Group #28389693 — subgroups

Subgroups — #28389693

Order 14464.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20299334	C₂	1	1	yes	maximal, characteristic
2	#20298890	C₂	1	2	yes	maximal, complemented
3	#20298342	C₂	1	1	yes	maximal, characteristic
4	#20298330	C₂	1	2	yes	maximal, complemented
5	#492×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#16813609	—	2	2	no	—
7	#16813603	—	2	2	no	—
8	C₄×Dic₂₂₆	C₂²	1	2	yes	—
9	D₄×Dic₁₁₃	—	2	2	no	—
10	#16813555	—	2	2	no	—
11	C₄⋊#11744751	C₂²	1	2	yes	—
12	#16813545	—	2	2	no	—
13	C₂×SD₁₆×C₁₁₃	—	2	2	no	—
14	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	#127×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₂₆	D₄	1	2	yes	—
18	C₂²×Dic₁₁₃	—	2	2	no	—
19	C₂×#11744751	—	2	2	no	—
20	C₄⋊Dic₁₁₃	D₄	1	2	yes	—
21	#14268626	—	2	2	no	—
22	C₂²⋊Dic₁₁₃	—	2	4	no	—
23	C₂²⋊Dic₁₁₃	—	2	2	no	—
24	C₄×Dic₁₁₃	—	2	2	no	—
25	C₂×Q₈×C₁₁₃	—	2	1	no	—
26	C₂×D₄×C₁₁₃	—	2	2	no	—
27	C₂×D₄×C₁₁₃	D₄	1	2	yes	—
28	SD₁₆×C₁₁₃	—	4	4	no	—
29	C₂×C₈×C₁₁₃	—	2	1	no	—
30	C₄⋊C₄×C₁₁₃	—	2	2	no	—
31	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂³×C₁₁₃	—	2	2	no	—
33	Q₈×C₁₁₃	—	4	2	no	—
34	D₄×C₁₁₃	—	4	2	no	—
35	D₄×C₁₁₃	—	4	2	no	—
36	D₄×C₁₁₃	—	4	2	no	—
37	C₂×C₄×C₁₁₃	—	4	1	no	—
38	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₁₃	—	2	2	no	—
41	C₂×Dic₁₁₃	—	2	2	no	—
42	C₂×Dic₁₁₃	—	8	2	no	—
43	Dic₂₂₆	—	2	2	no	—
44	Dic₂₂₆	—	2	2	no	—
45	C₈×C₁₁₃	—	2	2	no	—
46	#11744751	—	4	2	no	—
47	C₂²×C₁₁₃	—	4	4	no	—
48	C₂²×C₁₁₃	—	4	2	no	—
49	C₂²×C₁₁₃	#120	1	1	yes	characteristic
50	C₄×C₁₁₃	—	2	1	no	—
51	C₄×C₁₁₃	—	8	1	no	—
52	C₄×C₁₁₃	#135	1	2	yes	—
53	C₄×C₁₁₃	#137	1	2	yes	—
54	Dic₁₁₃	—	4	2	no	—
55	Dic₁₁₃	—	4	2	no	—
56	C₂×C₁₁₃	—	8	2	no	—
57	C₂×C₁₁₃	#452	1	2	yes	—
58	C₂×C₁₁₃	#453	1	1	yes	characteristic
59	#1630	—	113	1	no	maximal, Sylow
60	C₁₁₃	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	113	1	no	—
62	#492	D₁₁₃	1	1	yes	characteristic
63	#477	—	113	2	no	—
64	#331	—	113	1	no	—
65	Q₈⋊C₈	—	113	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
68	#127	C₂×D₁₁₃	1	1	yes	characteristic
69	#124	—	226	2	no	—
70	#123	—	226	2	no	—
71	C₄×Q₈	—	113	2	no	—
72	C₄×D₄	—	226	2	no	—
73	#107	—	226	2	no	—
74	C₄⋊C₈	—	113	2	no	—
75	#103	—	226	2	no	—
76	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	113	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₁₁₃	1	2	yes	—
81	C₂²×C₄	—	226	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	226	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	113	2	no	—
87	C₄⋊C₄	—	226	2	no	—
88	C₂²⋊C₄	—	226	4	no	—
89	C₂²⋊C₄	—	226	2	no	—
90	C₄²	—	226	2	no	—
91	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	226	2	no	—
95	Q₈	—	226	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	904	2	no	—
101	C₂×C₄	—	226	2	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	452	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#16813590	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	452	2	no	—
113	C₄	—	452	2	no	—
114	C₄	#16813643	1	2	yes	—
115	C₄	#16813651	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#20298757	1	2	yes	—
118	C₂	#20298765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299334

C₂

yes

maximal, characteristic

#20298890

C₂

yes

maximal, complemented

#20298342

C₂

yes

maximal, characteristic

#20298330

C₂

yes

maximal, complemented

#492×C₁₁₃

C₂

yes

maximal, characteristic

#16813609

—

#16813603

—

C₄×Dic₂₂₆

C₂²

yes

—

D₄×Dic₁₁₃

—

#16813555

—

C₄⋊#11744751

C₂²

yes

—

#16813545

—

C₂×SD₁₆×C₁₁₃

—

C₄⋊Q₈×C₁₁₃

C₂²

yes

characteristic

#127×C₁₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₁₃

C₂²

yes

characteristic

C₂×Dic₂₂₆

D₄

yes

—

C₂²×Dic₁₁₃

—

C₂×#11744751

—

C₄⋊Dic₁₁₃

D₄

yes

—

#14268626

—

C₂²⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₂×Q₈×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

D₄

yes

—

SD₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂³

yes

characteristic

C₂³×C₁₁₃

—

Q₈×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

#120

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#135

yes

—

C₄×C₁₁₃

#137

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#452

yes

—

C₂×C₁₁₃

#453

yes

characteristic

#1630

—

113

maximal, Sylow

C₁₁₃

#1630

yes

characteristic, complemented, Sylow

#559

—

113

—

#492

D₁₁₃

yes

characteristic

#477

—

113

—

#331

—

113

—

Q₈⋊C₈

—

113

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₁₁₃

yes

characteristic

#127

C₂×D₁₁₃

yes

characteristic

#124

—

226

—

#123

—

226

—

C₄×Q₈

—

113

—

C₄×D₄

—

226

—

#107

—

226

—

C₄⋊C₈

—

113

—

#103

—

226

—

C₄×C₈

C₂×D₁₁₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

113

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

—

226

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₄²

—

226

—

C₄²

C₂²×D₁₁₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

226

—

Q₈

—

226

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

904

—

101

C₂×C₄

—

226

—

102

C₂×C₄

—

226

—

103

C₂×C₄

D₄×D₁₁₃

yes

—

104

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

105

C₈

—

106

C₈

—

452

—

107

C₂²

—

108

C₂²

—

109

C₂²

#16813590

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

452

—

113

C₄

—

452

—

114

C₄

#16813643

yes

—

115

C₄

#16813651

yes

—

116

C₂

—

117

C₂

#20298757

yes

—

118

C₂

#20298765

yes

characteristic

	subgroup	G/H
center	C₂²	#16813590
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#492×C₁₁₃	C₂
socle	C₂²×C₁₁₃	#120
prime radical	#127×C₁₁₃	C₂²
hypercenter	#492	D₁₁₃
hypocenter	C₁₁₃	#1630
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#16813590

commutator, cosocle

C₄²×C₁₁₃

C₂³

Frattini

C₄²

C₂²×D₁₁₃

Fitting

#492×C₁₁₃

C₂

socle

C₂²×C₁₁₃

#120

prime radical

#127×C₁₁₃

C₂²

D₁₁₃

C₁₁₃

G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #20298890	C₂	semidirect
4. #20298330	C₂	semidirect
60. C₁₁₃	#1630	semidirect

H ≅ G/N

type

2. #20298890

C₂

semidirect

4. #20298330

C₂

semidirect

60. C₁₁₃

#1630

semidirect

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

subgroup	index
#20299334	2
#20298890	2
#20298342	2
#20298330	2
#492×C₁₁₃	2
#1630	113

subgroup

index

#492×C₁₁₃

113

Subgroups — #28389693

Conjugacy classes of subgroups

Complements

Maximal subgroups