Group #20295106 — subgroups

Subgroups — #20295106

Order 6976.

134 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 51 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×#13182014	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812661	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812621	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	—	2	1	no	—
10	C₂×C₄×D₁₀₉	—	2	1	no	—
11	#13182014	C₄	1	4	yes	complemented
12	C₂×#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²⋊Dic₁₀₉	—	2	1	no	—
15	#13182006	—	2	1	no	—
16	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂³×C₁₀₉	—	2	1	no	—
21	C₂²×D₁₀₉	—	2	1	no	—
22	D₄×C₁₀₉	—	2	1	no	—
23	D₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀₉	—	2	1	no	—
27	C₂×Dic₁₀₉	—	4	2	no	—
28	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
30	D₄₃₆	—	2	1	no	—
31	D₄₃₆	C₂×C₄	1	2	yes	complemented
32	C₄×D₁₀₉	—	4	2	no	—
33	C₈×C₁₀₉	—	2	1	no	—
34	#11724784	C₂×C₄	1	2	yes	complemented
35	C₂²×C₁₀₉	—	2	2	no	—
36	C₂²×C₁₀₉	—	4	1	no	—
37	C₂²×C₁₀₉	—	4	1	no	—
38	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	2	no	—
40	C₂×D₁₀₉	—	4	1	no	—
41	C₂×D₁₀₉	—	4	1	no	—
42	C₄×C₁₀₉	—	4	1	no	—
43	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₀₉	#41	1	1	yes	characteristic
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	4	1	no	—
47	Dic₁₀₉	D₈	1	2	yes	complemented
48	C₂×C₁₀₉	—	4	2	no	—
49	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₁₀₉	#135	1	1	yes	characteristic
52	D₁₀₉	—	4	2	no	—
53	C₁₀₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
54	C₄×D₈	—	109	1	no	maximal, Sylow
55	C₂×D₈	—	109	1	no	—
56	C₄×D₄	—	109	1	no	—
57	C₄×D₄	—	109	1	no	—
58	#107	—	109	1	no	—
59	#102	—	109	1	no	—
60	#102	D₁₀₉	1	1	yes	characteristic, complemented
61	C₄×C₈	—	109	1	no	—
62	C₂×D₄	—	109	1	no	—
63	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
64	C₂²×C₄	—	218	1	no	—
65	C₂²×C₄	—	218	1	no	—
66	D₈	—	109	4	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₄⋊C₄	—	109	1	no	—
70	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂²⋊C₄	—	218	1	no	—
72	C₂²⋊C₄	—	218	1	no	—
73	C₄²	—	109	1	no	—
74	C₂³	—	2	1	no	—
75	C₂³	—	218	1	no	—
76	D₄	—	2	1	no	—
77	D₄	—	109	2	no	—
78	D₄	—	218	1	no	—
79	D₄	C₄×D₁₀₉	1	2	yes	complemented
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	—	436	2	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
87	C₈	—	2	1	no	—
88	C₈	—	109	2	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	4	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	436	1	no	—
93	C₂²	—	436	1	no	—
94	C₂²	—	218	2	no	—
95	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
96	C₄	—	4	1	no	—
97	C₄	—	109	2	no	—
98	C₄	—	436	1	no	—
99	C₄	—	218	1	no	—
100	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
101	C₄	#13182028	1	1	yes	characteristic
102	C₂	—	4	2	no	—
103	C₂	—	436	2	no	—
104	C₂	#16812659	1	1	yes	characteristic
105	C₂	D₈×D₁₀₉	1	1	yes	characteristic
106	C₂	#16812720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182014

C₂

yes

maximal, characteristic

D₄×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812661

C₂

yes

maximal, characteristic, complemented

#16812632

C₂

yes

maximal, characteristic, complemented

#16812621

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#102×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄₃₆

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

—

#13182014

C₄

yes

complemented

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂²⋊Dic₁₀₉

—

#13182006

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₂×Dic₁₀₉

D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

complemented

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

C₂×C₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

D₈

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₁₀₉

#135

yes

characteristic

D₁₀₉

—

C₁₀₉

C₄×D₈

yes

characteristic, complemented, Sylow

C₄×D₈

—

109

maximal, Sylow

C₂×D₈

—

109

—

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#107

—

109

—

#102

—

109

—

#102

D₁₀₉

yes

characteristic, complemented

C₄×C₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

D₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

C₂³

—

218

—

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

C₄×D₁₀₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

D₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

100

C₄

C₂×C₄×D₁₀₉

yes

characteristic

101

C₄

#13182028

yes

characteristic

102

C₂

—

103

C₂

—

436

—

104

C₂

#16812659

yes

characteristic

105

C₂

D₈×D₁₀₉

yes

characteristic

106

C₂

#16812720

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#102×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#102	D₁₀₉
hypocenter	C₁₀₉	C₄×D₈
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#102×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

#102

D₁₀₉

hypocenter

C₁₀₉

C₄×D₈

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. D₄×Dic₁₀₉	C₂	semidirect
3. #16812661	C₂	semidirect
4. #16812632	C₂	semidirect
5. #16812621	C₂	semidirect
6. C₈×Dic₁₀₉	C₂	semidirect
7. #102×C₁₀₉	C₂	semidirect
11. #13182014	C₄	semidirect
23. D₄×C₁₀₉	C₂×C₄	semidirect
31. D₄₃₆	C₂×C₄	semidirect
34. #11724784	C₂×C₄	semidirect
47. Dic₁₀₉	D₈	semidirect
53. C₁₀₉	C₄×D₈	semidirect
60. #102	D₁₀₉	semidirect
79. D₄	C₄×D₁₀₉	semidirect

H ≅ G/N

type

2. D₄×Dic₁₀₉

C₂

semidirect

3. #16812661

C₂

semidirect

4. #16812632

C₂

semidirect

5. #16812621

C₂

semidirect

6. C₈×Dic₁₀₉

C₂

semidirect

7. #102×C₁₀₉

C₂

semidirect

11. #13182014

C₄

semidirect

23. D₄×C₁₀₉

C₂×C₄

semidirect

31. D₄₃₆

C₂×C₄

semidirect

34. #11724784

C₂×C₄

semidirect

47. Dic₁₀₉

D₈

semidirect

53. C₁₀₉

C₄×D₈

semidirect

60. #102

D₁₀₉

semidirect

79. D₄

C₄×D₁₀₉

semidirect

Subgroups — #20295106

Conjugacy classes of subgroups

Complements