Group #28364787 — subgroups

Subgroups — #28364787

Order 13952.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294844	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
7	#16812639	—	2	2	no	—
8	C₂×#13182004	C₂²	1	2	yes	—
9	C₈×Dic₁₀₉	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₀₉	—	2	2	no	—
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₉	C₈	1	2	yes	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
18	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
19	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
20	#13182004	—	2	2	no	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
25	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₀₉	—	2	2	no	—
27	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	2	4	no	—
33	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
34	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
35	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	#11724784	—	2	2	no	—
39	C₂²×C₁₀₉	—	2	2	no	—
40	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
43	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₀₉	—	2	2	no	—
46	Dic₁₀₉	M₃₂	1	4	yes	complemented
47	C₂×C₁₀₉	—	2	2	no	—
48	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	—
50	#2150	—	109	1	no	maximal, Sylow
51	C₁₀₉	#2150	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₃₂	—	109	1	no	—
53	C₂×M₃₂	D₁₀₉	1	1	yes	characteristic
54	C₂×C₄×C₈	—	109	1	no	—
55	C₄⋊C₁₆	—	109	4	no	—
56	C₂²×C₈	—	109	1	no	—
57	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
58	C₂×C₄²	—	109	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	218	2	no	—
61	C₂×C₁₆	—	109	2	no	—
62	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
63	C₄×C₈	—	109	2	no	—
64	C₄×C₈	—	109	2	no	—
65	C₂²×C₄	—	109	2	no	—
66	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
67	C₂×C₈	—	109	2	no	—
68	C₂×C₈	—	218	2	no	—
69	C₂×C₈	Dic₂₁₈	1	2	yes	—
70	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
72	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
73	C₄²	—	109	1	no	—
74	C₄²	—	109	1	no	—
75	C₄²	—	109	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	218	2	no	—
78	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	218	4	no	—
82	C₂×C₄	#13182008	1	2	yes	—
83	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
84	C₂×C₄	#13182012	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
86	C₈	—	218	2	no	—
87	C₈	#13182008	1	2	yes	—
88	C₈	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
89	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#16812628	1	2	yes	—
92	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
93	C₄	—	109	4	no	—
94	C₄	—	218	2	no	—
95	C₄	#16812628	1	2	yes	—
96	C₄	C₂×#13182008	1	1	yes	characteristic
97	C₄	C₂×#13182012	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×#16812628	1	1	yes	characteristic
100	C₂	M₃₂×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812639

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

#16812639

—

C₂×#13182004

C₂²

yes

—

C₈×Dic₁₀₉

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₈

yes

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

—

#13182004

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₈

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

M₃₂

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

—

#2150

—

109

maximal, Sylow

C₁₀₉

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×M₃₂

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

M₃₂

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₁₆

—

C₁₆

—

218

—

C₂³

C₈×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₈×D₁₀₉

yes

characteristic

C₂×C₄

#13182012

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₈

—

218

—

C₈

#13182008

yes

—

C₈

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

C₂×Dic₂₁₈

yes

characteristic

C₂²

—

C₂²

#16812628

yes

—

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

—

218

—

C₄

#16812628

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×#16812628

yes

characteristic

100

C₂

M₃₂×D₁₀₉

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×M₃₂×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂×M₃₂	D₁₀₉
hypocenter	C₁₀₉	#2150
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂×M₃₂×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₂×M₃₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #20294844	C₂	semidirect
46. Dic₁₀₉	M₃₂	semidirect
51. C₁₀₉	#2150	semidirect

H ≅ G/N

type

3. #20294844

C₂

semidirect

46. Dic₁₀₉

M₃₂

semidirect

51. C₁₀₉

#2150

semidirect

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

subgroup	index
C₂×#16812639	2
C₂×C₈×Dic₁₀₉	2
#20294844	2
C₂×M₃₂×C₁₀₉	2
#2150	109

subgroup

index

C₂×#16812639

C₂×C₈×Dic₁₀₉

#20294844

C₂×M₃₂×C₁₀₉

#2150

109

Subgroups — #28364787

Conjugacy classes of subgroups

Complements

Maximal subgroups