Group #28376606 — subgroups

Subgroups — Q₈²×D₁₀₉ #28376606

Order 13952.

1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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	Q₈×Dic₂₁₈	C₂	1	6	yes	maximal, complemented
2	#20295756	C₂	1	9	yes	maximal, complemented
3	C₄⋊Q₈×D₁₀₉	C₂	1	9	yes	maximal
4	C₄×Q₈×D₁₀₉	C₂	1	6	yes	maximal
5	Q₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₁₀₉	C₂²	1	6	yes	—
7	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
8	#16812699	C₂²	1	9	yes	—
9	#16812698	C₂²	1	36	yes	—
10	#16812697	C₂²	1	18	yes	—
11	Q₈×Dic₁₀₉	C₂²	1	6	yes	—
12	C₄×Dic₂₁₈	C₂²	1	18	yes	—
13	#16812664	C₂²	1	18	yes	—
14	C₄⋊C₄×D₁₀₉	C₂²	1	18	yes	—
15	C₄²×D₁₀₉	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	9	yes	—
17	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	—
18	C₂×Dic₂₁₈	C₂³	1	18	yes	—
19	C₂×Dic₂₁₈	C₂³	1	6	yes	—
20	Q₈×D₁₀₉	—	4	12	no	—
21	Q₈×D₁₀₉	Q₈	1	16	yes	direct product factor
22	C₂×C₄×D₁₀₉	C₂³	1	9	yes	—
23	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
24	C₄⋊Dic₁₀₉	C₂³	1	18	yes	—
25	#13182008	C₂³	1	36	yes	—
26	C₄×Dic₁₀₉	C₂³	1	9	yes	—
27	C₄×Dic₁₀₉	C₂³	1	18	yes	—
28	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
29	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
30	C₄⋊C₄×C₁₀₉	C₂³	1	18	yes	—
31	C₄²×C₁₀₉	C₂³	1	9	yes	—
32	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
33	Q₈×C₁₀₉	—	4	6	no	—
34	Q₈×C₁₀₉	C₂×Q₈	1	8	yes	complemented
35	C₂×C₄×C₁₀₉	C₂⁴	1	9	yes	—
36	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
37	C₂×Dic₁₀₉	C₂⁴	1	9	yes	—
38	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
39	C₄×D₁₀₉	—	2	18	no	—
40	C₄×D₁₀₉	C₂×Q₈	1	24	yes	—
41	Dic₂₁₈	—	4	18	no	—
42	Dic₂₁₈	C₂×Q₈	1	24	yes	complemented
43	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₀₉	C₂²×Q₈	1	4	yes	—
45	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
46	C₄×C₁₀₉	—	2	9	no	—
47	C₄×C₁₀₉	C₂²×Q₈	1	12	yes	—
48	Dic₁₀₉	—	2	9	no	—
49	Dic₁₀₉	C₂²×Q₈	1	12	yes	—
50	C₂×C₁₀₉	C₂³×Q₈	1	2	yes	—
51	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
52	D₁₀₉	Q₈²	1	4	yes	direct product factor
53	C₂×Q₈²	—	109	1	no	maximal, Sylow
54	C₁₀₉	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
55	Q₈²	—	109	9	no	—
56	Q₈²	—	109	6	no	—
57	Q₈²	D₁₀₉	1	1	yes	characteristic, direct product factor
58	C₂×C₄⋊Q₈	—	109	9	no	—
59	C₂×C₄×Q₈	—	109	6	no	—
60	C₂²×Q₈	—	109	2	no	—
61	C₂²×Q₈	—	109	6	no	—
62	C₄⋊Q₈	—	109	18	no	—
63	C₄⋊Q₈	—	109	36	no	—
64	C₄⋊Q₈	—	109	9	no	—
65	C₄⋊Q₈	C₂×D₁₀₉	1	9	yes	—
66	C₄×Q₈	—	109	6	no	—
67	C₄×Q₈	—	109	18	no	—
68	C₄×Q₈	—	109	18	no	—
69	C₄×Q₈	C₂×D₁₀₉	1	6	yes	—
70	C₂×C₄⋊C₄	—	109	18	no	—
71	C₂×C₄²	—	109	9	no	—
72	C₂×Q₈	—	109	16	no	—
73	C₂×Q₈	—	109	6	no	—
74	C₂×Q₈	—	109	18	no	—
75	C₂×Q₈	—	436	12	no	—
76	C₂×Q₈	C₂²×D₁₀₉	1	6	yes	—
77	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
78	C₂²×C₄	—	109	6	no	—
79	C₂²×C₄	—	109	9	no	—
80	C₄⋊C₄	—	109	36	no	—
81	C₄⋊C₄	—	109	18	no	—
82	C₄⋊C₄	C₂²×D₁₀₉	1	18	yes	—
83	C₄²	—	109	9	no	—
84	C₄²	—	109	18	no	—
85	C₄²	C₂²×D₁₀₉	1	9	yes	—
86	C₂³	—	109	1	no	—
87	Q₈	—	4	6	no	—
88	Q₈	—	109	24	no	—
89	Q₈	—	436	18	no	—
90	Q₈	Q₈×D₁₀₉	1	8	yes	direct product factor
91	C₂×C₄	—	109	6	no	—
92	C₂×C₄	—	109	24	no	—
93	C₂×C₄	—	109	9	no	—
94	C₂×C₄	—	218	18	no	—
95	C₂×C₄	C₂³×D₁₀₉	1	9	yes	—
96	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
97	C₂²	—	109	4	no	—
98	C₂²	—	109	2	no	—
99	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	109	12	no	—
102	C₄	—	218	9	no	—
103	C₄	C₂×Q₈×D₁₀₉	1	12	yes	—
104	C₂	—	109	4	no	—
105	C₂	C₂²×Q₈×D₁₀₉	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₁₈

C₂

yes

maximal, complemented

#20295756

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal

C₄×Q₈×D₁₀₉

C₂

yes

maximal

Q₈²×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₀₉

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₂²

yes

—

#16812699

C₂²

yes

—

#16812698

C₂²

yes

—

#16812697

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812664

C₂²

yes

—

C₄⋊C₄×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

Q₈

yes

direct product factor

C₂×C₄×D₁₀₉

C₂³

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

C₂⁴

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×Q₈

yes

complemented

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

Dic₂₁₈

—

Dic₂₁₈

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₂×D₁₀₉

C₂²×Q₈

yes

—

C₂×D₁₀₉

H(5,𝔽₂)

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×Q₈

yes

—

C₂×C₁₀₉

C₂³×Q₈

yes

—

C₂×C₁₀₉

C₂×H(5,𝔽₂)

yes

characteristic

D₁₀₉

Q₈²

yes

direct product factor

C₂×Q₈²

—

109

maximal, Sylow

C₁₀₉

C₂×Q₈²

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

Q₈²

—

109

—

Q₈²

D₁₀₉

yes

characteristic, direct product factor

C₂×C₄⋊Q₈

—

109

—

C₂×C₄×Q₈

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

C₂²×D₁₀₉

yes

—

C₂×Q₈

C₂²×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂²×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂²×D₁₀₉

yes

—

C₂³

—

109

—

Q₈

—

Q₈

—

109

—

Q₈

—

436

—

Q₈

Q₈×D₁₀₉

yes

direct product factor

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂³×D₁₀₉

yes

—

C₂×C₄

C₂³×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

C₂⁴×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

109

—

102

C₄

—

218

—

103

C₄

C₂×Q₈×D₁₀₉

yes

—

104

C₂

—

109

—

105

C₂

C₂²×Q₈×D₁₀₉

yes

—

106

C₂

H(5,𝔽₂)×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	Q₈²×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂⁴
hypercenter	Q₈²	D₁₀₉
hypocenter	C₁₀₉	C₂×Q₈²
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₉

commutator, socle, cosocle

C₂²×C₁₀₉

C₂⁵

Fitting

Q₈²×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₂⁴

hypercenter

Q₈²

D₁₀₉

hypocenter

C₁₀₉

C₂×Q₈²

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. Q₈×Dic₂₁₈	C₂	semidirect
2. #20295756	C₂	semidirect
5. Q₈²×C₁₀₉	C₂	semidirect
21. Q₈×D₁₀₉	Q₈	direct (N × H)
34. Q₈×C₁₀₉	C₂×Q₈	semidirect
42. Dic₂₁₈	C₂×Q₈	semidirect
52. D₁₀₉	Q₈²	direct (N × H)
54. C₁₀₉	C₂×Q₈²	semidirect
57. Q₈²	D₁₀₉	direct (N × H)
90. Q₈	Q₈×D₁₀₉	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₂₁₈

C₂

semidirect

2. #20295756

C₂

semidirect

5. Q₈²×C₁₀₉

C₂

semidirect

21. Q₈×D₁₀₉

Q₈

direct (N × H)

34. Q₈×C₁₀₉

C₂×Q₈

semidirect

42. Dic₂₁₈

C₂×Q₈

semidirect

52. D₁₀₉

Q₈²

direct (N × H)

54. C₁₀₉

C₂×Q₈²

semidirect

57. Q₈²

D₁₀₉

direct (N × H)

90. Q₈

Q₈×D₁₀₉

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Dic₂₁₈	2
#20295756	2
C₄⋊Q₈×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
Q₈²×C₁₀₉	2
C₂×Q₈²	109

subgroup

index

Q₈×Dic₂₁₈

#20295756

C₄⋊Q₈×D₁₀₉

C₄×Q₈×D₁₀₉

Q₈²×C₁₀₉

C₂×Q₈²

109

Subgroups — Q₈²×D₁₀₉ #28376606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q82×D109 #28376606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×D₁₀₉ #28376606