Group #20292699 — subgroups

Subgroups — #20292699

Order 6848.

124 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13133802	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812020	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	#13133802	—	2	2	no	—
9	C₂×C₁₀₇⋊C₈	—	2	2	no	—
10	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₀₇	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₀₇	C₄	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₀₇	C₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₀₇	C₄	1	1	yes	characteristic
20	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	D₄×C₁₀₇	—	2	2	no	—
24	C₂×C₄×C₁₀₇	—	2	1	no	—
25	C₂×C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄×C₁₀₇	—	2	1	no	—
28	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₁₀₇⋊C₈	—	2	1	no	—
34	C₁₀₇⋊C₈	—	2	1	no	—
35	C₁₀₇⋊C₈	—	2	1	no	—
36	C₁₀₇⋊C₈	D₄	1	2	yes	complemented
37	C₂²×C₁₀₇	—	2	1	no	—
38	C₂²×C₁₀₇	—	2	1	no	—
39	C₂²×C₁₀₇	—	2	1	no	—
40	C₂²×C₁₀₇	—	2	1	no	—
41	C₂²×C₁₀₇	—	2	1	no	—
42	C₂²×C₁₀₇	M₁₆	1	2	yes	complemented
43	C₂²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₀₇	—	2	1	no	—
45	C₄×C₁₀₇	—	2	1	no	—
46	C₄×C₁₀₇	—	2	1	no	—
47	C₄×C₁₀₇	—	2	1	no	—
48	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
49	C₄×C₁₀₇	#41	1	1	yes	characteristic
50	C₂×C₁₀₇	—	2	2	no	—
51	C₂×C₁₀₇	—	4	1	no	—
52	C₂×C₁₀₇	C₄×D₄	1	1	yes	characteristic
53	C₂×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₁₀₇	#131	1	1	yes	characteristic
55	C₁₀₇	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	107	1	no	maximal, Sylow
57	C₂×M₁₆	—	107	1	no	—
58	C₂²×C₈	—	107	1	no	—
59	C₄×D₄	D₁₀₇	1	1	yes	characteristic
60	C₄⋊C₈	—	107	1	no	—
61	C₂²⋊C₈	—	107	1	no	—
62	C₂²⋊C₈	—	107	1	no	—
63	#97	—	107	1	no	—
64	C₂×D₄	Dic₁₀₇	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
67	M₁₆	—	214	2	no	—
68	C₂×C₈	—	107	1	no	—
69	C₂×C₈	—	107	1	no	—
70	C₂×C₈	—	107	1	no	—
71	C₂×C₈	—	107	1	no	—
72	C₂×C₈	—	214	2	no	—
73	C₄⋊C₄	Dic₁₀₇	1	1	yes	characteristic
74	C₂²⋊C₄	Dic₁₀₇	1	1	yes	characteristic
75	C₂²⋊C₄	Dic₁₀₇	1	1	yes	characteristic
76	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
77	C₂³	C₂×Dic₁₀₇	1	1	yes	characteristic
78	C₂³	C₂×Dic₁₀₇	1	1	yes	characteristic
79	D₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
89	C₈	—	107	2	no	—
90	C₈	—	214	1	no	—
91	C₈	—	214	1	no	—
92	C₈	—	214	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#13133802	1	2	yes	complemented
99	C₂²	C₂²×Dic₁₀₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	D₄×D₁₀₇	1	1	yes	characteristic
105	C₄	#13133817	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	4	1	no	—
108	C₂	D₄×Dic₁₀₇	1	1	yes	characteristic
109	C₂	C₂×#13133802	1	1	yes	characteristic
110	C₂	#16812116	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133802

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#16812020

C₂

yes

maximal, characteristic, complemented

C₄×D₄×C₁₀₇

C₂

yes

maximal, characteristic

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×D₄×C₁₀₇

C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₄

yes

characteristic

C₂²⋊C₄×C₁₀₇

C₄

yes

characteristic

C₂²⋊C₄×C₁₀₇

C₄

yes

characteristic

C₄²×C₁₀₇

C₂²

yes

characteristic

C₂³×C₁₀₇

C₂×C₄

yes

characteristic

C₂³×C₁₀₇

C₂×C₄

yes

characteristic

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

D₄

yes

complemented

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

M₁₆

yes

complemented

C₂²×C₁₀₇

C₂²×C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₄×C₁₀₇

#41

yes

characteristic

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₄×D₄

yes

characteristic

C₂×C₁₀₇

C₂×M₁₆

yes

C₂×C₁₀₇

yes

C₁₀₇

yes

characteristic, complemented, Sylow

#435

—

107

maximal, Sylow

C₂×M₁₆

—

107

—

C₂²×C₈

—

107

—

C₄×D₄

D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₂²⋊C₈

—

107

—

C₂²⋊C₈

—

107

—

#97

—

107

—

C₂×D₄

Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₄⋊C₄

Dic₁₀₇

yes

characteristic

C₂²⋊C₄

Dic₁₀₇

yes

characteristic

C₂²⋊C₄

Dic₁₀₇

yes

characteristic

C₄²

C₂×D₁₀₇

yes

characteristic

C₂³

C₂×Dic₁₀₇

yes

characteristic

C₂³

C₂×Dic₁₀₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₈

—

107

—

C₈

—

214

—

C₈

—

214

—

C₈

—

214

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133802

yes

complemented

C₂²

C₂²×Dic₁₀₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

D₄×D₁₀₇

yes

characteristic

105

C₄

#13133817

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

D₄×Dic₁₀₇

yes

characteristic

109

C₂

C₂×#13133802

yes

characteristic

110

C₂

#16812116

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂²×C₄
Fitting	C₄×D₄×C₁₀₇	C₂
hypercenter	C₄×D₄	D₁₀₇
hypocenter	C₁₀₇	#435
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂²×C₄

Fitting

C₄×D₄×C₁₀₇

C₂

hypercenter

C₄×D₄

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#13133802	C₂	semidirect
2. C₂²×C₁₀₇⋊C₈	C₂	semidirect
3. C₄⋊(C₁₀₇⋊C₈)	C₂	semidirect
4. C₂²⋊(C₁₀₇⋊C₈)	C₂	semidirect
5. C₂²⋊(C₁₀₇⋊C₈)	C₂	semidirect
6. #16812020	C₂	semidirect
36. C₁₀₇⋊C₈	D₄	semidirect
42. C₂²×C₁₀₇	M₁₆	semidirect
55. C₁₀₇	#435	semidirect
98. C₂²	#13133802	semidirect

H ≅ G/N

type

1. C₂×#13133802

C₂

semidirect

2. C₂²×C₁₀₇⋊C₈

C₂

semidirect

3. C₄⋊(C₁₀₇⋊C₈)

C₂

semidirect

4. C₂²⋊(C₁₀₇⋊C₈)

C₂

semidirect

5. C₂²⋊(C₁₀₇⋊C₈)

C₂

semidirect

6. #16812020

C₂

semidirect

36. C₁₀₇⋊C₈

D₄

semidirect

42. C₂²×C₁₀₇

M₁₆

semidirect

55. C₁₀₇

#435

semidirect

98. C₂²

#13133802

semidirect

Subgroups — #20292699

Conjugacy classes of subgroups

Complements