Group #20294986 — subgroups

Subgroups — C₂×#16812611 #20294986

Order 6976.

202 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812611	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₀₉	—	2	2	no	—
8	C₂×C₄×D₁₀₉	—	2	2	no	—
9	C₂×C₄×D₁₀₉	C₄	1	2	yes	—
10	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
11	C₂×#11724784	—	2	2	no	—
12	C₂×#11724784	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₀₉	—	2	2	no	—
15	C₂×C₈×C₁₀₉	C₂²	1	2	yes	complemented
16	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₁₀₉	—	2	2	no	—
18	C₂²×D₁₀₉	—	2	2	no	—
19	C₂²×D₁₀₉	C₈	1	4	yes	complemented
20	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
24	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀₉	—	2	2	no	—
26	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
27	C₄×D₁₀₉	—	2	4	no	—
28	C₄×D₁₀₉	—	2	2	no	—
29	C₄×D₁₀₉	—	2	2	no	—
30	C₈×C₁₀₉	—	2	2	no	—
31	#11724784	—	2	2	no	—
32	C₂²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₁₀₉	C₂×C₈	1	2	yes	—
34	C₂²×C₁₀₉	M₁₆	1	2	yes	—
35	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₁₀₉	—	2	8	no	—
37	C₂×D₁₀₉	—	2	2	no	—
38	C₂×D₁₀₉	—	2	2	no	—
39	C₂×D₁₀₉	C₂×C₈	1	4	yes	complemented
40	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
41	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	Dic₁₀₉	—	2	2	no	—
44	C₂×C₁₀₉	C₂²⋊C₈	1	4	yes	complemented
45	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
48	D₁₀₉	—	2	4	no	—
49	C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₂²⋊C₈	—	109	1	no	maximal, Sylow
51	C₂³×C₄	—	109	1	no	—
52	C₂²×C₈	—	109	1	no	—
53	C₂²×C₈	D₁₀₉	1	1	yes	characteristic, complemented
54	C₂²⋊C₈	—	109	4	no	—
55	C₂⁴	—	109	1	no	—
56	C₂²×C₄	—	109	1	no	—
57	C₂²×C₄	—	109	2	no	—
58	C₂²×C₄	—	109	2	no	—
59	C₂²×C₄	—	218	2	no	—
60	C₂²×C₄	—	218	2	no	—
61	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	109	2	no	—
64	C₂×C₈	—	218	2	no	—
65	C₂×C₈	C₂×D₁₀₉	1	2	yes	complemented
66	C₂³	—	109	4	no	—
67	C₂³	—	109	2	no	—
68	C₂³	—	218	2	no	—
69	C₂³	—	218	2	no	—
70	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₄	—	109	2	no	—
72	C₂×C₄	—	218	2	no	—
73	C₂×C₄	—	218	4	no	—
74	C₂×C₄	—	218	2	no	—
75	C₂×C₄	—	218	2	no	—
76	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄	D₄₃₆	1	2	yes	—
78	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
79	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	218	2	no	—
82	C₂²	—	109	4	no	—
83	C₂²	—	218	8	no	—
84	C₂²	—	218	2	no	—
85	C₂²	—	218	2	no	—
86	C₂²	#13182006	1	2	yes	—
87	C₂²	C₈×D₁₀₉	1	2	yes	—
88	C₂²	#13182012	1	2	yes	—
89	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
90	C₄	—	218	2	no	—
91	C₄	#13182006	1	2	yes	—
92	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₄	C₂×D₄₃₆	1	1	yes	characteristic
94	C₂	—	218	4	no	—
95	C₂	#16812611	1	4	yes	direct product factor
96	C₂	C₂×#13182006	1	1	yes	characteristic
97	C₂	C₂×C₈×D₁₀₉	1	1	yes	characteristic
98	C₂	C₂×#13182012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, direct product factor

C₂²×C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₄

yes

characteristic

C₂²×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₄

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×#11724784

—

C₂×#11724784

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₈

yes

complemented

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₈

yes

—

C₂²×C₁₀₉

M₁₆

yes

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₈

yes

complemented

C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

C₂²⋊C₈

yes

complemented

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

characteristic

D₁₀₉

—

C₁₀₉

C₂×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₈

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

D₁₀₉

yes

characteristic, complemented

C₂²⋊C₈

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

complemented

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

D₄₃₆

yes

—

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

#13182006

yes

—

C₂²

C₈×D₁₀₉

yes

—

C₂²

#13182012

yes

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

C₄

—

218

—

C₄

#13182006

yes

—

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₄

C₂×D₄₃₆

yes

characteristic

C₂

—

218

—

C₂

#16812611

yes

direct product factor

C₂

C₂×#13182006

yes

characteristic

C₂

C₂×C₈×D₁₀₉

yes

characteristic

C₂

C₂×#13182012

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×C₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²×C₄

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

C₂²×C₈×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×C₄

hypercenter

C₂²×C₈

D₁₀₉

hypocenter

C₁₀₉

C₂×C₂²⋊C₈

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
2. C₂²×#11724784	C₂	semidirect
3. #16812611	C₂	direct (N × H)
4. C₂²×C₈×C₁₀₉	C₂	semidirect
12. C₂×#11724784	C₂²	semidirect
15. C₂×C₈×C₁₀₉	C₂²	semidirect
19. C₂²×D₁₀₉	C₈	semidirect
39. C₂×D₁₀₉	C₂×C₈	semidirect
44. C₂×C₁₀₉	C₂²⋊C₈	semidirect
49. C₁₀₉	C₂×C₂²⋊C₈	semidirect
53. C₂²×C₈	D₁₀₉	semidirect
65. C₂×C₈	C₂×D₁₀₉	semidirect
95. C₂	#16812611	direct (N × H)

H ≅ G/N

type

2. C₂²×#11724784

C₂

semidirect

3. #16812611

C₂

direct (N × H)

4. C₂²×C₈×C₁₀₉

C₂

semidirect

12. C₂×#11724784

C₂²

semidirect

15. C₂×C₈×C₁₀₉

C₂²

semidirect

19. C₂²×D₁₀₉

C₈

semidirect

39. C₂×D₁₀₉

C₂×C₈

semidirect

44. C₂×C₁₀₉

C₂²⋊C₈

semidirect

49. C₁₀₉

C₂×C₂²⋊C₈

semidirect

53. C₂²×C₈

D₁₀₉

semidirect

65. C₂×C₈

C₂×D₁₀₉

semidirect

95. C₂

#16812611

direct (N × H)

Subgroups — C₂×#16812611 #20294986

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812611 #20294986

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812611 #20294986