Group #28364804 — subgroups

Subgroups — #28364804

Order 13952.

144 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#16812639	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812635	C₂	1	1	yes	maximal, characteristic
3	#20294848	C₂	1	4	yes	maximal, complemented
4	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	1	no	—
6	#16812639	C₄	1	2	yes	complemented
7	#16812639	C₂²	1	2	yes	—
8	C₂×#13182004	C₄	1	2	yes	complemented
9	#16812635	—	2	1	no	—
10	#16812635	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	#117×C₁₀₉	—	2	1	no	—
13	#107×C₁₀₉	—	2	1	no	—
14	#106×C₁₀₉	C₂²	1	2	yes	—
15	#13182013	—	2	2	no	—
16	#13182013	—	4	1	no	—
17	C₂×#11724784	—	4	1	no	—
18	#13182004	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
28	C₄²×C₁₀₉	—	4	1	no	—
29	C₂³×C₁₀₉	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	4	2	no	—
31	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	#11724784	—	4	2	no	—
42	C₂²×C₁₀₉	—	4	1	no	—
43	C₂²×C₁₀₉	#106	1	1	yes	characteristic
44	C₂²×C₁₀₉	#106	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₄×C₁₀₉	—	8	2	no	—
47	C₄×C₁₀₉	#106	1	1	yes	characteristic
48	C₄×C₁₀₉	#106	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	2	1	no	—
53	C₂×C₁₀₉	—	2	1	no	—
54	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
55	#2155	—	109	1	no	maximal, Sylow
56	C₁₀₉	#2155	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	109	1	no	—
58	C₂×#108	—	109	1	no	—
59	#427	D₁₀₉	1	1	yes	characteristic
60	#365	—	109	4	no	—
61	C₂×M₁₆	—	218	1	no	—
62	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
63	#117	—	2	1	no	—
64	M₃₂	—	109	2	no	—
65	M₃₂	—	109	2	no	—
66	C₂×C₁₆	—	109	2	no	—
67	#108	—	109	2	no	—
68	#108	—	218	1	no	—
69	#107	—	2	1	no	—
70	#106	C₂×D₁₀₉	1	2	yes	—
71	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
72	M₁₆	—	436	1	no	—
73	M₁₆	—	218	2	no	—
74	C₂×C₈	—	436	1	no	—
75	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
76	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
77	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
78	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
81	C₄⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₄²	—	4	1	no	—
84	C₁₆	—	109	4	no	—
85	C₂³	#13182017	1	1	yes	characteristic
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	#13182008	1	1	yes	characteristic
88	C₂×C₄	#13182008	1	1	yes	characteristic
89	C₂×C₄	#13182016	1	1	yes	characteristic
90	C₂×C₄	#13182016	1	1	yes	characteristic
91	C₂×C₄	#13182017	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₈	—	436	2	no	—
94	C₈	#13182008	1	1	yes	characteristic
95	C₈	#13182008	1	1	yes	characteristic
96	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
97	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#16812630	1	1	yes	characteristic
100	C₂²	#16812630	1	1	yes	characteristic
101	C₂²	C₂×#13182017	1	1	yes	characteristic
102	C₄	—	8	2	no	—
103	C₄	#16812630	1	1	yes	characteristic
104	C₄	#16812630	1	1	yes	characteristic
105	C₄	C₂×#13182008	1	1	yes	characteristic
106	C₄	C₂×#13182016	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	C₂×#16812630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812639

C₂

yes

maximal, characteristic

C₂×#16812635

C₂

yes

maximal, characteristic

#20294848

C₂

yes

maximal, complemented

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

#16812639

C₄

yes

complemented

#16812639

C₂²

yes

—

C₂×#13182004

C₄

yes

complemented

#16812635

—

#16812635

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

—

#107×C₁₀₉

—

#106×C₁₀₉

C₂²

yes

—

—

—

—

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#106

yes

characteristic

#2155

—

109

maximal, Sylow

C₁₀₉

#2155

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

#365

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

M₃₂

—

109

—

M₃₂

—

109

—

C₂×C₁₆

—

109

—

#108

—

109

—

#108

—

218

—

#107

—

#106

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

436

—

C₂×C₈

Dic₂₁₈

yes

characteristic

C₂×C₈

Dic₂₁₈

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

109

—

C₂³

#13182017

yes

characteristic

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

—

436

—

C₈

yes

C₈

yes

C₈

C₂×C₄×D₁₀₉

yes

characteristic

C₈

C₂×Dic₂₁₈

yes

characteristic

C₂²

—

C₂²

#16812630

yes

characteristic

100

C₂²

#16812630

yes

characteristic

101

C₂²

C₂×#13182017

yes

characteristic

102

C₄

—

103

C₄

#16812630

yes

characteristic

104

C₄

#16812630

yes

characteristic

105

C₄

C₂×#13182008

yes

characteristic

106

C₄

C₂×#13182016

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

C₂×#16812630

yes

characteristic

	subgroup	G/H
center	C₄	#16812630
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#106
prime radical	C₂³×C₁₀₉	SD₁₆
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#2155
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#16812630

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×#106

prime radical

C₂³×C₁₀₉

SD₁₆

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #20294848	C₂	semidirect
6. #16812639	C₄	semidirect
8. C₂×#13182004	C₄	semidirect
56. C₁₀₉	#2155	semidirect

H ≅ G/N

type

3. #20294848

C₂

semidirect

6. #16812639

C₄

semidirect

8. C₂×#13182004

C₄

semidirect

56. C₁₀₉

#2155

semidirect

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

subgroup	index
C₂×#16812639	2
C₂×#16812635	2
#20294848	2
#427×C₁₀₉	2
#2155	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28364804

Conjugacy classes of subgroups

Complements

Maximal subgroups