Group #28390747 — subgroups

Subgroups — #28390747

Order 14464.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16813558	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813557	C₂	1	2	yes	maximal
3	#20298621	C₂	1	2	yes	maximal
4	C₂×#16813532	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268631	C₂²	1	2	yes	—
7	C₂²×#11744751	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
9	#16813558	—	2	2	no	—
10	#16813557	—	2	4	no	—
11	#16813551	—	2	4	no	—
12	#16813532	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	#108×C₁₁₃	—	2	2	no	—
16	C₂²×Dic₁₁₃	Q₈	1	2	yes	—
17	#14268631	—	2	2	no	—
18	#14268631	—	2	2	no	—
19	C₂×#11744751	—	2	2	no	—
20	C₂×#11744751	C₂×C₄	1	2	yes	—
21	C₂×#11744751	D₄	1	2	yes	—
22	C₂×#11744751	Q₈	1	2	yes	—
23	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₁₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₁₃	—	2	2	no	—
27	M₁₆×C₁₁₃	—	2	2	no	—
28	C₂×C₈×C₁₁₃	—	2	2	no	—
29	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
31	C₂×C₈×C₁₁₃	Q₈	1	2	yes	—
32	C₂³×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₁₁₃	—	4	2	no	—
40	C₂×Dic₁₁₃	C₄⋊C₄	1	4	yes	—
41	C₈×C₁₁₃	—	4	2	no	—
42	C₈×C₁₁₃	C₄⋊C₄	1	4	yes	—
43	#11744751	—	4	2	no	—
44	#11744751	C₄⋊C₄	1	4	yes	—
45	C₂²×C₁₁₃	—	2	2	no	—
46	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₁₃	#127	1	1	yes	characteristic
52	C₄×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₁₃	—	2	4	no	—
54	C₂×C₁₁₃	—	2	1	no	—
55	C₂×C₁₁₃	—	2	1	no	—
56	C₂×C₁₁₃	#384	1	1	yes	characteristic
57	C₂×C₁₁₃	#430	1	2	yes	—
58	#1849	—	113	1	no	maximal, Sylow
59	C₁₁₃	#1849	1	1	yes	characteristic, complemented, Sylow
60	C₂×#108	—	113	2	no	—
61	C₂×#108	—	113	1	no	—
62	C₂×#108	D₁₁₃	1	1	yes	characteristic
63	#423	—	113	2	no	—
64	C₂×#97	—	113	1	no	—
65	C₂×M₁₆	—	113	2	no	—
66	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
67	C₂²×C₈	—	113	1	no	—
68	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
69	C₂×C₄²	—	113	1	no	—
70	#108	—	2	2	no	—
71	#108	—	226	4	no	—
72	#108	—	226	2	no	—
73	C₄⋊C₈	—	226	4	no	—
74	#97	—	113	4	no	—
75	C₂²×C₄	—	113	2	no	—
76	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	226	2	no	—
80	M₁₆	—	226	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	113	2	no	—
83	C₂×C₈	—	113	2	no	—
84	C₂×C₈	—	113	2	no	—
85	C₂×C₈	—	226	2	no	—
86	C₂×C₈	Dic₂₂₆	1	2	yes	—
87	C₂×C₈	C₄×D₁₁₃	1	2	yes	—
88	C₂×C₈	C₂²⋊D₁₁₃	1	2	yes	—
89	C₄²	—	113	2	no	—
90	C₄²	—	113	2	no	—
91	C₂³	Q₈×D₁₁₃	1	1	yes	characteristic
92	C₂×C₄	—	113	4	no	—
93	C₂×C₄	—	452	2	no	—
94	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
95	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
96	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₁₁₃	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	113	4	no	—
102	C₈	—	452	2	no	—
103	C₈	#14268626	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₁₃	1	1	yes	characteristic
106	C₂²	#16813620	1	1	yes	characteristic
107	C₂²	#16813621	1	1	yes	characteristic
108	C₄	—	226	4	no	—
109	C₄	C₄⋊C₄×D₁₁₃	1	1	yes	characteristic
110	C₄	C₂×#14268626	1	1	yes	characteristic
111	C₄	#16813618	1	1	yes	characteristic
112	C₄	#16813621	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#20298467	1	1	yes	characteristic
116	C₂	#20298645	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813558

C₂

yes

maximal, characteristic

C₂×#16813557

C₂

yes

maximal

#20298621

C₂

yes

maximal

C₂×#16813532

C₂

yes

maximal, characteristic

C₂×#108×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

C₂²×#11744751

C₂²

yes

characteristic

C₂×C₄×Dic₁₁₃

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₁₁₃

C₂²

yes

—

C₂²×C₈×C₁₁₃

C₂²

yes

characteristic

#108×C₁₁₃

—

C₂²×Dic₁₁₃

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11744751

D₄

yes

—

C₂×#11744751

Q₈

yes

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₁₃

D₄

yes

—

C₂×C₈×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₈×C₁₁₃

—

C₈×C₁₁₃

C₄⋊C₄

yes

—

#11744751

—

#11744751

C₄⋊C₄

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁₃

#127

yes

characteristic

C₄×C₁₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#384

yes

characteristic

C₂×C₁₁₃

#430

yes

—

#1849

—

113

maximal, Sylow

C₁₁₃

#1849

yes

characteristic, complemented, Sylow

C₂×#108

—

113

—

C₂×#108

—

113

—

C₂×#108

D₁₁₃

yes

characteristic

#423

—

113

—

C₂×#97

—

113

—

C₂×M₁₆

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

—

C₂²×C₈

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

characteristic

C₂×C₄²

—

113

—

#108

—

#108

—

226

—

#108

—

226

—

C₄⋊C₈

—

226

—

#97

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

Dic₂₂₆

yes

—

C₂×C₈

C₄×D₁₁₃

yes

—

C₂×C₈

C₂²⋊D₁₁₃

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₂³

Q₈×D₁₁₃

yes

characteristic

C₂×C₄

—

113

—

C₂×C₄

—

452

—

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

D₄×D₁₁₃

yes

characteristic

C₂×C₄

D₄×D₁₁₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

Q₈×D₁₁₃

yes

characteristic

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

100

C₈

—

101

C₈

—

113

—

102

C₈

—

452

—

103

C₈

#14268626

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₁₃

yes

characteristic

106

C₂²

#16813620

yes

characteristic

107

C₂²

#16813621

yes

characteristic

108

C₄

—

226

—

109

C₄

C₄⋊C₄×D₁₁₃

yes

characteristic

110

C₄

C₂×#14268626

yes

characteristic

111

C₄

#16813618

yes

characteristic

112

C₄

#16813621

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#20298467

yes

characteristic

116

C₂

#20298645

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	C₂×#108×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	C₂×#108	D₁₁₃
hypocenter	C₁₁₃	#1849
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

C₂×#108×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₁₃

C₂×Q₈

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
59. C₁₁₃	#1849	semidirect

H ≅ G/N

type

59. C₁₁₃

#1849

semidirect

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

subgroup	index
C₂×#16813558	2
C₂×#16813557	2
#20298621	2
C₂×#16813532	2
C₂×#108×C₁₁₃	2
#1849	113

subgroup

index

C₂×#108×C₁₁₃

113

Subgroups — #28390747

Conjugacy classes of subgroups

Complements

Maximal subgroups