

characteristic

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

#117

yes

characteristic

C₂²×C₁₁₃

C₂×M₁₆

yes

—

C₂²×C₁₁₃

C₂×M₁₆

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×D₄

yes

—

C₄×C₁₁₃

C₄×Q₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

#97

yes

complemented

C₂×C₁₁₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₁₃

C₂×#97

yes

characteristic

C₂×C₁₁₃

C₄×M₁₆

yes

characteristic

C₂×C₁₁₃

#423

yes

—

C₂×C₁₁₃

#432

yes

—

#1767

—

113

maximal, Sylow

C₁₁₃

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

113

—

C₂×#97

—

113

—

C₄³

—

113

—

#336

—

113

—

#336

D₁₁₃

yes

characteristic

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

—

113

—

#97

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₄×D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

C₄×D₁₁₃

yes

—

C₄²

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

226

—

C₄²

—

226

—

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

113

—

105

C₂×C₄

—

226

—

106

C₂×C₄

—

226

—

107

C₂×C₄

—

226

—

108

C₂×C₄

#14268630

yes

—

109

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

110

C₂×C₄

D₄×D₁₁₃

yes

characteristic

111

C₂×C₄

Q₈×D₁₁₃

yes

characteristic

112

C₂×C₄

#14268646

yes

—

113

C₈

—

114

C₈

—

226

—

115

C₂²

C₄²×D₁₁₃

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₁₃

yes

characteristic

117

C₂²

#16813578

yes

characteristic

118

C₂²

M₁₆×D₁₁₃

yes

—

119

C₂²

C₂×#14268630

yes

—

120

C₄

—

121

C₄

—

113

—

122

C₄

—

226

—

123

C₄

—

226

—

124

C₄

#16813582

yes

—

125

C₄

#16813587

yes

—

126

C₂

#20298449

yes

characteristic

127

C₂

#97×D₁₁₃

yes

characteristic

128

C₂

C₄×#14268630

yes

characteristic

129

C₂

#20298620

yes

—

130

C₂

#20298652

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂²×C₁₁₃	C₂×C₄²
Fitting	#336×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#336	D₁₁₃
hypocenter	C₁₁₃	#1767
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁₃

commutator

C₂²×C₁₁₃

C₂×C₄²

Fitting

#336×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #16813551	C₄	semidirect
59. Dic₁₁₃	#97	semidirect
66. C₁₁₃	#1767	semidirect

H ≅ G/N

type

11. #16813551

C₄

semidirect

59. Dic₁₁₃

#97

semidirect

66. C₁₁₃

#1767

semidirect

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

subgroup	index
C₂×#16813551	2
C₂×#16813532	2
C₄²×Dic₁₁₃	2
#20298353	2
#336×C₁₁₃	2
#1767	113

subgroup

index

C₄²×Dic₁₁₃

#336×C₁₁₃

113

Subgroups — #28390426

Conjugacy classes of subgroups

Complements

Maximal subgroups