Group #20292825 — subgroups

Subgroups — #20292825

Order 6848.

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

normal subgroups only

This group has 128 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	C₂×#13133805	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812105	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812031	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812030	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
10	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
11	#13133805	—	2	2	no	—
12	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₇	—	2	1	no	—
14	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀₇	—	2	1	no	—
16	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₀₇	—	2	2	no	—
20	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₇	—	2	1	no	—
22	C₂²×D₁₀₇	—	2	1	no	—
23	Q₈×C₁₀₇	—	2	1	no	—
24	Q₈×C₁₀₇	D₄	1	2	yes	complemented
25	D₄×C₁₀₇	—	2	1	no	—
26	D₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄×C₁₀₇	—	2	1	no	—
28	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₇	—	4	2	no	—
30	C₂²⋊D₁₀₇	—	4	2	no	—
31	C₂×Dic₁₀₇	—	2	1	no	—
32	C₂×Dic₁₀₇	D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₇	D₄	1	1	yes	characteristic
34	D₄₂₈	—	2	1	no	—
35	D₄₂₈	—	2	1	no	—
36	C₈×C₁₀₇	—	2	1	no	—
37	C₁₀₇⋊C₈	—	2	1	no	—
38	C₂²×C₁₀₇	—	4	1	no	—
39	C₂²×C₁₀₇	—	4	1	no	—
40	C₂²×C₁₀₇	—	4	1	no	—
41	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₀₇	—	4	1	no	—
43	C₂×D₁₀₇	—	4	1	no	—
44	C₂×D₁₀₇	—	4	1	no	—
45	C₄×C₁₀₇	—	2	2	no	—
46	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₀₇	#41	1	1	yes	characteristic
48	Dic₁₀₇	—	2	1	no	—
49	Dic₁₀₇	—	4	1	no	—
50	Dic₁₀₇	SD₁₆	1	2	yes	complemented
51	C₂×C₁₀₇	—	8	1	no	—
52	C₂×C₁₀₇	#121	1	1	yes	characteristic
53	C₂×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	D₁₀₇	—	8	1	no	—
56	C₁₀₇	#460	1	1	yes	characteristic, complemented, Sylow
57	#460	—	107	1	no	maximal, Sylow
58	C₂×SD₁₆	—	107	1	no	—
59	C₂×SD₁₆	D₁₀₇	1	1	yes	characteristic, complemented
60	#127	—	107	1	no	—
61	C₄×Q₈	—	107	1	no	—
62	C₄⋊C₈	—	107	1	no	—
63	#102	—	107	1	no	—
64	#102	—	107	1	no	—
65	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
66	C₂×D₄	—	107	1	no	—
67	C₂×D₄	—	214	1	no	—
68	C₂×D₄	—	214	1	no	—
69	C₂×D₄	C₂×D₁₀₇	1	1	yes	characteristic
70	SD₁₆	—	2	2	no	—
71	SD₁₆	—	214	2	no	—
72	C₂×C₈	—	107	1	no	—
73	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
74	C₄⋊C₄	—	107	1	no	—
75	C₄⋊C₄	—	214	1	no	—
76	C₄²	—	107	1	no	—
77	C₄²	—	214	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	214	1	no	—
80	Q₈	—	2	1	no	—
81	Q₈	C₂²⋊D₁₀₇	1	2	yes	complemented
82	D₄	—	2	1	no	—
83	D₄	—	2	1	no	—
84	D₄	—	428	2	no	—
85	D₄	—	428	2	no	—
86	D₄	—	214	1	no	—
87	D₄	—	214	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	107	1	no	—
90	C₂×C₄	—	107	1	no	—
91	C₂×C₄	—	214	1	no	—
92	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	214	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	428	1	no	—
99	C₂²	—	428	1	no	—
100	C₂²	—	428	1	no	—
101	C₂²	D₄×D₁₀₇	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	107	2	no	—
104	C₄	—	428	1	no	—
105	C₄	—	214	1	no	—
106	C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
107	C₄	#13133817	1	1	yes	characteristic
108	C₂	—	8	1	no	—
109	C₂	—	856	1	no	—
110	C₂	#16812081	1	1	yes	characteristic
111	C₂	SD₁₆×D₁₀₇	1	1	yes	characteristic
112	C₂	#16812136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133805

C₂

yes

maximal, characteristic, complemented

#16812105

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

Dic₁₀₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16812031

C₂

yes

maximal, characteristic, complemented

#16812030

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₇

—

C₂×C₂²⋊D₁₀₇

—

#13133805

—

C₂×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

C₂²

yes

characteristic

C₂×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂×D₄×C₁₀₇

C₂²

yes

characteristic

SD₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂²

yes

characteristic

C₂³×C₁₀₇

—

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

D₄

yes

complemented

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

D₄

yes

characteristic

C₂×Dic₁₀₇

D₄

yes

characteristic

D₄₂₈

—

D₄₂₈

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₄×C₁₀₇

#41

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

SD₁₆

yes

complemented

C₂×C₁₀₇

—

C₂×C₁₀₇

#121

yes

characteristic

C₂×C₁₀₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₇

AGL(1,ℤ/8)

yes

characteristic

D₁₀₇

—

C₁₀₇

#460

yes

characteristic, complemented, Sylow

#460

—

107

maximal, Sylow

C₂×SD₁₆

—

107

—

C₂×SD₁₆

D₁₀₇

yes

characteristic, complemented

#127

—

107

—

C₄×Q₈

—

107

—

C₄⋊C₈

—

107

—

#102

—

107

—

#102

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

characteristic

C₂×D₄

—

107

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

C₂×D₄

C₂×D₁₀₇

yes

characteristic

SD₁₆

—

SD₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

214

—

C₄²

—

107

—

C₄²

—

214

—

C₂³

—

C₂³

—

214

—

Q₈

—

Q₈

C₂²⋊D₁₀₇

yes

complemented

D₄

—

D₄

—

D₄

—

428

—

D₄

—

428

—

D₄

—

214

—

D₄

—

214

—

C₂×C₄

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

214

—

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₈

—

C₈

—

214

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

428

—

C₂²

—

428

—

100

C₂²

—

428

—

101

C₂²

D₄×D₁₀₇

yes

characteristic

102

C₄

—

103

C₄

—

107

—

104

C₄

—

428

—

105

C₄

—

214

—

106

C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

107

C₄

#13133817

yes

characteristic

108

C₂

—

109

C₂

—

856

—

110

C₂

#16812081

yes

characteristic

111

C₂

SD₁₆×D₁₀₇

yes

characteristic

112

C₂

#16812136

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₂×SD₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₀₇
hypocenter	C₁₀₇	#460
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₇

Fitting

C₂×SD₁₆×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₀₇

hypocenter

C₁₀₇

#460

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#13133805	C₂	semidirect
3. Q₈×Dic₁₀₇	C₂	semidirect
4. Dic₁₀₇⋊C₈	C₂	semidirect
5. #16812031	C₂	semidirect
6. #16812030	C₂	semidirect
7. C₂×SD₁₆×C₁₀₇	C₂	semidirect
24. Q₈×C₁₀₇	D₄	semidirect
50. Dic₁₀₇	SD₁₆	semidirect
56. C₁₀₇	#460	semidirect
59. C₂×SD₁₆	D₁₀₇	semidirect
81. Q₈	C₂²⋊D₁₀₇	semidirect

H ≅ G/N

type

1. C₂×#13133805

C₂

semidirect

3. Q₈×Dic₁₀₇

C₂

semidirect

4. Dic₁₀₇⋊C₈

C₂

semidirect

5. #16812031

C₂

semidirect

6. #16812030

C₂

semidirect

7. C₂×SD₁₆×C₁₀₇

C₂

semidirect

24. Q₈×C₁₀₇

D₄

semidirect

50. Dic₁₀₇

SD₁₆

semidirect

56. C₁₀₇

#460

semidirect

59. C₂×SD₁₆

D₁₀₇

semidirect

81. Q₈

C₂²⋊D₁₀₇

semidirect

Subgroups — #20292825

Conjugacy classes of subgroups

Complements