Group #28378104 — subgroups

Subgroups — #28378104

Order 13952.

140 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 50 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	#20296036	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812767	C₂	1	1	yes	maximal, characteristic
3	M₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812778	—	2	1	no	—
5	C₂×C₄⋊#89137	—	2	1	no	—
6	#16812768	C₄	1	2	yes	complemented
7	#16812767	—	2	1	no	—
8	#16812767	—	2	1	no	—
9	#16812767	C₄	1	2	yes	complemented
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812639	C₄	1	1	yes	characteristic, complemented
12	#16812638	—	2	1	no	—
13	C₁₆×D₁₀₉	—	2	1	no	—
14	M₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	4	1	no	—
16	C₄⋊#89137	—	2	2	no	—
17	C₄⋊#89137	—	2	2	no	—
18	C₄⋊#89137	—	4	1	no	—
19	C₂²⋊#89137	—	4	1	no	—
20	C₄×#89137	—	4	1	no	—
21	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×#11724784	Q₈	1	1	yes	characteristic
23	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₈×D₁₀₉	D₄	1	1	yes	characteristic
26	C₈×D₁₀₉	D₄	1	1	yes	characteristic
27	#13182004	—	2	1	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₁₀₉	—	2	1	no	—
30	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
31	C₂×#89137	—	4	2	no	—
32	C₂×#89137	—	4	2	no	—
33	C₂×#89137	—	8	2	no	—
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
36	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
39	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
40	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
41	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	#11724784	C₄⋊C₄	1	1	yes	characteristic
43	#11724784	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₀₉	#107	1	1	yes	characteristic
45	C₂×D₁₀₉	—	2	2	no	—
46	C₂×D₁₀₉	#102	1	1	yes	characteristic
47	C₂×D₁₀₉	#102	1	1	yes	characteristic
48	#89137	—	8	2	no	—
49	#89137	—	8	2	no	—
50	C₄×C₁₀₉	#95	1	1	yes	characteristic
51	C₄×C₁₀₉	#106	1	1	yes	characteristic
52	Dic₁₀₉	#103	1	1	yes	characteristic
53	Dic₁₀₉	#103	1	1	yes	characteristic
54	C₂×C₁₀₉	—	2	1	no	—
55	C₂×C₁₀₉	#340	1	1	yes	characteristic
56	D₁₀₉	—	2	1	no	—
57	D₁₀₉	#360	1	2	yes	—
58	#1386	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1386	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₃₂	—	109	1	no	—
61	#427	—	109	1	no	—
62	C₂×#106	—	109	1	no	—
63	C₂²×C₈	—	109	1	no	—
64	#117	—	218	1	no	—
65	C₂×C₄⋊C₄	—	218	1	no	—
66	M₃₂	—	109	1	no	—
67	M₃₂	—	218	1	no	—
68	M₃₂	#89137	1	1	yes	characteristic, complemented
69	C₂×C₁₆	—	218	1	no	—
70	#107	—	109	2	no	—
71	#106	—	109	2	no	—
72	#106	—	218	1	no	—
73	#106	—	218	1	no	—
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	—	436	1	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	—	109	1	no	—
78	C₂×C₈	—	109	1	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	C₂×#89137	1	1	yes	characteristic
82	C₄⋊C₄	—	436	1	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄⋊C₄	—	218	2	no	—
85	C₂²⋊C₄	—	436	1	no	—
86	C₄²	—	436	1	no	—
87	C₁₆	—	2	1	no	—
88	C₁₆	—	218	1	no	—
89	C₂³	—	109	1	no	—
90	C₂×C₄	—	872	2	no	—
91	C₂×C₄	—	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	—	436	2	no	—
97	C₂×C₄	—	436	2	no	—
98	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
99	C₈	—	109	1	no	—
100	C₈	—	109	1	no	—
101	C₈	C₄×#89137	1	1	yes	characteristic
102	C₈	C₂²⋊#89137	1	1	yes	characteristic
103	C₂²	—	109	1	no	—
104	C₂²	—	109	1	no	—
105	C₂²	—	218	2	no	—
106	C₂²	#16812768	1	1	yes	characteristic
107	C₄	—	872	2	no	—
108	C₄	—	872	2	no	—
109	C₄	—	109	1	no	—
110	C₄	—	109	1	no	—
111	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
112	C₄	#16812767	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	109	2	no	—
115	C₂	—	218	1	no	—
116	C₂	#20295900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296036

C₂

yes

maximal, characteristic

C₂×#16812767

C₂

yes

maximal, characteristic

M₃₂×D₁₀₉

C₂

yes

maximal, characteristic

#16812778

—

C₂×C₄⋊#89137

—

#16812768

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812639

C₄

yes

characteristic, complemented

#16812638

—

C₁₆×D₁₀₉

—

M₃₂×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

—

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

D₈

yes

—

—

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

Q₁₆

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#107

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

#340

yes

characteristic

D₁₀₉

—

D₁₀₉

#360

yes

—

#1386

—

109

maximal, Sylow

C₁₀₉

#1386

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

#427

—

109

—

C₂×#106

—

109

—

C₂²×C₈

—

109

—

#117

—

218

—

C₂×C₄⋊C₄

—

218

—

M₃₂

—

109

—

M₃₂

—

218

—

M₃₂

#89137

yes

characteristic, complemented

C₂×C₁₆

—

218

—

#107

—

109

—

#106

—

109

—

#106

—

218

—

#106

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

436

—

C₄²

—

436

—

C₁₆

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

872

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

C₄⋊#89137

yes

characteristic

C₈

—

109

—

100

C₈

—

109

—

101

C₈

C₄×#89137

yes

characteristic

102

C₈

C₂²⋊#89137

yes

characteristic

103

C₂²

—

109

—

104

C₂²

—

109

—

105

C₂²

—

218

—

106

C₂²

#16812768

yes

characteristic

107

C₄

—

872

—

108

C₄

—

872

—

109

C₄

—

109

—

110

C₄

—

109

—

111

C₄

(C₂×#89137)⋊C₄

yes

characteristic

112

C₄

#16812767

yes

characteristic

113

C₂

—

114

C₂

—

109

—

115

C₂

—

218

—

116

C₂

#20295900

yes

characteristic

	subgroup	G/H
center	C₂	#20295900
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂×C₈	C₂×#89137
Fitting	M₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	#340
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	M₃₂	#89137
hypocenter	C₁₀₉	#1386
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₂

#20295900

commutator

C₈×C₁₀₉

C₄²

Frattini

C₂×C₈

C₂×#89137

Fitting

M₃₂×C₁₀₉

C₄

socle

C₂×C₁₀₉

#340

prime radical

C₂²×D₁₀₉

D₈

cosocle

C₂×C₈×D₁₀₉

C₂²

M₃₂

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
6. #16812768	C₄	semidirect
9. #16812767	C₄	semidirect
11. #16812639	C₄	semidirect
14. M₃₂×C₁₀₉	C₄	semidirect
59. C₁₀₉	#1386	semidirect
68. M₃₂	#89137	semidirect

H ≅ G/N

type

6. #16812768

C₄

semidirect

9. #16812767

C₄

semidirect

11. #16812639

C₄

semidirect

14. M₃₂×C₁₀₉

C₄

semidirect

59. C₁₀₉

#1386

semidirect

68. M₃₂

#89137

semidirect

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

subgroup	index
#20296036	2
C₂×#16812767	2
M₃₂×D₁₀₉	2
#1386	109

subgroup

index

#20296036

C₂×#16812767

M₃₂×D₁₀₉

#1386

109

Subgroups — #28378104

Conjugacy classes of subgroups

Complements

Maximal subgroups