Group #20295051 — subgroups

Subgroups — #20295051

Order 6976.

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

normal subgroups only

This group has 114 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×#13182012	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812656	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812628	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812628	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
9	#13182012	—	2	2	no	—
10	C₂×#11724784	C₂²	1	1	yes	characteristic
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₈×D₁₀₉	—	2	2	no	—
13	#13182008	C₄	1	1	yes	characteristic
14	#13182008	C₄	1	1	yes	characteristic
15	#13182006	C₄	1	1	yes	characteristic
16	#13182006	C₄	1	1	yes	characteristic
17	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×D₁₀₉	D₄	1	2	yes	—
29	C₄×D₁₀₉	Q₈	1	2	yes	—
30	C₈×C₁₀₉	—	2	1	no	—
31	C₈×C₁₀₉	—	2	1	no	—
32	#11724784	—	2	1	no	—
33	#11724784	—	2	1	no	—
34	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×D₁₀₉	—	2	1	no	—
36	C₂×D₁₀₉	—	2	1	no	—
37	C₂×D₁₀₉	C₄⋊C₄	1	2	yes	—
38	C₄×C₁₀₉	—	2	1	no	—
39	C₄×C₁₀₉	—	2	1	no	—
40	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	Dic₁₀₉	—	2	1	no	—
43	Dic₁₀₉	—	2	1	no	—
44	Dic₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₀₉	#131	1	1	yes	characteristic
47	C₂×C₁₀₉	#131	1	1	yes	characteristic
48	D₁₀₉	—	2	2	no	—
49	C₁₀₉	#424	1	1	yes	characteristic, complemented, Sylow
50	#424	—	109	1	no	maximal, Sylow
51	C₂×M₁₆	—	109	1	no	—
52	C₂²×C₈	—	109	1	no	—
53	#117	—	109	1	no	—
54	C₄⋊C₈	—	109	1	no	—
55	C₄⋊C₈	—	109	1	no	—
56	C₄⋊C₈	—	109	1	no	—
57	C₄⋊C₈	D₁₀₉	1	1	yes	characteristic, complemented
58	C₂²×C₄	—	109	1	no	—
59	M₁₆	—	218	2	no	—
60	C₂×C₈	—	109	1	no	—
61	C₂×C₈	—	109	1	no	—
62	C₂×C₈	—	218	2	no	—
63	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
64	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₄	—	109	1	no	—
66	C₄⋊C₄	—	109	1	no	—
67	C₂²⋊C₄	—	109	1	no	—
68	C₂²⋊C₄	—	109	1	no	—
69	C₄²	—	109	1	no	—
70	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂³	—	109	1	no	—
72	C₂×C₄	—	109	1	no	—
73	C₂×C₄	—	109	2	no	—
74	C₂×C₄	—	109	2	no	—
75	C₂×C₄	—	109	1	no	—
76	C₂×C₄	—	109	1	no	—
77	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
78	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	—	218	1	no	—
83	C₈	—	218	1	no	—
84	C₂²	—	109	2	no	—
85	C₂²	—	218	1	no	—
86	C₂²	—	218	1	no	—
87	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	109	2	no	—
91	C₄	—	218	1	no	—
92	C₄	—	218	1	no	—
93	C₄	D₄×D₁₀₉	1	1	yes	characteristic
94	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
95	C₂	—	218	2	no	—
96	C₂	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
97	C₂	#16812705	1	1	yes	characteristic
98	C₂	#16812707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182012

C₂

yes

maximal, characteristic

C₂×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812656

C₂

yes

maximal, characteristic

C₄⋊#11724784

C₂

yes

maximal, characteristic, complemented

#16812628

C₂

yes

maximal, characteristic, complemented

#16812628

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

#13182008

C₄

yes

characteristic

#13182008

C₄

yes

characteristic

#13182006

C₄

yes

characteristic

#13182006

C₄

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

D₄

yes

—

C₄×D₁₀₉

Q₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₄

yes

—

C₂×C₁₀₉

C₂×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₁₀₉

#424

yes

characteristic, complemented, Sylow

#424

—

109

maximal, Sylow

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

D₁₀₉

yes

characteristic, complemented

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

C₈

—

218

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

C₄

D₄×D₁₀₉

yes

characteristic

C₄

Q₈×D₁₀₉

yes

characteristic

C₂

—

218

—

C₂

C₄⋊C₄×D₁₀₉

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₄⋊C₈×C₁₀₉	C₂
hypercenter	C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#424
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₄⋊C₈×C₁₀₉

C₂

hypercenter

C₄⋊C₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
4. C₄⋊#11724784	C₂	semidirect
5. #16812628	C₂	semidirect
6. #16812628	C₂	semidirect
7. C₄⋊C₈×C₁₀₉	C₂	semidirect
49. C₁₀₉	#424	semidirect
57. C₄⋊C₈	D₁₀₉	semidirect

H ≅ G/N

type

4. C₄⋊#11724784

C₂

semidirect

5. #16812628

C₂

semidirect

6. #16812628

C₂

semidirect

7. C₄⋊C₈×C₁₀₉

C₂

semidirect

49. C₁₀₉

#424

semidirect

57. C₄⋊C₈

D₁₀₉

semidirect

Subgroups — #20295051

Conjugacy classes of subgroups

Complements