Group #28361806 — subgroups

Subgroups — #28361806

Order 13952.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294778	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	1	no	—
6	C₄⋊#11724784	—	2	1	no	—
7	C₄⋊#11724784	C₄	1	2	yes	—
8	C₄⋊#11724784	C₂²	1	2	yes	—
9	C₂²⋊#11724784	C₄	1	2	yes	—
10	C₄×#11724784	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	—	2	1	no	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₉	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
15	#13182013	—	4	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	—	2	2	no	—
22	C₂×C₈×C₁₀₉	—	4	2	no	—
23	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	—	4	1	no	—
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₁₀₉	—	4	2	no	—
39	#11724784	—	2	4	no	—
40	#11724784	—	4	2	no	—
41	C₂²×C₁₀₉	—	2	1	no	—
42	C₂²×C₁₀₉	—	2	1	no	—
43	C₂²×C₁₀₉	#107	1	2	yes	—
44	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	2	1	no	—
48	C₄×C₁₀₉	#107	1	2	yes	—
49	C₄×C₁₀₉	#131	1	2	yes	—
50	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	#424	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
55	C₂×C₁₀₉	#430	1	1	yes	characteristic
56	#1571	—	109	1	no	maximal, Sylow
57	C₁₀₉	#1571	1	1	yes	characteristic, complemented, Sylow
58	#431	—	109	1	no	—
59	#423	—	109	1	no	—
60	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
61	#335	—	109	4	no	—
62	C₂×M₁₆	—	218	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	109	2	no	—
67	C₄⋊C₈	—	109	2	no	—
68	C₄⋊C₈	—	218	1	no	—
69	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
70	C₂²⋊C₈	—	109	2	no	—
71	C₄×C₈	—	109	2	no	—
72	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
75	M₁₆	—	436	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	109	4	no	—
79	C₂×C₈	—	218	2	no	—
80	C₄²	Dic₂₁₈	1	1	yes	characteristic
81	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
83	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂³	#13182008	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	#13182008	1	1	yes	characteristic
88	C₂×C₄	#13182008	1	1	yes	characteristic
89	C₂×C₄	#13182008	1	1	yes	characteristic
90	C₂×C₄	#13182014	1	2	yes	—
91	C₂×C₄	#13182019	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	436	2	no	—
97	C₈	—	218	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#16812632	1	2	yes	—
101	C₂²	C₂×#13182008	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#16812632	1	2	yes	—
106	C₄	#16812705	1	2	yes	—
107	C₄	C₂×#13182014	1	1	yes	characteristic
108	C₄	C₂×#13182019	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#20295052	1	1	yes	characteristic
111	C₂	C₂×#16812632	1	1	yes	characteristic
112	C₂	#20295074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294778

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₄⋊#11724784

—

C₄⋊#11724784

C₄

yes

—

C₄⋊#11724784

C₂²

yes

—

C₂²⋊#11724784

C₄

yes

—

C₄×#11724784

C₂²

yes

—

C₂²×C₈×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#13182013

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

Q₈

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#107

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#107

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1571

yes

characteristic, complemented, Sylow

#431

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

#335

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄²

Dic₂₁₈

yes

characteristic

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

#13182008

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#13182019

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₈

—

C₈

—

436

—

C₈

—

218

—

C₂²

—

C₂²

—

100

C₂²

#16812632

yes

—

101

C₂²

C₂×#13182008

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#16812632

yes

—

106

C₄

#16812705

yes

—

107

C₄

C₂×#13182014

yes

characteristic

108

C₄

C₂×#13182019

yes

characteristic

109

C₂

—

110

C₂

#20295052

yes

characteristic

111

C₂

C₂×#16812632

yes

characteristic

112

C₂

#20295074

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#1571
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₉

hypocenter

C₁₀₉

#1571

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #20294778	C₂	semidirect
57. C₁₀₉	#1571	semidirect

H ≅ G/N

type

3. #20294778

C₂

semidirect

57. C₁₀₉

#1571

semidirect

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

subgroup	index
#20295079	2
#20295050	2
#20294778	2
C₂×C₄⋊C₈×C₁₀₉	2
#1571	109

subgroup

index

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28361806

Conjugacy classes of subgroups

Complements

Maximal subgroups