Group #20295124 — subgroups

Subgroups — #20295124

Order 6976.

120 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 51 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×#13182016	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic
3	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812619	C₂	1	1	yes	maximal, characteristic
6	#16812610	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₀₉	—	2	1	no	—
10	#13182016	C₄	1	4	yes	complemented
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₂×#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#13182006	—	2	1	no	—
15	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄²×C₁₀₉	—	2	1	no	—
19	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×D₁₀₉	—	2	1	no	—
21	Q₈×C₁₀₉	—	2	1	no	—
22	Q₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	1	no	—
29	D₄₃₆	—	2	1	no	—
30	D₄₃₆	C₂×C₄	1	2	yes	—
31	C₄×D₁₀₉	—	4	2	no	—
32	#11724784	—	2	1	no	—
33	#11724784	C₂×C₄	1	2	yes	complemented
34	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₀₉	—	2	2	no	—
36	C₂×D₁₀₉	—	4	1	no	—
37	C₂×D₁₀₉	—	4	1	no	—
38	C₄×C₁₀₉	—	2	2	no	—
39	C₄×C₁₀₉	—	2	1	no	—
40	C₄×C₁₀₉	—	2	1	no	—
41	C₄×C₁₀₉	—	4	1	no	—
42	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₁₀₉	#41	1	1	yes	characteristic
44	Dic₁₀₉	—	4	1	no	—
45	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₁₀₉	#137	1	1	yes	characteristic
48	D₁₀₉	—	4	2	no	—
49	C₁₀₉	#440	1	1	yes	characteristic, complemented, Sylow
50	#440	—	109	1	no	maximal, Sylow
51	C₂×SD₁₆	—	109	1	no	—
52	C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
53	C₄×D₄	—	109	1	no	—
54	#107	—	109	1	no	—
55	#103	—	109	1	no	—
56	#102	—	109	1	no	—
57	#97	—	109	1	no	—
58	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
59	C₂×D₄	—	109	1	no	—
60	C₂²×C₄	—	218	1	no	—
61	SD₁₆	—	109	4	no	—
62	C₂×C₈	—	109	1	no	—
63	C₂×C₈	—	109	1	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	109	1	no	—
66	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	218	1	no	—
68	C₄²	—	2	1	no	—
69	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂³	—	218	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₄×D₁₀₉	1	2	yes	complemented
73	D₄	—	109	2	no	—
74	D₄	—	218	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	436	2	no	—
78	C₂×C₄	—	218	1	no	—
79	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₈	—	109	2	no	—
83	C₈	—	218	1	no	—
84	C₂²	—	436	1	no	—
85	C₂²	—	436	1	no	—
86	C₂²	—	218	2	no	—
87	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	1	no	—
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	436	1	no	—
93	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
94	C₄	#13182029	1	1	yes	characteristic
95	C₂	—	436	2	no	—
96	C₂	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
97	C₂	#16812724	1	1	yes	characteristic
98	C₂	#16812730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182016

C₂

yes

maximal, characteristic

C₄×D₄₃₆

C₂

yes

maximal, characteristic

#16812632

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

#16812619

C₂

yes

maximal, characteristic

#16812610

C₂

yes

maximal, characteristic, complemented

C₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

—

#13182016

C₄

yes

complemented

C₂×#11724784

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182006

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

—

C₄×D₁₀₉

—

#11724784

—

#11724784

C₂×C₄

yes

complemented

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₁₀₉

#440

yes

characteristic, complemented, Sylow

#440

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₄×Q₈

D₁₀₉

yes

characteristic, complemented

C₄×D₄

—

109

—

#107

—

109

—

#103

—

109

—

#102

—

109

—

#97

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

218

—

SD₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₄²

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

218

—

Q₈

—

Q₈

C₄×D₁₀₉

yes

complemented

D₄

—

109

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

—

218

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

436

—

C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₄

#13182029

yes

characteristic

C₂

—

436

—

C₂

C₄×C₂²⋊D₁₀₉

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₄×Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	#440
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₄×Q₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₄×Q₈

D₁₀₉

hypocenter

C₁₀₉

#440

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #16812632	C₂	semidirect
4. #16812623	C₂	semidirect
6. #16812610	C₂	semidirect
7. C₄×Q₈×C₁₀₉	C₂	semidirect
10. #13182016	C₄	semidirect
22. Q₈×C₁₀₉	C₂×C₄	semidirect
33. #11724784	C₂×C₄	semidirect
49. C₁₀₉	#440	semidirect
52. C₄×Q₈	D₁₀₉	semidirect
72. Q₈	C₄×D₁₀₉	semidirect

H ≅ G/N

type

3. #16812632

C₂

semidirect

4. #16812623

C₂

semidirect

6. #16812610

C₂

semidirect

7. C₄×Q₈×C₁₀₉

C₂

semidirect

10. #13182016

C₄

semidirect

22. Q₈×C₁₀₉

C₂×C₄

semidirect

33. #11724784

C₂×C₄

semidirect

49. C₁₀₉

#440

semidirect

52. C₄×Q₈

D₁₀₉

semidirect

72. Q₈

C₄×D₁₀₉

semidirect

Subgroups — #20295124

Conjugacy classes of subgroups

Complements