Group #28363185 — subgroups

Subgroups — #28363185

Order 13952.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812628	C₂	1	4	yes	maximal
3	#20294872	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812628	—	2	8	no	—
9	#16812606	C₄	1	2	yes	—
10	#16812606	C₄	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
15	C₂×#11724784	—	2	4	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₄×Dic₁₀₉	D₄	1	4	yes	—
18	C₄×Dic₁₀₉	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	—	2	4	no	—
24	C₄²×C₁₀₉	D₄	1	2	yes	—
25	C₄²×C₁₀₉	Q₈	1	2	yes	—
26	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	2	4	no	—
36	C₂×Dic₁₀₉	—	2	4	no	—
37	C₂×Dic₁₀₉	C₄⋊C₄	1	8	yes	—
38	C₈×C₁₀₉	—	4	2	no	—
39	#11724784	—	4	2	no	—
40	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₀₉	#131	1	2	yes	—
43	C₂²×C₁₀₉	#131	1	2	yes	—
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	—	2	2	no	—
46	C₄×C₁₀₉	#127	1	1	yes	characteristic
47	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
49	Dic₁₀₉	—	2	8	no	—
50	C₂×C₁₀₉	#384	1	1	yes	characteristic
51	C₂×C₁₀₉	#424	1	4	yes	—
52	C₂×C₁₀₉	#424	1	1	yes	characteristic
53	C₂×C₁₀₉	#424	1	1	yes	characteristic
54	#1843	—	109	1	no	maximal, Sylow
55	C₁₀₉	#1843	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊C₈	—	109	4	no	—
57	C₂×C₄⋊C₈	—	109	1	no	—
58	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
59	#376	—	109	1	no	—
60	C₂²×C₈	—	109	2	no	—
61	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
62	C₂×C₄²	—	109	2	no	—
63	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	218	8	no	—
66	C₄⋊C₈	—	218	2	no	—
67	#95	—	109	2	no	—
68	#95	—	109	2	no	—
69	C₂²×C₄	—	109	4	no	—
70	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	218	4	no	—
76	C₂×C₈	—	218	2	no	—
77	C₄²	—	109	4	no	—
78	C₄²	—	109	4	no	—
79	C₄²	Dic₂₁₈	1	2	yes	—
80	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
81	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	109	8	no	—
85	C₂×C₄	—	218	4	no	—
86	C₂×C₄	—	218	4	no	—
87	C₂×C₄	#13182008	1	2	yes	—
88	C₂×C₄	#13182008	1	2	yes	—
89	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
92	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	436	2	no	—
95	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
96	C₂²	C₂×#13182008	1	1	yes	characteristic
97	C₂²	#16812705	1	2	yes	—
98	C₂²	#16812707	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	218	8	no	—
102	C₄	#16812695	1	1	yes	characteristic
103	C₄	#16812697	1	1	yes	characteristic
104	C₄	#16812698	1	2	yes	—
105	C₂	#20294895	1	1	yes	characteristic
106	C₂	#20295051	1	4	yes	—
107	C₂	#20295052	1	1	yes	characteristic
108	C₂	#20295054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

C₂×#16812628

C₂

yes

maximal

#20294872

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₄⋊#11724784

—

#16812628

—

#16812606

C₄

yes

—

#16812606

C₄

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#384

yes

characteristic

C₂×C₁₀₉

#424

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1843

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

#376

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#95

—

109

—

#95

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

#13182008

yes

—

C₂×C₄

#13182008

yes

—

C₂×C₄

D₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

—

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₈

—

C₈

—

436

—

C₂²

C₄⋊C₄×D₁₀₉

yes

—

C₂²

C₂×#13182008

yes

characteristic

C₂²

#16812705

yes

—

C₂²

#16812707

yes

—

C₄

—

100

C₄

—

101

C₄

—

218

—

102

C₄

#16812695

yes

characteristic

103

C₄

#16812697

yes

characteristic

104

C₄

#16812698

yes

—

105

C₂

#20294895

yes

characteristic

106

C₂

#20295051

yes

—

107

C₂

#20295052

yes

characteristic

108

C₂

#20295054

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator, socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#1843
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator, socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
55. C₁₀₉	#1843	semidirect

H ≅ G/N

type

55. C₁₀₉

#1843

semidirect

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

subgroup	index
C₂×C₄⋊#11724784	2
C₂×#16812628	2
#20294872	2
C₂×C₄⋊C₈×C₁₀₉	2
#1843	109

subgroup

index

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28363185

Conjugacy classes of subgroups

Complements

Maximal subgroups