Group #28361624 — subgroups

Subgroups — #28361624

Order 13952.

270 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×#13182013	C₂	1	2	yes	maximal, complemented
2	#20294766	C₂	1	4	yes	maximal, complemented
3	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	—	2	2	no	—
5	#16812610	C₂²	1	4	yes	—
6	C₄×#11724784	—	2	2	no	—
7	C₂×#41×C₁₀₉	C₄	1	2	yes	—
8	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
9	#127×C₁₀₉	C₄	1	1	yes	characteristic
10	#124×C₁₀₉	C₂²	1	2	yes	—
11	#121×C₁₀₉	—	2	2	no	—
12	C₄×D₄×C₁₀₉	—	2	2	no	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#13182013	—	2	8	no	—
15	C₂×#11724784	—	2	4	no	—
16	#41×C₁₀₉	—	4	4	no	—
17	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₁₀₉	—	2	2	no	—
19	C₂×D₄×C₁₀₉	—	2	2	no	—
20	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	—	2	2	no	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	D₄	1	2	yes	—
28	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂³×C₁₀₉	—	2	2	no	—
30	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	Q₈×C₁₀₉	—	4	2	no	—
32	D₄×C₁₀₉	—	4	2	no	—
33	D₄×C₁₀₉	—	4	4	no	—
34	D₄×C₁₀₉	—	4	4	no	—
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	—	4	4	no	—
39	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	#11724784	—	2	8	no	—
44	C₂²×C₁₀₉	—	2	1	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	4	2	no	—
47	C₂²×C₁₀₉	—	4	4	no	—
48	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	—	4	2	no	—
52	C₄×C₁₀₉	#100	1	2	yes	—
53	C₄×C₁₀₉	#104	1	4	yes	—
54	C₂×C₁₀₉	—	4	1	no	—
55	C₂×C₁₀₉	—	8	2	no	—
56	C₂×C₁₀₉	C₂×#100	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×#104	1	2	yes	—
58	#1525	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1525	1	1	yes	characteristic, complemented, Sylow
60	#532	D₁₀₉	1	1	yes	characteristic
61	C₄×M₁₆	—	109	2	no	—
62	#329	—	109	4	no	—
63	C₂×#41	Dic₁₀₉	1	2	yes	—
64	C₂×M₁₆	—	218	2	no	—
65	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
66	#127	Dic₁₀₉	1	1	yes	characteristic
67	#124	C₂×D₁₀₉	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	#97	—	109	4	no	—
72	C₄×C₈	—	218	2	no	—
73	#41	—	4	4	no	—
74	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
80	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
81	M₁₆	—	218	8	no	—
82	C₂×C₈	—	218	4	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₂²⋊C₄	—	2	4	no	—
85	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
87	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
99	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
102	C₈	—	218	8	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#16812616	1	2	yes	—
112	C₄	#16812626	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#16812616	1	1	yes	characteristic
116	C₂	C₂×#16812626	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13182013

C₂

yes

maximal, complemented

#20294766

C₂

yes

maximal, complemented

#532×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

#16812610

C₂²

yes

—

C₄×#11724784

—

C₂×#41×C₁₀₉

C₄

yes

—

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic

#127×C₁₀₉

C₄

yes

characteristic

#124×C₁₀₉

C₂²

yes

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#13182013

—

C₂×#11724784

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#100

yes

—

C₄×C₁₀₉

#104

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#100

yes

characteristic

C₂×C₁₀₉

C₂×#104

yes

—

#1525

—

109

maximal, Sylow

C₁₀₉

#1525

yes

characteristic, complemented, Sylow

#532

D₁₀₉

yes

characteristic

C₄×M₁₆

—

109

—

#329

—

109

—

C₂×#41

Dic₁₀₉

yes

—

C₂×M₁₆

—

218

—

C₄⋊Q₈

Dic₁₀₉

yes

characteristic

#127

Dic₁₀₉

yes

characteristic

#124

C₂×D₁₀₉

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#97

—

109

—

C₄×C₈

—

218

—

#41

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

100

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

102

C₈

—

218

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#16812616

yes

—

112

C₄

#16812626

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#16812616

yes

characteristic

116

C₂

C₂×#16812626

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting, prime radical	#532×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#1525
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting, prime radical

#532×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₄×#13182013	C₂	semidirect
2. #20294766	C₂	semidirect
59. C₁₀₉	#1525	semidirect

H ≅ G/N

type

1. C₄×#13182013

C₂

semidirect

2. #20294766

C₂

semidirect

59. C₁₀₉

#1525

semidirect

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

subgroup	index
C₄×#13182013	2
#20294766	2
#532×C₁₀₉	2
#1525	109

subgroup

index

C₄×#13182013

#20294766

#532×C₁₀₉

#1525

109

Subgroups — #28361624

Conjugacy classes of subgroups

Complements

Maximal subgroups