Group #28399509 — subgroups

Subgroups — #28399509

Order 14464.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#20298729	C₂	1	4	yes	maximal, complemented
2	Q₈×#11744751	C₂	1	4	yes	maximal, complemented
3	#20298625	C₂	1	2	yes	maximal
4	#20298622	C₂	1	1	yes	maximal, characteristic
5	#20298554	C₂	1	2	yes	maximal
6	C₂×C₄×#11744751	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14268631	C₂²	1	2	yes	—
9	C₂²×#11744751	C₂²	1	1	yes	characteristic
10	C₂²×#11744751	C₂²	1	1	yes	characteristic
11	C₄⋊#11744751	C₂²	1	4	yes	—
12	C₄⋊#11744751	C₂²	1	8	yes	—
13	#16813533	C₂²	1	4	yes	—
14	C₄×#11744751	C₂²	1	2	yes	—
15	C₄×#11744751	C₂²	1	2	yes	—
16	C₄×#11744751	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₁₃	C₄	1	2	yes	—
18	#125×C₁₁₃	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₁₁₃	C₄	1	4	yes	—
20	C₄×Q₈×C₁₁₃	C₂²	1	4	yes	—
21	#117×C₁₁₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
23	#14268631	—	2	4	no	—
24	C₂×#11744751	—	2	2	no	—
25	C₂×#11744751	Q₈	1	4	yes	complemented
26	C₂×#11744751	C₂³	1	2	yes	—
27	C₂×#11744751	C₂³	1	4	yes	—
28	C₂×#11744751	C₂³	1	1	yes	characteristic
29	C₂×#11744751	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
34	C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
37	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₁₃	C₂³	1	2	yes	—
40	C₄²×C₁₁₃	C₂³	1	4	yes	—
41	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₁₃	—	2	8	no	—
43	C₂×C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₁₃	#41	1	2	yes	—
50	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
51	#11744751	—	2	2	no	—
52	#11744751	—	2	4	no	—
53	#11744751	C₂×Q₈	1	2	yes	—
54	#11744751	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₁₃	—	2	2	no	—
56	C₂²×C₁₁₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₁₃	—	2	2	no	—
59	C₄×C₁₁₃	—	2	8	no	—
60	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
61	C₄×C₁₁₃	#131	1	4	yes	—
62	C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₁₃	—	2	2	no	—
65	C₂×C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₁₃	C₂×#131	1	2	yes	—
67	#2960	—	113	1	no	maximal, Sylow
68	C₁₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₁₃	1	1	yes	characteristic
70	#446	—	113	4	no	—
71	C₈×Q₈	—	113	4	no	—
72	#424	—	113	2	no	—
73	#423	—	113	1	no	—
74	#405	—	113	2	no	—
75	C₂×C₄×C₈	—	113	1	no	—
76	C₂×M₁₆	—	113	2	no	—
77	C₂²×C₈	—	113	1	no	—
78	C₂²×C₈	—	113	1	no	—
79	C₄⋊Q₈	Dic₁₁₃	1	2	yes	—
80	#125	Dic₁₁₃	1	2	yes	—
81	C₂²⋊Q₈	Dic₁₁₃	1	4	yes	—
82	C₄×Q₈	C₂×D₁₁₃	1	4	yes	—
83	#117	C₂×D₁₁₃	1	2	yes	—
84	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
85	C₄⋊C₈	—	113	4	no	—
86	C₄⋊C₈	—	113	8	no	—
87	#97	—	113	4	no	—
88	C₄×C₈	—	113	2	no	—
89	C₄×C₈	—	113	2	no	—
90	C₄×C₈	—	113	4	no	—
91	C₂×Q₈	C₂×Dic₁₁₃	1	4	yes	—
92	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
94	M₁₆	—	226	4	no	—
95	C₂×C₈	—	113	4	no	—
96	C₂×C₈	—	113	1	no	—
97	C₂×C₈	—	113	1	no	—
98	C₂×C₈	—	113	2	no	—
99	C₂×C₈	—	113	4	no	—
100	C₂×C₈	—	226	2	no	—
101	C₄⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₁₁₃	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
105	C₄²	C₂×Dic₁₁₃	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₁₃	1	1	yes	characteristic
107	C₄²	C₂²×D₁₁₃	1	2	yes	—
108	C₄²	C₂²×D₁₁₃	1	4	yes	—
109	C₂³	C₂²×Dic₁₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₁₃	1	8	yes	—
114	C₂×C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
116	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
117	C₂×C₄	#14268648	1	2	yes	—
118	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
119	C₈	—	113	2	no	—
120	C₈	—	113	2	no	—
121	C₈	—	226	2	no	—
122	C₈	—	226	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₁₁₃	1	2	yes	—
125	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₁₁₃	1	2	yes	—
129	C₄	#16813629	1	4	yes	—
130	C₄	C₂×Q₈×D₁₁₃	1	1	yes	characteristic
131	C₄	C₂×#14268648	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₁₁₃	1	1	yes	characteristic
134	C₂	C₂×#16813629	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20298729

C₂

yes

maximal, complemented

Q₈×#11744751

C₂

yes

maximal, complemented

#20298625

C₂

yes

maximal

#20298622

C₂

yes

maximal, characteristic

#20298554

C₂

yes

maximal

C₂×C₄×#11744751

C₂

yes

maximal, characteristic

#533×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

C₂²×#11744751

C₂²

yes

characteristic

C₂²×#11744751

C₂²

yes

characteristic

C₄⋊#11744751

C₂²

yes

—

C₄⋊#11744751

C₂²

yes

—

#16813533

C₂²

yes

—

C₄×#11744751

C₂²

yes

—

C₄×#11744751

C₂²

yes

—

C₄×#11744751

C₂²

yes

—

C₄⋊Q₈×C₁₁₃

C₄

yes

—

#125×C₁₁₃

C₄

yes

—

C₂²⋊Q₈×C₁₁₃

C₄

yes

—

C₄×Q₈×C₁₁₃

C₂²

yes

—

#117×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#11744751

C₂³

yes

—

C₂×#11744751

C₂³

yes

—

C₂×#11744751

C₂³

yes

characteristic

C₂×#11744751

C₂³

yes

characteristic

C₂×Q₈×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₂²⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₁₃

C₂³

yes

—

C₄²×C₁₁₃

C₂³

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#11744751

C₂×Q₈

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₄×Q₈

yes

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

C₄×C₁₁₃

#131

yes

—

C₄×C₁₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₁₃

C₂×#131

yes

—

#2960

—

113

maximal, Sylow

C₁₁₃

#2960

yes

characteristic, complemented, Sylow

#533

D₁₁₃

yes

characteristic

#446

—

113

—

C₈×Q₈

—

113

—

#424

—

113

—

#423

—

113

—

#405

—

113

—

C₂×C₄×C₈

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

—

113

—

C₄⋊Q₈

Dic₁₁₃

yes

—

#125

Dic₁₁₃

yes

—

C₂²⋊Q₈

Dic₁₁₃

yes

—

C₄×Q₈

C₂×D₁₁₃

yes

—

#117

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₂×Q₈

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

100

C₂×C₈

—

226

—

101

C₄⋊C₄

C₂×Dic₁₁₃

yes

—

102

C₄⋊C₄

C₂×Dic₁₁₃

yes

—

103

C₄⋊C₄

C₂×Dic₁₁₃

yes

—

104

C₂²⋊C₄

C₂×Dic₁₁₃

yes

—

105

C₄²

C₂×Dic₁₁₃

yes

characteristic

106

C₄²

C₂×Dic₁₁₃

yes

characteristic

107

C₄²

C₂²×D₁₁₃

yes

—

108

C₄²

C₂²×D₁₁₃

yes

—

109

C₂³

C₂²×Dic₁₁₃

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×Dic₁₁₃

yes

—

113

C₂×C₄

C₂²×Dic₁₁₃

yes

—

114

C₂×C₄

C₂²×Dic₁₁₃

yes

characteristic

115

C₂×C₄

C₂²×Dic₁₁₃

yes

—

116

C₂×C₄

Q₈×D₁₁₃

yes

—

117

C₂×C₄

#14268648

yes

—

118

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

119

C₈

—

113

—

120

C₈

—

113

—

121

C₈

—

226

—

122

C₈

—

226

—

123

C₂²

—

124

C₂²

Q₈×Dic₁₁₃

yes

—

125

C₂²

C₂³×Dic₁₁₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₁₁₃

yes

—

129

C₄

#16813629

yes

—

130

C₄

C₂×Q₈×D₁₁₃

yes

characteristic

131

C₄

C₂×#14268648

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₁₁₃

yes

characteristic

134

C₂

C₂×#16813629

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#533×C₁₁₃	C₂
prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#533	D₁₁₃
hypocenter	C₁₁₃	#2960
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂³×C₄

Fitting

#533×C₁₁₃

C₂

prime radical

C₂³×C₁₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₁₃

C₂⁴

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #20298729	C₂	semidirect
2. Q₈×#11744751	C₂	semidirect
25. C₂×#11744751	Q₈	semidirect
68. C₁₁₃	#2960	semidirect

H ≅ G/N

type

1. #20298729

C₂

semidirect

2. Q₈×#11744751

C₂

semidirect

25. C₂×#11744751

Q₈

semidirect

68. C₁₁₃

#2960

semidirect

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

subgroup	index
#20298729	2
Q₈×#11744751	2
#20298625	2
#20298622	2
#20298554	2
C₂×C₄×#11744751	2
#533×C₁₁₃	2
#2960	113

subgroup

index

#533×C₁₁₃

113

Subgroups — #28399509

Conjugacy classes of subgroups

Complements

Maximal subgroups