Group #26130950 — subgroups

Subgroups — #26130950

Order 12032.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20245670	C₂	1	1	yes	maximal, characteristic
2	#20245615	C₂	1	2	yes	maximal
3	#20242982	C₂	1	2	yes	maximal
4	#20241304	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689519	—	2	2	no	—
8	#15689475	—	2	2	no	—
9	#15688951	—	2	2	no	—
10	#15688927	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₄₇	—	2	2	no	—
14	C₄₇×#494	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	#13094024	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
18	#13093958	—	2	2	no	—
19	#13093958	—	2	2	no	—
20	#13093955	—	4	4	no	—
21	#13093947	—	4	4	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₄₇	—	2	2	no	—
24	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	Q₃₂×C₄₇	—	4	4	no	—
27	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×Dic₉₄	—	4	2	no	—
31	Q₁₆×C₄₇	—	4	4	no	—
32	Q₁₆×C₄₇	—	4	4	no	—
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₄⋊Dic₄₇	—	4	2	no	—
40	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
41	C₁₆×C₄₇	SD₁₆	1	4	yes	—
42	C₄₇⋊C₁₆	—	2	2	no	—
43	Q₈×C₄₇	—	8	2	no	—
44	Q₈×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	8	2	no	—
46	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₄₇	—	8	2	no	—
49	Dic₉₄	—	8	4	no	—
50	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
51	C₈×C₄₇	#135	1	2	yes	—
52	C₄₇⋊C₈	—	2	4	no	—
53	#12507	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	#493	1	1	yes	characteristic
55	C₄×C₄₇	—	16	2	no	—
56	C₄×C₄₇	#492	1	1	yes	characteristic
57	C₄×C₄₇	#495	1	1	yes	characteristic
58	C₄×C₄₇	#510	1	4	yes	—
59	Dic₄₇	—	16	2	no	—
60	#2254	—	47	1	no	—
61	#2247	D₄₇	1	1	yes	characteristic
62	#2241	—	47	2	no	—
63	#1707	—	47	2	no	—
64	#1311	—	47	1	no	—
65	C₂×C₄₇	#1706	1	1	yes	characteristic
66	C₂×C₄₇	#2251	1	2	yes	—
67	C₂×Q₃₂	—	2	2	no	—
68	#500	—	47	2	no	—
69	#498	—	94	2	no	—
70	#494	C₂×D₄₇	1	2	yes	—
71	#487	—	94	2	no	—
72	#367	—	94	2	no	—
73	#358	—	94	4	no	—
74	C₄×C₁₆	—	47	1	no	—
75	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
76	C₈²	—	47	1	no	—
77	C₄₇	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	94	2	no	—
82	Q₃₂	—	4	4	no	—
83	C₂×C₁₆	—	47	2	no	—
84	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
85	#107	—	94	2	no	—
86	#107	—	94	2	no	—
87	#106	—	188	4	no	—
88	#103	—	188	4	no	—
89	C₄×C₈	—	47	2	no	—
90	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	188	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	—	47	4	no	—
96	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	188	2	no	—
101	C₄⋊C₄	—	188	2	no	—
102	C₄²	D₄×D₄₇	1	1	yes	characteristic
103	C₁₆	—	94	2	no	—
104	C₁₆	#10625335	1	4	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	376	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	376	2	no	—
110	C₂×C₄	#13094020	1	1	yes	characteristic
111	C₂×C₄	D₈×D₄₇	1	2	yes	—
112	C₈	—	94	4	no	—
113	C₈	C₂×#10625335	1	2	yes	—
114	C₈	#13094043	1	2	yes	—
115	C₂²	#15689498	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	752	2	no	—
118	C₄	#15689495	1	1	yes	characteristic
119	C₄	#15689506	1	1	yes	characteristic
120	C₄	#15689565	1	4	yes	—
121	C₂	#20242977	1	1	yes	characteristic
122	C₂	#20245656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal, characteristic

#20245615

C₂

yes

maximal

#20242982

C₂

yes

maximal

#20241304

C₂

yes

maximal, characteristic

C₄₇×#2247

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₄₇

—

C₄₇×#494

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

Q₃₂×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

SD₁₆

yes

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2254

—

#2247

D₄₇

yes

—

—

—

C₂×C₄₇

#1706

yes

characteristic

C₂×C₄₇

#2251

yes

—

C₂×Q₃₂

—

#500

—

#498

—

#494

C₂×D₄₇

yes

—

#487

—

#367

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

#107

—

#106

—

188

—

#103

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

188

—

101

C₄⋊C₄

—

188

—

102

C₄²

D₄×D₄₇

yes

characteristic

103

C₁₆

—

104

C₁₆

#10625335

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

376

—

108

C₂×C₄

—

109

C₂×C₄

—

376

—

110

C₂×C₄

#13094020

yes

characteristic

111

C₂×C₄

D₈×D₄₇

yes

—

112

C₈

—

113

C₈

C₂×#10625335

yes

—

114

C₈

#13094043

yes

—

115

C₂²

#15689498

yes

characteristic

116

C₄

—

117

C₄

—

752

—

118

C₄

#15689495

yes

characteristic

119

C₄

#15689506

yes

characteristic

120

C₄

#15689565

yes

—

121

C₂

#20242977

yes

characteristic

122

C₂

#20245656

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2247	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#2247	D₄₇
hypocenter	C₄₇	#12507
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2247

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
77. C₄₇	#12507	semidirect

H ≅ G/N

type

77. C₄₇

#12507

semidirect

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

subgroup	index
#20245670	2
#20245615	2
#20242982	2
#20241304	2
C₄₇×#2247	2
#12507	47

subgroup

index

C₄₇×#2247

Subgroups — #26130950

Conjugacy classes of subgroups

Complements

Maximal subgroups