Group #28363151 — subgroups

Subgroups — #28363151

Order 13952.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20294872	C₂	1	1	yes	maximal, characteristic
3	#20294780	C₂	1	4	yes	maximal
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	#16812606	C₄	1	2	yes	—
9	#16812606	C₄	1	2	yes	—
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₉	—	2	2	no	—
13	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
14	C₂×#11724784	—	2	2	no	—
15	C₂×#11724784	—	2	4	no	—
16	C₄×Dic₁₀₉	—	2	2	no	—
17	C₄×Dic₁₀₉	—	2	2	no	—
18	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	—	2	2	no	—
22	C₂×C₈×C₁₀₉	—	2	4	no	—
23	C₄²×C₁₀₉	D₄	1	2	yes	—
24	C₄²×C₁₀₉	Q₈	1	2	yes	—
25	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	#41	1	2	yes	—
33	C₂×C₄×C₁₀₉	#41	1	2	yes	—
34	C₂×Dic₁₀₉	—	2	4	no	—
35	C₂×Dic₁₀₉	—	2	4	no	—
36	C₂×Dic₁₀₉	—	2	4	no	—
37	C₈×C₁₀₉	—	4	2	no	—
38	#11724784	—	4	2	no	—
39	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#117	1	2	yes	—
41	C₂²×C₁₀₉	#131	1	2	yes	—
42	C₂²×C₁₀₉	#131	1	2	yes	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	#124	1	1	yes	characteristic
46	C₄×C₁₀₉	#124	1	1	yes	characteristic
47	C₄×C₁₀₉	#125	1	2	yes	—
48	Dic₁₀₉	—	4	4	no	—
49	C₂×C₁₀₉	#382	1	1	yes	characteristic
50	C₂×C₁₀₉	#424	1	1	yes	characteristic
51	C₂×C₁₀₉	#424	1	1	yes	characteristic
52	C₂×C₁₀₉	#433	1	4	yes	—
53	#1832	—	109	1	no	maximal, Sylow
54	C₁₀₉	#1832	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄⋊C₈	—	109	1	no	—
56	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
57	#376	—	109	1	no	—
58	#336	—	109	4	no	—
59	C₂²×C₈	—	109	2	no	—
60	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
61	C₂×C₄²	—	109	2	no	—
62	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	218	2	no	—
65	#95	—	109	2	no	—
66	#95	—	109	2	no	—
67	C₂²×C₄	—	109	4	no	—
68	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	—	218	4	no	—
75	C₄²	—	218	2	no	—
76	C₄²	—	218	2	no	—
77	C₄²	Dic₂₁₈	1	2	yes	—
78	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
79	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	218	4	no	—
83	C₂×C₄	—	218	4	no	—
84	C₂×C₄	—	218	4	no	—
85	C₂×C₄	#13182008	1	2	yes	—
86	C₂×C₄	#13182008	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
89	C₂×C₄	#13182028	1	2	yes	—
90	C₂×C₄	#13182030	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	436	2	no	—
93	C₂²	C₂×#13182008	1	1	yes	characteristic
94	C₂²	#16812657	1	2	yes	—
95	C₂²	#16812705	1	2	yes	—
96	C₂²	#16812707	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	436	4	no	—
100	C₄	#16812686	1	1	yes	characteristic
101	C₄	#16812688	1	1	yes	characteristic
102	C₄	#16812690	1	2	yes	—
103	C₂	#20294891	1	1	yes	characteristic
104	C₂	#20295052	1	1	yes	characteristic
105	C₂	#20295054	1	1	yes	characteristic
106	C₂	#20295086	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

#20294872

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₄⋊#11724784

—

#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

—

C₄×Dic₁₀₉

—

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

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1832

—

109

maximal, Sylow

C₁₀₉

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

#376

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

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

—

218

—

C₄²

—

218

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

#13182008

yes

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₂×C₄

#13182028

yes

—

C₂×C₄

#13182030

yes

—

C₈

—

C₈

—

436

—

C₂²

C₂×#13182008

yes

characteristic

C₂²

#16812657

yes

—

C₂²

#16812705

yes

—

C₂²

#16812707

yes

—

C₄

—

C₄

—

C₄

—

436

—

100

C₄

#16812686

yes

characteristic

101

C₄

#16812688

yes

characteristic

102

C₄

#16812690

yes

—

103

C₂

#20294891

yes

characteristic

104

C₂

#20295052

yes

characteristic

105

C₂

#20295054

yes

characteristic

106

C₂

#20295086

yes

—

	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	#1832
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
54. C₁₀₉	#1832	semidirect

H ≅ G/N

type

54. C₁₀₉

#1832

semidirect

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

subgroup	index
C₂×C₄⋊#11724784	2
#20294872	2
#20294780	2
C₂×C₄⋊C₈×C₁₀₉	2
#1832	109

subgroup

index

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28363151

Conjugacy classes of subgroups

Complements

Maximal subgroups