Group #28390655 — subgroups

Subgroups — #28390655

Order 14464.

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

normal subgroups only

This group has 272 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	C₂×#16813581	C₂	1	1	yes	maximal, characteristic
2	#20298562	C₂	1	1	yes	maximal, characteristic
3	#20298366	C₂	1	4	yes	maximal, complemented
4	#407×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268631	C₂²	1	2	yes	—
6	#16813581	—	2	2	no	—
7	#16813581	—	2	2	no	—
8	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₁₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₁₃	C₄	1	2	yes	—
11	C₂²⋊#11744751	—	2	2	no	—
12	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₁₁₃	—	2	2	no	—
15	C₂²×Dic₁₁₃	C₂×C₄	1	4	yes	—
16	#14268631	—	2	4	no	—
17	C₂×#11744751	—	2	2	no	—
18	C₄⋊Dic₁₁₃	—	2	2	no	—
19	C₄⋊Dic₁₁₃	—	2	2	no	—
20	C₂²⋊Dic₁₁₃	—	2	4	no	—
21	C₄×Dic₁₁₃	—	2	4	no	—
22	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₁₃	—	2	4	no	—
24	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₁₃	—	2	4	no	—
29	C₂×C₈×C₁₁₃	—	2	2	no	—
30	C₂³×C₁₁₃	—	2	2	no	—
31	C₂³×C₁₁₃	—	4	2	no	—
32	C₂³×C₁₁₃	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄×C₁₁₃	—	2	4	no	—
36	C₂×C₄×C₁₁₃	—	2	4	no	—
37	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₁₃	#41	1	4	yes	—
41	C₂×Dic₁₁₃	—	2	4	no	—
42	C₂×Dic₁₁₃	—	4	4	no	—
43	C₈×C₁₁₃	—	4	2	no	—
44	#11744751	—	4	2	no	—
45	C₂²×C₁₁₃	—	2	2	no	—
46	C₂²×C₁₁₃	—	2	2	no	—
47	C₂²×C₁₁₃	—	2	4	no	—
48	C₂²×C₁₁₃	—	4	4	no	—
49	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₁₃	#117	1	2	yes	—
51	C₄×C₁₁₃	—	2	2	no	—
52	C₄×C₁₁₃	#123	1	2	yes	—
53	C₄×C₁₁₃	#123	1	2	yes	—
54	Dic₁₁₃	—	4	4	no	—
55	C₂×C₁₁₃	—	2	2	no	—
56	C₂×C₁₁₃	—	4	2	no	—
57	C₂×C₁₁₃	#381	1	1	yes	characteristic
58	C₂×C₁₁₃	#413	1	2	yes	—
59	#1822	—	113	1	no	maximal, Sylow
60	C₁₁₃	#1822	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	113	1	no	—
62	#407	—	113	1	no	—
63	#407	D₁₁₃	1	1	yes	characteristic
64	#343	—	113	4	no	—
65	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
66	C₂×M₁₆	—	113	2	no	—
67	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
68	#117	—	226	2	no	—
69	#117	—	226	2	no	—
70	C₂×C₄⋊C₄	—	113	2	no	—
71	C₂×C₂²⋊C₄	—	113	2	no	—
72	C₂×C₄²	—	113	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	226	2	no	—
75	C₂⁴	C₄×D₁₁₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	113	4	no	—
78	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
79	C₂²×C₄	D₄₅₂	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	226	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	226	2	no	—
86	C₄⋊C₄	—	226	2	no	—
87	C₄⋊C₄	—	226	2	no	—
88	C₂²⋊C₄	—	226	4	no	—
89	C₄²	—	226	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#14268624	1	2	yes	—
93	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	452	4	no	—
98	C₂×C₄	—	226	4	no	—
99	C₂×C₄	#14268624	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
102	C₂×C₄	#14268646	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	452	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#14268624	1	1	yes	characteristic
110	C₂²	#16813578	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	452	4	no	—
113	C₄	#16813603	1	2	yes	—
114	C₄	#16813606	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#20298459	1	1	yes	characteristic
118	C₂	#20298580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813581

C₂

yes

maximal, characteristic

#20298562

C₂

yes

maximal, characteristic

#20298366

C₂

yes

maximal, complemented

#407×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

#16813581

—

#16813581

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₁₃

C₄

yes

—

C₂×C₄×Dic₁₁₃

C₄

yes

—

C₂²⋊#11744751

—

C₂³×C₄×C₁₁₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₁₃

C₂²

yes

—

C₂²⋊C₈×C₁₁₃

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

#14268631

—

C₂×#11744751

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₂⁴×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₁₃

#117

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#123

yes

—

C₄×C₁₁₃

#123

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#381

yes

characteristic

C₂×C₁₁₃

#413

yes

—

#1822

—

113

maximal, Sylow

C₁₁₃

#1822

yes

characteristic, complemented, Sylow

C₂×#117

—

113

—

#407

—

113

—

#407

D₁₁₃

yes

characteristic

#343

—

113

—

C₂³×C₄

C₂×D₁₁₃

yes

characteristic

C₂×M₁₆

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

—

#117

—

226

—

#117

—

226

—

C₂×C₄⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

113

—

C₂×C₄²

—

113

—

C₂²⋊C₈

—

C₂²⋊C₈

—

226

—

C₂⁴

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

113

—

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

D₄₅₂

yes

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

M₁₆

—

226

—

C₂×C₈

—

C₂×C₈

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₄²

—

226

—

C₂³

—

C₂³

—

C₂³

#14268624

yes

—

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

452

—

C₂×C₄

—

226

—

C₂×C₄

#14268624

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

101

C₂×C₄

C₂×D₄₅₂

yes

characteristic

102

C₂×C₄

#14268646

yes

—

103

C₈

—

104

C₈

—

452

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#14268624

yes

characteristic

110

C₂²

#16813578

yes

—

111

C₄

—

112

C₄

—

452

—

113

C₄

#16813603

yes

—

114

C₄

#16813606

yes

—

115

C₂

—

116

C₂

—

117

C₂

#20298459

yes

characteristic

118

C₂

#20298580

yes

—

	subgroup	G/H
center	C₂²	C₂×#14268624
commutator	C₂³×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#407×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#407	D₁₁₃
hypocenter	C₁₁₃	#1822
LCS	G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#14268624

commutator

C₂³×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#407×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₁₃

C₂×C₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #20298366	C₂	semidirect
60. C₁₁₃	#1822	semidirect

H ≅ G/N

type

3. #20298366

C₂

semidirect

60. C₁₁₃

#1822

semidirect

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

subgroup	index
C₂×#16813581	2
#20298562	2
#20298366	2
#407×C₁₁₃	2
#1822	113

subgroup

index

#407×C₁₁₃

113

Subgroups — #28390655

Conjugacy classes of subgroups

Complements

Maximal subgroups