Group #28390740 — subgroups

Subgroups — #28390740

Order 14464.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16813555	C₂	1	2	yes	maximal
2	C₂×#16813553	C₂	1	2	yes	maximal
3	C₂×#16813533	C₂	1	1	yes	maximal, characteristic
4	#20298464	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11744751	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
9	#16813555	—	2	4	no	—
10	#16813553	—	2	4	no	—
11	#16813533	C₄	1	4	yes	complemented
12	#16813529	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₁₃	—	2	1	no	—
17	C₂²×Dic₁₁₃	—	2	1	no	—
18	C₂×#11744751	C₂×C₄	1	4	yes	—
19	C₂×#11744751	D₄	1	4	yes	—
20	C₂×#11744751	Q₈	1	4	yes	—
21	C₄⋊Dic₁₁₃	—	2	2	no	—
22	C₄⋊Dic₁₁₃	—	2	2	no	—
23	C₂²×C₄×C₁₁₃	—	2	2	no	—
24	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₁₃	—	2	2	no	—
27	C₄⋊C₄×C₁₁₃	—	2	2	no	—
28	C₄⋊C₄×C₁₁₃	—	4	4	no	—
29	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
30	C₄²×C₁₁₃	Q₈	1	2	yes	—
31	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	—	2	2	no	—
33	C₂×C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄×C₁₁₃	—	4	2	no	—
35	C₂×C₄×C₁₁₃	—	4	4	no	—
36	C₂×C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₁₃	—	4	1	no	—
42	C₂×Dic₁₁₃	—	4	1	no	—
43	C₂×Dic₁₁₃	—	4	2	no	—
44	C₂×Dic₁₁₃	—	4	2	no	—
45	#11744751	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₁₃	#127	1	1	yes	characteristic
48	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₁₃	#137	1	2	yes	—
51	C₄×C₁₁₃	—	2	4	no	—
52	C₄×C₁₁₃	—	8	2	no	—
53	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
55	Dic₁₁₃	—	8	1	no	—
56	Dic₁₁₃	—	8	1	no	—
57	C₂×C₁₁₃	#384	1	1	yes	characteristic
58	C₂×C₁₁₃	#428	1	2	yes	—
59	C₂×C₁₁₃	#496	1	1	yes	characteristic
60	C₂×C₁₁₃	#497	1	1	yes	characteristic
61	C₂×C₁₁₃	#501	1	2	yes	—
62	#1847	—	113	1	no	maximal, Sylow
63	C₁₁₃	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	113	2	no	—
65	C₂×#106	—	113	2	no	—
66	C₂×#97	—	113	1	no	—
67	#384	D₁₁₃	1	1	yes	characteristic
68	#382	—	113	1	no	—
69	C₂²×C₈	—	113	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	113	1	no	—
72	C₂×C₄⋊C₄	—	113	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
74	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
75	#107	—	226	4	no	—
76	#106	—	226	4	no	—
77	#97	—	113	4	no	—
78	#95	—	226	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	226	1	no	—
81	C₂²×C₄	—	226	1	no	—
82	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
84	C₂×C₈	—	113	4	no	—
85	C₂×C₈	—	113	4	no	—
86	C₂×C₈	—	113	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	226	2	no	—
91	C₄⋊C₄	—	226	2	no	—
92	C₄²	Dic₂₂₆	1	2	yes	—
93	C₄²	C₄×D₁₁₃	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	452	1	no	—
100	C₂×C₄	—	452	1	no	—
101	C₂×C₄	—	452	2	no	—
102	C₂×C₄	—	452	2	no	—
103	C₂×C₄	#14268626	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
106	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
108	C₈	—	113	8	no	—
109	C₂²	C₂×#14268626	1	1	yes	characteristic
110	C₂²	#16813618	1	1	yes	characteristic
111	C₂²	#16813620	1	1	yes	characteristic
112	C₂²	#16813648	1	2	yes	—
113	C₂²	#16813653	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	904	1	no	—
117	C₄	—	904	1	no	—
118	C₄	C₄⋊C₄×D₁₁₃	1	2	yes	—
119	C₄	#16813621	1	2	yes	—
120	C₂	#20298467	1	1	yes	characteristic
121	C₂	#20298638	1	2	yes	—
122	C₂	#20298985	1	1	yes	characteristic
123	C₂	#20298990	1	1	yes	characteristic
124	C₂	#20299004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813555

C₂

yes

maximal

C₂×#16813553

C₂

yes

maximal

C₂×#16813533

C₂

yes

maximal, characteristic

#20298464

C₂

yes

maximal, characteristic

#384×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

—

C₄

yes

complemented

#16813529

—

C₂×C₄⋊C₄×C₁₁₃

—

C₂×C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₂×#11744751

C₂×C₄

yes

—

C₂×#11744751

D₄

yes

—

C₂×#11744751

Q₈

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×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×Q₈

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

#11744751

C₄⋊C₄

yes

complemented

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

#127

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₁₃

#137

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₁₃

C₄⋊Q₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

#384

yes

characteristic

C₂×C₁₁₃

#428

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

#1847

—

113

maximal, Sylow

C₁₁₃

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

113

—

C₂×#106

—

113

—

C₂×#97

—

113

—

#384

D₁₁₃

yes

characteristic

#382

—

113

—

C₂²×C₈

—

113

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#107

—

226

—

#106

—

226

—

#97

—

113

—

#95

—

226

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄²

Dic₂₂₆

yes

—

C₄²

C₄×D₁₁₃

yes

—

C₂³

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

452

—

100

C₂×C₄

—

452

—

101

C₂×C₄

—

452

—

102

C₂×C₄

—

452

—

103

C₂×C₄

#14268626

yes

—

104

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

105

C₂×C₄

D₄×D₁₁₃

yes

—

106

C₂×C₄

Q₈×D₁₁₃

yes

—

107

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

108

C₈

—

113

—

109

C₂²

C₂×#14268626

yes

characteristic

110

C₂²

#16813618

yes

characteristic

111

C₂²

#16813620

yes

characteristic

112

C₂²

#16813648

yes

—

113

C₂²

#16813653

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

904

—

117

C₄

—

904

—

118

C₄

C₄⋊C₄×D₁₁₃

yes

—

119

C₄

#16813621

yes

—

120

C₂

#20298467

yes

characteristic

121

C₂

#20298638

yes

—

122

C₂

#20298985

yes

characteristic

123

C₂

#20298990

yes

characteristic

124

C₂

#20299004

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#384×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#384	D₁₁₃
hypocenter	C₁₁₃	#1847
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#384×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #16813533	C₄	semidirect
45. #11744751	C₄⋊C₄	semidirect
63. C₁₁₃	#1847	semidirect

H ≅ G/N

type

11. #16813533

C₄

semidirect

45. #11744751

C₄⋊C₄

semidirect

63. C₁₁₃

#1847

semidirect

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

subgroup	index
C₂×#16813555	2
C₂×#16813553	2
C₂×#16813533	2
#20298464	2
#384×C₁₁₃	2
#1847	113

subgroup

index

#384×C₁₁₃

113

Subgroups — #28390740

Conjugacy classes of subgroups

Complements

Maximal subgroups