Group #20295818 — subgroups

Subgroups — C₂×#16812729 #20295818

Order 6976.

258 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812729	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13182019	C₂	1	2	yes	maximal, complemented
4	C₂×#13182017	C₂	1	2	yes	maximal, complemented
5	C₂×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₂₁₈	—	2	2	no	—
8	C₂×Dic₂₁₈	—	2	1	no	—
9	C₂×Dic₂₁₈	C₂²	1	2	yes	—
10	C₂×Dic₂₁₈	C₂²	1	4	yes	complemented
11	C₂²×Dic₁₀₉	—	2	1	no	—
12	#13182019	C₂²	1	8	yes	complemented
13	#13182017	C₂²	1	8	yes	complemented
14	#13182013	C₂²	1	4	yes	complemented
15	C₂×#11724784	C₂²	1	2	yes	—
16	#41×C₁₀₉	—	2	2	no	—
17	#41×C₁₀₉	C₂²	1	4	yes	—
18	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₁₀₉	—	2	1	no	—
20	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	—	2	1	no	—
22	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂³×C₁₀₉	—	2	1	no	—
24	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
25	Q₈×C₁₀₉	—	2	1	no	—
26	Q₈×C₁₀₉	C₂³	1	2	yes	—
27	D₄×C₁₀₉	—	2	4	no	—
28	D₄×C₁₀₉	—	2	1	no	—
29	D₄×C₁₀₉	C₂³	1	2	yes	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	4	no	—
32	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	—	2	4	no	—
38	Dic₂₁₈	—	2	2	no	—
39	Dic₂₁₈	—	2	4	no	—
40	Dic₂₁₈	C₂³	1	4	yes	—
41	#11724784	C₂³	1	4	yes	—
42	C₂²×C₁₀₉	—	2	2	no	—
43	C₂²×C₁₀₉	—	2	2	no	—
44	C₂²×C₁₀₉	—	4	2	no	—
45	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	Dic₁₀₉	—	2	4	no	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	—	4	2	no	—
54	C₂×C₁₀₉	#137	1	2	yes	complemented
55	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
56	C₁₀₉	C₂×#137	1	1	yes	characteristic, complemented, Sylow
57	C₂×#137	—	109	1	no	maximal, Sylow
58	C₂×#41	D₁₀₉	1	1	yes	characteristic
59	C₂²×Q₈	—	109	1	no	—
60	#137	—	109	8	no	—
61	C₂×Q₁₆	—	109	2	no	—
62	C₂×SD₁₆	—	109	2	no	—
63	C₂×M₁₆	—	109	1	no	—
64	#41	—	2	2	no	—
65	#41	C₂×D₁₀₉	1	4	yes	—
66	C₂×Q₈	—	109	2	no	—
67	C₂×Q₈	—	109	4	no	—
68	C₂×Q₈	—	218	2	no	—
69	C₂×Q₈	—	218	1	no	—
70	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	218	1	no	—
75	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
76	Q₁₆	—	109	8	no	—
77	SD₁₆	—	109	8	no	—
78	M₁₆	—	109	4	no	—
79	C₂×C₈	—	109	2	no	—
80	C₂³	—	2	1	no	—
81	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
82	Q₈	—	2	1	no	—
83	Q₈	—	109	4	no	—
84	Q₈	—	218	4	no	—
85	Q₈	—	218	2	no	—
86	Q₈	C₂²×D₁₀₉	1	2	yes	—
87	D₄	—	2	4	no	—
88	D₄	—	2	1	no	—
89	D₄	C₂²×D₁₀₉	1	2	yes	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	218	4	no	—
93	C₂×C₄	—	218	2	no	—
94	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
95	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
97	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₈	—	109	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	4	2	no	—
102	C₂²	C₂×C₂²⋊D₁₀₉	1	2	yes	—
103	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	218	4	no	—
106	C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
107	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
108	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	—	4	2	no	—
111	C₂	#16812729	1	2	yes	direct product factor
112	C₂	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₈

C₂

yes

maximal, characteristic, complemented

#16812729

C₂

yes

maximal, direct product factor

C₂×#13182019

C₂

yes

maximal, complemented

C₂×#13182017

C₂

yes

maximal, complemented

C₂×#13182013

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

—

C₂×Dic₂₁₈

C₂²

yes

complemented

C₂²×Dic₁₀₉

—

#13182019

C₂²

yes

complemented

#13182017

C₂²

yes

complemented

#13182013

C₂²

yes

complemented

C₂×#11724784

C₂²

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂³

yes

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂³

yes

—

#11724784

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂⁴

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#137

yes

complemented

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₁₀₉

C₂×#137

yes

characteristic, complemented, Sylow

C₂×#137

—

109

maximal, Sylow

C₂×#41

D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

#137

—

109

—

C₂×Q₁₆

—

109

—

C₂×SD₁₆

—

109

—

C₂×M₁₆

—

109

—

#41

—

#41

C₂×D₁₀₉

yes

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

Q₁₆

—

109

—

SD₁₆

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂³

—

C₂³

C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

C₂²×D₁₀₉

yes

—

D₄

—

D₄

—

D₄

C₂²×D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

109

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₂²⋊D₁₀₉

yes

—

103

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

104

C₄

—

105

C₄

—

218

—

106

C₄

C₂×C₂²⋊D₁₀₉

yes

—

107

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

108

C₄

C₂³×D₁₀₉

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#16812729

yes

direct product factor

112

C₂

C₂²×C₂²⋊D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂⁴
Frattini	C₄	C₂³×D₁₀₉
Fitting	C₂×#41×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×#41	D₁₀₉
hypocenter	C₁₀₉	C₂×#137
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂⁴

Frattini

C₄

C₂³×D₁₀₉

Fitting

C₂×#41×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂×#41

D₁₀₉

hypocenter

C₁₀₉

C₂×#137

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×Dic₂₁₈	C₂	semidirect
2. #16812729	C₂	direct (N × H)
3. C₂×#13182019	C₂	semidirect
4. C₂×#13182017	C₂	semidirect
5. C₂×#13182013	C₂	semidirect
10. C₂×Dic₂₁₈	C₂²	semidirect
12. #13182019	C₂²	semidirect
13. #13182017	C₂²	semidirect
14. #13182013	C₂²	semidirect
54. C₂×C₁₀₉	#137	semidirect
56. C₁₀₉	C₂×#137	semidirect
111. C₂	#16812729	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₂₁₈

C₂

semidirect

2. #16812729

C₂

direct (N × H)

3. C₂×#13182019

C₂

semidirect

4. C₂×#13182017

C₂

semidirect

5. C₂×#13182013

C₂

semidirect

10. C₂×Dic₂₁₈

C₂²

semidirect

12. #13182019

C₂²

semidirect

13. #13182017

C₂²

semidirect

14. #13182013

C₂²

semidirect

54. C₂×C₁₀₉

#137

semidirect

56. C₁₀₉

C₂×#137

semidirect

111. C₂

#16812729

direct (N × H)

Subgroups — C₂×#16812729 #20295818

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812729 #20295818

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812729 #20295818