Group #20293201 — subgroups

Subgroups — #20293201

Order 6848.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×#13133818	C₂	1	1	yes	maximal, characteristic
2	#16812107	C₂	1	2	yes	maximal, complemented
3	#16812098	C₂	1	2	yes	maximal, complemented
4	#16812092	C₂	1	4	yes	maximal, complemented
5	#16812085	C₂	1	4	yes	maximal, complemented
6	#16812068	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133818	—	2	4	no	—
9	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
10	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
14	Dic₁₀₇⋊C₄	C₂²	1	8	yes	—
15	C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
16	(C₂×D₁₀₇)⋊C₄	C₂²	1	8	yes	—
17	C₄×Dic₁₀₇	C₂²	1	2	yes	—
18	C₂×Q₈×C₁₀₇	—	2	4	no	—
19	C₂×Q₈×C₁₀₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₀₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₁₀₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
23	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₇	C₂³	1	2	yes	—
25	Q₈×C₁₀₇	—	2	4	no	—
26	Q₈×C₁₀₇	—	2	4	no	—
27	C₂×C₄×C₁₀₇	—	2	4	no	—
28	C₂×C₄×C₁₀₇	D₄	1	4	yes	—
29	C₂×C₄×C₁₀₇	C₂³	1	4	yes	—
30	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂²⋊D₁₀₇	—	2	4	no	—
33	C₂×Dic₁₀₇	C₂³	1	4	yes	—
34	C₂×Dic₁₀₇	C₂³	1	2	yes	—
35	D₄₂₈	—	2	2	no	—
36	C₄×D₁₀₇	—	2	4	no	—
37	Dic₂₁₄	—	2	2	no	—
38	C₂²×C₁₀₇	—	2	2	no	—
39	C₂²×C₁₀₇	C₂×D₄	1	2	yes	—
40	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₀₇	—	2	2	no	—
42	C₂×D₁₀₇	—	2	4	no	—
43	C₄×C₁₀₇	—	2	4	no	—
44	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
45	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
46	Dic₁₀₇	—	2	2	no	—
47	Dic₁₀₇	—	2	4	no	—
48	C₂×C₁₀₇	—	2	2	no	—
49	C₂×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₀₇	2₋¹⁺⁴	1	2	yes	—
51	D₁₀₇	—	4	2	no	—
52	C₁₀₇	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	107	1	no	maximal, Sylow
54	C₂×#41	—	107	1	no	—
55	C₂²×Q₈	D₁₀₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	107	2	no	—
57	#124	—	107	2	no	—
58	#123	—	107	4	no	—
59	C₂²⋊Q₈	—	107	4	no	—
60	#117	—	107	1	no	—
61	#41	—	214	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	107	1	no	—
64	C₂×Q₈	C₂×D₁₀₇	1	2	yes	—
65	C₂×Q₈	C₂×D₁₀₇	1	2	yes	—
66	C₂×D₄	—	107	2	no	—
67	C₂×D₄	—	107	1	no	—
68	C₂²×C₄	—	107	2	no	—
69	C₂²×C₄	C₂×D₁₀₇	1	2	yes	—
70	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₄	—	107	2	no	—
72	C₄⋊C₄	—	107	8	no	—
73	C₂²⋊C₄	—	107	2	no	—
74	C₂²⋊C₄	—	107	8	no	—
75	C₄²	—	107	2	no	—
76	C₂³	—	107	2	no	—
77	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	214	2	no	—
81	D₄	—	214	4	no	—
82	D₄	—	214	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	107	4	no	—
85	C₂×C₄	—	107	2	no	—
86	C₂×C₄	—	214	4	no	—
87	C₂×C₄	C₂²⋊D₁₀₇	1	4	yes	—
88	C₂×C₄	C₂²×D₁₀₇	1	4	yes	—
89	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	214	2	no	—
93	C₂²	—	214	4	no	—
94	C₂²	C₂×C₂²⋊D₁₀₇	1	2	yes	—
95	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	214	2	no	—
98	C₄	—	214	4	no	—
99	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
100	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	428	2	no	—
103	C₂	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
104	C₂	#16812157	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133818

C₂

yes

maximal, characteristic

#16812107

C₂

yes

maximal, complemented

#16812098

C₂

yes

maximal, complemented

#16812092

C₂

yes

maximal, complemented

#16812085

C₂

yes

maximal, complemented

#16812068

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#13133818

—

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂²

yes

—

C₂×Q₈×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

D₄

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

—

C₄×C₁₀₇

C₂×D₄

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×D₄

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

—

D₁₀₇

—

C₁₀₇

#536

yes

characteristic, complemented, Sylow

#536

—

107

maximal, Sylow

C₂×#41

—

107

—

C₂²×Q₈

D₁₀₇

yes

characteristic, complemented

C₄⋊Q₈

—

107

—

#124

—

107

—

#123

—

107

—

C₂²⋊Q₈

—

107

—

#117

—

107

—

#41

—

214

—

C₂×Q₈

—

C₂×Q₈

—

107

—

C₂×Q₈

C₂×D₁₀₇

yes

—

C₂×Q₈

C₂×D₁₀₇

yes

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₂³

—

107

—

C₂³

C₂²×D₁₀₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

214

—

D₄

—

214

—

D₄

—

214

—

C₂×C₄

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

214

—

C₂×C₄

C₂²⋊D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂²

—

C₂²

—

214

—

C₂²

—

214

—

C₂²

C₂×C₂²⋊D₁₀₇

yes

—

C₂²

C₂³×D₁₀₇

yes

characteristic

C₄

—

C₄

—

214

—

C₄

—

214

—

C₄

C₂×C₂²⋊D₁₀₇

yes

—

100

C₄

C₂×C₂²⋊D₁₀₇

yes

—

101

C₂

—

102

C₂

—

428

—

103

C₂

C₂²×C₂²⋊D₁₀₇

yes

characteristic

104

C₂

#16812157

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	C₂²×Q₈×C₁₀₇	C₂
hypercenter	C₂²×Q₈	D₁₀₇
hypocenter	C₁₀₇	#536
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

C₂²×Q₈×C₁₀₇

C₂

hypercenter

C₂²×Q₈

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #16812107	C₂	semidirect
3. #16812098	C₂	semidirect
4. #16812092	C₂	semidirect
5. #16812085	C₂	semidirect
6. #16812068	C₂	semidirect
7. C₂²×Q₈×C₁₀₇	C₂	semidirect
52. C₁₀₇	#536	semidirect
55. C₂²×Q₈	D₁₀₇	semidirect

H ≅ G/N

type

2. #16812107

C₂

semidirect

3. #16812098

C₂

semidirect

4. #16812092

C₂

semidirect

5. #16812085

C₂

semidirect

6. #16812068

C₂

semidirect

7. C₂²×Q₈×C₁₀₇

C₂

semidirect

52. C₁₀₇

#536

semidirect

55. C₂²×Q₈

D₁₀₇

semidirect

Subgroups — #20293201

Conjugacy classes of subgroups

Complements