Group #20295033 — subgroups

Subgroups — #20295033

Order 6976.

150 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 55 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×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182012	C₂	1	1	yes	maximal, characteristic
3	#16812657	C₂	1	1	yes	maximal, characteristic
4	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812622	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	—	2	1	no	—
9	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	Q₈×D₁₀₉	—	2	2	no	—
11	Q₈×D₁₀₉	C₄	1	4	yes	complemented
12	C₂×C₄×D₁₀₉	—	2	1	no	—
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#13182012	—	2	2	no	—
15	C₂×#11724784	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
17	#13182006	—	2	1	no	—
18	C₄×Dic₁₀₉	—	2	1	no	—
19	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
23	Q₈×C₁₀₉	—	2	1	no	—
24	Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	4	no	—
32	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×D₁₀₉	D₄	1	2	yes	—
34	Dic₂₁₈	—	2	1	no	—
35	Dic₂₁₈	—	2	4	no	—
36	Dic₂₁₈	C₂×C₄	1	2	yes	—
37	C₈×C₁₀₉	—	2	1	no	—
38	#11724784	—	2	1	no	—
39	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	4	1	no	—
45	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	—	4	1	no	—
49	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₁₀₉	#137	1	1	yes	characteristic
52	C₂×C₁₀₉	#137	1	1	yes	characteristic
53	D₁₀₉	—	2	2	no	—
54	C₁₀₉	#419	1	1	yes	characteristic, complemented, Sylow
55	#419	—	109	1	no	maximal, Sylow
56	C₂²×Q₈	—	109	1	no	—
57	C₂×M₁₆	—	109	1	no	—
58	#117	—	109	1	no	—
59	#103	—	109	1	no	—
60	#103	—	109	1	no	—
61	#103	—	109	1	no	—
62	#103	D₁₀₉	1	1	yes	characteristic, complemented
63	C₂×Q₈	—	109	4	no	—
64	C₂×Q₈	—	109	1	no	—
65	C₂×Q₈	—	218	1	no	—
66	C₂×Q₈	—	218	2	no	—
67	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
68	C₂²×C₄	—	109	1	no	—
69	C₂²×C₄	—	218	1	no	—
70	M₁₆	—	218	2	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₄⋊C₄	—	109	1	no	—
74	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂²⋊C₄	—	218	1	no	—
76	C₄²	—	218	1	no	—
77	C₂³	—	109	1	no	—
78	Q₈	—	2	1	no	—
79	Q₈	—	109	2	no	—
80	Q₈	—	218	4	no	—
81	Q₈	—	218	1	no	—
82	Q₈	C₄×D₁₀₉	1	2	yes	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	109	2	no	—
87	C₂×C₄	—	109	2	no	—
88	C₂×C₄	—	218	4	no	—
89	C₂×C₄	—	218	1	no	—
90	C₂×C₄	—	218	1	no	—
91	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	218	1	no	—
94	C₂²	—	109	2	no	—
95	C₂²	—	218	1	no	—
96	C₂²	—	218	1	no	—
97	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	4	1	no	—
100	C₄	—	109	2	no	—
101	C₄	—	436	1	no	—
102	C₄	—	218	2	no	—
103	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
104	C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₂	—	218	2	no	—
106	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
107	C₂	#16812728	1	1	yes	characteristic
108	C₂	#16812731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182012

C₂

yes

maximal, characteristic

#16812657

C₂

yes

maximal, characteristic

#16812624

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

#16812622

C₂

yes

maximal, characteristic, complemented

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

Q₈×D₁₀₉

C₄

yes

complemented

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182006

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₁₀₉

#419

yes

characteristic, complemented, Sylow

#419

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂×M₁₆

—

109

—

#117

—

109

—

#103

—

109

—

#103

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, complemented

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

—

109

—

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

C₄×D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

D₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

100

C₄

—

109

—

101

C₄

—

436

—

102

C₄

—

218

—

103

C₄

C₂×C₄×D₁₀₉

yes

characteristic

104

C₄

D₄×D₁₀₉

yes

characteristic

105

C₂

—

218

—

106

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

107

C₂

#16812728

yes

characteristic

108

C₂

#16812731

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#103	D₁₀₉
hypocenter	C₁₀₉	#419
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#103×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
4. #16812624	C₂	semidirect
5. #16812623	C₂	semidirect
6. #16812622	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
11. Q₈×D₁₀₉	C₄	semidirect
54. C₁₀₉	#419	semidirect
62. #103	D₁₀₉	semidirect

H ≅ G/N

type

4. #16812624

C₂

semidirect

5. #16812623

C₂

semidirect

6. #16812622

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

11. Q₈×D₁₀₉

C₄

semidirect

54. C₁₀₉

#419

semidirect

62. #103

D₁₀₉

semidirect

Subgroups — #20295033

Conjugacy classes of subgroups

Complements