Group #28402378 — subgroups

Subgroups — #28402378

Order 14464.

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

normal subgroups only

This group has 444 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	#20299411	C₂	1	4	yes	maximal, complemented
2	#20298870	C₂	1	4	yes	maximal, complemented
3	#20298780	C₂	1	4	yes	maximal, complemented
4	#20298716	C₂	1	2	yes	maximal, complemented
5	#494×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813670	C₂²	1	4	yes	—
7	#16813654	—	2	8	no	—
8	#16813653	C₂²	1	8	yes	—
9	#16813645	—	2	4	no	—
10	#16813644	C₂²	1	2	yes	—
11	Q₁₆×D₁₁₃	—	2	4	no	—
12	#16813628	C₂²	1	4	yes	—
13	#16813558	C₂²	1	2	yes	—
14	#16813550	C₂²	1	4	yes	—
15	#16813541	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₁₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
18	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	#14268648	—	2	8	no	—
20	#14268647	—	4	4	no	—
21	#14268647	C₂³	1	4	yes	—
22	Q₈×D₁₁₃	—	2	8	no	—
23	Dic₄₅₂	D₄	1	2	yes	—
24	#14268637	—	2	8	no	—
25	#14268636	D₄	1	4	yes	—
26	#14268634	—	2	8	no	—
27	D₉₀₄	D₄	1	2	yes	—
28	#14268631	C₂³	1	4	yes	—
29	#14268630	—	2	4	no	—
30	C₈×D₁₁₃	—	2	4	no	—
31	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
32	Q₁₆×C₁₁₃	—	2	4	no	—
33	Q₁₆×C₁₁₃	—	2	4	no	—
34	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₁₃	—	4	2	no	—
36	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
37	Q₈×C₁₁₃	—	2	8	no	—
38	C₂×C₄×C₁₁₃	—	2	1	no	—
39	C₂×C₄×C₁₁₃	—	2	4	no	—
40	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₁₁₃	—	2	4	no	—
42	D₄₅₂	—	4	4	no	—
43	D₄₅₂	C₂×D₄	1	4	yes	—
44	C₄×D₁₁₃	—	2	4	no	—
45	C₄×D₁₁₃	—	4	8	no	—
46	Dic₂₂₆	—	4	4	no	—
47	Dic₂₂₆	C₂×D₄	1	4	yes	—
48	C₈×C₁₁₃	C₂×D₄	1	2	yes	—
49	C₈×C₁₁₃	C₂×D₄	1	2	yes	—
50	#11744751	—	2	4	no	—
51	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₁₃	—	2	4	no	—
53	C₄×C₁₁₃	—	2	2	no	—
54	C₄×C₁₁₃	—	4	4	no	—
55	C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
56	Dic₁₁₃	—	2	4	no	—
57	C₂×C₁₁₃	—	2	1	no	—
58	C₂×C₁₁₃	D₄²	1	1	yes	characteristic
59	D₁₁₃	—	4	4	no	—
60	#3293	—	113	1	no	maximal, Sylow
61	C₁₁₃	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	113	4	no	—
63	#494	D₁₁₃	1	1	yes	characteristic, complemented
64	#473	—	113	4	no	—
65	#456	—	113	4	no	—
66	#443	—	113	2	no	—
67	2₋¹⁺⁴	—	113	4	no	—
68	#137	—	113	8	no	—
69	#137	—	226	8	no	—
70	#135	—	113	2	no	—
71	#135	—	226	4	no	—
72	C₂×Q₁₆	—	226	4	no	—
73	C₂×Q₁₆	C₂×D₁₁₃	1	4	yes	—
74	#131	—	113	4	no	—
75	C₄⋊Q₈	C₂×D₁₁₃	1	2	yes	—
76	#108	—	113	2	no	—
77	#104	—	113	4	no	—
78	#101	—	113	4	no	—
79	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
80	#41	—	113	4	no	—
81	#41	—	452	4	no	—
82	#41	—	226	8	no	—
83	C₂×Q₈	—	226	8	no	—
84	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	113	2	no	—
88	Q₁₆	—	226	8	no	—
89	SD₁₆	—	113	4	no	—
90	SD₁₆	—	226	8	no	—
91	D₈	—	113	2	no	—
92	M₁₆	—	113	4	no	—
93	M₁₆	—	226	4	no	—
94	C₂×C₈	—	226	4	no	—
95	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	113	4	no	—
100	Q₈	—	452	4	no	—
101	D₄	—	113	4	no	—
102	D₄	—	452	4	no	—
103	D₄	—	226	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	452	8	no	—
107	C₂×C₄	—	226	4	no	—
108	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
109	C₈	—	226	4	no	—
110	C₈	D₄×D₁₁₃	1	2	yes	—
111	C₈	D₄×D₁₁₃	1	2	yes	—
112	C₂²	—	226	4	no	—
113	C₂²	#16813667	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	226	4	no	—
117	C₄	C₂×D₄×D₁₁₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	452	4	no	—
120	C₂	#20299233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20299411

C₂

yes

maximal, complemented

#20298870

C₂

yes

maximal, complemented

#20298780

C₂

yes

maximal, complemented

#20298716

C₂

yes

maximal, complemented

#494×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#16813670

C₂²

yes

—

#16813654

—

#16813653

C₂²

yes

—

#16813645

—

#16813644

C₂²

yes

—

Q₁₆×D₁₁₃

—

#16813628

C₂²

yes

—

#16813558

C₂²

yes

—

#16813550

C₂²

yes

—

#16813541

C₂²

yes

—

C₂×Q₁₆×C₁₁₃

C₂²

yes

—

C₄⋊Q₈×C₁₁₃

C₂²

yes

—

C₄×C₈×C₁₁₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₁₃

—

Dic₄₅₂

D₄

yes

—

#14268637

—

#14268636

D₄

yes

—

#14268634

—

D₉₀₄

D₄

yes

—

#14268631

C₂³

yes

—

#14268630

—

C₈×D₁₁₃

—

C₂×Q₈×C₁₁₃

C₂³

yes

—

Q₁₆×C₁₁₃

—

Q₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂³

yes

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂³

yes

characteristic

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₁₃

—

D₄₅₂

—

D₄₅₂

C₂×D₄

yes

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

C₂×D₄

yes

—

C₈×C₁₁₃

C₂×D₄

yes

—

C₈×C₁₁₃

C₂×D₄

yes

—

#11744751

—

C₂²×C₁₁₃

H(5,𝔽₂)

yes

characteristic

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂²×D₄

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

D₄²

yes

characteristic

D₁₁₃

—

#3293

—

113

maximal, Sylow

C₁₁₃

#3293

yes

characteristic, complemented, Sylow

#578

—

113

—

#494

D₁₁₃

yes

characteristic, complemented

#473

—

113

—

#456

—

113

—

#443

—

113

—

2₋¹⁺⁴

—

113

—

#137

—

113

—

#137

—

226

—

#135

—

113

—

#135

—

226

—

C₂×Q₁₆

—

226

—

C₂×Q₁₆

C₂×D₁₁₃

yes

—

#131

—

113

—

C₄⋊Q₈

C₂×D₁₁₃

yes

—

#108

—

113

—

#104

—

113

—

#101

—

113

—

C₄×C₈

C₂×D₁₁₃

yes

characteristic

#41

—

113

—

#41

—

452

—

#41

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

C₂²×D₁₁₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

113

—

Q₁₆

—

226

—

SD₁₆

—

113

—

SD₁₆

—

226

—

D₈

—

113

—

M₁₆

—

113

—

M₁₆

—

226

—

C₂×C₈

—

226

—

C₂×C₈

C₂²×D₁₁₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₁₃

yes

characteristic

Q₈

—

Q₈

—

113

—

100

Q₈

—

452

—

101

D₄

—

113

—

102

D₄

—

452

—

103

D₄

—

226

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

452

—

107

C₂×C₄

—

226

—

108

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

109

C₈

—

226

—

110

C₈

D₄×D₁₁₃

yes

—

111

C₈

D₄×D₁₁₃

yes

—

112

C₂²

—

226

—

113

C₂²

#16813667

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

226

—

117

C₄

C₂×D₄×D₁₁₃

yes

—

118

C₂

—

119

C₂

—

452

—

120

C₂

#20299233

yes

characteristic

	subgroup	G/H
center	C₂	#20299233
commutator, cosocle	C₂×C₄×C₁₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	#494×C₁₁₃	C₂
socle	C₂×C₁₁₃	D₄²
hypercenter	#494	D₁₁₃
hypocenter	C₁₁₃	#3293
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20299233

commutator, cosocle

C₂×C₄×C₁₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁₃

Fitting

#494×C₁₁₃

C₂

socle

C₂×C₁₁₃

D₄²

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20299411	C₂	semidirect
2. #20298870	C₂	semidirect
3. #20298780	C₂	semidirect
4. #20298716	C₂	semidirect
5. #494×C₁₁₃	C₂	semidirect
61. C₁₁₃	#3293	semidirect
63. #494	D₁₁₃	semidirect

H ≅ G/N

type

1. #20299411

C₂

semidirect

2. #20298870

C₂

semidirect

3. #20298780

C₂

semidirect

4. #20298716

C₂

semidirect

5. #494×C₁₁₃

C₂

semidirect

61. C₁₁₃

#3293

semidirect

63. #494

D₁₁₃

semidirect

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

subgroup	index
#20299411	2
#20298870	2
#20298780	2
#20298716	2
#494×C₁₁₃	2
#3293	113

subgroup

index

#494×C₁₁₃

113

Subgroups — #28402378

Conjugacy classes of subgroups

Complements

Maximal subgroups