Group #28406786 — subgroups

Subgroups — #28406786

Order 14464.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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×#16813695	C₂	1	1	yes	maximal, characteristic
2	#20299561	C₂	1	1	yes	maximal, characteristic
3	#20299485	C₂	1	2	yes	maximal, complemented
4	C₁₆×#90673	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₁₁₃	C₂	1	1	yes	maximal, characteristic
6	D₁₁₃⋊M₁₆	C₄	1	1	yes	characteristic
7	#16813701	C₄	1	1	yes	characteristic
8	#16813695	C₄	1	2	yes	complemented
9	#16813695	C₂²	1	1	yes	characteristic
10	#16813695	C₂²	1	1	yes	characteristic
11	C₂×#14268650	C₄	1	2	yes	complemented
12	#16813676	C₄	1	2	yes	—
13	C₈×#90673	C₂²	1	2	yes	—
14	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
15	#16813562	C₄	1	1	yes	characteristic, complemented
16	#16813561	C₄	1	2	yes	complemented
17	C₁₆×D₁₁₃	C₂²	1	2	yes	—
18	M₃₂×C₁₁₃	C₄	1	1	yes	characteristic, complemented
19	#14268657	C₈	1	2	yes	—
20	#14268656	C₈	1	2	yes	—
21	#14268655	C₂×C₄	1	2	yes	—
22	C₄⋊#90673	C₈	1	2	yes	—
23	C₂²⋊#90673	C₈	1	2	yes	—
24	C₄×#90673	C₂×C₄	1	2	yes	—
25	#14268650	C₂×C₄	1	2	yes	—
26	#14268650	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11744751	C₂×C₄	1	1	yes	characteristic
29	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
31	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
33	#14268622	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₁₆×C₁₁₃	C₂×C₄	1	2	yes	—
36	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂×#90673	C₂×C₈	1	4	yes	—
38	C₂×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
39	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
40	C₄×D₁₁₃	C₄²	1	1	yes	characteristic
41	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
42	C₄×D₁₁₃	C₂×C₈	1	1	yes	characteristic
43	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
44	#11744753	C₂×C₈	1	4	yes	—
45	C₈×C₁₁₃	C₄²	1	1	yes	characteristic
46	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
47	#11744751	C₄²	1	1	yes	characteristic
48	#11744751	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₁₁₃	C₄×C₈	1	1	yes	characteristic
50	C₂×D₁₁₃	—	2	2	no	—
51	C₂×D₁₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×D₁₁₃	C₂²×C₈	1	1	yes	characteristic
53	#90673	—	2	4	no	—
54	C₄×C₁₁₃	C₄×C₈	1	1	yes	characteristic
55	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
56	Dic₁₁₃	C₄×C₈	1	1	yes	characteristic
57	Dic₁₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₁₁₃	—	2	1	no	—
59	C₂×C₁₁₃	C₂×C₄×C₈	1	1	yes	characteristic
60	D₁₁₃	—	2	1	no	—
61	D₁₁₃	#504	1	2	yes	—
62	#2108	—	113	1	no	maximal, Sylow
63	C₁₁₃	#2108	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	113	1	no	—
65	C₂²×C₁₆	—	113	1	no	—
66	#405	—	113	1	no	—
67	#346	—	113	2	no	—
68	C₄×C₁₆	—	113	2	no	—
69	C₂×M₁₆	—	113	1	no	—
70	C₂²×C₈	—	113	1	no	—
71	#117	—	113	1	no	—
72	M₃₂	—	113	1	no	—
73	M₃₂	—	113	2	no	—
74	M₃₂	#90673	1	1	yes	characteristic, complemented
75	C₂×C₁₆	—	113	1	no	—
76	C₂×C₁₆	—	113	1	no	—
77	C₂×C₁₆	—	113	2	no	—
78	C₂×C₁₆	—	113	2	no	—
79	C₂×C₁₆	—	113	2	no	—
80	#97	—	113	2	no	—
81	C₄×C₈	—	113	2	no	—
82	C₂²×C₄	—	113	1	no	—
83	M₁₆	—	113	2	no	—
84	M₁₆	—	113	2	no	—
85	C₂×C₈	—	113	2	no	—
86	C₂×C₈	—	113	1	no	—
87	C₂×C₈	—	113	1	no	—
88	C₂×C₈	—	113	1	no	—
89	C₂×C₈	—	113	1	no	—
90	C₂×C₈	—	113	1	no	—
91	C₂×C₈	C₂×#90673	1	1	yes	characteristic
92	C₄⋊C₄	—	113	2	no	—
93	C₂²⋊C₄	—	113	2	no	—
94	C₄²	—	113	2	no	—
95	C₁₆	—	113	2	no	—
96	C₁₆	—	113	2	no	—
97	C₁₆	—	113	2	no	—
98	C₁₆	C₂×#90673	1	2	yes	—
99	C₂³	—	113	1	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	113	1	no	—
103	C₂×C₄	—	113	1	no	—
104	C₂×C₄	—	113	1	no	—
105	C₂×C₄	—	113	4	no	—
106	C₂×C₄	C₄×#90673	1	1	yes	characteristic
107	C₈	—	113	4	no	—
108	C₈	—	113	1	no	—
109	C₈	—	113	1	no	—
110	C₈	C₄×#90673	1	1	yes	characteristic
111	C₈	C₂²×#90673	1	1	yes	characteristic
112	C₂²	—	113	1	no	—
113	C₂²	—	113	1	no	—
114	C₂²	—	226	2	no	—
115	C₂²	C₈×#90673	1	1	yes	characteristic
116	C₄	—	113	1	no	—
117	C₄	—	113	1	no	—
118	C₄	—	226	4	no	—
119	C₄	C₈×#90673	1	1	yes	characteristic
120	C₄	C₂×C₄×#90673	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	113	2	no	—
123	C₂	—	226	1	no	—
124	C₂	C₂×C₈×#90673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813695

C₂

yes

maximal, characteristic

#20299561

C₂

yes

maximal, characteristic

#20299485

C₂

yes

maximal, complemented

C₁₆×#90673

C₂

yes

maximal, complemented

M₃₂×D₁₁₃

C₂

yes

maximal, characteristic

D₁₁₃⋊M₁₆

C₄

yes

characteristic

#16813701

C₄

yes

characteristic

#16813695

C₄

yes

complemented

#16813695

C₂²

yes

characteristic

#16813695

C₂²

yes

characteristic

C₂×#14268650

C₄

yes

complemented

#16813676

C₄

yes

—

C₈×#90673

C₂²

yes

—

C₂×C₈×D₁₁₃

C₂²

yes

characteristic

#16813562

C₄

yes

characteristic, complemented

#16813561

C₄

yes

complemented

C₁₆×D₁₁₃

C₂²

yes

—

M₃₂×C₁₁₃

C₄

yes

characteristic, complemented

#14268657

C₈

yes

—

#14268656

C₈

yes

—

#14268655

C₂×C₄

yes

—

C₄⋊#90673

C₈

yes

—

C₂²⋊#90673

C₈

yes

—

C₄×#90673

C₂×C₄

yes

—

#14268650

C₂×C₄

yes

—

#14268650

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

characteristic

C₂×#11744751

C₂×C₄

yes

characteristic

C₈×D₁₁₃

C₂×C₄

yes

characteristic

C₈×D₁₁₃

C₂×C₄

yes

characteristic

C₈×D₁₁₃

C₂×C₄

yes

characteristic

C₈×D₁₁₃

C₂³

yes

characteristic

#14268622

C₂×C₄

yes

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

characteristic

C₁₆×C₁₁₃

C₂×C₄

yes

—

C₂²×D₁₁₃

C₂×C₈

yes

characteristic

C₂×#90673

C₂×C₈

yes

—

C₂×C₄×C₁₁₃

C₄²

yes

characteristic

C₂×Dic₁₁₃

C₂×C₈

yes

characteristic

C₄×D₁₁₃

C₄²

yes

characteristic

C₄×D₁₁₃

C₂×C₈

yes

characteristic

C₄×D₁₁₃

C₂×C₈

yes

characteristic

C₄×D₁₁₃

C₂²×C₄

yes

characteristic

#11744753

C₂×C₈

yes

—

C₈×C₁₁₃

C₄²

yes

characteristic

C₈×C₁₁₃

C₂²×C₄

yes

characteristic

#11744751

C₄²

yes

characteristic

#11744751

C₂²×C₄

yes

characteristic

C₂²×C₁₁₃

C₄×C₈

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

C₄×C₈

yes

characteristic

C₂×D₁₁₃

C₂²×C₈

yes

characteristic

#90673

—

C₄×C₁₁₃

C₄×C₈

yes

characteristic

C₄×C₁₁₃

C₂×C₄²

yes

characteristic

Dic₁₁₃

C₄×C₈

yes

characteristic

Dic₁₁₃

C₂²×C₈

yes

characteristic

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×C₄×C₈

yes

characteristic

D₁₁₃

—

D₁₁₃

#504

yes

—

#2108

—

113

maximal, Sylow

C₁₁₃

#2108

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

113

—

C₂²×C₁₆

—

113

—

#405

—

113

—

#346

—

113

—

C₄×C₁₆

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

#117

—

113

—

M₃₂

—

113

—

M₃₂

—

113

—

M₃₂

#90673

yes

characteristic, complemented

C₂×C₁₆

—

113

—

C₂×C₁₆

—

113

—

C₂×C₁₆

—

113

—

C₂×C₁₆

—

113

—

C₂×C₁₆

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₂²×C₄

—

113

—

M₁₆

—

113

—

M₁₆

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

C₂×#90673

yes

characteristic

C₄⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₄²

—

113

—

C₁₆

—

113

—

C₁₆

—

113

—

C₁₆

—

113

—

C₁₆

C₂×#90673

yes

—

C₂³

—

113

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

113

—

105

C₂×C₄

—

113

—

106

C₂×C₄

C₄×#90673

yes

characteristic

107

C₈

—

113

—

108

C₈

—

113

—

109

C₈

—

113

—

110

C₈

C₄×#90673

yes

characteristic

111

C₈

C₂²×#90673

yes

characteristic

112

C₂²

—

113

—

113

C₂²

—

113

—

114

C₂²

—

226

—

115

C₂²

C₈×#90673

yes

characteristic

116

C₄

—

113

—

117

C₄

—

113

—

118

C₄

—

226

—

119

C₄

C₈×#90673

yes

characteristic

120

C₄

C₂×C₄×#90673

yes

characteristic

121

C₂

—

122

C₂

—

113

—

123

C₂

—

226

—

124

C₂

C₂×C₈×#90673

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#90673
commutator, socle	C₂×C₁₁₃	C₂×C₄×C₈
Fitting	M₃₂×C₁₁₃	C₄
prime radical	C₂²×D₁₁₃	C₂×C₈
cosocle	C₈×D₁₁₃	C₂³
hypercenter	M₃₂	#90673
hypocenter	C₁₁₃	#2108
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×#90673

commutator, socle

C₂×C₁₁₃

C₂×C₄×C₈

Fitting

M₃₂×C₁₁₃

C₄

prime radical

C₂²×D₁₁₃

C₂×C₈

cosocle

C₈×D₁₁₃

C₂³

M₃₂

C₁₁₃

G ▷ C₂×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #20299485	C₂	semidirect
4. C₁₆×#90673	C₂	semidirect
8. #16813695	C₄	semidirect
11. C₂×#14268650	C₄	semidirect
15. #16813562	C₄	semidirect
16. #16813561	C₄	semidirect
18. M₃₂×C₁₁₃	C₄	semidirect
63. C₁₁₃	#2108	semidirect
74. M₃₂	#90673	semidirect

H ≅ G/N

type

3. #20299485

C₂

semidirect

4. C₁₆×#90673

C₂

semidirect

8. #16813695

C₄

semidirect

11. C₂×#14268650

C₄

semidirect

15. #16813562

C₄

semidirect

16. #16813561

C₄

semidirect

18. M₃₂×C₁₁₃

C₄

semidirect

63. C₁₁₃

#2108

semidirect

74. M₃₂

#90673

semidirect

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

subgroup	index
C₂×#16813695	2
#20299561	2
#20299485	2
C₁₆×#90673	2
M₃₂×D₁₁₃	2
#2108	113

subgroup

index

C₂×#16813695

#20299561

#20299485

C₁₆×#90673

M₃₂×D₁₁₃

#2108

113

Subgroups — #28406786

Conjugacy classes of subgroups

Complements

Maximal subgroups