Group #28348265 — subgroups

Subgroups — #28348265

Order 13696.

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	#20293440	C₂	1	4	yes	maximal, complemented
2	#20292899	C₂	1	4	yes	maximal, complemented
3	#20292809	C₂	1	4	yes	maximal, complemented
4	#20292745	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812157	C₂²	1	4	yes	—
7	#16812141	—	2	8	no	—
8	#16812140	C₂²	1	8	yes	—
9	#16812132	—	2	4	no	—
10	#16812131	C₂²	1	2	yes	—
11	Q₁₆×D₁₀₇	—	2	4	no	—
12	#16812115	C₂²	1	4	yes	—
13	#16812045	C₂²	1	2	yes	—
14	#16812037	C₂²	1	4	yes	—
15	#16812028	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₀₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
19	#13133819	—	2	8	no	—
20	#13133818	—	4	4	no	—
21	#13133818	C₂³	1	4	yes	—
22	Q₈×D₁₀₇	—	2	8	no	—
23	Dic₄₂₈	D₄	1	2	yes	—
24	#13133808	—	2	8	no	—
25	#13133807	D₄	1	4	yes	—
26	#13133805	—	2	8	no	—
27	D₈₅₆	D₄	1	2	yes	—
28	#13133802	C₂³	1	4	yes	—
29	#13133801	—	2	4	no	—
30	C₈×D₁₀₇	—	2	4	no	—
31	C₂×Q₈×C₁₀₇	C₂³	1	4	yes	—
32	Q₁₆×C₁₀₇	—	2	4	no	—
33	Q₁₆×C₁₀₇	—	2	4	no	—
34	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₇	—	4	2	no	—
36	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₇	—	2	8	no	—
38	C₂×C₄×C₁₀₇	—	2	1	no	—
39	C₂×C₄×C₁₀₇	—	2	4	no	—
40	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₁₀₇	—	2	4	no	—
42	D₄₂₈	—	4	4	no	—
43	D₄₂₈	C₂×D₄	1	4	yes	—
44	C₄×D₁₀₇	—	2	4	no	—
45	C₄×D₁₀₇	—	4	8	no	—
46	Dic₂₁₄	—	4	4	no	—
47	Dic₂₁₄	C₂×D₄	1	4	yes	—
48	C₈×C₁₀₇	C₂×D₄	1	2	yes	—
49	C₈×C₁₀₇	C₂×D₄	1	2	yes	—
50	C₁₀₇⋊C₈	—	2	4	no	—
51	C₂²×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₀₇	—	2	4	no	—
53	C₄×C₁₀₇	—	2	2	no	—
54	C₄×C₁₀₇	—	4	4	no	—
55	C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
56	Dic₁₀₇	—	2	4	no	—
57	C₂×C₁₀₇	—	2	1	no	—
58	C₂×C₁₀₇	D₄²	1	1	yes	characteristic
59	D₁₀₇	—	4	4	no	—
60	#3293	—	107	1	no	maximal, Sylow
61	C₁₀₇	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	107	4	no	—
63	#494	D₁₀₇	1	1	yes	characteristic, complemented
64	#473	—	107	4	no	—
65	#456	—	107	4	no	—
66	#443	—	107	2	no	—
67	2₋¹⁺⁴	—	107	4	no	—
68	#137	—	107	8	no	—
69	#137	—	214	8	no	—
70	#135	—	107	2	no	—
71	#135	—	214	4	no	—
72	C₂×Q₁₆	—	214	4	no	—
73	C₂×Q₁₆	C₂×D₁₀₇	1	4	yes	—
74	#131	—	107	4	no	—
75	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
76	#108	—	107	2	no	—
77	#104	—	107	4	no	—
78	#101	—	107	4	no	—
79	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
80	#41	—	107	4	no	—
81	#41	—	428	4	no	—
82	#41	—	214	8	no	—
83	C₂×Q₈	—	214	8	no	—
84	C₂×Q₈	C₂²×D₁₀₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	107	2	no	—
88	Q₁₆	—	214	8	no	—
89	SD₁₆	—	107	4	no	—
90	SD₁₆	—	214	8	no	—
91	D₈	—	107	2	no	—
92	M₁₆	—	107	4	no	—
93	M₁₆	—	214	4	no	—
94	C₂×C₈	—	214	4	no	—
95	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	107	4	no	—
100	Q₈	—	428	4	no	—
101	D₄	—	107	4	no	—
102	D₄	—	428	4	no	—
103	D₄	—	214	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	428	8	no	—
107	C₂×C₄	—	214	4	no	—
108	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
109	C₈	—	214	4	no	—
110	C₈	D₄×D₁₀₇	1	2	yes	—
111	C₈	D₄×D₁₀₇	1	2	yes	—
112	C₂²	—	214	4	no	—
113	C₂²	#16812154	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	214	4	no	—
117	C₄	C₂×D₄×D₁₀₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	428	4	no	—
120	C₂	#20293262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293440

C₂

yes

maximal, complemented

#20292899

C₂

yes

maximal, complemented

#20292809

C₂

yes

maximal, complemented

#20292745

C₂

yes

maximal, complemented

#494×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#16812157

C₂²

yes

—

#16812141

—

#16812140

C₂²

yes

—

#16812132

—

#16812131

C₂²

yes

—

Q₁₆×D₁₀₇

—

#16812115

C₂²

yes

—

#16812045

C₂²

yes

—

#16812037

C₂²

yes

—

#16812028

C₂²

yes

—

C₂×Q₁₆×C₁₀₇

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₀₇

—

Dic₄₂₈

D₄

yes

—

#13133808

—

#13133807

D₄

yes

—

#13133805

—

D₈₅₆

D₄

yes

—

#13133802

C₂³

yes

—

#13133801

—

C₈×D₁₀₇

—

C₂×Q₈×C₁₀₇

C₂³

yes

—

Q₁₆×C₁₀₇

—

Q₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂³

yes

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂²⋊D₁₀₇

—

D₄₂₈

—

D₄₂₈

C₂×D₄

yes

—

C₄×D₁₀₇

—

C₄×D₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

C₂×D₄

yes

—

C₈×C₁₀₇

C₂×D₄

yes

—

C₈×C₁₀₇

C₂×D₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²×D₄

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

D₄²

yes

characteristic

D₁₀₇

—

#3293

—

107

maximal, Sylow

C₁₀₇

#3293

yes

characteristic, complemented, Sylow

#578

—

107

—

#494

D₁₀₇

yes

characteristic, complemented

#473

—

107

—

#456

—

107

—

#443

—

107

—

2₋¹⁺⁴

—

107

—

#137

—

107

—

#137

—

214

—

#135

—

107

—

#135

—

214

—

C₂×Q₁₆

—

214

—

C₂×Q₁₆

C₂×D₁₀₇

yes

—

#131

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

#108

—

107

—

#104

—

107

—

#101

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

#41

—

107

—

#41

—

428

—

#41

—

214

—

C₂×Q₈

—

214

—

C₂×Q₈

C₂²×D₁₀₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

107

—

Q₁₆

—

214

—

SD₁₆

—

107

—

SD₁₆

—

214

—

D₈

—

107

—

M₁₆

—

107

—

M₁₆

—

214

—

C₂×C₈

—

214

—

C₂×C₈

C₂²×D₁₀₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₇

yes

characteristic

Q₈

—

Q₈

—

107

—

100

Q₈

—

428

—

101

D₄

—

107

—

102

D₄

—

428

—

103

D₄

—

214

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

428

—

107

C₂×C₄

—

214

—

108

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

109

C₈

—

214

—

110

C₈

D₄×D₁₀₇

yes

—

111

C₈

D₄×D₁₀₇

yes

—

112

C₂²

—

214

—

113

C₂²

#16812154

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

214

—

117

C₄

C₂×D₄×D₁₀₇

yes

—

118

C₂

—

119

C₂

—

428

—

120

C₂

#20293262

yes

characteristic

	subgroup	G/H
center	C₂	#20293262
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	#494×C₁₀₇	C₂
socle	C₂×C₁₀₇	D₄²
hypercenter	#494	D₁₀₇
hypocenter	C₁₀₇	#3293
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20293262

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

#494×C₁₀₇

C₂

socle

C₂×C₁₀₇

D₄²

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20293440	C₂	semidirect
2. #20292899	C₂	semidirect
3. #20292809	C₂	semidirect
4. #20292745	C₂	semidirect
5. #494×C₁₀₇	C₂	semidirect
61. C₁₀₇	#3293	semidirect
63. #494	D₁₀₇	semidirect

H ≅ G/N

type

1. #20293440

C₂

semidirect

2. #20292899

C₂

semidirect

3. #20292809

C₂

semidirect

4. #20292745

C₂

semidirect

5. #494×C₁₀₇

C₂

semidirect

61. C₁₀₇

#3293

semidirect

63. #494

D₁₀₇

semidirect

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

subgroup	index
#20293440	2
#20292899	2
#20292809	2
#20292745	2
#494×C₁₀₇	2
#3293	107

subgroup

index

#494×C₁₀₇

107

Subgroups — #28348265

Conjugacy classes of subgroups

Complements

Maximal subgroups