Group #28390727 — subgroups

Subgroups — #28390727

Order 14464.

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⋊#11744751	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813551	C₂	1	4	yes	maximal
3	#20298444	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11744751	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
7	C₄⋊#11744751	—	2	2	no	—
8	#16813551	—	2	8	no	—
9	#16813529	C₄	1	2	yes	—
10	#16813529	C₄	1	2	yes	—
11	C₂²×C₈×C₁₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₁₃	—	2	2	no	—
14	C₂²×Dic₁₁₃	C₂×C₄	1	4	yes	—
15	C₂×#11744751	—	2	4	no	—
16	C₂×#11744751	—	2	2	no	—
17	C₄×Dic₁₁₃	D₄	1	4	yes	—
18	C₄×Dic₁₁₃	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₁₃	—	2	2	no	—
23	C₂×C₈×C₁₁₃	—	2	4	no	—
24	C₄²×C₁₁₃	D₄	1	2	yes	—
25	C₄²×C₁₁₃	Q₈	1	2	yes	—
26	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₁₃	—	2	2	no	—
28	C₂×C₄×C₁₁₃	—	2	2	no	—
29	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₁₁₃	—	2	4	no	—
36	C₂×Dic₁₁₃	—	2	4	no	—
37	C₂×Dic₁₁₃	C₄⋊C₄	1	8	yes	—
38	C₈×C₁₁₃	—	4	2	no	—
39	#11744751	—	4	2	no	—
40	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₁₃	#131	1	2	yes	—
43	C₂²×C₁₁₃	#131	1	2	yes	—
44	C₄×C₁₁₃	—	2	2	no	—
45	C₄×C₁₁₃	—	2	2	no	—
46	C₄×C₁₁₃	#127	1	1	yes	characteristic
47	C₄×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
49	Dic₁₁₃	—	2	8	no	—
50	C₂×C₁₁₃	#384	1	1	yes	characteristic
51	C₂×C₁₁₃	#424	1	4	yes	—
52	C₂×C₁₁₃	#424	1	1	yes	characteristic
53	C₂×C₁₁₃	#424	1	1	yes	characteristic
54	#1843	—	113	1	no	maximal, Sylow
55	C₁₁₃	#1843	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊C₈	—	113	4	no	—
57	C₂×C₄⋊C₈	—	113	1	no	—
58	C₂×C₄⋊C₈	D₁₁₃	1	1	yes	characteristic
59	#376	—	113	1	no	—
60	C₂²×C₈	—	113	2	no	—
61	C₂²×C₈	C₂×D₁₁₃	1	2	yes	—
62	C₂×C₄²	—	113	2	no	—
63	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	226	8	no	—
66	C₄⋊C₈	—	226	2	no	—
67	#95	—	113	2	no	—
68	#95	—	113	2	no	—
69	C₂²×C₄	—	113	4	no	—
70	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	226	4	no	—
76	C₂×C₈	—	226	2	no	—
77	C₄²	—	113	4	no	—
78	C₄²	—	113	4	no	—
79	C₄²	Dic₂₂₆	1	2	yes	—
80	C₄²	C₂²⋊D₁₁₃	1	2	yes	—
81	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	113	8	no	—
85	C₂×C₄	—	226	4	no	—
86	C₂×C₄	—	226	4	no	—
87	C₂×C₄	#14268626	1	2	yes	—
88	C₂×C₄	#14268626	1	2	yes	—
89	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
92	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	452	2	no	—
95	C₂²	C₄⋊C₄×D₁₁₃	1	2	yes	—
96	C₂²	C₂×#14268626	1	1	yes	characteristic
97	C₂²	#16813628	1	2	yes	—
98	C₂²	#16813630	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	226	8	no	—
102	C₄	#16813618	1	1	yes	characteristic
103	C₄	#16813620	1	1	yes	characteristic
104	C₄	#16813621	1	2	yes	—
105	C₂	#20298467	1	1	yes	characteristic
106	C₂	#20298623	1	4	yes	—
107	C₂	#20298624	1	1	yes	characteristic
108	C₂	#20298626	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11744751

C₂

yes

maximal, characteristic

C₂×#16813551

C₂

yes

maximal

#20298444

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄×Dic₁₁₃

C₂²

yes

—

C₄⋊#11744751

—

#16813551

—

#16813529

C₄

yes

—

#16813529

C₄

yes

—

C₂²×C₈×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₁₃

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

C₂×#11744751

—

C₂×#11744751

—

C₄×Dic₁₁₃

D₄

yes

—

C₄×Dic₁₁₃

Q₈

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₄²×C₁₁₃

D₄

yes

—

C₄²×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

#131

yes

—

C₂²×C₁₁₃

#131

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#127

yes

characteristic

C₄×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₁₃

C₄⋊Q₈

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

#384

yes

characteristic

C₂×C₁₁₃

#424

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

#1843

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

113

—

C₂×C₄⋊C₈

—

113

—

C₂×C₄⋊C₈

D₁₁₃

yes

characteristic

#376

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

—

C₂×C₄²

—

113

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

226

—

C₄⋊C₈

—

226

—

#95

—

113

—

#95

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

Dic₂₂₆

yes

—

C₄²

C₂²⋊D₁₁₃

yes

—

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

#14268626

yes

—

C₂×C₄

#14268626

yes

—

C₂×C₄

D₄×D₁₁₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

Q₈×D₁₁₃

yes

—

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

C₈

—

C₈

—

452

—

C₂²

C₄⋊C₄×D₁₁₃

yes

—

C₂²

C₂×#14268626

yes

characteristic

C₂²

#16813628

yes

—

C₂²

#16813630

yes

—

C₄

—

100

C₄

—

101

C₄

—

226

—

102

C₄

#16813618

yes

characteristic

103

C₄

#16813620

yes

characteristic

104

C₄

#16813621

yes

—

105

C₂

#20298467

yes

characteristic

106

C₂

#20298623

yes

—

107

C₂

#20298624

yes

characteristic

108

C₂

#20298626

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁₃
commutator, socle, prime radical	C₂³×C₁₁₃	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₁₃	C₂
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₁₃
hypocenter	C₁₁₃	#1843
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁₃

commutator, socle, prime radical

C₂³×C₁₁₃

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₁₃

C₂

cosocle

C₂²×C₄×C₁₁₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
55. C₁₁₃	#1843	semidirect

H ≅ G/N

type

55. C₁₁₃

#1843

semidirect

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

subgroup	index
C₂×C₄⋊#11744751	2
C₂×#16813551	2
#20298444	2
C₂×C₄⋊C₈×C₁₁₃	2
#1843	113

subgroup

index

C₂×C₄⋊C₈×C₁₁₃

113

Subgroups — #28390727

Conjugacy classes of subgroups

Complements

Maximal subgroups