Group #20293368 — subgroups

Subgroups — #20293368

Order 6848.

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

normal subgroups only

This group has 252 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	#16812106	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812103	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812094	C₂	1	3	yes	maximal, complemented
4	#16812084	C₂	1	3	yes	maximal, complemented
5	#16812082	C₂	1	3	yes	maximal, complemented
6	#16812079	C₂	1	3	yes	maximal, complemented
7	#126×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×D₄₂₈	C₂²	1	3	yes	—
10	C₂×D₄₂₈	C₂²	1	3	yes	—
11	C₂×D₄₂₈	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₁₀₇	C₂²	1	3	yes	—
13	C₂×C₄×D₁₀₇	C₂²	1	3	yes	complemented
14	Dic₁₀₇⋊C₄	C₂²	1	3	yes	complemented
15	(C₂×D₁₀₇)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₁₀₇)⋊C₄	C₂²	1	3	yes	—
17	(C₂×D₁₀₇)⋊C₄	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₁₀₇	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₁₀₇	C₂²	1	3	yes	—
20	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂²×D₁₀₇	—	4	3	no	—
23	C₂²×D₁₀₇	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₇	C₂³	1	1	yes	characteristic
25	C₂²×D₁₀₇	C₂³	1	3	yes	—
26	C₂×C₄×C₁₀₇	C₂³	1	3	yes	—
27	C₂×C₄×C₁₀₇	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₇	—	4	3	no	—
29	C₂×Dic₁₀₇	C₂³	1	3	yes	—
30	D₄₂₈	—	4	3	no	—
31	D₄₂₈	—	4	3	no	—
32	D₄₂₈	—	4	3	no	—
33	C₄×D₁₀₇	—	4	3	no	—
34	C₂²×C₁₀₇	—	2	3	no	—
35	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₀₇	—	2	3	no	—
37	C₂×D₁₀₇	—	2	3	no	—
38	C₂×D₁₀₇	—	2	3	no	—
39	C₂×D₁₀₇	—	2	3	no	—
40	C₂×D₁₀₇	—	2	3	no	—
41	C₂×D₁₀₇	—	4	4	no	—
42	C₄×C₁₀₇	—	2	3	no	—
43	C₄×C₁₀₇	—	2	3	no	—
44	Dic₁₀₇	—	2	3	no	—
45	C₂×C₁₀₇	—	4	1	no	—
46	C₂×C₁₀₇	H(5,𝔽₂)	1	3	yes	—
47	D₁₀₇	—	4	3	no	—
48	D₁₀₇	—	4	1	no	—
49	D₁₀₇	—	4	1	no	—
50	C₁₀₇	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	107	1	no	maximal, Sylow
52	#127	—	107	1	no	—
53	#126	—	107	1	no	—
54	#126	D₁₀₇	1	1	yes	characteristic, complemented
55	#123	—	107	3	no	—
56	#121	—	107	3	no	—
57	#121	—	107	3	no	—
58	#120	—	107	3	no	—
59	C₂⁴	—	107	1	no	—
60	C₂×D₄	—	107	3	no	—
61	C₂×D₄	—	107	3	no	—
62	C₂×D₄	—	107	3	no	—
63	C₂×D₄	—	107	3	no	—
64	C₂²×C₄	—	107	3	no	—
65	C₄⋊C₄	—	107	3	no	—
66	C₄⋊C₄	C₂×D₁₀₇	1	3	yes	complemented
67	C₂²⋊C₄	—	107	3	no	—
68	C₂²⋊C₄	—	107	3	no	—
69	C₂²⋊C₄	—	107	3	no	—
70	C₂²⋊C₄	C₂×D₁₀₇	1	3	yes	—
71	C₄²	C₂×D₁₀₇	1	1	yes	characteristic, complemented
72	C₂³	—	107	1	no	—
73	C₂³	—	107	1	no	—
74	C₂³	—	107	3	no	—
75	C₂³	—	428	3	no	—
76	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
77	D₄	—	428	3	no	—
78	D₄	—	428	3	no	—
79	D₄	—	428	3	no	—
80	D₄	—	428	3	no	—
81	C₂×C₄	—	107	3	no	—
82	C₂×C₄	—	428	3	no	—
83	C₂×C₄	C₂²×D₁₀₇	1	3	yes	—
84	C₂×C₄	C₂²×D₁₀₇	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	428	4	no	—
87	C₂²	—	214	3	no	—
88	C₂²	—	214	3	no	—
89	C₂²	—	214	3	no	—
90	C₂²	—	214	3	no	—
91	C₂²	—	214	3	no	—
92	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	214	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	428	1	no	—
98	C₂	—	428	1	no	—
99	C₂	—	428	3	no	—
100	C₂	#16812155	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16812106

C₂

yes

maximal, characteristic, complemented

#16812103

C₂

yes

maximal, characteristic, complemented

#16812094

C₂

yes

maximal, complemented

#16812084

C₂

yes

maximal, complemented

#16812082

C₂

yes

maximal, complemented

#16812079

C₂

yes

maximal, complemented

#126×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₇

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

—

C₂×D₄₂₈

C₂²

yes

—

C₂×D₄₂₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

complemented

Dic₁₀₇⋊C₄

C₂²

yes

complemented

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

complemented

C₄⋊C₄×C₁₀₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

characteristic, complemented

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

—

C₂²×D₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

C₄×D₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

H(5,𝔽₂)

yes

—

D₁₀₇

—

D₁₀₇

—

D₁₀₇

—

C₁₀₇

#560

yes

characteristic, complemented, Sylow

#560

—

107

maximal, Sylow

#127

—

107

—

#126

—

107

—

#126

D₁₀₇

yes

characteristic, complemented

#123

—

107

—

#121

—

107

—

#121

—

107

—

#120

—

107

—

C₂⁴

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

complemented

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

C₂×D₁₀₇

yes

—

C₄²

C₂×D₁₀₇

yes

characteristic, complemented

C₂³

—

107

—

C₂³

—

107

—

C₂³

—

107

—

C₂³

—

428

—

C₂³

C₂²×D₁₀₇

yes

characteristic

D₄

—

428

—

D₄

—

428

—

D₄

—

428

—

D₄

—

428

—

C₂×C₄

—

107

—

C₂×C₄

—

428

—

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂²

—

C₂²

—

428

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

C₂³×D₁₀₇

yes

characteristic

C₄

—

C₄

—

C₄

—

214

—

C₂

—

C₂

—

428

—

C₂

—

428

—

C₂

—

428

—

100

C₂

#16812155

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#126×C₁₀₇	C₂
hypercenter	#126	D₁₀₇
hypocenter	C₁₀₇	#560
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

#126×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #16812106	C₂	semidirect
2. #16812103	C₂	semidirect
3. #16812094	C₂	semidirect
4. #16812084	C₂	semidirect
5. #16812082	C₂	semidirect
6. #16812079	C₂	semidirect
7. #126×C₁₀₇	C₂	semidirect
11. C₂×D₄₂₈	C₂²	semidirect
13. C₂×C₄×D₁₀₇	C₂²	semidirect
14. Dic₁₀₇⋊C₄	C₂²	semidirect
17. (C₂×D₁₀₇)⋊C₄	C₂²	semidirect
18. C₄⋊C₄×C₁₀₇	C₂²	semidirect
20. C₄²×C₁₀₇	C₂²	semidirect
50. C₁₀₇	#560	semidirect
54. #126	D₁₀₇	semidirect
66. C₄⋊C₄	C₂×D₁₀₇	semidirect
71. C₄²	C₂×D₁₀₇	semidirect

H ≅ G/N

type

1. #16812106

C₂

semidirect

2. #16812103

C₂

semidirect

3. #16812094

C₂

semidirect

4. #16812084

C₂

semidirect

5. #16812082

C₂

semidirect

6. #16812079

C₂

semidirect

7. #126×C₁₀₇

C₂

semidirect

11. C₂×D₄₂₈

C₂²

semidirect

13. C₂×C₄×D₁₀₇

C₂²

semidirect

14. Dic₁₀₇⋊C₄

C₂²

semidirect

17. (C₂×D₁₀₇)⋊C₄

C₂²

semidirect

18. C₄⋊C₄×C₁₀₇

C₂²

semidirect

20. C₄²×C₁₀₇

C₂²

semidirect

50. C₁₀₇

#560

semidirect

54. #126

D₁₀₇

semidirect

66. C₄⋊C₄

C₂×D₁₀₇

semidirect

71. C₄²

C₂×D₁₀₇

semidirect

Subgroups — #20293368

Conjugacy classes of subgroups

Complements