Group #20293203 — subgroups

Subgroups — #20293203

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	C₂²×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812109	C₂	1	2	yes	maximal, complemented
4	#16812097	C₂	1	2	yes	maximal, complemented
5	#16812093	C₂	1	4	yes	maximal, complemented
6	#16812089	C₂	1	4	yes	maximal, complemented
7	#117×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133818	—	2	4	no	—
9	C₂×Dic₂₁₄	—	2	4	no	—
10	C₂×Dic₂₁₄	C₂²	1	2	yes	—
11	C₂×Dic₂₁₄	C₂²	1	2	yes	—
12	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic
13	C₂×D₄₂₈	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
15	C₂²×Dic₁₀₇	C₂²	1	2	yes	—
16	C₂×C₄×D₁₀₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₇	C₂²	1	8	yes	—
18	(C₂×D₁₀₇)⋊C₄	C₂²	1	8	yes	—
19	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
22	C₄²×C₁₀₇	C₂²	1	2	yes	—
23	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₇	C₂³	1	2	yes	—
25	C₂×C₄×C₁₀₇	D₄	1	4	yes	—
26	C₂×C₄×C₁₀₇	C₂³	1	4	yes	—
27	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₇	—	2	4	no	—
30	C₂×Dic₁₀₇	—	2	4	no	—
31	C₂×Dic₁₀₇	C₂³	1	4	yes	—
32	C₂×Dic₁₀₇	C₂³	1	2	yes	—
33	D₄₂₈	—	2	2	no	—
34	C₄×D₁₀₇	—	2	4	no	—
35	Dic₂₁₄	—	2	4	no	—
36	Dic₂₁₄	—	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	4	no	—
47	Dic₁₀₇	—	2	2	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	—	107	1	no	—
56	C₄⋊Q₈	—	107	2	no	—
57	#124	—	107	2	no	—
58	#123	—	107	4	no	—
59	C₂²⋊Q₈	—	107	4	no	—
60	#117	D₁₀₇	1	1	yes	characteristic, complemented
61	#41	—	214	4	no	—
62	C₂×Q₈	—	107	2	no	—
63	C₂×Q₈	—	107	2	no	—
64	C₂×Q₈	—	107	1	no	—
65	C₂×Q₈	—	214	4	no	—
66	C₂×D₄	—	107	2	no	—
67	C₂×D₄	—	107	1	no	—
68	C₂²×C₄	—	107	2	no	—
69	C₂²×C₄	—	107	2	no	—
70	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₄	—	107	8	no	—
72	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
73	C₂²⋊C₄	—	107	8	no	—
74	C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
75	C₄²	C₂×D₁₀₇	1	2	yes	—
76	C₂³	—	107	2	no	—
77	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
78	Q₈	—	214	4	no	—
79	Q₈	—	214	4	no	—
80	Q₈	—	214	2	no	—
81	D₄	—	214	4	no	—
82	D₄	—	214	2	no	—
83	C₂×C₄	—	107	4	no	—
84	C₂×C₄	—	107	2	no	—
85	C₂×C₄	—	214	4	no	—
86	C₂×C₄	—	214	4	no	—
87	C₂×C₄	D₄₂₈	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×D₄₂₈	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×D₄₂₈	1	2	yes	—
100	C₄	C₂×D₄₂₈	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	428	2	no	—
103	C₂	C₂²×D₄₂₈	1	1	yes	characteristic
104	C₂	#16812156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133818

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂

yes

maximal, characteristic, complemented

#16812109

C₂

yes

maximal, complemented

#16812097

C₂

yes

maximal, complemented

#16812093

C₂

yes

maximal, complemented

#16812089

C₂

yes

maximal, complemented

#117×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#13133818

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂²

yes

—

C₂×Dic₂₁₄

C₂²

yes

—

C₂×Dic₂₁₄

C₂²

yes

characteristic

C₂×D₄₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂²×Dic₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

C₂³

yes

—

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

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

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

—

107

—

C₄⋊Q₈

—

107

—

#124

—

107

—

#123

—

107

—

C₂²⋊Q₈

—

107

—

#117

D₁₀₇

yes

characteristic, complemented

#41

—

214

—

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

C₂×D₁₀₇

yes

—

C₄²

C₂×D₁₀₇

yes

—

C₂³

—

107

—

C₂³

C₂²×D₁₀₇

yes

characteristic

Q₈

—

214

—

Q₈

—

214

—

Q₈

—

214

—

D₄

—

214

—

D₄

—

214

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

D₄₂₈

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×D₄₂₈

yes

—

C₂²

C₂³×D₁₀₇

yes

characteristic

C₄

—

C₄

—

214

—

C₄

—

214

—

C₄

C₂×D₄₂₈

yes

—

100

C₄

C₂×D₄₂₈

yes

—

101

C₂

—

102

C₂

—

428

—

103

C₂

C₂²×D₄₂₈

yes

characteristic

104

C₂

#16812156

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#117×C₁₀₇	C₂
hypercenter	#117	D₁₀₇
hypocenter	C₁₀₇	#536
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

#117×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₂²×Dic₂₁₄	C₂	semidirect
3. #16812109	C₂	semidirect
4. #16812097	C₂	semidirect
5. #16812093	C₂	semidirect
6. #16812089	C₂	semidirect
7. #117×C₁₀₇	C₂	semidirect
52. C₁₀₇	#536	semidirect
60. #117	D₁₀₇	semidirect

H ≅ G/N

type

2. C₂²×Dic₂₁₄

C₂

semidirect

3. #16812109

C₂

semidirect

4. #16812097

C₂

semidirect

5. #16812093

C₂

semidirect

6. #16812089

C₂

semidirect

7. #117×C₁₀₇

C₂

semidirect

52. C₁₀₇

#536

semidirect

60. #117

D₁₀₇

semidirect

Subgroups — #20293203

Conjugacy classes of subgroups

Complements