Group #28378962 — subgroups

Subgroups — #28378962

Order 13952.

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

normal subgroups only

This group has 162 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	#20295908	C₂	1	2	yes	maximal
2	#20295897	C₂	1	2	yes	maximal
3	#20295895	C₂	1	1	yes	maximal, characteristic
4	#20295894	C₂	1	1	yes	maximal, characteristic
5	C₄×#13182012	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
7	#16812778	C₂²	1	2	yes	—
8	C₂×#13182012	C₂²	1	1	yes	characteristic
9	C₂×#13182012	C₂²	1	1	yes	characteristic
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812609	C₄	1	1	yes	characteristic, complemented
12	#16812609	C₄	1	1	yes	characteristic, complemented
13	C₄×#11724784	C₄	1	1	yes	characteristic, complemented
14	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	#13182039	—	2	2	no	—
16	#13182038	—	2	2	no	—
17	D₁₀₉⋊C₈	—	2	2	no	—
18	C₄⋊#89137	—	2	2	no	—
19	C₂²⋊#89137	—	2	2	no	—
20	C₄×#89137	—	2	2	no	—
21	C₂×C₄×D₁₀₉	—	2	1	no	—
22	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
23	#13182012	—	2	2	no	—
24	#13182012	—	2	2	no	—
25	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	—	2	1	no	—
28	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	#41	1	1	yes	characteristic
33	C₂×#89137	—	4	2	no	—
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
38	C₄×D₁₀₉	—	4	2	no	—
39	C₄×D₁₀₉	#41	1	2	yes	—
40	C₄×D₁₀₉	#41	1	2	yes	—
41	#11724786	—	4	2	no	—
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	—	2	1	no	—
44	#11724784	—	2	1	no	—
45	#11724784	—	2	1	no	—
46	C₂²×C₁₀₉	#117	1	1	yes	characteristic
47	C₂×D₁₀₉	—	4	1	no	—
48	C₂×D₁₀₉	#126	1	2	yes	—
49	#89137	—	8	2	no	—
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	#117	1	1	yes	characteristic
52	C₄×C₁₀₉	#117	1	1	yes	characteristic
53	Dic₁₀₉	—	4	1	no	—
54	Dic₁₀₉	#126	1	2	yes	—
55	C₂×C₁₀₉	—	2	1	no	—
56	C₂×C₁₀₉	#383	1	1	yes	characteristic
57	D₁₀₉	—	2	2	no	—
58	#1841	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1841	1	1	yes	characteristic, complemented, Sylow
60	C₄×M₁₆	—	109	1	no	—
61	#344	—	109	2	no	—
62	#339	—	109	2	no	—
63	#338	—	109	1	no	—
64	H(3,ℤ/4)	—	109	1	no	—
65	C₂×M₁₆	—	109	1	no	—
66	C₂×M₁₆	—	109	1	no	—
67	C₂×M₁₆	—	109	2	no	—
68	#117	—	109	2	no	—
69	C₂×C₄²	—	109	1	no	—
70	#97	—	109	1	no	—
71	#97	—	109	1	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	#89137	1	1	yes	characteristic, complemented
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	—	218	1	no	—
76	M₁₆	—	218	2	no	—
77	M₁₆	—	218	2	no	—
78	M₁₆	—	218	2	no	—
79	M₁₆	—	218	2	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	C₂×#89137	1	1	yes	characteristic
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₄⋊C₄	—	218	2	no	—
86	C₂²⋊C₄	—	218	2	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	218	1	no	—
89	C₄²	—	218	2	no	—
90	C₄²	C₂×#89137	1	1	yes	characteristic
91	C₂³	—	109	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	109	2	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	436	2	no	—
97	C₂×C₄	—	436	2	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	436	2	no	—
103	C₈	—	218	1	no	—
104	C₈	—	218	1	no	—
105	C₂²	—	109	2	no	—
106	C₂²	—	436	1	no	—
107	C₂²	#16812778	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	872	2	no	—
110	C₄	—	109	2	no	—
111	C₄	—	436	1	no	—
112	C₄	#16812778	1	1	yes	characteristic
113	C₄	#16812778	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	218	2	no	—
116	C₂	#20295971	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295908

C₂

yes

maximal

#20295897

C₂

yes

maximal

#20295895

C₂

yes

maximal, characteristic

#20295894

C₂

yes

maximal, characteristic

C₄×#13182012

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₂²

yes

—

#16812778

C₂²

yes

—

C₂×#13182012

C₂²

yes

characteristic

C₂×#13182012

C₂²

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812609

C₄

yes

characteristic, complemented

#16812609

C₄

yes

characteristic, complemented

C₄×#11724784

C₄

yes

characteristic, complemented

C₄×C₈×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

—

#13182038

—

D₁₀₉⋊C₈

—

C₄⋊#89137

—

C₂²⋊#89137

—

C₄×#89137

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂³

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

—

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

C₂²×D₁₀₉

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

#41

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

#41

yes

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

#117

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#126

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

#126

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#383

yes

characteristic

D₁₀₉

—

#1841

—

109

maximal, Sylow

C₁₀₉

#1841

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

109

—

#344

—

109

—

#339

—

109

—

#338

—

109

—

H(3,ℤ/4)

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

436

—

103

C₈

—

218

—

104

C₈

—

218

—

105

C₂²

—

109

—

106

C₂²

—

436

—

107

C₂²

#16812778

yes

characteristic

108

C₄

—

109

C₄

—

872

—

110

C₄

—

109

—

111

C₄

—

436

—

112

C₄

#16812778

yes

characteristic

113

C₄

#16812778

yes

characteristic

114

C₂

—

115

C₂

—

218

—

116

C₂

#20295971

yes

characteristic

	subgroup	G/H
center	C₄	#16812778
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#383
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1841
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#16812778

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

C₂×C₁₀₉

C₂²×D₁₀₉

C₂×C₄×D₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
11. #16812609	C₄	semidirect
12. #16812609	C₄	semidirect
13. C₄×#11724784	C₄	semidirect
14. C₄×C₈×C₁₀₉	C₄	semidirect
59. C₁₀₉	#1841	semidirect
73. C₄×C₈	#89137	semidirect

H ≅ G/N

type

11. #16812609

C₄

semidirect

12. #16812609

C₄

semidirect

13. C₄×#11724784

C₄

semidirect

14. C₄×C₈×C₁₀₉

C₄

semidirect

59. C₁₀₉

#1841

semidirect

73. C₄×C₈

#89137

semidirect

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

subgroup	index
#20295908	2
#20295897	2
#20295895	2
#20295894	2
C₄×#13182012	2
#1841	109

subgroup

index

109

Subgroups — #28378962

Conjugacy classes of subgroups

Complements

Maximal subgroups