Group #28364833 — subgroups

Subgroups — #28364833

Order 13952.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20294844	C₂	1	2	yes	maximal, complemented
3	#20294805	C₂	1	2	yes	maximal
4	#20294802	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
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	C₂×#13182004	C₂²	1	2	yes	—
10	C₄×#11724784	C₄	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₀₉	—	2	2	no	—
21	C₈×D₁₀₉	—	2	2	no	—
22	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
23	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
24	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	#13182004	—	2	2	no	—
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₁₀₉	—	2	2	no	—
30	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₄×D₁₀₉	—	2	2	no	—
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	2	1	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₈×C₁₀₉	M₁₆	1	2	yes	—
42	C₈×C₁₀₉	#41	1	2	yes	—
43	#11724784	—	2	1	no	—
44	#11724784	—	2	1	no	—
45	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	M₃₂	1	2	yes	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	#117	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	M₃₂	1	2	yes	—
56	C₂×C₁₀₉	#431	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
59	D₁₀₉	—	2	2	no	—
60	#2163	—	109	1	no	maximal, Sylow
61	C₁₀₉	#2163	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	109	1	no	—
63	C₄⋊C₁₆	—	109	2	no	—
64	C₂²⋊C₁₆	—	109	2	no	—
65	#346	—	109	1	no	—
66	#346	D₁₀₉	1	1	yes	characteristic, complemented
67	C₂²×C₈	—	109	1	no	—
68	C₂²×C₈	—	109	1	no	—
69	C₂×C₄²	—	109	1	no	—
70	C₂×C₁₆	—	109	2	no	—
71	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	—	109	1	no	—
75	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	218	2	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
85	C₄²	—	109	1	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
89	C₁₆	—	2	2	no	—
90	C₁₆	—	218	2	no	—
91	C₂³	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	—	218	1	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
102	C₈	—	218	1	no	—
103	C₈	—	218	1	no	—
104	C₈	#13182012	1	2	yes	—
105	C₈	#13182029	1	2	yes	—
106	C₂²	—	109	2	no	—
107	C₂²	—	218	1	no	—
108	C₂²	—	218	1	no	—
109	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	109	2	no	—
113	C₄	—	218	1	no	—
114	C₄	—	218	1	no	—
115	C₄	#16812656	1	1	yes	characteristic
116	C₄	C₂×#13182012	1	1	yes	characteristic
117	C₂	—	218	2	no	—
118	C₂	#20295077	1	1	yes	characteristic
119	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic
120	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, complemented

#20294805

C₂

yes

maximal

#20294802

C₂

yes

maximal, characteristic, complemented

#346×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

C₂×#13182004

C₂²

yes

—

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

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

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

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

—

C₈×C₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

#41

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

M₃₂

yes

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

#2163

—

109

maximal, Sylow

C₁₀₉

#2163

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

#346

—

109

—

#346

D₁₀₉

yes

characteristic, complemented

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

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

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₈×D₁₀₉

yes

characteristic

100

C₂×C₄

C₈×D₁₀₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

102

C₈

—

218

—

103

C₈

—

218

—

104

C₈

#13182012

yes

—

105

C₈

#13182029

yes

—

106

C₂²

—

109

—

107

C₂²

—

218

—

108

C₂²

—

218

—

109

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

109

—

113

C₄

—

218

—

114

C₄

—

218

—

115

C₄

#16812656

yes

characteristic

116

C₄

C₂×#13182012

yes

characteristic

117

C₂

—

218

—

118

C₂

#20295077

yes

characteristic

119

C₂

M₃₂×D₁₀₉

yes

characteristic

120

C₂

M₃₂×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	#346×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#346	D₁₀₉
hypocenter	C₁₀₉	#2163
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

#346×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. #20294844	C₂	semidirect
4. #20294802	C₂	semidirect
5. #346×C₁₀₉	C₂	semidirect
61. C₁₀₉	#2163	semidirect
66. #346	D₁₀₉	semidirect

H ≅ G/N

type

2. #20294844

C₂

semidirect

4. #20294802

C₂

semidirect

5. #346×C₁₀₉

C₂

semidirect

61. C₁₀₉

#2163

semidirect

66. #346

D₁₀₉

semidirect

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

subgroup	index
C₄×C₈×D₁₀₉	2
#20294844	2
#20294805	2
#20294802	2
#346×C₁₀₉	2
#2163	109

subgroup

index

C₄×C₈×D₁₀₉

#346×C₁₀₉

109

Subgroups — #28364833

Conjugacy classes of subgroups

Complements

Maximal subgroups