Group #28390251 — subgroups

Subgroups — #28390251

Order 14464.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20299003	C₂	1	1	yes	maximal, characteristic
2	#20299003	C₂	1	1	yes	maximal, characteristic
3	#20298996	C₂	1	1	yes	maximal, characteristic
4	#20298978	C₂	1	1	yes	maximal, characteristic
5	#20298949	C₂	1	1	yes	maximal, characteristic
6	#20298949	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₅₂	—	2	2	no	—
9	#16813622	C₂²	1	1	yes	characteristic
10	#16813622	C₂²	1	1	yes	characteristic
11	#16813622	C₂²	1	1	yes	characteristic
12	#16813622	C₂²	1	1	yes	characteristic
13	#16813556	—	2	2	no	—
14	#16813556	—	2	2	no	—
15	#16813554	—	2	2	no	—
16	#16813547	—	2	2	no	—
17	#16813547	—	2	2	no	—
18	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₂₆	—	2	1	no	—
22	C₂×Dic₂₂₆	—	2	1	no	—
23	C₂×Dic₂₂₆	—	2	1	no	—
24	C₂×Dic₂₂₆	—	2	1	no	—
25	Dic₄₅₂	—	4	4	no	—
26	C₄⋊Dic₁₁₃	—	2	1	no	—
27	C₄⋊Dic₁₁₃	—	2	1	no	—
28	C₄⋊Dic₁₁₃	—	2	1	no	—
29	C₄⋊Dic₁₁₃	—	2	1	no	—
30	C₄⋊Dic₁₁₃	—	2	1	no	—
31	C₄⋊Dic₁₁₃	—	2	1	no	—
32	C₄⋊Dic₁₁₃	—	2	1	no	—
33	C₄⋊Dic₁₁₃	—	2	1	no	—
34	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
35	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
36	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
37	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₁₃	—	4	1	no	—
42	C₂×Dic₁₁₃	—	4	1	no	—
43	C₂×Dic₁₁₃	—	4	1	no	—
44	C₂×Dic₁₁₃	—	4	1	no	—
45	Dic₂₂₆	—	4	2	no	—
46	Dic₂₂₆	—	4	2	no	—
47	Dic₂₂₆	—	4	2	no	—
48	Dic₂₂₆	—	4	2	no	—
49	C₈×C₁₁₃	—	2	2	no	—
50	C₈×C₁₁₃	—	2	2	no	—
51	C₈×C₁₁₃	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₁₃	#127	1	1	yes	characteristic
53	C₄×C₁₁₃	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₁₃	#137	1	2	yes	—
56	Dic₁₁₃	—	8	1	no	—
57	Dic₁₁₃	—	8	1	no	—
58	Dic₁₁₃	—	8	1	no	—
59	Dic₁₁₃	—	8	1	no	—
60	C₂×C₁₁₃	#494	1	1	yes	characteristic
61	C₂×C₁₁₃	#496	1	1	yes	characteristic
62	C₂×C₁₁₃	#497	1	1	yes	characteristic
63	#1723	—	113	1	no	maximal, Sylow
64	C₁₁₃	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	113	1	no	—
66	#500	—	113	1	no	—
67	#498	—	113	1	no	—
68	#494	—	113	1	no	—
69	#487	—	113	1	no	—
70	#487	—	113	1	no	—
71	#322	D₁₁₃	1	1	yes	characteristic
72	C₂×Q₁₆	—	226	2	no	—
73	C₄⋊Q₈	—	113	1	no	—
74	C₄⋊Q₈	—	113	1	no	—
75	C₄⋊Q₈	—	113	1	no	—
76	C₄⋊Q₈	—	113	1	no	—
77	#107	—	226	2	no	—
78	#107	—	226	2	no	—
79	#106	—	226	2	no	—
80	#103	—	226	2	no	—
81	#103	—	226	2	no	—
82	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
85	C₂×Q₈	—	226	1	no	—
86	C₂×Q₈	—	226	1	no	—
87	C₂×Q₈	—	226	1	no	—
88	C₂×Q₈	—	226	1	no	—
89	Q₁₆	—	452	4	no	—
90	C₂×C₈	D₄₅₂	1	2	yes	—
91	C₂×C₈	D₄₅₂	1	2	yes	—
92	C₂×C₈	D₄₅₂	1	2	yes	—
93	C₄⋊C₄	—	226	1	no	—
94	C₄⋊C₄	—	226	1	no	—
95	C₄⋊C₄	—	226	1	no	—
96	C₄⋊C₄	—	226	1	no	—
97	C₄⋊C₄	—	226	1	no	—
98	C₄⋊C₄	—	226	1	no	—
99	C₄⋊C₄	—	226	1	no	—
100	C₄⋊C₄	—	226	1	no	—
101	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
102	Q₈	—	452	2	no	—
103	Q₈	—	452	2	no	—
104	Q₈	—	452	2	no	—
105	Q₈	—	452	2	no	—
106	C₂×C₄	—	452	1	no	—
107	C₂×C₄	—	452	1	no	—
108	C₂×C₄	—	452	1	no	—
109	C₂×C₄	—	452	1	no	—
110	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₄₅₂	1	4	yes	complemented
116	C₂²	#16813619	1	1	yes	characteristic
117	C₄	—	904	1	no	—
118	C₄	—	904	1	no	—
119	C₄	—	904	1	no	—
120	C₄	—	904	1	no	—
121	C₄	C₂×Dic₄₅₂	1	2	yes	—
122	C₄	#16813650	1	2	yes	—
123	C₄	#16813655	1	2	yes	—
124	C₂	#20298978	1	1	yes	characteristic
125	C₂	#20298987	1	1	yes	characteristic
126	C₂	#20298992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299003

C₂

yes

maximal, characteristic

#20299003

C₂

yes

maximal, characteristic

#20298996

C₂

yes

maximal, characteristic

#20298978

C₂

yes

maximal, characteristic

#20298949

C₂

yes

maximal, characteristic

#20298949

C₂

yes

maximal, characteristic

#322×C₁₁₃

C₂

yes

maximal, characteristic

C₂×Dic₄₅₂

—

#16813622

C₂²

yes

characteristic

#16813622

C₂²

yes

characteristic

#16813622

C₂²

yes

characteristic

#16813622

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₁₃

C₂²

yes

characteristic

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

Dic₄₅₂

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂×C₈×C₁₁₃

D₄

yes

—

C₂×C₈×C₁₁₃

D₄

yes

—

C₂×C₈×C₁₁₃

D₄

yes

—

C₄²×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

—

Dic₂₂₆

—

Dic₂₂₆

—

C₈×C₁₁₃

—

C₈×C₁₁₃

—

C₈×C₁₁₃

Q₁₆

yes

complemented

C₂²×C₁₁₃

#127

yes

characteristic

C₄×C₁₁₃

C₂×Q₁₆

yes

—

C₄×C₁₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₁₃

#137

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

#1723

yes

characteristic, complemented, Sylow

#500

—

113

—

#500

—

113

—

#498

—

113

—

#494

—

113

—

#487

—

113

—

#487

—

113

—

#322

D₁₁₃

yes

characteristic

C₂×Q₁₆

—

226

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

#107

—

226

—

#107

—

226

—

#106

—

226

—

#103

—

226

—

#103

—

226

—

C₄×C₈

C₂×D₁₁₃

yes

characteristic

C₄×C₈

C₂×D₁₁₃

yes

characteristic

C₄×C₈

C₂×D₁₁₃

yes

characteristic

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

Q₁₆

—

452

—

C₂×C₈

D₄₅₂

yes

—

C₂×C₈

D₄₅₂

yes

—

C₂×C₈

D₄₅₂

yes

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

100

C₄⋊C₄

—

226

—

101

C₄²

C₂²×D₁₁₃

yes

characteristic

102

Q₈

—

452

—

103

Q₈

—

452

—

104

Q₈

—

452

—

105

Q₈

—

452

—

106

C₂×C₄

—

452

—

107

C₂×C₄

—

452

—

108

C₂×C₄

—

452

—

109

C₂×C₄

—

452

—

110

C₂×C₄

C₂×D₄₅₂

yes

characteristic

111

C₂×C₄

C₂×D₄₅₂

yes

characteristic

112

C₂×C₄

C₂×D₄₅₂

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₄₅₂

yes

complemented

116

C₂²

#16813619

yes

characteristic

117

C₄

—

904

—

118

C₄

—

904

—

119

C₄

—

904

—

120

C₄

—

904

—

121

C₄

C₂×Dic₄₅₂

yes

—

122

C₄

#16813650

yes

—

123

C₄

#16813655

yes

—

124

C₂

#20298978

yes

characteristic

125

C₂

#20298987

yes

characteristic

126

C₂

#20298992

yes

characteristic

	subgroup	G/H
center	C₂²	#16813619
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#322×C₁₁₃	C₂
socle, prime radical	C₂²×C₁₁₃	#127
hypercenter	#322	D₁₁₃
hypocenter	C₁₁₃	#1723
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16813619

commutator, cosocle

C₄²×C₁₁₃

C₂³

Frattini

C₄²

C₂²×D₁₁₃

Fitting

#322×C₁₁₃

C₂

C₂²×C₁₁₃

D₁₁₃

C₁₁₃

G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₁₃	Q₁₆	semidirect
64. C₁₁₃	#1723	semidirect
115. C₈	Dic₄₅₂	semidirect

H ≅ G/N

type

51. C₈×C₁₁₃

Q₁₆

semidirect

64. C₁₁₃

#1723

semidirect

115. C₈

Dic₄₅₂

semidirect

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

subgroup	index
#20299003	2
#20298996	2
#20298978	2
#20298949	2
#322×C₁₁₃	2
#1723	113

subgroup

index

#322×C₁₁₃

113

Subgroups — #28390251

Conjugacy classes of subgroups

Complements

Maximal subgroups