Group #20295028 — subgroups

Subgroups — #20295028

Order 6976.

162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 63 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×#13182029	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812624	C₂	1	2	yes	maximal, complemented
4	#16812621	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#13182029	—	2	2	no	—
7	#13182029	C₄	1	4	yes	complemented
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
11	C₂²×Dic₁₀₉	—	2	1	no	—
12	C₂×C₄×D₁₀₉	—	2	1	no	—
13	C₄⋊Dic₁₀₉	—	2	1	no	—
14	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₀₉	—	2	2	no	—
17	C₂×C₈×C₁₀₉	C₂²	1	2	yes	—
18	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×D₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²⋊D₁₀₉	—	2	4	no	—
25	C₂×Dic₁₀₉	—	2	1	no	—
26	C₂×Dic₁₀₉	—	2	2	no	—
27	C₂×Dic₁₀₉	—	4	2	no	—
28	D₄₃₆	—	2	1	no	—
29	D₄₃₆	C₂×C₄	1	2	yes	complemented
30	C₄×D₁₀₉	—	2	4	no	—
31	Dic₂₁₈	—	2	1	no	—
32	Dic₂₁₈	C₂×C₄	1	2	yes	complemented
33	C₈×C₁₀₉	—	2	2	no	—
34	C₂²×C₁₀₉	—	2	1	no	—
35	C₂²×C₁₀₉	—	2	1	no	—
36	C₂²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₀₉	—	2	2	no	—
39	C₂×D₁₀₉	—	4	1	no	—
40	C₂×D₁₀₉	—	4	1	no	—
41	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	—	4	2	no	—
46	C₂×C₁₀₉	—	2	2	no	—
47	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₂×C₁₀₉	#137	1	1	yes	characteristic
50	D₁₀₉	—	4	2	no	—
51	C₁₀₉	#417	1	1	yes	characteristic, complemented, Sylow
52	#417	—	109	1	no	maximal, Sylow
53	C₂×#41	—	109	1	no	—
54	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic, complemented
55	C₂×C₄⋊C₄	—	109	1	no	—
56	#103	—	109	2	no	—
57	#102	—	109	2	no	—
58	#41	—	109	4	no	—
59	#41	—	218	2	no	—
60	C₂×Q₈	—	109	1	no	—
61	C₂×D₄	—	109	1	no	—
62	C₂×D₄	—	218	1	no	—
63	C₂²×C₄	—	218	1	no	—
64	C₂²×C₄	—	218	1	no	—
65	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
66	M₁₆	—	2	2	no	—
67	C₂×C₈	C₂×D₁₀₉	1	2	yes	—
68	C₄⋊C₄	—	109	2	no	—
69	C₄⋊C₄	—	218	1	no	—
70	C₂³	—	218	1	no	—
71	C₂³	D₄₃₆	1	1	yes	characteristic
72	Q₈	—	109	2	no	—
73	Q₈	—	218	1	no	—
74	D₄	—	109	2	no	—
75	D₄	—	218	4	no	—
76	D₄	—	218	1	no	—
77	C₂×C₄	—	436	2	no	—
78	C₂×C₄	—	218	2	no	—
79	C₂×C₄	—	218	1	no	—
80	C₂×C₄	—	218	4	no	—
81	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
82	C₂×C₄	D₄₃₆	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
84	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	436	1	no	—
89	C₂²	—	436	1	no	—
90	C₂²	—	218	2	no	—
91	C₂²	#13182006	1	2	yes	—
92	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
93	C₄	—	436	2	no	—
94	C₄	—	218	2	no	—
95	C₄	#13182006	1	2	yes	—
96	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
97	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	—	436	2	no	—
100	C₂	C₂×#13182006	1	1	yes	characteristic
101	C₂	#16812727	1	1	yes	characteristic
102	C₂	#16812732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812624

C₂

yes

maximal, complemented

#16812621

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

—

#13182029

C₄

yes

complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

complemented

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₄

yes

complemented

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₁₀₉

#417

yes

characteristic, complemented, Sylow

#417

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂×M₁₆

D₁₀₉

yes

characteristic, complemented

C₂×C₄⋊C₄

—

109

—

#103

—

109

—

#102

—

109

—

#41

—

109

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

C₂×C₈

C₂×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₂³

—

218

—

C₂³

D₄₃₆

yes

characteristic

Q₈

—

109

—

Q₈

—

218

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

—

C₂×C₄

D₄₃₆

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

—

436

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

#13182006

yes

—

C₂²

C₂×D₄₃₆

yes

characteristic

C₄

—

436

—

C₄

—

218

—

C₄

#13182006

yes

—

C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂

—

C₂

—

436

—

100

C₂

C₂×#13182006

yes

characteristic

101

C₂

#16812727

yes

characteristic

102

C₂

#16812732

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₃₆
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#417
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂²

C₂×D₄₃₆

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

hypercenter

C₂×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#417

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
3. #16812624	C₂	semidirect
4. #16812621	C₂	semidirect
5. C₂×M₁₆×C₁₀₉	C₂	semidirect
7. #13182029	C₄	semidirect
29. D₄₃₆	C₂×C₄	semidirect
32. Dic₂₁₈	C₂×C₄	semidirect
51. C₁₀₉	#417	semidirect
54. C₂×M₁₆	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

3. #16812624

C₂

semidirect

4. #16812621

C₂

semidirect

5. C₂×M₁₆×C₁₀₉

C₂

semidirect

7. #13182029

C₄

semidirect

29. D₄₃₆

C₂×C₄

semidirect

32. Dic₂₁₈

C₂×C₄

semidirect

51. C₁₀₉

#417

semidirect

54. C₂×M₁₆

D₁₀₉

semidirect

Subgroups — #20295028

Conjugacy classes of subgroups

Complements