Group #28379391 — subgroups

Subgroups — #28379391

Order 13952.

160 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20295947	C₂	1	2	yes	maximal, complemented
2	#20295921	C₂	1	2	yes	maximal
3	C₄×#13182032	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182032	C₂²	1	4	yes	—
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
9	C₄×#11724784	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	#13182032	—	2	4	no	—
14	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
15	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂³	1	1	yes	characteristic
19	C₈×D₁₀₉	—	2	2	no	—
20	C₈×D₁₀₉	—	2	2	no	—
21	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
22	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₄×D₁₀₉	—	2	2	no	—
35	C₄×D₁₀₉	—	2	1	no	—
36	C₄×D₁₀₉	—	2	1	no	—
37	C₄×D₁₀₉	C₁₆	1	4	yes	complemented
38	C₈×C₁₀₉	—	2	1	no	—
39	C₈×C₁₀₉	—	2	1	no	—
40	#11724784	M₁₆	1	2	yes	—
41	#11724784	#41	1	2	yes	—
42	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	—	2	1	no	—
45	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	—
46	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
47	C₄×C₁₀₉	M₃₂	1	2	yes	—
48	C₄×C₁₀₉	#117	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
50	Dic₁₀₉	—	2	1	no	—
51	Dic₁₀₉	—	2	1	no	—
52	Dic₁₀₉	C₂×C₁₆	1	2	yes	—
53	C₂×C₁₀₉	#431	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
56	D₁₀₉	—	2	2	no	—
57	#2162	—	109	1	no	maximal, Sylow
58	C₁₀₉	#2162	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄×C₈	—	109	1	no	—
60	C₄⋊C₁₆	—	109	2	no	—
61	C₂²⋊C₁₆	—	109	2	no	—
62	C₄×C₁₆	—	109	2	no	—
63	C₂²×C₈	—	109	1	no	—
64	C₂²×C₈	—	109	1	no	—
65	C₂×C₄²	—	109	1	no	—
66	C₂×C₁₆	—	109	4	no	—
67	C₄×C₈	—	109	1	no	—
68	C₄×C₈	—	109	1	no	—
69	C₄×C₈	—	109	1	no	—
70	C₄×C₈	#89137	1	1	yes	characteristic
71	C₂²×C₄	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	109	1	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	1	no	—
76	C₂×C₈	—	218	2	no	—
77	C₂×C₈	—	218	2	no	—
78	C₂×C₈	C₂×#89137	1	1	yes	characteristic
79	C₂×C₈	C₂×#89137	1	1	yes	characteristic
80	C₄²	—	109	1	no	—
81	C₄²	—	109	1	no	—
82	C₄²	—	109	1	no	—
83	C₄²	C₂×#89137	1	1	yes	characteristic
84	C₁₆	—	218	4	no	—
85	C₂³	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	4	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	109	1	no	—
90	C₂×C₄	—	218	2	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	—	218	1	no	—
93	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
94	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
95	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	109	2	no	—
99	C₈	—	109	2	no	—
100	C₂²	—	109	2	no	—
101	C₂²	—	218	1	no	—
102	C₂²	—	218	1	no	—
103	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
104	C₄	—	109	2	no	—
105	C₄	—	218	1	no	—
106	C₄	—	218	1	no	—
107	C₄	D₁₀₉⋊C₁₆	1	2	yes	—
108	C₄	#16812774	1	2	yes	—
109	C₄	#16812778	1	1	yes	characteristic
110	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
111	C₂	—	218	2	no	—
112	C₂	#20296044	1	1	yes	characteristic
113	C₂	C₂×D₁₀₉⋊C₁₆	1	1	yes	characteristic
114	C₂	C₂×#16812774	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295947

C₂

yes

maximal, complemented

#20295921

C₂

yes

maximal

C₄×#13182032

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182032

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

—

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₁₆

yes

complemented

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

M₁₆

yes

—

#11724784

#41

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₁₆

yes

—

C₄×C₁₀₉

C₂×C₁₆

yes

—

C₄×C₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×C₁₆

yes

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

#2162

—

109

maximal, Sylow

C₁₀₉

#2162

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₁₆

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

C₈

—

109

—

C₈

—

109

—

100

C₂²

—

109

—

101

C₂²

—

218

—

102

C₂²

—

218

—

103

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

104

C₄

—

109

—

105

C₄

—

218

—

106

C₄

—

218

—

107

C₄

D₁₀₉⋊C₁₆

yes

—

108

C₄

#16812774

yes

—

109

C₄

#16812778

yes

characteristic

110

C₄

D₁₀₉⋊M₁₆

yes

characteristic

111

C₂

—

218

—

112

C₂

#20296044

yes

characteristic

113

C₂

C₂×D₁₀₉⋊C₁₆

yes

characteristic

114

C₂

C₂×#16812774

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#89137
commutator	C₂×C₁₀₉	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#2162
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#89137

commutator

C₂×C₁₀₉

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
1. #20295947	C₂	semidirect
3. C₄×#13182032	C₂	semidirect
37. C₄×D₁₀₉	C₁₆	semidirect
58. C₁₀₉	#2162	semidirect

H ≅ G/N

type

1. #20295947

C₂

semidirect

3. C₄×#13182032

C₂

semidirect

37. C₄×D₁₀₉

C₁₆

semidirect

58. C₁₀₉

#2162

semidirect

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

subgroup	index
#20295947	2
#20295921	2
C₄×#13182032	2
C₄×C₈×D₁₀₉	2
#2162	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28379391

Conjugacy classes of subgroups

Complements

Maximal subgroups