Group #28364864 — subgroups

Subgroups — #28364864

Order 13952.

162 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 87 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	2	yes	maximal, complemented
2	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#13182004	C₂	1	2	yes	maximal, complemented
4	C₄×#13182004	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812639	—	2	4	no	—
7	C₂×#13182004	C₂²	1	2	yes	—
8	C₂×#13182004	C₂²	1	1	yes	characteristic
9	C₂×#13182004	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#13182004	—	2	2	no	—
16	#13182004	—	2	1	no	—
17	#13182004	D₄	1	2	yes	complemented
18	C₂×D₄×C₁₀₉	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₀₉	—	2	2	no	—
21	C₂×C₈×C₁₀₉	—	2	2	no	—
22	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
26	C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₉	C₂×C₈	1	2	yes	—
29	D₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₀₉	—	2	2	no	—
37	C₈×C₁₀₉	—	2	1	no	—
38	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₀₉	#41	1	1	yes	characteristic
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	—	2	2	no	—
42	C₂²×C₁₀₉	—	2	2	no	—
43	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	M₃₂	1	2	yes	complemented
47	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₀₉	#131	1	1	yes	characteristic
49	C₂×C₁₀₉	—	4	2	no	—
50	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₁₀₉	#504	1	1	yes	characteristic
53	#2169	—	109	1	no	maximal, Sylow
54	C₁₀₉	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	109	2	no	—
56	C₈×D₄	D₁₀₉	1	1	yes	characteristic
57	C₄⋊C₁₆	—	109	1	no	—
58	C₂²⋊C₁₆	—	109	2	no	—
59	C₄×C₁₆	—	109	1	no	—
60	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
61	C₄×D₄	Dic₁₀₉	1	1	yes	characteristic
62	M₃₂	—	218	4	no	—
63	C₂×C₁₆	—	109	1	no	—
64	C₂×C₁₆	—	109	1	no	—
65	C₂×C₁₆	—	109	2	no	—
66	C₄⋊C₈	Dic₁₀₉	1	1	yes	characteristic
67	C₂²⋊C₈	Dic₁₀₉	1	2	yes	—
68	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂×D₄	#11724784	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
75	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₄⋊C₄	#11724784	1	1	yes	characteristic
77	C₂²⋊C₄	#11724784	1	2	yes	—
78	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
79	C₁₆	—	109	2	no	—
80	C₁₆	—	218	2	no	—
81	C₁₆	—	218	1	no	—
82	C₂³	C₂×#11724784	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#11724784	1	1	yes	characteristic
87	C₂×C₄	C₂×#11724784	1	2	yes	—
88	C₂×C₄	C₂×#11724784	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	D₄×D₁₀₉	1	1	yes	characteristic
93	C₈	#13182028	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#11724784	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	2	no	—
100	C₄	#16812639	1	2	yes	complemented
101	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
102	C₄	#16812706	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×#11724784	1	1	yes	characteristic
105	C₂	C₂×#16812639	1	1	yes	characteristic
106	C₂	#20295443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812639

C₂

yes

maximal, complemented

C₄⋊#13182004

C₂

yes

maximal, characteristic, complemented

C₂²⋊#13182004

C₂

yes

maximal, complemented

C₄×#13182004

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₁₀₉

C₂

yes

maximal, characteristic

#16812639

—

C₂×#13182004

C₂²

yes

—

C₂×#13182004

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₄×D₄×C₁₀₉

C₄

yes

characteristic

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic

C₂²⋊C₈×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

—

D₄

yes

complemented

C₂×D₄×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₈

yes

characteristic

C₂²⋊C₄×C₁₀₉

C₈

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

#41

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

M₃₂

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₈×D₄

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₈×D₄

D₁₀₉

yes

characteristic

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄×D₄

Dic₁₀₉

yes

characteristic

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₄⋊C₈

Dic₁₀₉

yes

characteristic

C₂²⋊C₈

Dic₁₀₉

yes

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

#11724784

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

C₄⋊C₄

yes

C₂²⋊C₄

yes

—

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

C₂×#11724784

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#11724784

yes

characteristic

C₂×C₄

C₂×#11724784

yes

—

C₂×C₄

C₂×#11724784

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₁₀₉

yes

characteristic

C₈

#13182028

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#11724784

yes

characteristic

C₄

—

C₄

—

100

C₄

#16812639

yes

complemented

101

C₄

D₄×Dic₁₀₉

yes

characteristic

102

C₄

#16812706

yes

characteristic

103

C₂

—

104

C₂

D₄×#11724784

yes

characteristic

105

C₂

C₂×#16812639

yes

characteristic

106

C₂

#20295443

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₈×D₄×C₁₀₉	C₂
prime radical	C₂×D₄×C₁₀₉	C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₈×D₄	D₁₀₉
hypocenter	C₁₀₉	#2169
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₈×D₄×C₁₀₉

C₂

prime radical

C₂×D₄×C₁₀₉

C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₈×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂×#16812639	C₂	semidirect
2. C₄⋊#13182004	C₂	semidirect
3. C₂²⋊#13182004	C₂	semidirect
4. C₄×#13182004	C₂	semidirect
17. #13182004	D₄	semidirect
46. C₄×C₁₀₉	M₃₂	semidirect
54. C₁₀₉	#2169	semidirect
100. C₄	#16812639	semidirect

H ≅ G/N

type

1. C₂×#16812639

C₂

semidirect

2. C₄⋊#13182004

C₂

semidirect

3. C₂²⋊#13182004

C₂

semidirect

4. C₄×#13182004

C₂

semidirect

17. #13182004

D₄

semidirect

46. C₄×C₁₀₉

M₃₂

semidirect

54. C₁₀₉

#2169

semidirect

100. C₄

#16812639

semidirect

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

subgroup	index
C₂×#16812639	2
C₄⋊#13182004	2
C₂²⋊#13182004	2
C₄×#13182004	2
C₈×D₄×C₁₀₉	2
#2169	109

subgroup

index

C₈×D₄×C₁₀₉

109

Subgroups — #28364864

Conjugacy classes of subgroups

Complements

Maximal subgroups