Group #20294901 — subgroups

Subgroups — #20294901

Order 6976.

234 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂³×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812606	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₉	—	2	2	no	—
7	C₂²×Dic₁₀₉	—	2	4	no	—
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Dic₁₀₉	—	2	4	no	—
12	C₂²⋊Dic₁₀₉	—	2	2	no	—
13	C₂⁴×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂²⋊C₄×C₁₀₉	C₄	1	4	yes	complemented
16	C₂³×C₁₀₉	—	2	2	no	—
17	C₂³×C₁₀₉	—	2	2	no	—
18	C₂³×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂³×C₁₀₉	D₄	1	2	yes	—
20	C₂³×C₁₀₉	Q₈	1	2	yes	—
21	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
25	C₂×Dic₁₀₉	—	2	8	no	—
26	C₂×Dic₁₀₉	—	2	2	no	—
27	C₂×Dic₁₀₉	—	2	2	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	2	no	—
32	C₂×Dic₁₀₉	D₄	1	4	yes	—
33	C₂²×C₁₀₉	—	2	4	no	—
34	C₂²×C₁₀₉	—	2	2	no	—
35	C₂²×C₁₀₉	—	2	4	no	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
38	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₉	#41	1	2	yes	—
43	C₄×C₁₀₉	—	2	4	no	—
44	Dic₁₀₉	—	2	4	no	—
45	Dic₁₀₉	—	4	1	no	—
46	Dic₁₀₉	—	4	1	no	—
47	C₂×C₁₀₉	—	2	4	no	—
48	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₀₉	C₄×D₄	1	2	yes	—
50	C₂×C₁₀₉	#120	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
52	C₂×C₁₀₉	#123	1	1	yes	characteristic
53	C₁₀₉	#385	1	1	yes	characteristic, complemented, Sylow
54	#385	—	109	1	no	maximal, Sylow
55	C₂³×C₄	—	109	1	no	—
56	C₂×C₄⋊C₄	—	109	2	no	—
57	C₂×C₂²⋊C₄	—	109	1	no	—
58	C₂×C₂²⋊C₄	D₁₀₉	1	1	yes	characteristic
59	#95	—	109	2	no	—
60	C₂⁴	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂²×C₄	—	109	2	no	—
62	C₂²×C₄	—	109	1	no	—
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	218	4	no	—
65	C₂²×C₄	—	218	2	no	—
66	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
67	C₄⋊C₄	—	218	4	no	—
68	C₂²⋊C₄	—	218	2	no	—
69	C₂²⋊C₄	Dic₁₀₉	1	4	yes	complemented
70	C₂³	—	2	2	no	—
71	C₂³	—	2	2	no	—
72	C₂³	Dic₂₁₈	1	2	yes	—
73	C₂³	D₄₃₆	1	2	yes	—
74	C₂³	C₂×Dic₁₀₉	1	2	yes	—
75	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	—	218	2	no	—
80	C₂×C₄	—	218	1	no	—
81	C₂×C₄	—	218	1	no	—
82	C₂×C₄	—	218	2	no	—
83	C₂×C₄	—	218	8	no	—
84	C₂×C₄	—	218	2	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	C₂×Dic₁₀₉	1	4	yes	complemented
87	C₂²	—	2	4	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₄⋊Dic₁₀₉	1	4	yes	complemented
92	C₂²	C₂²×Dic₁₀₉	1	1	yes	characteristic
93	C₂²	D₄×D₁₀₉	1	2	yes	—
94	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
95	C₂²	C₂×Dic₂₁₈	1	1	yes	characteristic
96	C₂²	#13182028	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	436	1	no	—
99	C₄	—	436	1	no	—
100	C₄	—	218	4	no	—
101	C₂	—	2	4	no	—
102	C₂	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
103	C₂	D₄×Dic₁₀₉	1	2	yes	—
104	C₂	#16812669	1	1	yes	characteristic
105	C₂	#16812676	1	2	yes	—
106	C₂	#16812683	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₀₉

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, complemented

C₂×C₂²⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₂⁴×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²⋊C₄×C₁₀₉

C₄

yes

complemented

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₄

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

#41

yes

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₄×D₄

yes

—

C₂×C₁₀₉

#120

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

#123

yes

characteristic

C₁₀₉

#385

yes

characteristic, complemented, Sylow

#385

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

D₁₀₉

yes

characteristic

#95

—

109

—

C₂⁴

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

Dic₁₀₉

yes

complemented

C₂³

—

C₂³

—

C₂³

Dic₂₁₈

yes

—

C₂³

D₄₃₆

yes

—

C₂³

C₂×Dic₁₀₉

yes

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×Dic₁₀₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₄⋊Dic₁₀₉

yes

complemented

C₂²

C₂²×Dic₁₀₉

yes

characteristic

C₂²

D₄×D₁₀₉

yes

—

C₂²

C₂×D₄₃₆

yes

characteristic

C₂²

C₂×Dic₂₁₈

yes

characteristic

C₂²

#13182028

yes

—

C₄

—

C₄

—

436

—

C₄

—

436

—

100

C₄

—

218

—

101

C₂

—

102

C₂

C₂×C₄⋊Dic₁₀₉

yes

characteristic

103

C₂

D₄×Dic₁₀₉

yes

—

104

C₂

#16812669

yes

characteristic

105

C₂

#16812676

yes

—

106

C₂

#16812683

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#385
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
4. #16812606	C₂	semidirect
15. C₂²⋊C₄×C₁₀₉	C₄	semidirect
24. C₂×C₄×C₁₀₉	C₂×C₄	semidirect
37. C₂²×C₁₀₉	C₄⋊C₄	semidirect
53. C₁₀₉	#385	semidirect
69. C₂²⋊C₄	Dic₁₀₉	semidirect
86. C₂×C₄	C₂×Dic₁₀₉	semidirect
91. C₂²	C₄⋊Dic₁₀₉	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

4. #16812606

C₂

semidirect

15. C₂²⋊C₄×C₁₀₉

C₄

semidirect

24. C₂×C₄×C₁₀₉

C₂×C₄

semidirect

37. C₂²×C₁₀₉

C₄⋊C₄

semidirect

53. C₁₀₉

#385

semidirect

69. C₂²⋊C₄

Dic₁₀₉

semidirect

86. C₂×C₄

C₂×Dic₁₀₉

semidirect

91. C₂²

C₄⋊Dic₁₀₉

semidirect

Subgroups — #20294901

Conjugacy classes of subgroups

Complements