Group #28378682 — subgroups

Subgroups — #28378682

Order 13952.

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20296061	C₂	1	2	yes	maximal, complemented
2	#20295887	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11724786	C₂	1	2	yes	maximal, complemented
4	#20295867	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182038	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	C₂²⋊#11724786	—	2	2	no	—
9	C₄×#11724786	C₂²	1	1	yes	characteristic
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	D₄×Dic₁₀₉	C₂²	1	2	yes	—
12	#16812633	C₄	1	1	yes	characteristic
13	#16812620	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182038	—	4	4	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×#11724786	—	2	1	no	—
19	C₂×#11724786	D₄	1	2	yes	—
20	C₂²×Dic₁₀₉	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	D₈×C₁₀₉	—	2	2	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₉	—	2	2	no	—
29	D₄×C₁₀₉	—	2	2	no	—
30	D₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	2	2	no	—
33	C₂×Dic₁₀₉	—	4	2	no	—
34	C₂×Dic₁₀₉	—	4	2	no	—
35	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
37	#11724786	—	2	2	no	—
38	#11724786	—	4	1	no	—
39	#11724786	—	4	2	no	—
40	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
41	#11724784	—	2	1	no	—
42	C₂²×C₁₀₉	—	4	2	no	—
43	C₂²×C₁₀₉	—	4	2	no	—
44	C₂²×C₁₀₉	—	4	2	no	—
45	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
47	C₄×C₁₀₉	#131	1	1	yes	characteristic
48	Dic₁₀₉	—	2	1	no	—
49	Dic₁₀₉	—	4	2	no	—
50	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₁₀₉	—	8	2	no	—
52	C₂×C₁₀₉	#436	1	1	yes	characteristic
53	C₂×C₁₀₉	#442	1	1	yes	characteristic
54	C₂×C₁₀₉	#444	1	1	yes	characteristic
55	#1594	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	109	1	no	—
58	#436	—	109	2	no	—
59	#334	—	109	1	no	—
60	D₄⋊C₈	—	109	2	no	—
61	#322	—	109	1	no	—
62	C₂×D₈	#89137	1	1	yes	characteristic
63	C₂×M₁₆	—	218	2	no	—
64	C₄×D₄	—	109	2	no	—
65	#107	—	109	1	no	—
66	C₄⋊C₈	—	109	2	no	—
67	#102	—	109	2	no	—
68	C₂²⋊C₈	—	218	2	no	—
69	C₄×C₈	—	109	1	no	—
70	C₄×C₈	—	109	1	no	—
71	C₄×C₈	—	109	1	no	—
72	C₂×D₄	C₂×#89137	1	2	yes	—
73	C₂²×C₄	—	218	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	436	4	no	—
76	C₂×C₈	—	109	2	no	—
77	C₂×C₈	—	109	1	no	—
78	C₂×C₈	—	218	1	no	—
79	C₂×C₈	—	218	2	no	—
80	C₂×C₈	C₂×#89137	1	1	yes	characteristic
81	C₄⋊C₄	—	109	2	no	—
82	C₂²⋊C₄	—	218	2	no	—
83	C₄²	—	109	1	no	—
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	436	2	no	—
90	C₂×C₄	—	436	2	no	—
91	C₂×C₄	—	218	2	no	—
92	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
93	C₈	—	436	2	no	—
94	C₈	—	436	1	no	—
95	C₈	—	218	2	no	—
96	C₈	—	218	1	no	—
97	C₈	#13182038	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×#89137	1	1	yes	characteristic
102	C₄	—	109	2	no	—
103	C₄	—	436	2	no	—
104	C₄	—	218	1	no	—
105	C₄	C₂×#13182038	1	1	yes	characteristic
106	C₄	#16812786	1	1	yes	characteristic
107	C₂	—	8	2	no	—
108	C₂	#20296061	1	1	yes	characteristic
109	C₂	#20296067	1	1	yes	characteristic
110	C₂	#20296073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296061

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, characteristic, complemented

D₄⋊#11724786

C₂

yes

maximal, complemented

#20295867

C₂

yes

maximal, characteristic, complemented

D₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182038

—

C₄⋊#11724786

C₂²

yes

—

C₂²⋊#11724786

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

D₄×Dic₁₀₉

C₂²

yes

—

#16812633

C₄

yes

characteristic

#16812620

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₄

yes

—

—

—

D₄

yes

—

C₂²×Dic₁₀₉

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

C₂×Dic₁₀₉

yes

—

—

—

C₈×C₁₀₉

M₁₆

yes

complemented

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1594

yes

characteristic, complemented, Sylow

C₄×D₈

—

109

—

#436

—

109

—

#334

—

109

—

D₄⋊C₈

—

109

—

#322

—

109

—

C₂×D₈

#89137

yes

characteristic

C₂×M₁₆

—

218

—

C₄×D₄

—

109

—

#107

—

109

—

C₄⋊C₈

—

109

—

#102

—

109

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂×D₄

C₂×#89137

yes

—

C₂²×C₄

—

218

—

D₈

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

436

—

C₈

—

436

—

C₈

—

218

—

C₈

—

218

—

C₈

#13182038

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×#89137

yes

characteristic

102

C₄

—

109

—

103

C₄

—

436

—

104

C₄

—

218

—

105

C₄

C₂×#13182038

yes

characteristic

106

C₄

#16812786

yes

characteristic

107

C₂

—

108

C₂

#20296061

yes

characteristic

109

C₂

#20296067

yes

characteristic

110

C₂

#20296073

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting, prime radical	C₂×D₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×D₈	#89137
hypocenter	C₁₀₉	#1594
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting, prime radical

C₂×D₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂×D₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. #20296061	C₂	semidirect
2. #20295887	C₂	semidirect
3. D₄⋊#11724786	C₂	semidirect
4. #20295867	C₂	semidirect
40. C₈×C₁₀₉	M₁₆	semidirect
56. C₁₀₉	#1594	semidirect
97. C₈	#13182038	semidirect

H ≅ G/N

type

1. #20296061

C₂

semidirect

2. #20295887

C₂

semidirect

3. D₄⋊#11724786

C₂

semidirect

4. #20295867

C₂

semidirect

40. C₈×C₁₀₉

M₁₆

semidirect

56. C₁₀₉

#1594

semidirect

97. C₈

#13182038

semidirect

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

subgroup	index
#20296061	2
#20295887	2
D₄⋊#11724786	2
#20295867	2
D₈×Dic₁₀₉	2
#1594	109

subgroup

index

D₈×Dic₁₀₉

109

Subgroups — #28378682

Conjugacy classes of subgroups

Complements

Maximal subgroups