Group #20295043 — subgroups

Subgroups — #20295043

Order 6976.

154 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 67 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×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812658	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812626	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#13182013	—	2	1	no	—
6	#13182013	C₂²	1	2	yes	complemented
7	C₂×#11724784	—	2	1	no	—
8	C₄⋊Dic₁₀₉	—	2	1	no	—
9	C₂²⋊Dic₁₀₉	—	2	1	no	—
10	C₄×Dic₁₀₉	C₂²	1	2	yes	complemented
11	#41×C₁₀₉	—	2	1	no	—
12	#41×C₁₀₉	—	2	1	no	—
13	#41×C₁₀₉	C₄	1	2	yes	complemented
14	#41×C₁₀₉	C₂²	1	2	yes	—
15	C₂×Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂×D₄×C₁₀₉	—	2	1	no	—
17	C₂×D₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×C₄×C₁₀₉	—	2	1	no	—
19	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂³×C₁₀₉	—	2	1	no	—
21	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
22	Q₈×C₁₀₉	—	2	1	no	—
23	Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	D₄×C₁₀₉	—	2	2	no	—
25	D₄×C₁₀₉	—	2	2	no	—
26	D₄×C₁₀₉	—	2	1	no	—
27	D₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₀₉	—	2	2	no	—
38	#11724784	—	2	2	no	—
39	C₂²×C₁₀₉	—	2	2	no	—
40	C₂²×C₁₀₉	—	4	1	no	—
41	C₂²×C₁₀₉	—	4	2	no	—
42	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₄×C₁₀₉	—	2	2	no	—
46	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	Dic₁₀₉	—	4	2	no	—
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	2	1	no	—
53	C₂×C₁₀₉	—	2	1	no	—
54	C₂×C₁₀₉	—	4	2	no	—
55	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₁₀₉	#421	1	1	yes	characteristic, complemented, Sylow
57	#421	—	109	1	no	maximal, Sylow
58	C₂×#41	D₁₀₉	1	1	yes	characteristic
59	C₂×M₁₆	—	109	1	no	—
60	#117	—	109	1	no	—
61	#104	—	109	4	no	—
62	#41	—	2	1	no	—
63	#41	—	2	1	no	—
64	#41	Dic₁₀₉	1	2	yes	complemented
65	#41	C₂×D₁₀₉	1	2	yes	—
66	C₂×Q₈	Dic₁₀₉	1	1	yes	characteristic, complemented
67	C₂×D₄	—	2	1	no	—
68	C₂×D₄	Dic₁₀₉	1	1	yes	characteristic, complemented
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	M₁₆	—	109	2	no	—
72	M₁₆	—	218	1	no	—
73	C₂×C₈	—	218	1	no	—
74	C₄⋊C₄	—	218	1	no	—
75	C₂²⋊C₄	—	218	1	no	—
76	C₄²	—	109	2	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
79	Q₈	—	2	1	no	—
80	Q₈	C₂×Dic₁₀₉	1	2	yes	—
81	D₄	—	2	2	no	—
82	D₄	—	2	2	no	—
83	D₄	—	2	1	no	—
84	D₄	C₂×Dic₁₀₉	1	2	yes	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	218	2	no	—
89	C₂×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
95	C₈	—	218	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	4	2	no	—
99	C₂²	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
100	C₂²	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
101	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	436	2	no	—
104	C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
105	C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
106	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
107	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	4	2	no	—
112	C₂	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182013

C₂

yes

maximal, characteristic, complemented

#16812658

C₂

yes

maximal, characteristic, complemented

#16812626

C₂

yes

maximal, complemented

C₂×#41×C₁₀₉

C₂

yes

maximal, characteristic

#13182013

—

#13182013

C₂²

yes

complemented

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂²

yes

complemented

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

C₄

yes

complemented

#41×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₄

yes

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₁₀₉

#421

yes

characteristic, complemented, Sylow

#421

—

109

maximal, Sylow

C₂×#41

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

#117

—

109

—

#104

—

109

—

#41

—

#41

—

#41

Dic₁₀₉

yes

complemented

#41

C₂×D₁₀₉

yes

—

C₂×Q₈

Dic₁₀₉

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

Dic₁₀₉

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

C₂³

C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

Q₈

C₂×Dic₁₀₉

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂×Dic₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊Dic₁₀₉

yes

characteristic

100

C₂²

C₂²⋊Dic₁₀₉

yes

characteristic

101

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

C₄

—

103

C₄

—

436

—

104

C₄

C₂²⋊Dic₁₀₉

yes

characteristic

105

C₄

C₂²⋊Dic₁₀₉

yes

characteristic

106

C₄

C₂²×Dic₁₀₉

yes

characteristic

107

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×#41×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₂²⋊C₄
hypercenter	C₂×#41	D₁₀₉
hypocenter	C₁₀₉	#421
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂×#41×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×C₂²⋊C₄

hypercenter

C₂×#41

D₁₀₉

hypocenter

C₁₀₉

#421

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#13182013	C₂	semidirect
2. #16812658	C₂	semidirect
3. #16812626	C₂	semidirect
6. #13182013	C₂²	semidirect
10. C₄×Dic₁₀₉	C₂²	semidirect
13. #41×C₁₀₉	C₄	semidirect
15. C₂×Q₈×C₁₀₉	C₄	semidirect
17. C₂×D₄×C₁₀₉	C₄	semidirect
56. C₁₀₉	#421	semidirect
64. #41	Dic₁₀₉	semidirect
66. C₂×Q₈	Dic₁₀₉	semidirect
68. C₂×D₄	Dic₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182013

C₂

semidirect

2. #16812658

C₂

semidirect

3. #16812626

C₂

semidirect

6. #13182013

C₂²

semidirect

10. C₄×Dic₁₀₉

C₂²

semidirect

13. #41×C₁₀₉

C₄

semidirect

15. C₂×Q₈×C₁₀₉

C₄

semidirect

17. C₂×D₄×C₁₀₉

C₄

semidirect

56. C₁₀₉

#421

semidirect

64. #41

Dic₁₀₉

semidirect

66. C₂×Q₈

Dic₁₀₉

semidirect

68. C₂×D₄

Dic₁₀₉

semidirect

Subgroups — #20295043

Conjugacy classes of subgroups

Complements