Group #26103000 — subgroups

Subgroups — #26103000

Order 12032.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#20242998	C₂	1	1	yes	maximal, characteristic
2	#20241452	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1367	C₂	1	1	yes	maximal, characteristic
4	#15689528	C₄	1	1	yes	characteristic
5	#15689528	C₄	1	1	yes	characteristic
6	#15689474	—	2	1	no	—
7	#15689474	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄₇×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₄₇	C₂²	1	1	yes	characteristic
11	#13094024	—	2	1	no	—
12	#13094024	—	2	1	no	—
13	C₂×C₄₇⋊C₁₆	—	4	1	no	—
14	#13093958	—	4	1	no	—
15	#13093958	—	4	1	no	—
16	#13093949	—	4	1	no	—
17	#13093949	—	4	1	no	—
18	#13093949	—	4	1	no	—
19	#13093949	—	4	1	no	—
20	C₂×C₁₆×C₄₇	—	4	1	no	—
21	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×Dic₉₄	—	4	1	no	—
25	C₂×Dic₉₄	—	4	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
29	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
30	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
31	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄⋊Dic₄₇	—	4	1	no	—
33	C₄⋊Dic₄₇	—	4	1	no	—
34	C₄⋊Dic₄₇	—	4	1	no	—
35	C₄⋊Dic₄₇	—	4	1	no	—
36	C₁₆×C₄₇	—	8	1	no	—
37	C₄₇⋊C₁₆	—	8	1	no	—
38	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
41	C₂×Dic₄₇	—	8	1	no	—
42	C₂×Dic₄₇	—	8	1	no	—
43	Dic₉₄	—	8	1	no	—
44	Dic₉₄	—	8	1	no	—
45	Dic₉₄	—	8	1	no	—
46	Dic₉₄	—	8	1	no	—
47	C₈×C₄₇	—	2	2	no	—
48	C₈×C₄₇	—	2	1	no	—
49	C₈×C₄₇	—	2	1	no	—
50	C₈×C₄₇	—	2	1	no	—
51	C₈×C₄₇	—	2	1	no	—
52	#7093	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	#331	1	1	yes	characteristic
54	C₄×C₄₇	—	2	1	no	—
55	C₄×C₄₇	#359	1	1	yes	characteristic
56	C₄×C₄₇	#359	1	1	yes	characteristic
57	C₄×C₄₇	#360	1	1	yes	characteristic
58	C₄×C₄₇	#360	1	1	yes	characteristic
59	Dic₄₇	—	16	1	no	—
60	Dic₄₇	—	16	1	no	—
61	#1710	—	47	1	no	—
62	#1367	—	47	1	no	—
63	#1367	D₄₇	1	1	yes	characteristic
64	C₂×C₄₇	#1358	1	1	yes	characteristic
65	C₂×C₄₇	#1360	1	1	yes	characteristic
66	C₂×C₄₇	#1360	1	1	yes	characteristic
67	#500	—	47	1	no	—
68	#500	—	47	1	no	—
69	#487	—	94	1	no	—
70	#487	—	94	1	no	—
71	C₄⋊C₁₆	—	2	1	no	—
72	C₄⋊C₁₆	—	94	1	no	—
73	C₈²	C₂×D₄₇	1	1	yes	characteristic
74	C₄₇	#7093	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	94	1	no	—
76	C₄⋊Q₈	—	94	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	188	1	no	—
79	#107	—	188	1	no	—
80	#107	—	188	1	no	—
81	#103	—	188	1	no	—
82	#103	—	188	1	no	—
83	#103	—	188	1	no	—
84	#103	—	188	1	no	—
85	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
86	C₄×C₈	D₁₈₈	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
88	C₂×Q₈	—	188	1	no	—
89	C₂×Q₈	—	188	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	#10625325	1	1	yes	characteristic
92	C₂×C₈	D₃₇₆	1	1	yes	characteristic
93	C₂×C₈	#10625334	1	1	yes	characteristic
94	C₂×C₈	#10625336	1	1	yes	characteristic
95	C₄⋊C₄	—	188	1	no	—
96	C₄⋊C₄	—	188	1	no	—
97	C₄⋊C₄	—	188	1	no	—
98	C₄⋊C₄	—	188	1	no	—
99	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	376	1	no	—
102	Q₈	—	376	1	no	—
103	Q₈	—	376	1	no	—
104	Q₈	—	376	1	no	—
105	Q₈	—	376	1	no	—
106	C₂×C₄	—	376	1	no	—
107	C₂×C₄	—	376	1	no	—
108	C₂×C₄	#13093940	1	1	yes	characteristic
109	C₂×C₄	#13093944	1	1	yes	characteristic
110	C₂×C₄	#13093946	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	1	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₂²	#15688866	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	752	1	no	—
119	C₄	—	752	1	no	—
120	C₄	#15688929	1	1	yes	characteristic
121	C₄	#15688931	1	1	yes	characteristic
122	C₄	#15688932	1	1	yes	characteristic
123	C₄	#15688934	1	1	yes	characteristic
124	C₂	#20241426	1	1	yes	characteristic
125	C₂	#20241430	1	1	yes	characteristic
126	C₂	#20241432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242998

C₂

yes

maximal, characteristic

#20241452

C₂

yes

maximal, characteristic

C₄₇×#1367

C₂

yes

maximal, characteristic

#15689528

C₄

yes

characteristic

#15689528

C₄

yes

characteristic

#15689474

—

#15689474

—

C₄⋊(C₄₇⋊C₁₆)

—

C₄₇×C₄⋊C₁₆

—

C₈²×C₄₇

C₂²

yes

characteristic

#13094024

—

#13094024

—

C₂×C₄₇⋊C₁₆

—

—

—

—

—

—

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1710

—

#1367

—

#1367

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#7093

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

#107

—

188

—

#107

—

188

—

#103

—

188

—

#103

—

188

—

#103

—

188

—

#103

—

188

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂×C₈

—

C₂×C₈

#10625325

yes

characteristic

C₂×C₈

D₃₇₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

100

C₁₆

—

101

C₁₆

—

376

—

102

Q₈

—

376

—

103

Q₈

—

376

—

104

Q₈

—

376

—

105

Q₈

—

376

—

106

C₂×C₄

—

376

—

107

C₂×C₄

—

376

—

108

C₂×C₄

#13093940

yes

characteristic

109

C₂×C₄

#13093944

yes

characteristic

110

C₂×C₄

#13093946

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

#15688866

yes

characteristic

117

C₄

—

118

C₄

—

752

—

119

C₄

—

752

—

120

C₄

#15688929

yes

characteristic

121

C₄

#15688931

yes

characteristic

122

C₄

#15688932

yes

characteristic

123

C₄

#15688934

yes

characteristic

124

C₂

#20241426

yes

characteristic

125

C₂

#20241430

yes

characteristic

126

C₂

#20241432

yes

characteristic

	subgroup	G/H
center	C₂²	#15688866
commutator	C₄×C₈×C₄₇	C₂×C₄
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1367	C₂
socle, prime radical	C₂²×C₄₇	#331
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1367	D₄₇
hypocenter	C₄₇	#7093
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#15688866

commutator

C₄×C₈×C₄₇

C₂×C₄

Frattini

C₈²

C₂×D₄₇

Fitting

C₄₇×#1367

C₂

socle, prime radical

C₂²×C₄₇

#331

cosocle

C₈²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
74. C₄₇	#7093	semidirect

H ≅ G/N

type

74. C₄₇

#7093

semidirect

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

subgroup	index
#20242998	2
#20241452	2
C₄₇×#1367	2
#7093	47

subgroup

index

#20242998

#20241452

C₄₇×#1367

#7093

Subgroups — #26103000

Conjugacy classes of subgroups

Complements

Maximal subgroups