Group #28377853 — subgroups

Subgroups — #28377853

Order 13952.

166 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 54 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	#20295892	C₂	1	2	yes	maximal
2	#20294931	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724786	—	2	2	no	—
4	C₂²×Dic₂₁₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
6	#16812606	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×#11724786	—	2	4	no	—
9	C₂×#11724786	—	4	2	no	—
10	C₂×#11724786	—	4	2	no	—
11	C₂×Dic₂₁₈	—	2	1	no	—
12	C₂×Dic₂₁₈	—	4	1	no	—
13	C₂×Dic₂₁₈	—	4	1	no	—
14	C₂×Dic₂₁₈	C₈	1	2	yes	complemented
15	C₂²×Dic₁₀₉	—	2	1	no	—
16	C₂²×Dic₁₀₉	D₄	1	2	yes	—
17	C₄×Dic₁₀₉	—	4	1	no	—
18	C₂²×C₄×C₁₀₉	—	2	1	no	—
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	—	4	1	no	—
21	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	4	1	no	—
25	C₂×C₄×C₁₀₉	—	4	2	no	—
26	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	2	no	—
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	4	1	no	—
32	C₂×Dic₁₀₉	—	4	1	no	—
33	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
34	C₂×Dic₁₀₉	Q₁₆	1	2	yes	—
35	Dic₂₁₈	—	4	2	no	—
36	Dic₂₁₈	—	4	2	no	—
37	#11724786	—	4	4	no	—
38	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂²×C₁₀₉	#99	1	1	yes	characteristic
40	C₂²×C₁₀₉	#100	1	1	yes	characteristic
41	C₂²×C₁₀₉	#103	1	2	yes	—
42	C₂²×C₁₀₉	#104	1	2	yes	—
43	C₄×C₁₀₉	—	2	1	no	—
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	—	8	1	no	—
46	Dic₁₀₉	—	2	2	no	—
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	—	4	2	no	—
49	C₂×C₁₀₉	#323	1	1	yes	characteristic
50	C₂×C₁₀₉	Q₈⋊C₈	1	2	yes	—
51	C₂×C₁₀₉	#328	1	2	yes	—
52	C₂×C₁₀₉	#329	1	1	yes	characteristic
53	C₂×C₁₀₉	#333	1	1	yes	characteristic
54	#1272	—	109	1	no	maximal, Sylow
55	C₁₀₉	#1272	1	1	yes	characteristic, complemented, Sylow
56	#391	—	109	1	no	—
57	#336	—	109	2	no	—
58	C₂²×Q₈	—	109	1	no	—
59	C₂²×C₈	—	218	2	no	—
60	C₂×C₄⋊C₄	#89137	1	1	yes	characteristic
61	C₂×C₄²	—	109	1	no	—
62	#95	—	218	2	no	—
63	C₂×Q₈	—	109	2	no	—
64	C₂×Q₈	—	436	1	no	—
65	C₂×Q₈	—	436	1	no	—
66	C₂×Q₈	—	218	1	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	109	2	no	—
69	C₂²×C₄	—	218	1	no	—
70	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
71	C₂×C₈	—	436	2	no	—
72	C₂×C₈	—	436	2	no	—
73	C₂×C₈	—	218	4	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄²	—	436	1	no	—
76	C₂³	C₂²⋊#89137	1	1	yes	characteristic
77	Q₈	—	436	2	no	—
78	Q₈	—	436	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	436	1	no	—
86	C₂×C₄	—	436	1	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	—	218	2	no	—
89	C₂×C₄	—	218	2	no	—
90	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
91	C₂×C₄	#13182039	1	1	yes	characteristic
92	C₈	—	436	4	no	—
93	C₂²	#16812756	1	1	yes	characteristic
94	C₂²	#16812758	1	1	yes	characteristic
95	C₂²	#16812759	1	1	yes	characteristic
96	C₂²	#16812762	1	2	yes	—
97	C₂²	#16812763	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	8	1	no	—
101	C₄	—	436	2	no	—
102	C₄	—	218	2	no	—
103	C₄	—	218	2	no	—
104	C₂	#20295870	1	1	yes	characteristic
105	C₂	#20295876	1	2	yes	—
106	C₂	#20295879	1	2	yes	—
107	C₂	#20295881	1	1	yes	characteristic
108	C₂	#20295886	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295892

C₂

yes

maximal

#20294931

C₂

yes

maximal, characteristic

C₂²×#11724786

—

C₂²×Dic₂₁₈

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

—

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₈

yes

complemented

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂²⋊C₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂²×C₁₀₉

#104

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#323

yes

characteristic

C₂×C₁₀₉

Q₈⋊C₈

yes

—

C₂×C₁₀₉

#328

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1272

yes

characteristic, complemented, Sylow

#391

—

109

—

#336

—

109

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

218

—

C₂×C₄⋊C₄

#89137

yes

characteristic

C₂×C₄²

—

109

—

#95

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄²

—

436

—

C₂³

C₂²⋊#89137

yes

characteristic

Q₈

—

436

—

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

#13182039

yes

characteristic

C₈

—

436

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16812763

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

436

—

102

C₄

—

218

—

103

C₄

—

218

—

104

C₂

#20295870

yes

characteristic

105

C₂

#20295876

yes

—

106

C₂

#20295879

yes

—

107

C₂

#20295881

yes

characteristic

108

C₂

#20295886

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂×C₈
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂×C₄⋊C₄	#89137
hypocenter	C₁₀₉	#1272
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂×C₈

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂×C₄⋊C₄

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. C₂×Dic₂₁₈	C₈	semidirect
55. C₁₀₉	#1272	semidirect

H ≅ G/N

type

14. C₂×Dic₂₁₈

C₈

semidirect

55. C₁₀₉

#1272

semidirect

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

subgroup	index
#20295892	2
#20294931	2
#1272	109

subgroup

index

#20295892

#20294931

#1272

109

Subgroups — #28377853

Conjugacy classes of subgroups

Complements

Maximal subgroups