Group #28338631 — subgroups

Subgroups — #28338631

Order 13696.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#20293024	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292743	C₂	1	1	yes	maximal, characteristic
3	#20292451	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292447	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292438	C₂	1	1	yes	maximal, characteristic
6	#20292436	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16812131	C₂²	1	1	yes	characteristic
9	#16812117	—	2	1	no	—
10	#16812108	—	2	1	no	—
11	#16812049	C₂²	1	1	yes	characteristic
12	#16812041	C₂²	1	1	yes	characteristic
13	#16812040	—	2	1	no	—
14	#16812035	—	2	1	no	—
15	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
17	#108×C₁₀₇	C₂²	1	1	yes	characteristic
18	#106×C₁₀₇	C₂²	1	1	yes	characteristic
19	#13133818	—	2	1	no	—
20	C₂×Dic₂₁₄	—	4	1	no	—
21	Dic₄₂₈	Q₈	1	1	yes	characteristic, complemented
22	#13133807	—	2	1	no	—
23	D₈₅₆	Q₈	1	1	yes	characteristic, complemented
24	#13133801	—	4	1	no	—
25	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₁₀₇	—	4	1	no	—
27	C₄⋊Dic₁₀₇	—	2	1	no	—
28	Dic₁₀₇⋊C₄	—	4	1	no	—
29	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
30	C₁₀₇⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₀₇	—	2	1	no	—
32	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₇	—	2	1	no	—
34	C₁₆×C₁₀₇	—	2	1	no	—
35	C₂×C₄×C₁₀₇	—	4	1	no	—
36	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₁₀₇	—	4	1	no	—
38	C₂×Dic₁₀₇	—	2	1	no	—
39	C₂×Dic₁₀₇	—	4	1	no	—
40	D₄₂₈	—	2	1	no	—
41	C₄×D₁₀₇	—	4	1	no	—
42	Dic₂₁₄	—	2	1	no	—
43	Dic₂₁₄	—	8	2	no	—
44	C₈×C₁₀₇	—	4	1	no	—
45	C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₀₇	#41	1	1	yes	characteristic
47	C₁₀₇⋊C₈	SD₁₆	1	1	yes	characteristic
48	C₁₀₇⋊C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₀₇	—	4	1	no	—
51	C₄×C₁₀₇	—	8	1	no	—
52	C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₀₇	—	2	2	no	—
55	Dic₁₀₇	—	4	1	no	—
56	Dic₁₀₇	—	8	1	no	—
57	C₂×C₁₀₇	—	2	1	no	—
58	C₂×C₁₀₇	#476	1	1	yes	characteristic
59	D₁₀₇	—	8	1	no	—
60	#2230	—	107	1	no	maximal, Sylow
61	C₁₀₇	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	107	1	no	—
63	#443	—	107	1	no	—
64	#365	—	107	1	no	—
65	#365	D₁₀₇	1	1	yes	characteristic, complemented
66	#364	—	107	1	no	—
67	#360	—	107	1	no	—
68	#360	—	107	1	no	—
69	#135	—	107	1	no	—
70	#131	—	214	1	no	—
71	C₄⋊Q₈	—	214	1	no	—
72	M₃₂	—	107	1	no	—
73	M₃₂	C₂×D₁₀₇	1	1	yes	characteristic
74	#108	C₂×D₁₀₇	1	1	yes	characteristic
75	#106	—	107	1	no	—
76	#106	—	214	1	no	—
77	#106	C₂×D₁₀₇	1	1	yes	characteristic
78	#104	—	214	1	no	—
79	C₄×C₈	—	107	1	no	—
80	#41	—	214	1	no	—
81	C₂×Q₈	—	428	1	no	—
82	Q₁₆	—	107	1	no	—
83	SD₁₆	—	214	1	no	—
84	D₈	—	107	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	428	1	no	—
87	C₂×C₈	—	107	1	no	—
88	C₂×C₈	—	428	1	no	—
89	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	428	1	no	—
92	C₄⋊C₄	—	214	1	no	—
93	C₄²	—	107	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	214	1	no	—
96	Q₈	—	856	2	no	—
97	Q₈	—	214	1	no	—
98	D₄	—	428	1	no	—
99	D₄	—	214	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	428	1	no	—
102	C₂×C₄	—	428	1	no	—
103	C₂×C₄	—	214	1	no	—
104	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	107	1	no	—
107	C₈	—	107	1	no	—
108	C₈	Q₈×D₁₀₇	1	1	yes	characteristic
109	C₈	#13133818	1	1	yes	characteristic
110	C₂²	—	428	1	no	—
111	C₂²	#16812136	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	856	1	no	—
114	C₄	—	428	1	no	—
115	C₄	—	214	2	no	—
116	C₄	#16812087	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₀₇	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	856	1	no	—
120	C₂	#20292916	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293024

C₂

yes

maximal, characteristic, complemented

#20292743

C₂

yes

maximal, characteristic

#20292451

C₂

yes

maximal, characteristic, complemented

#20292447

C₂

yes

maximal, characteristic, complemented

#20292438

C₂

yes

maximal, characteristic

#20292436

C₂

yes

maximal, characteristic

#365×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#16812131

C₂²

yes

—

—

C₂²

yes

characteristic

#16812041

C₂²

yes

characteristic

#16812040

—

#16812035

—

C₈×Dic₁₀₇

C₂²

yes

characteristic

M₃₂×C₁₀₇

C₂²

yes

characteristic

#108×C₁₀₇

C₂²

yes

characteristic

#106×C₁₀₇

C₂²

yes

characteristic

#13133818

—

C₂×Dic₂₁₄

—

Dic₄₂₈

Q₈

yes

characteristic, complemented

#13133807

—

D₈₅₆

Q₈

yes

characteristic, complemented

#13133801

—

C₂×C₁₀₇⋊C₈

D₄

yes

characteristic

C₈×D₁₀₇

—

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

D₄

yes

characteristic

C₁₀₇⋊C₁₆

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₁₆×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂×Q₈

yes

characteristic

C₈×C₁₀₇

#41

yes

characteristic

C₁₀₇⋊C₈

SD₁₆

yes

characteristic

C₁₀₇⋊C₈

SD₁₆

yes

characteristic

C₂²×C₁₀₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₇

C₂×SD₁₆

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#476

yes

characteristic

D₁₀₇

—

#2230

—

107

maximal, Sylow

C₁₀₇

#2230

yes

characteristic, complemented, Sylow

#498

—

107

—

#443

—

107

—

#365

—

107

—

#365

D₁₀₇

yes

characteristic, complemented

#364

—

107

—

#360

—

107

—

#360

—

107

—

#135

—

107

—

#131

—

214

—

C₄⋊Q₈

—

214

—

M₃₂

—

107

—

M₃₂

C₂×D₁₀₇

yes

characteristic

#108

C₂×D₁₀₇

yes

characteristic

#106

—

107

—

#106

—

214

—

#106

C₂×D₁₀₇

yes

characteristic

#104

—

214

—

C₄×C₈

—

107

—

#41

—

214

—

C₂×Q₈

—

428

—

Q₁₆

—

107

—

SD₁₆

—

214

—

D₈

—

107

—

M₁₆

—

M₁₆

—

428

—

C₂×C₈

—

107

—

C₂×C₈

—

428

—

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

428

—

C₄⋊C₄

—

214

—

C₄²

—

107

—

C₁₆

—

C₁₆

—

214

—

Q₈

—

856

—

Q₈

—

214

—

D₄

—

428

—

D₄

—

214

—

100

C₂×C₄

—

101

C₂×C₄

—

428

—

102

C₂×C₄

—

428

—

103

C₂×C₄

—

214

—

104

C₂×C₄

D₄×D₁₀₇

yes

characteristic

105

C₈

—

106

C₈

—

107

—

107

C₈

—

107

—

108

C₈

Q₈×D₁₀₇

yes

characteristic

109

C₈

#13133818

yes

characteristic

110

C₂²

—

428

—

111

C₂²

#16812136

yes

characteristic

112

C₄

—

113

C₄

—

856

—

114

C₄

—

428

—

115

C₄

—

214

—

116

C₄

#16812087

yes

characteristic

117

C₄

SD₁₆×D₁₀₇

yes

characteristic

118

C₂

—

119

C₂

—

856

—

120

C₂

#20292916

yes

characteristic

	subgroup	G/H
center	C₂	#20292916
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#365×C₁₀₇	C₂
socle	C₂×C₁₀₇	#476
prime radical	#16812131	C₂²
hypercenter	#365	D₁₀₇
hypocenter	C₁₀₇	#2230
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#20292916

commutator, cosocle

C₂×C₈×C₁₀₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₇

Fitting

#365×C₁₀₇

C₂

C₂×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #20293024	C₂	semidirect
3. #20292451	C₂	semidirect
4. #20292447	C₂	semidirect
7. #365×C₁₀₇	C₂	semidirect
21. Dic₄₂₈	Q₈	semidirect
23. D₈₅₆	Q₈	semidirect
61. C₁₀₇	#2230	semidirect
65. #365	D₁₀₇	semidirect

H ≅ G/N

type

1. #20293024

C₂

semidirect

3. #20292451

C₂

semidirect

4. #20292447

C₂

semidirect

7. #365×C₁₀₇

C₂

semidirect

21. Dic₄₂₈

Q₈

semidirect

23. D₈₅₆

Q₈

semidirect

61. C₁₀₇

#2230

semidirect

65. #365

D₁₀₇

semidirect

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

subgroup	index
#20293024	2
#20292743	2
#20292451	2
#20292447	2
#20292438	2
#20292436	2
#365×C₁₀₇	2
#2230	107

subgroup

index

#365×C₁₀₇

107

Subgroups — #28338631

Conjugacy classes of subgroups

Complements

Maximal subgroups