Group #20295412 — subgroups

Subgroups — #20295412

Order 6976.

144 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 43 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×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812695	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812688	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812609	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
11	#13182016	—	2	2	no	—
12	D₈₇₂	—	2	2	no	—
13	#13182014	—	2	2	no	—
14	C₂×#11724784	C₂²	1	1	yes	characteristic
15	#13182006	—	2	2	no	—
16	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₀₉	—	2	2	no	—
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	—	2	1	no	—
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂²×D₁₀₉	—	2	1	no	—
24	Q₈×C₁₀₉	—	2	1	no	—
25	Q₈×C₁₀₉	—	2	1	no	—
26	D₄×C₁₀₉	—	2	1	no	—
27	D₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²⋊D₁₀₉	—	4	4	no	—
31	C₂×Dic₁₀₉	D₄	1	2	yes	—
32	D₄₃₆	—	2	1	no	—
33	D₄₃₆	—	2	1	no	—
34	D₄₃₆	—	2	1	no	—
35	D₄₃₆	—	2	1	no	—
36	C₈×C₁₀₉	D₄	1	2	yes	complemented
37	#11724784	D₄	1	2	yes	complemented
38	C₂²×C₁₀₉	—	4	1	no	—
39	C₂²×C₁₀₉	—	4	2	no	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₀₉	—	4	1	no	—
42	C₂×D₁₀₉	—	4	1	no	—
43	C₂×D₁₀₉	—	4	2	no	—
44	C₂×D₁₀₉	—	4	2	no	—
45	C₄×C₁₀₉	—	4	1	no	—
46	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	Dic₁₀₉	—	2	2	no	—
49	C₂×C₁₀₉	—	8	1	no	—
50	C₂×C₁₀₉	#127	1	1	yes	characteristic
51	C₂×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
52	D₁₀₉	—	8	1	no	—
53	D₁₀₉	—	8	1	no	—
54	C₁₀₉	#496	1	1	yes	characteristic, complemented, Sylow
55	#496	—	109	1	no	maximal, Sylow
56	C₂×SD₁₆	—	109	1	no	—
57	C₂×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
58	C₂×D₈	—	109	1	no	—
59	C₂×D₈	—	109	1	no	—
60	#127	—	109	1	no	—
61	#124	—	109	1	no	—
62	#97	—	109	1	no	—
63	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
64	C₂×D₄	—	109	1	no	—
65	C₂×D₄	—	109	1	no	—
66	C₂×D₄	—	218	2	no	—
67	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
68	SD₁₆	—	2	2	no	—
69	SD₁₆	—	218	2	no	—
70	D₈	—	218	2	no	—
71	D₈	—	218	2	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
74	C₂²⋊C₄	—	218	2	no	—
75	C₄²	—	109	1	no	—
76	C₂³	—	2	1	no	—
77	C₂³	—	218	1	no	—
78	C₂³	—	218	1	no	—
79	Q₈	—	2	1	no	—
80	Q₈	—	2	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	436	4	no	—
84	D₄	—	218	1	no	—
85	D₄	—	218	1	no	—
86	D₄	—	218	1	no	—
87	D₄	—	218	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	109	2	no	—
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₈	—	109	2	no	—
92	C₈	C₂²⋊D₁₀₉	1	2	yes	complemented
93	C₂²	—	4	1	no	—
94	C₂²	—	4	2	no	—
95	C₂²	—	436	1	no	—
96	C₂²	—	436	1	no	—
97	C₂²	—	436	2	no	—
98	C₂²	—	436	2	no	—
99	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	218	2	no	—
102	C₄	D₄×D₁₀₉	1	1	yes	characteristic
103	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
104	C₂	—	8	1	no	—
105	C₂	—	872	1	no	—
106	C₂	—	872	1	no	—
107	C₂	#16812695	1	1	yes	characteristic
108	C₂	#16812726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182016

C₂

yes

maximal, characteristic, complemented

C₂×D₈₇₂

C₂

yes

maximal, characteristic, complemented

C₂×#13182014

C₂

yes

maximal, characteristic, complemented

#16812695

C₂

yes

maximal, characteristic, complemented

#16812688

C₂

yes

maximal, characteristic, complemented

#16812609

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

#13182016

—

D₈₇₂

—

#13182014

—

C₂×#11724784

C₂²

yes

characteristic

#13182006

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

D₄

yes

complemented

#11724784

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#127

yes

characteristic

C₂×C₁₀₉

AGL(1,ℤ/8)

yes

—

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#496

yes

characteristic, complemented, Sylow

#496

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₂×SD₁₆

D₁₀₉

yes

characteristic, complemented

C₂×D₈

—

109

—

C₂×D₈

—

109

—

#127

—

109

—

#124

—

109

—

#97

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

SD₁₆

—

SD₁₆

—

218

—

D₈

—

218

—

D₈

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

C₂³

—

218

—

C₂³

—

218

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

436

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

C₂²⋊D₁₀₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

D₄×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

218

—

102

C₄

D₄×D₁₀₉

yes

characteristic

103

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

104

C₂

—

105

C₂

—

872

—

106

C₂

—

872

—

107

C₂

#16812695

yes

characteristic

108

C₂

#16812726

yes

—

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#496
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂×SD₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₀₉

hypocenter

C₁₀₉

#496

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#13182016	C₂	semidirect
2. C₂×D₈₇₂	C₂	semidirect
3. C₂×#13182014	C₂	semidirect
4. #16812695	C₂	semidirect
5. #16812688	C₂	semidirect
6. #16812609	C₂	semidirect
7. C₂×SD₁₆×C₁₀₉	C₂	semidirect
36. C₈×C₁₀₉	D₄	semidirect
37. #11724784	D₄	semidirect
54. C₁₀₉	#496	semidirect
57. C₂×SD₁₆	D₁₀₉	semidirect
92. C₈	C₂²⋊D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182016

C₂

semidirect

2. C₂×D₈₇₂

C₂

semidirect

3. C₂×#13182014

C₂

semidirect

4. #16812695

C₂

semidirect

5. #16812688

C₂

semidirect

6. #16812609

C₂

semidirect

7. C₂×SD₁₆×C₁₀₉

C₂

semidirect

36. C₈×C₁₀₉

D₄

semidirect

37. #11724784

D₄

semidirect

54. C₁₀₉

#496

semidirect

57. C₂×SD₁₆

D₁₀₉

semidirect

92. C₈

C₂²⋊D₁₀₉

semidirect

Subgroups — #20295412

Conjugacy classes of subgroups

Complements