Group #20293441 — subgroups

Subgroups — #20293441

Order 6848.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16812156	C₂	1	2	yes	maximal, complemented
2	#16812139	C₂	1	2	yes	maximal, complemented
3	#16812138	C₂	1	4	yes	maximal, complemented
4	#16812128	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₁₀₇	C₂	1	2	yes	maximal, complemented
7	#16812115	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133818	—	2	2	no	—
10	#13133818	C₂²	1	1	yes	characteristic
11	#13133817	—	2	4	no	—
12	#13133817	C₂²	1	4	yes	—
13	C₂×Dic₂₁₄	—	2	2	no	—
14	C₂×Dic₂₁₄	C₂²	1	2	yes	—
15	Q₈×D₁₀₇	C₂²	1	4	yes	—
16	Dic₄₂₈	C₂²	1	2	yes	—
17	Dic₄₂₈	C₂²	1	2	yes	—
18	#13133808	C₂²	1	4	yes	—
19	#13133806	C₂²	1	4	yes	—
20	#13133802	C₂²	1	1	yes	characteristic
21	#13133801	C₂²	1	2	yes	—
22	C₈×D₁₀₇	C₂²	1	2	yes	—
23	#41×C₁₀₇	C₂²	1	2	yes	—
24	Q₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
25	SD₁₆×C₁₀₇	C₂²	1	2	yes	—
26	D₈×C₁₀₇	C₂²	1	1	yes	characteristic
27	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
28	Q₈×C₁₀₇	C₂³	1	2	yes	—
29	D₄×C₁₀₇	—	2	2	no	—
30	D₄×C₁₀₇	C₂³	1	2	yes	—
31	C₂×C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₇	—	2	4	no	—
34	C₂²⋊D₁₀₇	D₄	1	2	yes	—
35	C₂×Dic₁₀₇	—	2	2	no	—
36	C₂×Dic₁₀₇	—	2	4	no	—
37	D₄₂₈	D₄	1	1	yes	characteristic
38	C₄×D₁₀₇	—	2	4	no	—
39	C₄×D₁₀₇	C₂³	1	2	yes	—
40	Dic₂₁₄	—	2	2	no	—
41	Dic₂₁₄	—	2	4	no	—
42	Dic₂₁₄	D₄	1	1	yes	characteristic
43	Dic₂₁₄	C₂³	1	4	yes	—
44	C₈×C₁₀₇	C₂³	1	1	yes	characteristic
45	C₈×C₁₀₇	C₂³	1	1	yes	characteristic
46	C₁₀₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₁₀₇	—	2	2	no	—
48	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₀₇	C₂×D₄	1	2	yes	—
50	C₄×C₁₀₇	—	2	2	no	—
51	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
52	C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₀₇	—	2	4	no	—
54	Dic₁₀₇	C₂×D₄	1	2	yes	—
55	C₂×C₁₀₇	—	2	1	no	—
56	C₂×C₁₀₇	—	4	2	no	—
57	C₂×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
58	D₁₀₇	—	2	2	no	—
59	C₁₀₇	#578	1	1	yes	characteristic, complemented, Sylow
60	#578	—	107	1	no	maximal, Sylow
61	2₋¹⁺⁴	—	107	2	no	—
62	#137	—	107	2	no	—
63	#137	—	107	4	no	—
64	#135	—	107	2	no	—
65	#135	D₁₀₇	1	1	yes	characteristic, complemented
66	C₂×Q₁₆	—	107	2	no	—
67	C₂×Q₁₆	—	107	1	no	—
68	#131	—	107	1	no	—
69	#41	—	107	1	no	—
70	#41	—	107	4	no	—
71	#41	—	214	2	no	—
72	#41	—	214	4	no	—
73	#41	C₂×D₁₀₇	1	2	yes	—
74	C₂×Q₈	—	107	4	no	—
75	C₂×Q₈	—	107	2	no	—
76	C₂×Q₈	—	214	2	no	—
77	Q₁₆	—	107	4	no	—
78	Q₁₆	—	107	2	no	—
79	Q₁₆	—	107	2	no	—
80	Q₁₆	C₂×D₁₀₇	1	1	yes	characteristic
81	SD₁₆	—	107	4	no	—
82	SD₁₆	C₂×D₁₀₇	1	2	yes	—
83	D₈	C₂×D₁₀₇	1	1	yes	characteristic
84	M₁₆	—	107	1	no	—
85	M₁₆	—	107	2	no	—
86	C₂×C₈	—	107	2	no	—
87	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
88	Q₈	—	107	1	no	—
89	Q₈	—	107	4	no	—
90	Q₈	—	214	4	no	—
91	Q₈	—	214	2	no	—
92	Q₈	C₂²×D₁₀₇	1	2	yes	—
93	D₄	—	2	2	no	—
94	D₄	—	107	2	no	—
95	D₄	—	107	1	no	—
96	D₄	—	214	4	no	—
97	D₄	C₂²×D₁₀₇	1	2	yes	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	107	2	no	—
100	C₂×C₄	—	214	4	no	—
101	C₂×C₄	—	214	4	no	—
102	C₂×C₄	—	214	2	no	—
103	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
104	C₈	—	107	2	no	—
105	C₈	C₂²×D₁₀₇	1	1	yes	characteristic
106	C₈	C₂²×D₁₀₇	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	107	2	no	—
109	C₂²	D₄×D₁₀₇	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	107	2	no	—
112	C₄	—	214	4	no	—
113	C₄	D₄×D₁₀₇	1	1	yes	characteristic
114	C₄	C₂³×D₁₀₇	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	4	2	no	—
117	C₂	—	214	2	no	—
118	C₂	C₂×D₄×D₁₀₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812156

C₂

yes

maximal, complemented

#16812139

C₂

yes

maximal, complemented

#16812138

C₂

yes

maximal, complemented

#16812128

C₂

yes

maximal, complemented

C₂×Dic₄₂₈

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₁₀₇

C₂

yes

maximal, complemented

#16812115

C₂

yes

maximal, characteristic, complemented

#135×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#13133818

—

#13133818

C₂²

yes

characteristic

#13133817

—

#13133817

C₂²

yes

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂²

yes

—

Q₈×D₁₀₇

C₂²

yes

—

Dic₄₂₈

C₂²

yes

—

Dic₄₂₈

C₂²

yes

—

#13133808

C₂²

yes

—

#13133806

C₂²

yes

—

#13133802

C₂²

yes

characteristic

#13133801

C₂²

yes

—

C₈×D₁₀₇

C₂²

yes

—

#41×C₁₀₇

C₂²

yes

—

Q₁₆×C₁₀₇

C₂²

yes

characteristic

SD₁₆×C₁₀₇

C₂²

yes

—

D₈×C₁₀₇

C₂²

yes

characteristic

C₂×C₈×C₁₀₇

C₂²

yes

characteristic

Q₈×C₁₀₇

C₂³

yes

—

D₄×C₁₀₇

—

D₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂²⋊D₁₀₇

D₄

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

D₄₂₈

D₄

yes

characteristic

C₄×D₁₀₇

—

C₄×D₁₀₇

C₂³

yes

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

D₄

yes

characteristic

Dic₂₁₄

C₂³

yes

—

C₈×C₁₀₇

C₂³

yes

characteristic

C₈×C₁₀₇

C₂³

yes

characteristic

C₁₀₇⋊C₈

C₂³

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

characteristic

C₂×D₁₀₇

C₂×D₄

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₄×C₁₀₇

C₂⁴

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

C₂×D₄

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×D₄

yes

characteristic

D₁₀₇

—

C₁₀₇

#578

yes

characteristic, complemented, Sylow

#578

—

107

maximal, Sylow

2₋¹⁺⁴

—

107

—

#137

—

107

—

#137

—

107

—

#135

—

107

—

#135

D₁₀₇

yes

characteristic, complemented

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

#131

—

107

—

#41

—

107

—

#41

—

107

—

#41

—

214

—

#41

—

214

—

#41

C₂×D₁₀₇

yes

—

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

Q₁₆

—

107

—

Q₁₆

—

107

—

Q₁₆

—

107

—

Q₁₆

C₂×D₁₀₇

yes

characteristic

SD₁₆

—

107

—

SD₁₆

C₂×D₁₀₇

yes

—

D₈

C₂×D₁₀₇

yes

characteristic

M₁₆

—

107

—

M₁₆

—

107

—

C₂×C₈

—

107

—

C₂×C₈

C₂×D₁₀₇

yes

characteristic

Q₈

—

107

—

Q₈

—

107

—

Q₈

—

214

—

Q₈

—

214

—

Q₈

C₂²×D₁₀₇

yes

—

D₄

—

D₄

—

107

—

D₄

—

107

—

D₄

—

214

—

D₄

C₂²×D₁₀₇

yes

—

C₂×C₄

—

C₂×C₄

—

107

—

100

C₂×C₄

—

214

—

101

C₂×C₄

—

214

—

102

C₂×C₄

—

214

—

103

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

104

C₈

—

107

—

105

C₈

C₂²×D₁₀₇

yes

characteristic

106

C₈

C₂²×D₁₀₇

yes

characteristic

107

C₂²

—

108

C₂²

—

107

—

109

C₂²

D₄×D₁₀₇

yes

characteristic

110

C₄

—

111

C₄

—

107

—

112

C₄

—

214

—

113

C₄

D₄×D₁₀₇

yes

characteristic

114

C₄

C₂³×D₁₀₇

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

214

—

118

C₂

C₂×D₄×D₁₀₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₀₇
commutator	C₄×C₁₀₇	C₂⁴
Frattini	C₄	C₂³×D₁₀₇
Fitting	#135×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂²×D₄
hypercenter	#135	D₁₀₇
hypocenter	C₁₀₇	#578
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₁₀₇

commutator

C₄×C₁₀₇

C₂⁴

Frattini

C₄

C₂³×D₁₀₇

Fitting

#135×C₁₀₇

C₂

socle

C₂×C₁₀₇

C₂²×D₄

D₁₀₇

C₁₀₇

G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #16812156	C₂	semidirect
2. #16812139	C₂	semidirect
3. #16812138	C₂	semidirect
4. #16812128	C₂	semidirect
5. C₂×Dic₄₂₈	C₂	semidirect
6. Q₁₆×D₁₀₇	C₂	semidirect
7. #16812115	C₂	semidirect
8. #135×C₁₀₇	C₂	semidirect
59. C₁₀₇	#578	semidirect
65. #135	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812156

C₂

semidirect

2. #16812139

C₂

semidirect

3. #16812138

C₂

semidirect

4. #16812128

C₂

semidirect

5. C₂×Dic₄₂₈

C₂

semidirect

6. Q₁₆×D₁₀₇

C₂

semidirect

7. #16812115

C₂

semidirect

8. #135×C₁₀₇

C₂

semidirect

59. C₁₀₇

#578

semidirect

65. #135

D₁₀₇

semidirect

Subgroups — #20293441

Conjugacy classes of subgroups

Complements