Group #20295690 — subgroups

Subgroups — Q₈×C₂²⋊D₁₀₉ #20295690

Order 6976.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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, complemented
2	#16812697	C₂	1	3	yes	maximal, complemented
3	#16812678	C₂	1	3	yes	maximal, complemented
4	#16812675	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₀₉	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	C₂²	1	3	yes	—
9	Q₈×D₁₀₉	—	2	4	no	—
10	C₂×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₀₉	C₂²	1	3	yes	—
12	C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
13	#13182008	C₂²	1	3	yes	—
14	#13182008	C₂²	1	6	yes	—
15	C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
16	#13182006	C₂²	1	3	yes	—
17	C₄×Dic₁₀₉	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₀₉	—	2	4	no	—
19	C₂×Q₈×C₁₀₉	C₂²	1	3	yes	—
20	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂²	1	3	yes	—
22	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
24	Q₈×C₁₀₉	—	2	6	no	—
25	Q₈×C₁₀₉	D₄	1	4	yes	complemented
26	C₂×C₄×C₁₀₉	—	2	6	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
29	C₂²⋊D₁₀₉	Q₈	1	4	yes	direct product factor
30	C₂×Dic₁₀₉	C₂³	1	3	yes	—
31	C₂×Dic₁₀₉	C₂³	1	3	yes	—
32	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
33	C₄×D₁₀₉	—	2	6	no	—
34	Dic₂₁₈	—	2	6	no	—
35	C₂²×C₁₀₉	—	2	1	no	—
36	C₂²×C₁₀₉	—	2	1	no	—
37	C₂²×C₁₀₉	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	1	no	—
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	C₂×Q₈	1	2	yes	complemented
42	C₄×C₁₀₉	—	2	3	no	—
43	C₄×C₁₀₉	C₂×D₄	1	6	yes	—
44	Dic₁₀₉	—	2	3	no	—
45	Dic₁₀₉	—	2	3	no	—
46	Dic₁₀₉	C₂×Q₈	1	2	yes	complemented
47	C₂×C₁₀₉	—	2	2	no	—
48	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₀₉	—	2	2	no	—
52	C₁₀₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
53	D₄×Q₈	—	109	1	no	maximal, Sylow
54	C₂²×Q₈	—	109	1	no	—
55	C₂²×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	109	3	no	—
57	C₂²⋊Q₈	—	109	3	no	—
58	C₂²⋊Q₈	—	109	3	no	—
59	C₄×Q₈	—	109	1	no	—
60	C₄×D₄	—	109	3	no	—
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	109	3	no	—
63	C₂×Q₈	—	218	4	no	—
64	C₂×Q₈	C₂×D₁₀₉	1	3	yes	—
65	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
66	C₂×D₄	—	109	1	no	—
67	C₂²×C₄	—	109	3	no	—
68	C₂²×C₄	C₂×D₁₀₉	1	3	yes	—
69	C₄⋊C₄	—	109	3	no	—
70	C₄⋊C₄	—	109	6	no	—
71	C₄⋊C₄	—	109	3	no	—
72	C₂²⋊C₄	—	109	3	no	—
73	C₂²⋊C₄	—	109	3	no	—
74	C₄²	—	109	3	no	—
75	C₂³	—	109	1	no	—
76	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	218	6	no	—
79	Q₈	C₂²⋊D₁₀₉	1	4	yes	direct product factor
80	D₄	—	109	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	109	3	no	—
84	C₂×C₄	—	109	3	no	—
85	C₂×C₄	—	218	6	no	—
86	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
87	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	109	2	no	—
91	C₂²	—	218	1	no	—
92	C₂²	—	218	1	no	—
93	C₂²	Q₈×D₁₀₉	1	2	yes	complemented
94	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	109	2	no	—
97	C₄	—	218	3	no	—
98	C₄	—	218	3	no	—
99	C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	218	2	no	—
102	C₂	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
103	C₂	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
104	C₂	#16812747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812697

C₂

yes

maximal, complemented

#16812678

C₂

yes

maximal, complemented

#16812675

C₂

yes

maximal, complemented

Q₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₉

C₂

yes

maximal

C₂²×Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

—

Q₈×D₁₀₉

—

C₂×C₂²⋊D₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

C₂²⋊Dic₁₀₉

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

Q₈

yes

direct product factor

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₈

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₈

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

characteristic

D₁₀₉

—

C₁₀₉

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂²×Q₈

D₁₀₉

yes

characteristic, complemented

C₄⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

C₂×Q₈

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₂³

—

109

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

218

—

Q₈

C₂²⋊D₁₀₉

yes

direct product factor

D₄

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

Q₈×D₁₀₉

yes

complemented

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂×C₂²⋊D₁₀₉

yes

—

100

C₂

—

101

C₂

—

218

—

102

C₂

C₂²×C₂²⋊D₁₀₉

yes

characteristic

103

C₂

C₂×Q₈×D₁₀₉

yes

characteristic

104

C₂

#16812747

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	C₂²×Q₈×C₁₀₉	C₂
hypercenter	C₂²×Q₈	D₁₀₉
hypocenter	C₁₀₉	D₄×Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

C₂²×Q₈×C₁₀₉

C₂

hypercenter

C₂²×Q₈

D₁₀₉

hypocenter

C₁₀₉

D₄×Q₈

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₀₉	C₂	semidirect
2. #16812697	C₂	semidirect
3. #16812678	C₂	semidirect
4. #16812675	C₂	semidirect
5. Q₈×Dic₁₀₉	C₂	semidirect
7. C₂²×Q₈×C₁₀₉	C₂	semidirect
25. Q₈×C₁₀₉	D₄	semidirect
29. C₂²⋊D₁₀₉	Q₈	direct (N × H)
37. C₂²×C₁₀₉	C₂×Q₈	semidirect
41. C₂×D₁₀₉	C₂×Q₈	semidirect
46. Dic₁₀₉	C₂×Q₈	semidirect
52. C₁₀₉	D₄×Q₈	semidirect
55. C₂²×Q₈	D₁₀₉	semidirect
79. Q₈	C₂²⋊D₁₀₉	direct (N × H)
93. C₂²	Q₈×D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₀₉

C₂

semidirect

2. #16812697

C₂

semidirect

3. #16812678

C₂

semidirect

4. #16812675

C₂

semidirect

5. Q₈×Dic₁₀₉

C₂

semidirect

7. C₂²×Q₈×C₁₀₉

C₂

semidirect

25. Q₈×C₁₀₉

D₄

semidirect

29. C₂²⋊D₁₀₉

Q₈

direct (N × H)

37. C₂²×C₁₀₉

C₂×Q₈

semidirect

41. C₂×D₁₀₉

C₂×Q₈

semidirect

46. Dic₁₀₉

C₂×Q₈

semidirect

52. C₁₀₉

D₄×Q₈

semidirect

55. C₂²×Q₈

D₁₀₉

semidirect

79. Q₈

C₂²⋊D₁₀₉

direct (N × H)

93. C₂²

Q₈×D₁₀₉

semidirect

Subgroups — Q₈×C₂²⋊D₁₀₉ #20295690

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C22⋊D109 #20295690

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₂²⋊D₁₀₉ #20295690