Group #20295271 — subgroups

Subgroups — #20295271

Order 6976.

120 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 39 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×#13182017	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812679	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812670	C₂	1	1	yes	maximal, characteristic
5	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
10	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
11	#13182017	—	2	2	no	—
12	#13182012	—	2	2	no	—
13	C₂×#11724784	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₉	—	2	1	no	—
15	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²⋊Dic₁₀₉	—	2	1	no	—
17	#13182006	—	2	1	no	—
18	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	—	2	1	no	—
22	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
23	D₄×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₀₉	—	4	2	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	1	no	—
32	C₄×D₁₀₉	—	2	1	no	—
33	Dic₂₁₈	—	2	1	no	—
34	Dic₂₁₈	—	2	1	no	—
35	C₈×C₁₀₉	—	2	1	no	—
36	#11724784	D₄	1	2	yes	complemented
37	C₂²×C₁₀₉	—	4	1	no	—
38	C₂²×C₁₀₉	—	4	1	no	—
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	1	no	—
44	C₄×C₁₀₉	—	4	1	no	—
45	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₀₉	#41	1	1	yes	characteristic
47	Dic₁₀₉	—	2	1	no	—
48	Dic₁₀₉	—	4	1	no	—
49	Dic₁₀₉	—	4	1	no	—
50	C₂×C₁₀₉	—	8	1	no	—
51	C₂×C₁₀₉	#121	1	1	yes	characteristic
52	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂×C₁₀₉	#137	1	1	yes	characteristic
54	D₁₀₉	—	4	1	no	—
55	C₁₀₉	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	109	1	no	maximal, Sylow
57	C₂×SD₁₆	—	109	1	no	—
58	C₂×M₁₆	—	109	1	no	—
59	C₂²⋊Q₈	—	109	1	no	—
60	#121	—	109	1	no	—
61	#107	—	109	1	no	—
62	#103	—	109	1	no	—
63	#102	D₁₀₉	1	1	yes	characteristic, complemented
64	C₂×Q₈	—	109	1	no	—
65	C₂×D₄	—	218	1	no	—
66	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂²×C₄	—	109	1	no	—
68	SD₁₆	—	218	2	no	—
69	M₁₆	—	218	2	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₄⋊C₄	—	109	1	no	—
73	C₄⋊C₄	—	218	1	no	—
74	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂²⋊C₄	—	218	1	no	—
76	C₂²⋊C₄	—	218	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	109	1	no	—
79	Q₈	—	218	1	no	—
80	Q₈	—	218	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	436	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	—	218	1	no	—
88	C₂×C₄	—	218	1	no	—
89	C₂×C₄	—	218	1	no	—
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	—	109	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	218	1	no	—
97	C₂²	—	218	1	no	—
98	C₂²	—	218	1	no	—
99	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	436	1	no	—
102	C₄	—	436	1	no	—
103	C₄	—	218	1	no	—
104	C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₄	#13182029	1	1	yes	characteristic
106	C₂	—	8	1	no	—
107	C₂	—	436	1	no	—
108	C₂	#16812672	1	1	yes	characteristic
109	C₂	#16812723	1	1	yes	characteristic
110	C₂	#16812728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182017

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

C₂

yes

maximal, characteristic, complemented

#16812679

C₂

yes

maximal, characteristic, complemented

#16812670

C₂

yes

maximal, characteristic

#16812632

C₂

yes

maximal, characteristic, complemented

#16812624

C₂

yes

maximal, characteristic, complemented

#102×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂²⋊Dic₁₀₉

—

#13182006

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₁₀₉

#468

yes

characteristic, complemented, Sylow

#468

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²⋊Q₈

—

109

—

#121

—

109

—

#107

—

109

—

#103

—

109

—

#102

D₁₀₉

yes

characteristic, complemented

C₂×Q₈

—

109

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

SD₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

C₂³

—

109

—

Q₈

—

218

—

Q₈

—

218

—

D₄

—

D₄

—

D₄

—

436

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

D₄×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

436

—

102

C₄

—

436

—

103

C₄

—

218

—

104

C₄

D₄×D₁₀₉

yes

characteristic

105

C₄

#13182029

yes

characteristic

106

C₂

—

107

C₂

—

436

—

108

C₂

#16812672

yes

characteristic

109

C₂

#16812723

yes

characteristic

110

C₂

#16812728

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#102×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#102	D₁₀₉
hypocenter	C₁₀₉	#468
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#102×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×#13182017	C₂	semidirect
2. C₂×#13182012	C₂	semidirect
3. #16812679	C₂	semidirect
5. #16812632	C₂	semidirect
6. #16812624	C₂	semidirect
7. #102×C₁₀₉	C₂	semidirect
36. #11724784	D₄	semidirect
55. C₁₀₉	#468	semidirect
63. #102	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182017

C₂

semidirect

2. C₂×#13182012

C₂

semidirect

3. #16812679

C₂

semidirect

5. #16812632

C₂

semidirect

6. #16812624

C₂

semidirect

7. #102×C₁₀₉

C₂

semidirect

36. #11724784

D₄

semidirect

55. C₁₀₉

#468

semidirect

63. #102

D₁₀₉

semidirect

Subgroups — #20295271

Conjugacy classes of subgroups

Complements