Group #20295546 — subgroups

Subgroups — C₂×#16812689 #20295546

Order 6976.

226 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 119 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	#16812689	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182008	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182008	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182008	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182008	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
14	#13182008	C₂²	1	4	yes	—
15	#13182008	C₂²	1	4	yes	—
16	#13182008	C₂²	1	4	yes	—
17	#13182008	C₂²	1	4	yes	—
18	C₄×Dic₁₀₉	C₂²	1	4	yes	—
19	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
23	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	2	no	—
32	C₂×Dic₁₀₉	—	2	2	no	—
33	C₂×Dic₁₀₉	Q₈	1	4	yes	—
34	C₂×Dic₁₀₉	C₂³	1	2	yes	—
35	C₂×Dic₁₀₉	C₂³	1	2	yes	—
36	C₂×Dic₁₀₉	C₂³	1	2	yes	—
37	C₂×Dic₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₁₀₉	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₀₉	#41	1	2	yes	—
40	C₂²×C₁₀₉	#41	1	2	yes	—
41	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	2	2	no	—
45	Dic₁₀₉	—	2	2	no	—
46	Dic₁₀₉	—	2	2	no	—
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	C₂×Q₈	1	4	yes	—
49	C₂×C₁₀₉	#125	1	4	yes	complemented
50	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
53	C₁₀₉	C₂×#125	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	—	109	1	no	maximal, Sylow
55	#125	—	109	8	no	—
56	C₂×C₄⋊C₄	—	109	1	no	—
57	C₂×C₄⋊C₄	—	109	1	no	—
58	C₂×C₄⋊C₄	—	109	1	no	—
59	C₂×C₄⋊C₄	—	109	1	no	—
60	C₂×C₄⋊C₄	—	109	1	no	—
61	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
62	C₂×C₄²	—	109	1	no	—
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
68	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
70	C₄⋊C₄	—	109	4	no	—
71	C₄⋊C₄	—	109	4	no	—
72	C₄⋊C₄	—	109	4	no	—
73	C₄⋊C₄	—	109	4	no	—
74	C₄⋊C₄	—	109	4	no	—
75	C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
76	C₄²	—	109	4	no	—
77	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	109	4	no	—
82	C₂×C₄	—	109	2	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	109	2	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	—	218	2	no	—
89	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
90	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
91	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
92	C₂²	Q₈×D₁₀₉	1	2	yes	—
93	C₂²	#13182028	1	2	yes	—
94	C₂²	#13182029	1	2	yes	—
95	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	109	4	no	—
100	C₄	—	218	2	no	—
101	C₄	—	218	2	no	—
102	C₄	—	218	2	no	—
103	C₂	#16812689	1	4	yes	direct product factor
104	C₂	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
105	C₂	C₂×#13182028	1	1	yes	characteristic
106	C₂	C₂×#13182029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812689

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182008

C₂

yes

maximal, characteristic

C₂×#13182008

C₂

yes

maximal, characteristic

C₂×#13182008

C₂

yes

maximal, characteristic

C₂×#13182008

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

Q₈

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂²×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₈

yes

—

C₂×C₁₀₉

#125

yes

complemented

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₁₀₉

C₂×#125

yes

characteristic, complemented, Sylow

C₂×#125

—

109

maximal, Sylow

#125

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₄²

—

109

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

Q₈×D₁₀₉

yes

—

C₂²

#13182028

yes

—

C₂²

#13182029

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

109

—

100

C₄

—

218

—

101

C₄

—

218

—

102

C₄

—

218

—

103

C₂

#16812689

yes

direct product factor

104

C₂

C₂×Q₈×D₁₀₉

yes

characteristic

105

C₂

C₂×#13182028

yes

characteristic

106

C₂

C₂×#13182029

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂⁴
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#125
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂⁴

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #16812689	C₂	direct (N × H)
49. C₂×C₁₀₉	#125	semidirect
53. C₁₀₉	C₂×#125	semidirect
103. C₂	#16812689	direct (N × H)

H ≅ G/N

type

1. #16812689

C₂

direct (N × H)

49. C₂×C₁₀₉

#125

semidirect

53. C₁₀₉

C₂×#125

semidirect

103. C₂

#16812689

direct (N × H)

Subgroups — C₂×#16812689 #20295546

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812689 #20295546

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812689 #20295546