Group #20293381 — subgroups

Subgroups — #20293381

Order 6848.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16812108	C₂	1	2	yes	maximal, complemented
2	#16812104	C₂	1	2	yes	maximal, complemented
3	#16812104	C₂	1	2	yes	maximal, complemented
4	#16812101	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812097	C₂	1	1	yes	maximal, characteristic
6	#16812094	C₂	1	2	yes	maximal
7	#16812089	C₂	1	2	yes	maximal, complemented
8	#16812087	C₂	1	2	yes	maximal, complemented
9	#125×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₄	C₂²	1	2	yes	—
11	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
12	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
17	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
18	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₀₇⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₁₀₇	C₂²	1	2	yes	—
26	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
28	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
30	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
31	C₂²×D₁₀₇	C₂³	1	2	yes	—
32	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
36	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
37	C₂×Dic₁₀₇	C₂³	1	2	yes	—
38	C₂×Dic₁₀₇	C₂³	1	2	yes	—
39	C₂×Dic₁₀₇	C₂³	1	2	yes	—
40	D₄₂₈	—	4	1	no	—
41	C₄×D₁₀₇	—	4	2	no	—
42	Dic₂₁₄	—	4	1	no	—
43	Dic₂₁₄	—	4	2	no	—
44	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₀₇	—	2	2	no	—
46	C₂×D₁₀₇	—	2	2	no	—
47	C₂×D₁₀₇	—	2	2	no	—
48	C₄×C₁₀₇	—	2	2	no	—
49	C₄×C₁₀₇	—	2	1	no	—
50	C₄×C₁₀₇	—	2	1	no	—
51	C₄×C₁₀₇	—	2	2	no	—
52	C₄×C₁₀₇	—	2	1	no	—
53	Dic₁₀₇	—	2	2	no	—
54	Dic₁₀₇	—	2	2	no	—
55	Dic₁₀₇	—	2	2	no	—
56	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
59	D₁₀₇	—	4	2	no	—
60	C₁₀₇	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	107	1	no	maximal, Sylow
62	C₄⋊Q₈	—	107	2	no	—
63	#126	—	107	2	no	—
64	#126	—	107	2	no	—
65	#125	—	107	1	no	—
66	#125	D₁₀₇	1	1	yes	characteristic, complemented
67	#124	—	107	1	no	—
68	#123	—	107	2	no	—
69	C₂²⋊Q₈	—	107	2	no	—
70	C₂²⋊Q₈	—	107	2	no	—
71	C₂×Q₈	—	107	2	no	—
72	C₂×Q₈	—	107	1	no	—
73	C₂×D₄	—	107	1	no	—
74	C₂²×C₄	—	107	2	no	—
75	C₄⋊C₄	—	107	1	no	—
76	C₄⋊C₄	—	107	2	no	—
77	C₄⋊C₄	—	107	2	no	—
78	C₄⋊C₄	—	107	2	no	—
79	C₄⋊C₄	—	107	2	no	—
80	C₄⋊C₄	—	107	1	no	—
81	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
83	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
84	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
85	C₂²⋊C₄	—	107	2	no	—
86	C₂²⋊C₄	—	107	2	no	—
87	C₂²⋊C₄	—	107	2	no	—
88	C₂²⋊C₄	—	107	2	no	—
89	C₂²⋊C₄	—	107	2	no	—
90	C₄²	—	107	2	no	—
91	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
92	C₂³	—	107	2	no	—
93	Q₈	—	428	2	no	—
94	Q₈	—	428	1	no	—
95	D₄	—	428	1	no	—
96	C₂×C₄	—	107	2	no	—
97	C₂×C₄	—	107	2	no	—
98	C₂×C₄	—	107	2	no	—
99	C₂×C₄	—	428	2	no	—
100	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
101	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
105	C₂²	—	214	2	no	—
106	C₂²	—	214	2	no	—
107	C₂²	—	214	2	no	—
108	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	214	2	no	—
115	C₄	—	214	2	no	—
116	C₄	—	214	2	no	—
117	C₂	—	428	2	no	—
118	C₂	#16812155	1	1	yes	characteristic
119	C₂	#16812156	1	1	yes	characteristic
120	C₂	#16812157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812108

C₂

yes

maximal, complemented

#16812104

C₂

yes

maximal, complemented

#16812104

C₂

yes

maximal, complemented

#16812101

C₂

yes

maximal, characteristic, complemented

#16812097

C₂

yes

maximal, characteristic

#16812094

C₂

yes

maximal

#16812089

C₂

yes

maximal, complemented

#16812087

C₂

yes

maximal, complemented

#125×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

—

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄²×C₁₀₇

C₂²

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#563

yes

characteristic, complemented, Sylow

#563

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

#126

—

107

—

#126

—

107

—

#125

—

107

—

#125

D₁₀₇

yes

characteristic, complemented

#124

—

107

—

#123

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₄²

C₂×D₁₀₇

yes

characteristic

C₂³

—

107

—

Q₈

—

428

—

Q₈

—

428

—

D₄

—

428

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

428

—

100

C₂×C₄

C₂²×D₁₀₇

yes

—

101

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

102

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

103

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

104

C₂×C₄

C₂²×D₁₀₇

yes

—

105

C₂²

—

214

—

106

C₂²

—

214

—

107

C₂²

—

214

—

108

C₂²

C₂³×D₁₀₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

214

—

115

C₄

—

214

—

116

C₄

—

214

—

117

C₂

—

428

—

118

C₂

#16812155

yes

characteristic

119

C₂

#16812156

yes

characteristic

120

C₂

#16812157

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#125×C₁₀₇	C₂
hypercenter	#125	D₁₀₇
hypocenter	C₁₀₇	#563
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

#125×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #16812108	C₂	semidirect
2. #16812104	C₂	semidirect
3. #16812104	C₂	semidirect
4. #16812101	C₂	semidirect
7. #16812089	C₂	semidirect
8. #16812087	C₂	semidirect
9. #125×C₁₀₇	C₂	semidirect
60. C₁₀₇	#563	semidirect
66. #125	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812108

C₂

semidirect

2. #16812104

C₂

semidirect

3. #16812104

C₂

semidirect

4. #16812101

C₂

semidirect

7. #16812089

C₂

semidirect

8. #16812087

C₂

semidirect

9. #125×C₁₀₇

C₂

semidirect

60. C₁₀₇

#563

semidirect

66. #125

D₁₀₇

semidirect

Subgroups — #20293381

Conjugacy classes of subgroups

Complements