Group #26125314 — subgroups

Subgroups — #26125314

Order 12032.

260 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20245185	C₂	1	2	yes	maximal
2	#20241670	C₂	1	1	yes	maximal, characteristic
3	#20241307	C₂	1	2	yes	maximal, complemented
4	#20241305	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689174	C₄	1	2	yes	—
7	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
8	Dic₄₇⋊C₁₆	C₄	1	4	yes	—
9	(C₂×D₄₇)⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	#15688845	C₄	1	2	yes	—
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
16	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	Dic₄₇⋊C₈	C₈	1	4	yes	—
19	(C₂×D₄₇)⋊C₈	C₈	1	4	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
30	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
32	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
33	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
34	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
35	C₈×D₄₇	—	2	4	no	—
36	C₁₆×C₄₇	—	2	4	no	—
37	C₄₇⋊C₁₆	—	2	4	no	—
38	C₂²×D₄₇	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
41	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
42	C₂×Dic₄₇	C₄×C₈	1	1	yes	characteristic
43	C₄×D₄₇	—	2	2	no	—
44	C₄×D₄₇	—	2	4	no	—
45	C₈×C₄₇	—	2	4	no	—
46	C₈×C₄₇	#131	1	4	yes	—
47	C₄₇⋊C₈	—	2	2	no	—
48	C₄₇⋊C₈	—	2	4	no	—
49	#11412	—	47	1	no	maximal, Sylow
50	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×D₄₇	—	2	2	no	—
52	C₂×D₄₇	—	2	1	no	—
53	C₄×C₄₇	#405	1	2	yes	—
54	C₄×C₄₇	#504	1	4	yes	—
55	Dic₄₇	—	2	2	no	—
56	Dic₄₇	—	2	1	no	—
57	#2162	—	47	2	no	—
58	#1450	—	47	1	no	—
59	#1312	—	47	2	no	—
60	#1311	—	47	1	no	—
61	C₈×C₁₆	D₄₇	1	1	yes	characteristic, complemented
62	C₂×C₄₇	#1450	1	1	yes	characteristic
63	C₂×C₄₇	#2108	1	2	yes	—
64	D₄₇	—	4	1	no	—
65	#431	—	47	2	no	—
66	C₂×C₄×C₈	—	47	1	no	—
67	C₄⋊C₁₆	—	47	4	no	—
68	C₂²⋊C₁₆	—	47	4	no	—
69	C₄×C₁₆	—	47	2	no	—
70	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
71	#322	—	47	2	no	—
72	C₈²	—	47	1	no	—
73	C₈²	C₂×D₄₇	1	1	yes	characteristic
74	C₄₇	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	47	2	no	—
76	C₂×C₄²	—	47	1	no	—
77	C₂×C₁₆	—	47	4	no	—
78	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
79	C₄⋊C₈	—	47	4	no	—
80	C₂²⋊C₈	—	47	4	no	—
81	C₄×C₈	—	47	2	no	—
82	C₄×C₈	—	47	2	no	—
83	C₄×C₈	—	47	1	no	—
84	C₄×C₈	C₄×D₄₇	1	2	yes	—
85	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
86	C₂²×C₄	—	47	1	no	—
87	C₂²×C₄	—	47	2	no	—
88	C₂×C₈	—	47	2	no	—
89	C₂×C₈	—	47	4	no	—
90	C₂×C₈	—	94	4	no	—
91	C₂×C₈	C₈×D₄₇	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
93	C₄²	—	47	1	no	—
94	C₄²	—	47	2	no	—
95	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	94	4	no	—
98	C₂³	—	47	1	no	—
99	C₂×C₄	—	47	1	no	—
100	C₂×C₄	—	47	2	no	—
101	C₂×C₄	—	94	4	no	—
102	C₂×C₄	—	94	2	no	—
103	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	94	4	no	—
107	C₈	—	94	2	no	—
108	C₈	#13094030	1	4	yes	—
109	C₂²	—	94	1	no	—
110	C₂²	—	94	2	no	—
111	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
112	C₄	—	94	1	no	—
113	C₄	—	94	2	no	—
114	C₄	#15689077	1	2	yes	—
115	C₄	#15689539	1	4	yes	—
116	C₂	—	188	1	no	—
117	C₂	#20241670	1	1	yes	characteristic
118	C₂	#20244983	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245185

C₂

yes

maximal

#20241670

C₂

yes

maximal, characteristic

#20241307

C₂

yes

maximal, complemented

#20241305

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

#15689174

C₄

yes

—

C₄×C₈×D₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₁₆

C₄

yes

—

(C₂×D₄₇)⋊C₁₆

C₄

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

#15688845

C₄

yes

—

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₂×C₄

yes

—

C₄²×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

Dic₄₇⋊C₈

C₈

yes

—

(C₂×D₄₇)⋊C₈

C₈

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

C₄²

yes

characteristic

C₂×C₄×D₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄×Dic₄₇

C₄²

yes

characteristic

C₄×Dic₄₇

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

C₄²

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂²×D₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×Dic₄₇

C₄×C₈

yes

—

C₂×Dic₄₇

C₄×C₈

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#131

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#11412

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#405

yes

—

C₄×C₄₇

#504

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

C₈×C₁₆

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1450

yes

characteristic

C₂×C₄₇

#2108

yes

—

D₄₇

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

—

#322

—

C₈²

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄₇

yes

—

C₂×C₈

C₂×C₄×D₄₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₄₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄²×D₄₇

yes

characteristic

104

C₂×C₄

C₂×C₈×D₄₇

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#13094030

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₄×C₈×D₄₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#15689077

yes

—

115

C₄

#15689539

yes

—

116

C₂

—

188

—

117

C₂

#20241670

yes

characteristic

118

C₂

#20244983

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₈×C₁₆×C₄₇	C₂
prime radical	C₂²×D₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#11412
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₄₇

commutator, socle

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₈×C₁₆×C₄₇

C₂

prime radical

C₂²×D₄₇

C₄×C₈

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

C₈×C₁₆

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

N	H ≅ G/N	type
3. #20241307	C₂	semidirect
4. #20241305	C₂	semidirect
5. C₈×C₁₆×C₄₇	C₂	semidirect
61. C₈×C₁₆	D₄₇	semidirect
74. C₄₇	#11412	semidirect

H ≅ G/N

type

3. #20241307

C₂

semidirect

4. #20241305

C₂

semidirect

5. C₈×C₁₆×C₄₇

C₂

semidirect

61. C₈×C₁₆

D₄₇

semidirect

74. C₄₇

#11412

semidirect

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

subgroup	index
#20245185	2
#20241670	2
#20241307	2
#20241305	2
C₈×C₁₆×C₄₇	2
#11412	47

subgroup

index

C₈×C₁₆×C₄₇

Subgroups — #26125314

Conjugacy classes of subgroups

Complements

Maximal subgroups