Group #20293350 — subgroups

Subgroups — #20293350

Order 6848.

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

normal subgroups only

This group has 212 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	#16812109	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812104	C₂	1	4	yes	maximal, complemented
3	#16812097	C₂	1	2	yes	maximal
4	#16812085	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₁₄	C₂	1	2	yes	maximal, complemented
6	C₄×D₄₂₈	C₂	1	1	yes	maximal, characteristic
7	#16812067	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₁₄	C₂²	1	2	yes	—
10	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
12	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
15	Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
16	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
17	(C₂×D₁₀₇)⋊C₄	C₂²	1	2	yes	—
18	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₀₇	C₂²	1	4	yes	—
20	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
23	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
24	C₄²×C₁₀₇	C₂²	1	2	yes	—
25	C₂²×D₁₀₇	C₂³	1	2	yes	—
26	Q₈×C₁₀₇	—	2	2	no	—
27	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
30	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₀₇	C₂³	1	2	yes	—
33	C₂×Dic₁₀₇	C₂³	1	4	yes	—
34	D₄₂₈	—	2	2	no	—
35	C₄×D₁₀₇	—	2	4	no	—
36	Dic₂₁₄	—	2	4	no	—
37	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₀₇	—	2	2	no	—
39	C₂×D₁₀₇	—	2	2	no	—
40	C₂×D₁₀₇	—	2	2	no	—
41	C₄×C₁₀₇	—	2	2	no	—
42	C₄×C₁₀₇	—	2	2	no	—
43	C₄×C₁₀₇	—	2	1	no	—
44	C₄×C₁₀₇	#41	1	2	yes	—
45	C₄×C₁₀₇	#41	1	2	yes	—
46	Dic₁₀₇	—	2	4	no	—
47	Dic₁₀₇	—	2	2	no	—
48	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
49	C₂×C₁₀₇	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₀₇	—	4	2	no	—
52	C₁₀₇	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	107	1	no	maximal, Sylow
54	C₄⋊Q₈	—	107	1	no	—
55	#126	—	107	4	no	—
56	#124	—	107	2	no	—
57	C₂²⋊Q₈	—	107	2	no	—
58	C₄×Q₈	—	107	2	no	—
59	C₄×Q₈	D₁₀₇	1	1	yes	characteristic, complemented
60	C₄×D₄	—	107	1	no	—
61	#117	—	107	2	no	—
62	C₂×Q₈	—	107	2	no	—
63	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
64	C₂×D₄	—	107	1	no	—
65	C₂²×C₄	—	107	2	no	—
66	C₄⋊C₄	—	107	2	no	—
67	C₄⋊C₄	—	107	1	no	—
68	C₄⋊C₄	—	107	4	no	—
69	C₄⋊C₄	—	107	2	no	—
70	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
72	C₂²⋊C₄	—	107	4	no	—
73	C₂²⋊C₄	—	107	2	no	—
74	C₂²⋊C₄	—	107	4	no	—
75	C₄²	—	107	4	no	—
76	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
77	C₄²	C₂×D₁₀₇	1	2	yes	—
78	C₂³	—	107	2	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	214	4	no	—
81	D₄	—	214	2	no	—
82	C₂×C₄	—	107	2	no	—
83	C₂×C₄	—	107	4	no	—
84	C₂×C₄	—	214	4	no	—
85	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
86	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
88	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₂²	—	214	2	no	—
91	C₂²	—	214	2	no	—
92	C₂²	—	214	2	no	—
93	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	—	214	2	no	—
98	C₄	—	214	4	no	—
99	C₄	#13133818	1	2	yes	—
100	C₄	#13133819	1	2	yes	—
101	C₂	—	428	2	no	—
102	C₂	C₂×#13133818	1	1	yes	characteristic
103	C₂	C₂×#13133819	1	1	yes	characteristic
104	C₂	#16812156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812109

C₂

yes

maximal, characteristic, complemented

#16812104

C₂

yes

maximal, complemented

#16812097

C₂

yes

maximal

#16812085

C₂

yes

maximal, complemented

C₄×Dic₂₁₄

C₂

yes

maximal, complemented

C₄×D₄₂₈

C₂

yes

maximal, characteristic

#16812067

C₂

yes

maximal, complemented

C₄×Q₈×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₄

C₂²

yes

—

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₂×Q₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

characteristic

C₄²×C₁₀₇

C₂²

yes

—

C₂²×D₁₀₇

C₂³

yes

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#41

yes

—

C₄×C₁₀₇

#41

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

C₂×#41

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

D₁₀₇

—

C₁₀₇

#556

yes

characteristic, complemented, Sylow

#556

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

#126

—

107

—

#124

—

107

—

C₂²⋊Q₈

—

107

—

C₄×Q₈

—

107

—

C₄×Q₈

D₁₀₇

yes

characteristic, complemented

C₄×D₄

—

107

—

#117

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

characteristic

C₂×D₄

—

107

—

C₂²×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

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₄²

C₂×D₁₀₇

yes

characteristic

C₄²

C₂×D₁₀₇

yes

—

C₂³

—

107

—

Q₈

—

Q₈

—

214

—

D₄

—

214

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

214

—

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂²

—

214

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

C₂³×D₁₀₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

214

—

C₄

—

214

—

C₄

#13133818

yes

—

100

C₄

#13133819

yes

—

101

C₂

—

428

—

102

C₂

C₂×#13133818

yes

characteristic

103

C₂

C₂×#13133819

yes

characteristic

104

C₂

#16812156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₄×Q₈×C₁₀₇	C₂
hypercenter	C₄×Q₈	D₁₀₇
hypocenter	C₁₀₇	#556
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₄×Q₈×C₁₀₇

C₂

hypercenter

C₄×Q₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #16812109	C₂	semidirect
2. #16812104	C₂	semidirect
4. #16812085	C₂	semidirect
5. C₄×Dic₂₁₄	C₂	semidirect
7. #16812067	C₂	semidirect
8. C₄×Q₈×C₁₀₇	C₂	semidirect
52. C₁₀₇	#556	semidirect
59. C₄×Q₈	D₁₀₇	semidirect

H ≅ G/N

type

1. #16812109

C₂

semidirect

2. #16812104

C₂

semidirect

4. #16812085

C₂

semidirect

5. C₄×Dic₂₁₄

C₂

semidirect

7. #16812067

C₂

semidirect

8. C₄×Q₈×C₁₀₇

C₂

semidirect

52. C₁₀₇

#556

semidirect

59. C₄×Q₈

D₁₀₇

semidirect

Subgroups — #20293350

Conjugacy classes of subgroups

Complements