Group #28393160 — subgroups

Subgroups — #28393160

Order 14464.

784 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 309 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×#16813581	C₂	1	2	yes	maximal
2	#20298505	C₂	1	8	yes	maximal, complemented
3	#20298461	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	#16813581	—	2	8	no	—
6	C₂×C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
8	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	—
9	#16813529	C₂²	1	16	yes	—
10	C₂²×Q₈×C₁₁₃	—	2	8	no	—
11	C₂²×Q₈×C₁₁₃	C₄	1	8	yes	complemented
12	C₂²×Q₈×C₁₁₃	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
16	C₄⋊Dic₁₁₃	—	2	8	no	—
17	C₂²⋊Dic₁₁₃	—	2	8	no	—
18	C₄×Dic₁₁₃	—	2	8	no	—
19	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₁₁₃	—	2	32	no	—
21	C₂×Q₈×C₁₁₃	—	2	8	no	—
22	C₂×Q₈×C₁₁₃	—	2	8	no	—
23	C₂×Q₈×C₁₁₃	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₁₁₃	—	2	8	no	—
25	C₂²×C₄×C₁₁₃	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₁₃	D₄	1	8	yes	—
28	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
30	C₂³×C₁₁₃	—	2	4	no	—
31	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₁₁₃	C₂×D₄	1	4	yes	—
33	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
34	Q₈×C₁₁₃	—	2	32	no	—
35	C₂×C₄×C₁₁₃	—	2	16	no	—
36	C₂×C₄×C₁₁₃	—	2	8	no	—
37	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	16	yes	—
38	C₂×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
42	C₂×Dic₁₁₃	—	2	8	no	—
43	C₂×Dic₁₁₃	—	2	16	no	—
44	C₂²×C₁₁₃	—	2	4	no	—
45	C₂²×C₁₁₃	—	2	8	no	—
46	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
47	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
49	C₂²×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
50	C₄×C₁₁₃	—	2	8	no	—
51	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
52	Dic₁₁₃	—	4	8	no	—
53	C₂×C₁₁₃	—	2	4	no	—
54	C₂×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₁₁₃	#519	1	2	yes	—
56	C₂×C₁₁₃	#536	1	4	yes	—
57	#2310	—	113	1	no	maximal, Sylow
58	C₁₁₃	#2310	1	1	yes	characteristic, complemented, Sylow
59	C₂³×Q₈	D₁₁₃	1	1	yes	characteristic
60	C₂×#117	—	113	2	no	—
61	#391	—	113	8	no	—
62	#381	—	113	4	no	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	Dic₁₁₃	1	8	yes	complemented
65	C₂²×Q₈	C₂×D₁₁₃	1	4	yes	—
66	C₂³×C₄	C₂×D₁₁₃	1	2	yes	—
67	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
68	#117	—	226	8	no	—
69	C₂×C₄⋊C₄	—	113	4	no	—
70	C₂×C₂²⋊C₄	—	113	4	no	—
71	C₂×C₄²	—	113	4	no	—
72	#95	—	113	16	no	—
73	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×Dic₁₁₃	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	—	113	8	no	—
80	C₂²×C₄	C₂×Dic₁₁₃	1	8	yes	—
81	C₂²×C₄	C₂×Dic₁₁₃	1	4	yes	—
82	C₂²×C₄	C₂²⋊D₁₁₃	1	8	yes	—
83	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
84	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
85	C₄⋊C₄	—	226	8	no	—
86	C₂²⋊C₄	—	226	8	no	—
87	C₄²	—	226	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
90	C₂³	C₂×C₂²⋊D₁₁₃	1	4	yes	—
91	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	226	8	no	—
96	C₂×C₄	—	226	16	no	—
97	C₂×C₄	C₂²⋊Dic₁₁₃	1	16	yes	—
98	C₂×C₄	C₂²×Dic₁₁₃	1	8	yes	—
99	C₂×C₄	C₂²×Dic₁₁₃	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
102	C₂²	—	2	4	no	—
103	C₂²	—	2	8	no	—
104	C₂²	C₂×C₂²⋊Dic₁₁₃	1	4	yes	—
105	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
106	C₂²	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
107	C₂²	#16813670	1	4	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	452	8	no	—
110	C₄	C₂×C₂²⋊Dic₁₁₃	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
113	C₂	#20299076	1	2	yes	—
114	C₂	#20299172	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813581

C₂

yes

maximal

#20298505

C₂

yes

maximal, complemented

#20298461

C₂

yes

maximal

C₂³×Q₈×C₁₁₃

C₂

yes

maximal, characteristic

#16813581

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₁₃

C₂²

yes

—

C₂×C₄×Dic₁₁₃

C₂²

yes

—

#16813529

C₂²

yes

—

C₂²×Q₈×C₁₁₃

—

C₂²×Q₈×C₁₁₃

C₄

yes

complemented

C₂²×Q₈×C₁₁₃

C₂²

yes

—

C₂³×C₄×C₁₁₃

C₂²

yes

—

C₂³×C₄×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

C₂³

yes

—

C₄⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₄×Dic₁₁₃

—

C₂⁴×C₁₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₁₃

—

C₂×Q₈×C₁₁₃

—

C₂×Q₈×C₁₁₃

—

C₂×Q₈×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₂²×C₄

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

—

C₂³×C₁₁₃

C₂⁴

yes

characteristic

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₁₃

C₂²×D₄

yes

—

C₂²×C₁₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×C₂²⋊C₄

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₁₁₃

#519

yes

—

C₂×C₁₁₃

#536

yes

—

#2310

—

113

maximal, Sylow

C₁₁₃

#2310

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₁₁₃

yes

characteristic

C₂×#117

—

113

—

#391

—

113

—

#381

—

113

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₁₃

yes

complemented

C₂²×Q₈

C₂×D₁₁₃

yes

—

C₂³×C₄

C₂×D₁₁₃

yes

—

C₂³×C₄

C₂×D₁₁₃

yes

characteristic

#117

—

226

—

C₂×C₄⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

113

—

C₂×C₄²

—

113

—

#95

—

113

—

C₂⁴

C₂²×D₁₁₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

—

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

226

—

C₄²

—

226

—

C₂³

—

C₂³

C₂²×Dic₁₁₃

yes

—

C₂³

C₂×C₂²⋊D₁₁₃

yes

—

C₂³

C₂³×D₁₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²⋊Dic₁₁₃

yes

—

C₂×C₄

C₂²×Dic₁₁₃

yes

—

C₂×C₄

C₂²×Dic₁₁₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊Dic₁₁₃

yes

—

105

C₂²

C₂³×Dic₁₁₃

yes

characteristic

106

C₂²

C₂²×C₂²⋊D₁₁₃

yes

—

107

C₂²

#16813670

yes

—

108

C₄

—

109

C₄

—

452

—

110

C₄

C₂×C₂²⋊Dic₁₁₃

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊Dic₁₁₃

yes

characteristic

113

C₂

#20299076

yes

—

114

C₂

#20299172

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂³×Q₈×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂⁴×C₁₁₃	C₂³
hypercenter	C₂³×Q₈	D₁₁₃
hypocenter	C₁₁₃	#2310
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₁₃

commutator

C₂²×C₁₁₃

C₂³×C₄

Fitting

C₂³×Q₈×C₁₁₃

C₂

socle, cosocle

C₂³×C₁₁₃

C₂⁴

prime radical

C₂⁴×C₁₁₃

C₂³

hypercenter

C₂³×Q₈

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #20298505	C₂	semidirect
11. C₂²×Q₈×C₁₁₃	C₄	semidirect
58. C₁₁₃	#2310	semidirect
64. C₂²×Q₈	Dic₁₁₃	semidirect

H ≅ G/N

type

2. #20298505

C₂

semidirect

11. C₂²×Q₈×C₁₁₃

C₄

semidirect

58. C₁₁₃

#2310

semidirect

64. C₂²×Q₈

Dic₁₁₃

semidirect

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

subgroup	index
C₂×#16813581	2
#20298505	2
#20298461	2
C₂³×Q₈×C₁₁₃	2
#2310	113

subgroup

index

C₂³×Q₈×C₁₁₃

113

Subgroups — #28393160

Conjugacy classes of subgroups

Complements

Maximal subgroups