Group #20295255 — subgroups

Subgroups — #20295255

Order 6976.

124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 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×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812678	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812673	C₂	1	1	yes	maximal, characteristic
4	#16812631	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812621	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
11	#13182018	—	2	2	no	—
12	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#13182008	—	2	1	no	—
15	C₂²⋊Dic₁₀₉	—	2	1	no	—
16	#13182006	—	2	1	no	—
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	—	2	2	no	—
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
31	D₄₃₆	—	2	1	no	—
32	D₄₃₆	—	2	1	no	—
33	Dic₂₁₈	—	2	1	no	—
34	Dic₂₁₈	—	2	1	no	—
35	C₈×C₁₀₉	—	2	1	no	—
36	C₈×C₁₀₉	D₄	1	2	yes	complemented
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	—	2	1	no	—
39	C₂²×C₁₀₉	SD₁₆	1	2	yes	complemented
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	1	no	—
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₀₉	#41	1	1	yes	characteristic
47	Dic₁₀₉	—	4	1	no	—
48	Dic₁₀₉	—	4	1	no	—
49	Dic₁₀₉	—	4	1	no	—
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	#121	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
53	C₂×C₁₀₉	#135	1	1	yes	characteristic
54	D₁₀₉	—	8	1	no	—
55	C₁₀₉	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	109	1	no	maximal, Sylow
57	C₂×SD₁₆	—	109	1	no	—
58	C₂²×C₈	D₁₀₉	1	1	yes	characteristic, complemented
59	C₂²⋊Q₈	—	109	1	no	—
60	#121	—	109	1	no	—
61	#106	—	109	1	no	—
62	#103	—	109	1	no	—
63	#102	—	109	1	no	—
64	C₂×Q₈	—	109	1	no	—
65	C₂×D₄	—	109	1	no	—
66	C₂×D₄	—	218	1	no	—
67	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
68	SD₁₆	—	218	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₄⋊C₄	—	109	1	no	—
73	C₄⋊C₄	—	109	1	no	—
74	C₄⋊C₄	—	218	1	no	—
75	C₂²⋊C₄	—	218	1	no	—
76	C₂²⋊C₄	—	218	1	no	—
77	C₂³	—	218	1	no	—
78	C₂³	D₄₃₆	1	1	yes	characteristic
79	Q₈	—	218	1	no	—
80	Q₈	—	218	1	no	—
81	D₄	—	436	2	no	—
82	D₄	—	218	1	no	—
83	D₄	—	218	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	—	218	1	no	—
88	C₂×C₄	—	218	1	no	—
89	C₂×C₄	D₄₃₆	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	C₂²⋊D₁₀₉	1	2	yes	complemented
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	436	1	no	—
96	C₂²	—	436	1	no	—
97	C₂²	—	436	1	no	—
98	C₂²	#13182018	1	2	yes	complemented
99	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	436	1	no	—
102	C₄	—	436	1	no	—
103	C₄	—	436	1	no	—
104	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₄	#13182029	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	872	1	no	—
108	C₂	#16812673	1	1	yes	characteristic
109	C₂	C₂×#13182018	1	1	yes	characteristic
110	C₂	#16812721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182018

C₂

yes

maximal, characteristic, complemented

#16812678

C₂

yes

maximal, characteristic, complemented

#16812673

C₂

yes

maximal, characteristic

#16812631

C₂

yes

maximal, characteristic, complemented

#16812624

C₂

yes

maximal, characteristic, complemented

#16812621

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

#13182018

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

C₂²⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

SD₁₆

yes

complemented

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

#121

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

#135

yes

characteristic

D₁₀₉

—

C₁₀₉

#465

yes

characteristic, complemented, Sylow

#465

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₂²×C₈

D₁₀₉

yes

characteristic, complemented

C₂²⋊Q₈

—

109

—

#121

—

109

—

#106

—

109

—

#103

—

109

—

#102

—

109

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

SD₁₆

—

218

—

C₂×C₈

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

218

—

C₂³

D₄₃₆

yes

characteristic

Q₈

—

218

—

Q₈

—

218

—

D₄

—

436

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄₃₆

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

C₂²⋊D₁₀₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

#13182018

yes

complemented

C₂²

C₂×D₄₃₆

yes

characteristic

100

C₄

—

101

C₄

—

436

—

102

C₄

—

436

—

103

C₄

—

436

—

104

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

105

C₄

#13182029

yes

characteristic

106

C₂

—

107

C₂

—

872

—

108

C₂

#16812673

yes

characteristic

109

C₂

C₂×#13182018

yes

characteristic

110

C₂

#16812721

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	#465
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂²×C₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂²×C₈

D₁₀₉

hypocenter

C₁₀₉

#465

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#13182018	C₂	semidirect
2. #16812678	C₂	semidirect
4. #16812631	C₂	semidirect
5. #16812624	C₂	semidirect
6. #16812621	C₂	semidirect
7. C₂²×C₈×C₁₀₉	C₂	semidirect
36. C₈×C₁₀₉	D₄	semidirect
39. C₂²×C₁₀₉	SD₁₆	semidirect
55. C₁₀₉	#465	semidirect
58. C₂²×C₈	D₁₀₉	semidirect
92. C₈	C₂²⋊D₁₀₉	semidirect
98. C₂²	#13182018	semidirect

H ≅ G/N

type

1. C₂×#13182018

C₂

semidirect

2. #16812678

C₂

semidirect

4. #16812631

C₂

semidirect

5. #16812624

C₂

semidirect

6. #16812621

C₂

semidirect

7. C₂²×C₈×C₁₀₉

C₂

semidirect

36. C₈×C₁₀₉

D₄

semidirect

39. C₂²×C₁₀₉

SD₁₆

semidirect

55. C₁₀₉

#465

semidirect

58. C₂²×C₈

D₁₀₉

semidirect

92. C₈

C₂²⋊D₁₀₉

semidirect

98. C₂²

#13182018

semidirect

Subgroups — #20295255

Conjugacy classes of subgroups

Complements