Group #28390793 — subgroups

Subgroups — #28390793

Order 14464.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20299365	C₂	1	1	yes	maximal, characteristic
2	#20298646	C₂	1	2	yes	maximal, complemented
3	#20298582	C₂	1	1	yes	maximal, characteristic
4	#20298363	C₂	1	2	yes	maximal, complemented
5	#408×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#16813629	—	2	4	no	—
7	C₂×#14268631	—	2	2	no	—
8	C₂×#14268631	C₂²	1	2	yes	—
9	C₂×#14268631	C₂²	1	2	yes	—
10	#16813557	—	2	4	no	—
11	#16813541	C₄	1	2	yes	—
12	#16813539	C₄	1	2	yes	—
13	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₁₁₃	—	2	2	no	—
17	#14268631	—	2	4	no	—
18	#14268631	—	2	2	no	—
19	#14268631	—	2	2	no	—
20	#14268631	C₂×C₄	1	4	yes	—
21	#14268631	D₄	1	4	yes	—
22	C₂×#11744751	—	2	4	no	—
23	C₂×#11744751	—	2	2	no	—
24	C₂×#11744751	—	2	2	no	—
25	#41×C₁₁₃	—	2	2	no	—
26	#41×C₁₁₃	—	2	2	no	—
27	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₁₁₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
31	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₁₃	—	2	2	no	—
33	C₂×C₈×C₁₁₃	—	2	1	no	—
34	C₂×C₈×C₁₁₃	—	2	1	no	—
35	C₂×C₈×C₁₁₃	—	2	2	no	—
36	C₂³×C₁₁₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₁₃	—	2	2	no	—
39	D₄×C₁₁₃	—	2	4	no	—
40	D₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄×C₁₁₃	—	2	2	no	—
42	C₂×C₄×C₁₁₃	—	2	2	no	—
43	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁₃	#41	1	2	yes	—
48	C₈×C₁₁₃	—	4	1	no	—
49	C₈×C₁₁₃	—	4	1	no	—
50	#11744751	—	2	4	no	—
51	#11744751	—	2	4	no	—
52	C₂²×C₁₁₃	—	2	2	no	—
53	C₂²×C₁₁₃	—	4	1	no	—
54	C₂²×C₁₁₃	—	4	2	no	—
55	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₁₃	—	2	2	no	—
58	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
59	C₄×C₁₁₃	#120	1	1	yes	characteristic
60	C₄×C₁₁₃	#123	1	1	yes	characteristic
61	C₂×C₁₁₃	—	2	1	no	—
62	C₂×C₁₁₃	—	2	2	no	—
63	C₂×C₁₁₃	—	4	2	no	—
64	C₂×C₁₁₃	#385	1	1	yes	characteristic
65	#1858	—	113	1	no	maximal, Sylow
66	C₁₁₃	#1858	1	1	yes	characteristic, complemented, Sylow
67	#568	—	113	1	no	—
68	#430	—	113	2	no	—
69	#413	—	113	1	no	—
70	#408	D₁₁₃	1	1	yes	characteristic
71	#341	—	113	2	no	—
72	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
73	#131	—	226	4	no	—
74	C₂×M₁₆	—	113	2	no	—
75	C₂×M₁₆	—	113	2	no	—
76	C₂×M₁₆	—	226	2	no	—
77	C₂×M₁₆	C₂×D₁₁₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
79	#108	—	226	4	no	—
80	#101	—	113	2	no	—
81	#100	—	113	2	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₁₃	1	1	yes	characteristic
86	C₂×D₄	Dic₂₂₆	1	2	yes	—
87	C₂×D₄	C₄×D₁₁₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	113	4	no	—
92	M₁₆	—	113	4	no	—
93	M₁₆	—	226	4	no	—
94	M₁₆	—	226	2	no	—
95	M₁₆	—	226	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	226	4	no	—
100	C₂×C₈	—	226	2	no	—
101	C₂×C₈	—	226	2	no	—
102	C₂³	#14268626	1	2	yes	—
103	C₂³	C₂×Dic₂₂₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#14268626	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
113	C₂×C₄	#14268646	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	226	4	no	—
117	C₈	—	226	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#14268626	1	1	yes	characteristic
122	C₂²	#16813599	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	#16813582	1	2	yes	—
125	C₄	#16813590	1	1	yes	characteristic
126	C₄	#16813603	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#20298470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299365

C₂

yes

maximal, characteristic

#20298646

C₂

yes

maximal, complemented

#20298582

C₂

yes

maximal, characteristic

#20298363

C₂

yes

maximal, complemented

#408×C₁₁₃

C₂

yes

maximal, characteristic

#16813629

—

C₂×#14268631

—

C₂×#14268631

C₂²

yes

—

C₂×#14268631

C₂²

yes

—

#16813557

—

#16813541

C₄

yes

—

#16813539

C₄

yes

—

C₂×#41×C₁₁₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₈×C₁₁₃

—

—

—

—

C₂×C₄

yes

—

#14268631

D₄

yes

—

C₂×#11744751

—

C₂×#11744751

—

C₂×#11744751

—

#41×C₁₁₃

—

#41×C₁₁₃

—

C₂×Q₈×C₁₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₁₃

Q₈

yes

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂³×C₁₁₃

C₄⋊C₄

yes

—

C₂³×C₁₁₃

C₂×Q₈

yes

characteristic

Q₈×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₈×C₁₁₃

—

C₈×C₁₁₃

—

#11744751

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×D₄

yes

—

C₄×C₁₁₃

yes

C₄×C₁₁₃

yes

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#385

yes

characteristic

#1858

—

113

maximal, Sylow

C₁₁₃

#1858

yes

characteristic, complemented, Sylow

#568

—

113

—

#430

—

113

—

#413

—

113

—

#408

D₁₁₃

yes

characteristic

#341

—

113

—

C₂×#41

C₂×D₁₁₃

yes

characteristic

#131

—

226

—

C₂×M₁₆

—

113

—

C₂×M₁₆

—

113

—

C₂×M₁₆

—

226

—

C₂×M₁₆

C₂×D₁₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₁₃

yes

characteristic

#108

—

226

—

#101

—

113

—

#100

—

113

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₁₃

yes

characteristic

C₂×D₄

Dic₂₂₆

yes

—

C₂×D₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

M₁₆

—

113

—

M₁₆

—

113

—

M₁₆

—

226

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

226

—

100

C₂×C₈

—

226

—

101

C₂×C₈

—

226

—

102

C₂³

#14268626

yes

—

103

C₂³

C₂×Dic₂₂₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#14268626

yes

—

110

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

111

C₂×C₄

D₄×D₁₁₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

113

C₂×C₄

#14268646

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

226

—

117

C₈

—

226

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#14268626

yes

characteristic

122

C₂²

#16813599

yes

—

123

C₄

—

124

C₄

#16813582

yes

—

125

C₄

#16813590

yes

characteristic

126

C₄

#16813603

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20298470

yes

characteristic

	subgroup	G/H
center	C₄	#16813590
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#408×C₁₁₃	C₂
socle	C₂×C₁₁₃	#385
prime radical	C₂×#41×C₁₁₃	C₂²
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#408	D₁₁₃
hypocenter	C₁₁₃	#1858
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#16813590

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#408×C₁₁₃

C₂

socle

C₂×C₁₁₃

#385

prime radical

C₂×#41×C₁₁₃

C₂²

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #20298646	C₂	semidirect
4. #20298363	C₂	semidirect
66. C₁₁₃	#1858	semidirect

H ≅ G/N

type

2. #20298646

C₂

semidirect

4. #20298363

C₂

semidirect

66. C₁₁₃

#1858

semidirect

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

subgroup	index
#20299365	2
#20298646	2
#20298582	2
#20298363	2
#408×C₁₁₃	2
#1858	113

subgroup

index

#408×C₁₁₃

113

Subgroups — #28390793

Conjugacy classes of subgroups

Complements

Maximal subgroups