Group #28406294 — subgroups

Subgroups — #28406294

Order 14464.

240 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₄×#14268655	C₂	1	1	yes	maximal, characteristic
2	#20299534	C₂	1	1	yes	maximal, characteristic
3	#20299463	C₂	1	4	yes	maximal
4	#97×D₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268655	C₂²	1	2	yes	—
6	C₂×C₄×#90673	C₂²	1	2	yes	—
7	C₄×#11744753	C₄	1	4	yes	complemented
8	(C₂×#90673)⋊C₄	C₄	1	4	yes	—
9	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
10	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
11	#16813533	C₄	1	1	yes	characteristic, complemented
12	#16813532	C₄	1	2	yes	complemented
13	#97×C₁₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂²×#90673	C₂×C₄	1	4	yes	—
15	#14268655	—	2	4	no	—
16	C₂×#11744753	C₂×C₄	1	4	yes	—
17	C₄×#90673	—	2	4	no	—
18	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×#11744751	C₂×C₄	1	2	yes	—
21	C₈×D₁₁₃	—	2	4	no	—
22	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×#90673	—	2	4	no	—
28	C₂×#90673	—	2	8	no	—
29	C₂×C₄×C₁₁₃	C₄²	1	2	yes	—
30	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₁₃	C₄²	1	2	yes	—
32	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×D₁₁₃	—	2	4	no	—
34	C₄×D₁₁₃	#41	1	4	yes	—
35	#11744753	—	2	4	no	—
36	C₈×C₁₁₃	—	2	2	no	—
37	#11744751	—	2	2	no	—
38	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×D₁₁₃	#117	1	2	yes	—
40	C₂×D₁₁₃	#131	1	4	yes	—
41	#90673	—	4	4	no	—
42	C₄×C₁₁₃	—	2	2	no	—
43	C₄×C₁₁₃	#117	1	2	yes	—
44	Dic₁₁₃	—	2	2	no	—
45	Dic₁₁₃	#117	1	2	yes	—
46	C₂×C₁₁₃	#376	1	1	yes	characteristic
47	C₂×C₁₁₃	#405	1	2	yes	—
48	D₁₁₃	#433	1	4	yes	—
49	#1746	—	113	1	no	maximal, Sylow
50	C₁₁₃	#1746	1	1	yes	characteristic, complemented, Sylow
51	C₂×#97	—	113	1	no	—
52	C₂×C₄×C₈	—	113	1	no	—
53	#376	—	113	1	no	—
54	#336	—	113	4	no	—
55	C₂²×C₈	—	113	2	no	—
56	C₂²×C₈	—	113	2	no	—
57	C₂×C₄²	—	113	1	no	—
58	C₂×C₄²	—	113	2	no	—
59	#97	—	113	2	no	—
60	#97	—	113	1	no	—
61	#97	#90673	1	1	yes	characteristic, complemented
62	C₄×C₈	—	113	4	no	—
63	#95	—	113	4	no	—
64	C₂²×C₄	—	113	1	no	—
65	C₂²×C₄	—	113	2	no	—
66	C₂²×C₄	—	113	4	no	—
67	C₂×C₈	—	113	4	no	—
68	C₂×C₈	—	113	2	no	—
69	C₂×C₈	—	226	4	no	—
70	C₂×C₈	—	226	4	no	—
71	C₂×C₈	C₂×#90673	1	2	yes	—
72	C₄²	—	113	2	no	—
73	C₄²	—	113	1	no	—
74	C₄²	—	226	4	no	—
75	C₄²	C₂×#90673	1	1	yes	characteristic
76	C₂³	—	113	1	no	—
77	C₂×C₄	—	113	1	no	—
78	C₂×C₄	—	113	4	no	—
79	C₂×C₄	—	113	2	no	—
80	C₂×C₄	—	226	4	no	—
81	C₂×C₄	—	226	4	no	—
82	C₂×C₄	—	226	8	no	—
83	C₂×C₄	C₄×#90673	1	2	yes	—
84	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	226	4	no	—
87	C₈	—	226	2	no	—
88	C₂²	—	113	2	no	—
89	C₂²	—	113	4	no	—
90	C₂²	C₂×C₄×#90673	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	113	2	no	—
93	C₄	—	452	4	no	—
94	C₄	—	226	2	no	—
95	C₄	#16813701	1	2	yes	—
96	C₂	—	113	4	no	—
97	C₂	#20299534	1	1	yes	characteristic
98	C₂	#20299564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14268655

C₂

yes

maximal, characteristic

#20299534

C₂

yes

maximal, characteristic

#20299463

C₂

yes

maximal

#97×D₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268655

C₂²

yes

—

C₂×C₄×#90673

C₂²

yes

—

C₄×#11744753

C₄

yes

complemented

(C₂×#90673)⋊C₄

C₄

yes

—

C₂×C₈×D₁₁₃

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#16813533

C₄

yes

characteristic, complemented

#16813532

C₄

yes

complemented

#97×C₁₁₃

C₄

yes

characteristic, complemented

C₂²×#90673

C₂×C₄

yes

—

#14268655

—

C₂×#11744753

C₂×C₄

yes

—

C₄×#90673

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₂×#11744751

C₂×C₄

yes

—

C₈×D₁₁₃

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×D₁₁₃

C₂²×C₄

yes

characteristic

C₂×#90673

—

C₂×#90673

—

C₂×C₄×C₁₁₃

C₄²

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₃

C₄²

yes

—

C₂×Dic₁₁₃

C₂²×C₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

#41

yes

—

#11744753

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄²

yes

characteristic

C₂×D₁₁₃

#117

yes

—

C₂×D₁₁₃

#131

yes

—

#90673

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#117

yes

—

Dic₁₁₃

—

Dic₁₁₃

#117

yes

—

C₂×C₁₁₃

#376

yes

characteristic

C₂×C₁₁₃

#405

yes

—

D₁₁₃

#433

yes

—

#1746

—

113

maximal, Sylow

C₁₁₃

#1746

yes

characteristic, complemented, Sylow

C₂×#97

—

113

—

C₂×C₄×C₈

—

113

—

#376

—

113

—

#336

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

—

113

—

#97

—

113

—

#97

—

113

—

#97

#90673

yes

characteristic, complemented

C₄×C₈

—

113

—

#95

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

C₂×#90673

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

226

—

C₄²

C₂×#90673

yes

characteristic

C₂³

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₄×#90673

yes

—

C₂×C₄

C₂²×#90673

yes

characteristic

C₈

—

C₈

—

226

—

C₈

—

226

—

C₂²

—

113

—

C₂²

—

113

—

C₂²

C₂×C₄×#90673

yes

characteristic

C₄

—

C₄

—

113

—

C₄

—

452

—

C₄

—

226

—

C₄

#16813701

yes

—

C₂

—

113

—

C₂

#20299534

yes

characteristic

C₂

#20299564

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂×C₄²
Fitting	#97×C₁₁₃	C₄
prime radical	C₂²×D₁₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	#97	#90673
hypocenter	C₁₁₃	#1746
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#90673

commutator, socle

C₂²×C₁₁₃

C₂×C₄²

Fitting

#97×C₁₁₃

C₄

prime radical

C₂²×D₁₁₃

C₂²×C₄

cosocle

C₂×C₄×D₁₁₃

C₂³

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
7. C₄×#11744753	C₄	semidirect
11. #16813533	C₄	semidirect
12. #16813532	C₄	semidirect
13. #97×C₁₁₃	C₄	semidirect
50. C₁₁₃	#1746	semidirect
61. #97	#90673	semidirect

H ≅ G/N

type

7. C₄×#11744753

C₄

semidirect

11. #16813533

C₄

semidirect

12. #16813532

C₄

semidirect

13. #97×C₁₁₃

C₄

semidirect

50. C₁₁₃

#1746

semidirect

61. #97

#90673

semidirect

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

subgroup	index
C₄×#14268655	2
#20299534	2
#20299463	2
#97×D₁₁₃	2
#1746	113

subgroup

index

#97×D₁₁₃

113

Subgroups — #28406294

Conjugacy classes of subgroups

Complements

Maximal subgroups