Group #28378081 — subgroups

Subgroups — #28378081

Order 13952.

132 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 52 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	D₁₀₉⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	#20295989	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812638	C₂	1	1	yes	maximal, characteristic
4	D₁₀₉⋊M₁₆	C₄	1	1	yes	characteristic
5	#16812778	C₄	1	1	yes	characteristic
6	#16812774	—	2	1	no	—
7	#16812773	C₄	1	2	yes	complemented
8	D₁₀₉⋊C₁₆	—	2	1	no	—
9	#16812753	—	2	1	no	—
10	C₈×#89137	—	2	1	no	—
11	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812638	—	2	1	no	—
13	#16812638	—	2	1	no	—
14	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	#13182039	—	2	1	no	—
17	#13182038	C₈	1	2	yes	—
18	D₁₀₉⋊C₈	—	2	1	no	—
19	C₄⋊#89137	—	2	1	no	—
20	C₂²⋊#89137	C₈	1	2	yes	—
21	C₄×#89137	—	2	1	no	—
22	#13182032	—	2	2	no	—
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
25	C₈×D₁₀₉	D₄	1	1	yes	characteristic
26	C₈×D₁₀₉	D₄	1	1	yes	characteristic
27	C₈×D₁₀₉	D₄	1	1	yes	characteristic
28	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
29	#13182004	—	2	1	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	1	no	—
32	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×#89137	—	2	2	no	—
34	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	M₁₆	1	1	yes	characteristic
39	C₄×D₁₀₉	M₁₆	1	1	yes	characteristic
40	#11724786	—	2	2	no	—
41	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	#11724784	C₂²⋊C₄	1	1	yes	characteristic
44	#11724784	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
46	C₂×D₁₀₉	—	4	1	no	—
47	C₂×D₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×D₁₀₉	C₄⋊C₈	1	1	yes	characteristic
49	#89137	—	4	2	no	—
50	C₄×C₁₀₉	#95	1	1	yes	characteristic
51	C₄×C₁₀₉	#97	1	1	yes	characteristic
52	Dic₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
53	Dic₁₀₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₁₀₉	—	2	1	no	—
55	C₂×C₁₀₉	#336	1	1	yes	characteristic
56	D₁₀₉	—	2	1	no	—
57	D₁₀₉	—	2	1	no	—
58	#1378	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1378	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₃₂	—	109	1	no	—
61	C₂×M₃₂	—	109	1	no	—
62	#405	—	109	1	no	—
63	C₂×M₁₆	—	109	1	no	—
64	C₂²×C₈	—	109	1	no	—
65	#117	—	109	1	no	—
66	M₃₂	—	109	2	no	—
67	M₃₂	—	218	1	no	—
68	M₃₂	—	218	1	no	—
69	M₃₂	—	218	1	no	—
70	C₂×C₁₆	—	109	1	no	—
71	C₂×C₁₆	—	218	1	no	—
72	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
73	#97	—	218	1	no	—
74	C₄×C₈	—	218	1	no	—
75	C₂²×C₄	—	109	1	no	—
76	M₁₆	—	109	2	no	—
77	M₁₆	—	218	1	no	—
78	C₂×C₈	—	109	1	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	218	1	no	—
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₄⋊C₄	—	218	1	no	—
86	C₂²⋊C₄	—	109	2	no	—
87	C₄²	—	218	1	no	—
88	C₁₆	—	2	1	no	—
89	C₁₆	—	218	2	no	—
90	C₁₆	—	218	1	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	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	218	2	no	—
98	C₂×C₄	C₄×#89137	1	1	yes	characteristic
99	C₈	—	109	1	no	—
100	C₈	—	109	1	no	—
101	C₈	—	218	2	no	—
102	C₈	C₂²⋊#89137	1	1	yes	characteristic
103	C₈	C₄⋊#89137	1	1	yes	characteristic
104	C₂²	—	109	1	no	—
105	C₂²	—	109	1	no	—
106	C₂²	—	436	1	no	—
107	C₂²	C₈×#89137	1	1	yes	characteristic
108	C₄	—	109	1	no	—
109	C₄	—	109	1	no	—
110	C₄	—	436	2	no	—
111	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
112	C₄	#16812753	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	218	1	no	—
115	C₂	—	218	1	no	—
116	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊M₃₂

C₂

yes

maximal, characteristic

#20295989

C₂

yes

maximal, characteristic

C₂×#16812638

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₄

yes

characteristic

#16812778

C₄

yes

characteristic

#16812774

—

#16812773

C₄

yes

complemented

D₁₀₉⋊C₁₆

—

#16812753

—

C₈×#89137

—

C₂×C₈×D₁₀₉

C₂²

yes

—

—

C₄

yes

characteristic, complemented

C₂×C₁₆×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

—

#13182038

C₈

yes

—

D₁₀₉⋊C₈

—

C₄⋊#89137

—

C₂²⋊#89137

C₈

yes

—

C₄×#89137

—

#13182032

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×#89137

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

M₁₆

yes

characteristic

C₄×D₁₀₉

M₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

characteristic

C₂×D₁₀₉

C₄⋊C₈

yes

characteristic

#89137

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

C₂²⋊C₈

yes

characteristic

Dic₁₀₉

C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#336

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#1378

—

109

maximal, Sylow

C₁₀₉

#1378

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×M₃₂

—

109

—

#405

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

M₃₂

—

109

—

M₃₂

—

218

—

M₃₂

—

218

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

#89137

yes

characteristic, complemented

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

109

—

C₄²

—

218

—

C₁₆

—

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

characteristic

C₈

—

109

—

100

C₈

—

109

—

101

C₈

—

218

—

102

C₈

C₂²⋊#89137

yes

characteristic

103

C₈

C₄⋊#89137

yes

characteristic

104

C₂²

—

109

—

105

C₂²

—

109

—

106

C₂²

—

436

—

107

C₂²

C₈×#89137

yes

characteristic

108

C₄

—

109

—

109

C₄

—

109

—

110

C₄

—

436

—

111

C₄

(C₂×#89137)⋊C₄

yes

characteristic

112

C₄

#16812753

yes

characteristic

113

C₂

—

114

C₂

—

218

—

115

C₂

—

218

—

116

C₂

(C₂×#89137)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#89137
commutator	C₂²×C₁₀₉	C₄×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#336
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1378
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₈

C₂²⋊#89137

commutator

C₂²×C₁₀₉

C₄×C₈

Frattini

C₂×C₈

C₂×#89137

Fitting

C₂×C₁₆×C₁₀₉

C₄

socle

C₂×C₁₀₉

#336

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×D₁₀₉

C₂²

C₂×C₁₆

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
7. #16812773	C₄	semidirect
14. C₂×#13182004	C₄	semidirect
15. C₂×C₁₆×C₁₀₉	C₄	semidirect
59. C₁₀₉	#1378	semidirect
72. C₂×C₁₆	#89137	semidirect

H ≅ G/N

type

7. #16812773

C₄

semidirect

14. C₂×#13182004

C₄

semidirect

15. C₂×C₁₆×C₁₀₉

C₄

semidirect

59. C₁₀₉

#1378

semidirect

72. C₂×C₁₆

#89137

semidirect

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

subgroup	index
D₁₀₉⋊M₃₂	2
#20295989	2
C₂×#16812638	2
#1378	109

subgroup

index

D₁₀₉⋊M₃₂

#20295989

C₂×#16812638

#1378

109

Subgroups — #28378081

Conjugacy classes of subgroups

Complements

Maximal subgroups