Group #28361946 — subgroups

Subgroups — #28361946

Order 13952.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20295104	C₂	1	2	yes	maximal, complemented
2	#20294777	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11724784	C₂	1	2	yes	maximal, complemented
4	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	—	2	2	no	—
7	C₄⋊#11724784	C₂²	1	2	yes	—
8	C₂²⋊#11724784	—	2	2	no	—
9	C₄×#11724784	C₂²	1	1	yes	characteristic
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₀₉	C₄	1	1	yes	characteristic
12	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
13	#107×C₁₀₉	C₄	1	1	yes	characteristic
14	#102×C₁₀₉	C₄	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#13182013	—	4	4	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	2	1	no	—
19	C₂×#11724784	D₄	1	2	yes	—
20	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	—	2	2	no	—
22	D₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄²×C₁₀₉	C₂³	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	—	2	2	no	—
32	C₂×C₄×C₁₀₉	—	4	2	no	—
33	C₂×C₄×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
37	C₈×C₁₀₉	—	2	1	no	—
38	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
39	#11724784	—	2	2	no	—
40	#11724784	—	4	1	no	—
41	#11724784	—	4	2	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	—	2	1	no	—
47	C₄×C₁₀₉	—	4	2	no	—
48	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₁₀₉	#131	1	1	yes	characteristic
50	C₄×C₁₀₉	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	D₁₀₉	1	1	yes	characteristic
58	#436	—	109	2	no	—
59	#334	—	109	1	no	—
60	D₄⋊C₈	—	109	2	no	—
61	#322	—	109	1	no	—
62	C₂×D₈	Dic₁₀₉	1	1	yes	characteristic
63	C₂×M₁₆	—	218	2	no	—
64	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
65	#107	Dic₁₀₉	1	1	yes	characteristic
66	C₄⋊C₈	—	109	2	no	—
67	#102	Dic₁₀₉	1	2	yes	—
68	C₂²⋊C₈	—	218	2	no	—
69	C₄×C₈	—	109	1	no	—
70	C₄×C₈	—	109	1	no	—
71	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	436	4	no	—
76	C₂×C₈	—	109	2	no	—
77	C₂×C₈	—	218	2	no	—
78	C₂×C₈	—	218	1	no	—
79	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
91	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
92	C₂×C₄	#13182028	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	436	1	no	—
95	C₈	—	436	2	no	—
96	C₈	—	218	2	no	—
97	C₈	#13182013	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂×#13182013	1	1	yes	characteristic
105	C₄	#16812706	1	1	yes	characteristic
106	C₄	#16812723	1	2	yes	—
107	C₂	—	8	2	no	—
108	C₂	#20295104	1	1	yes	characteristic
109	C₂	#20295138	1	1	yes	characteristic
110	C₂	#20295148	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295104

C₂

yes

maximal, complemented

#20294777

C₂

yes

maximal, characteristic, complemented

D₄⋊#11724784

C₂

yes

maximal, complemented

#20294747

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₄⋊#11724784

C₂²

yes

—

C₂²⋊#11724784

—

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₄

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₄

yes

characteristic

#102×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

—

—

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1594

yes

characteristic, complemented, Sylow

C₄×D₈

D₁₀₉

yes

characteristic

#436

—

109

—

#334

—

109

—

D₄⋊C₈

—

109

—

#322

—

109

—

C₂×D₈

Dic₁₀₉

yes

characteristic

C₂×M₁₆

—

218

—

C₄×D₄

C₂×D₁₀₉

yes

—

#107

Dic₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

#102

Dic₁₀₉

yes

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

D₄×D₁₀₉

yes

characteristic

C₂×C₄

#13182028

yes

characteristic

C₈

—

C₈

—

436

—

C₈

—

436

—

C₈

—

218

—

C₈

#13182013

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₁₀₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#13182013

yes

characteristic

105

C₄

#16812706

yes

characteristic

106

C₄

#16812723

yes

—

107

C₂

—

108

C₂

#20295104

yes

characteristic

109

C₂

#20295138

yes

characteristic

110

C₂

#20295148

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₄×D₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂×D₈×C₁₀₉	C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₄×D₈	D₁₀₉
hypocenter	C₁₀₉	#1594
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₄×D₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂×D₈×C₁₀₉

C₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₄×D₈

D₁₀₉

hypocenter

C₁₀₉

#1594

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #20295104	C₂	semidirect
2. #20294777	C₂	semidirect
3. D₄⋊#11724784	C₂	semidirect
4. #20294747	C₂	semidirect
38. C₈×C₁₀₉	M₁₆	semidirect
56. C₁₀₉	#1594	semidirect
97. C₈	#13182013	semidirect

H ≅ G/N

type

1. #20295104

C₂

semidirect

2. #20294777

C₂

semidirect

3. D₄⋊#11724784

C₂

semidirect

4. #20294747

C₂

semidirect

38. C₈×C₁₀₉

M₁₆

semidirect

56. C₁₀₉

#1594

semidirect

97. C₈

#13182013

semidirect

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

subgroup	index
#20295104	2
#20294777	2
D₄⋊#11724784	2
#20294747	2
C₄×D₈×C₁₀₉	2
#1594	109

subgroup

index

C₄×D₈×C₁₀₉

109

Subgroups — #28361946

Conjugacy classes of subgroups

Complements

Maximal subgroups