Group #28362172 — subgroups

Subgroups — #28362172

Order 13952.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20295405	C₂	1	1	yes	maximal, characteristic
2	#20295321	C₂	1	2	yes	maximal
3	#20294775	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11724784	C₂	1	2	yes	maximal
5	Q₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182019	—	2	2	no	—
7	#16812699	C₂²	1	1	yes	characteristic
8	#16812632	—	2	2	no	—
9	C₄⋊#11724784	C₂²	1	2	yes	—
10	#16812623	—	2	2	no	—
11	C₄×#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
13	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₀₉	—	2	2	no	—
16	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂×Dic₂₁₈	—	2	1	no	—
18	#13182019	—	4	4	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₂×Q₈×C₁₀₉	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
24	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₉	—	2	4	no	—
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄⋊C₄×C₁₀₉	—	2	2	no	—
28	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
29	C₄²×C₁₀₉	—	2	2	no	—
30	C₄²×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	Q₈×C₁₀₉	—	2	2	no	—
33	Q₈×C₁₀₉	—	4	2	no	—
34	Q₈×C₁₀₉	—	8	2	no	—
35	Q₈×C₁₀₉	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
41	C₂×Dic₁₀₉	—	4	1	no	—
42	Dic₂₁₈	—	4	2	no	—
43	#11724784	—	2	2	no	—
44	#11724784	—	4	2	no	—
45	C₂²×C₁₀₉	#120	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	4	no	—
47	C₄×C₁₀₉	—	2	1	no	—
48	C₄×C₁₀₉	—	4	2	no	—
49	C₄×C₁₀₉	—	4	2	no	—
50	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₀₉	#137	1	2	yes	—
52	Dic₁₀₉	—	8	1	no	—
53	C₂×C₁₀₉	#451	1	2	yes	—
54	C₂×C₁₀₉	#453	1	1	yes	characteristic
55	#1633	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	D₁₀₉	1	1	yes	characteristic
58	#494	—	109	1	no	—
59	#477	—	109	2	no	—
60	#333	—	109	1	no	—
61	Q₈⋊C₈	—	109	2	no	—
62	C₂×Q₁₆	—	218	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	109	1	no	—
66	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
69	#107	—	218	2	no	—
70	C₄⋊C₈	—	109	2	no	—
71	#103	—	218	2	no	—
72	C₄×C₈	—	109	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	218	1	no	—
75	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
77	Q₁₆	—	436	4	no	—
78	C₂×C₈	—	218	2	no	—
79	C₂×C₈	—	218	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	436	2	no	—
92	Q₈	#13182019	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	436	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
99	C₈	—	436	2	no	—
100	C₈	—	218	2	no	—
101	C₂²	#16812668	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	872	1	no	—
107	C₄	C₂×#13182019	1	2	yes	—
108	C₄	#16812730	1	2	yes	—
109	C₂	#20295182	1	2	yes	—
110	C₂	#20295194	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295405

C₂

yes

maximal, characteristic

#20295321

C₂

yes

maximal

#20294775

C₂

yes

maximal, characteristic

Q₈⋊#11724784

C₂

yes

maximal

Q₈²×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

—

#16812699

C₂²

yes

characteristic

#16812632

—

C₄⋊#11724784

C₂²

yes

—

#16812623

—

C₄×#11724784

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

#13182019

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

Q₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#451

yes

—

C₂×C₁₀₉

#453

yes

characteristic

#1633

—

109

maximal, Sylow

C₁₀₉

#1633

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₉

yes

characteristic

#494

—

109

—

#477

—

109

—

#333

—

109

—

Q₈⋊C₈

—

109

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₀₉

yes

—

#107

—

218

—

C₄⋊C₈

—

109

—

#103

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

Q₁₆

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

436

—

Q₈

#13182019

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

C₈

—

436

—

100

C₈

—

218

—

101

C₂²

#16812668

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

872

—

107

C₄

C₂×#13182019

yes

—

108

C₄

#16812730

yes

—

109

C₂

#20295182

yes

—

110

C₂

#20295194

yes

characteristic

	subgroup	G/H
center	C₂²	#16812668
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	Q₈²×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#120
hypercenter	Q₈²	D₁₀₉
hypocenter	C₁₀₉	#1633
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#16812668

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

Q₈²×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

#120

hypercenter

Q₈²

D₁₀₉

hypocenter

C₁₀₉

#1633

LCS

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₁₀₉	Q₁₆	semidirect
56. C₁₀₉	#1633	semidirect
92. Q₈	#13182019	semidirect

H ≅ G/N

type

35. Q₈×C₁₀₉

Q₁₆

semidirect

56. C₁₀₉

#1633

semidirect

92. Q₈

#13182019

semidirect

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

subgroup	index
#20295405	2
#20295321	2
#20294775	2
Q₈⋊#11724784	2
Q₈²×C₁₀₉	2
#1633	109

subgroup

index

Q₈²×C₁₀₉

109

Subgroups — #28362172

Conjugacy classes of subgroups

Complements

Maximal subgroups