Group #26106710 — subgroups

Subgroups — #26106710

Order 12032.

414 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20243134	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#1273	C₂	1	4	yes	maximal
4	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093935	C₄	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688878	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
10	C₄³×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#336	C₄	1	4	yes	—
12	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
13	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
14	#13093935	—	2	8	no	—
15	C₄×C₄₇⋊C₈	—	2	4	no	—
16	C₄×C₄₇⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	8	no	—
29	C₄²×C₄₇	—	2	4	no	—
30	C₄²×C₄₇	—	2	4	no	—
31	C₄²×C₄₇	—	2	4	no	—
32	C₄²×C₄₇	#41	1	4	yes	—
33	C₂×C₄₇⋊C₈	—	2	4	no	—
34	C₂×C₄₇⋊C₈	—	2	4	no	—
35	C₂×C₄₇⋊C₈	—	2	4	no	—
36	C₂×C₄₇⋊C₈	—	2	8	no	—
37	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	—
38	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	#117	1	2	yes	—
43	C₂×C₄×C₄₇	#117	1	2	yes	—
44	C₂×C₄×C₄₇	#117	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
46	C₂×C₄×C₄₇	#131	1	4	yes	—
47	C₂×C₄×C₄₇	#131	1	4	yes	—
48	C₈×C₄₇	—	4	4	no	—
49	C₄₇⋊C₈	—	2	8	no	—
50	C₄₇⋊C₈	—	4	4	no	—
51	#8134	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	#376	1	1	yes	characteristic
53	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
54	C₂²×C₄₇	#405	1	2	yes	—
55	C₂²×C₄₇	#405	1	2	yes	—
56	C₄×C₄₇	—	4	4	no	—
57	C₄×C₄₇	#432	1	4	yes	—
58	C₄×C₄₇	#433	1	4	yes	—
59	C₄×C₄₇	#433	1	4	yes	—
60	#1744	—	47	1	no	—
61	#1744	D₄₇	1	1	yes	characteristic
62	C₄×#97	—	47	1	no	—
63	#1273	—	47	4	no	—
64	C₂×C₄₇	#1453	1	4	yes	—
65	C₂×C₄₇	#1745	1	1	yes	characteristic
66	C₂×C₄₇	#1746	1	1	yes	characteristic
67	C₂×C₄₇	#1746	1	1	yes	characteristic
68	C₂×#97	—	47	4	no	—
69	C₂×C₄×C₈	—	47	2	no	—
70	C₂×C₄×C₈	—	47	2	no	—
71	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
72	C₄³	C₂×D₄₇	1	1	yes	characteristic
73	#336	—	47	4	no	—
74	#336	Dic₄₇	1	4	yes	—
75	C₄₇	#8134	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	47	4	no	—
77	C₂²×C₈	—	47	4	no	—
78	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
79	C₂×C₄²	C₄×D₄₇	1	2	yes	—
80	C₂×C₄²	C₄×D₄₇	1	2	yes	—
81	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
82	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
83	#97	—	94	8	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	—	94	4	no	—
86	C₄×C₈	—	94	4	no	—
87	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
88	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	8	no	—
93	C₂×C₈	—	47	8	no	—
94	C₂×C₈	—	94	4	no	—
95	C₂×C₈	—	94	4	no	—
96	C₂×C₈	—	94	4	no	—
97	C₂×C₈	—	94	8	no	—
98	C₄²	—	2	4	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	#10625350	1	4	yes	—
102	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	#13093981	1	2	yes	—
107	C₂×C₄	#13093981	1	2	yes	—
108	C₂×C₄	#13093983	1	2	yes	—
109	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
110	C₂×C₄	#13094030	1	4	yes	—
111	C₂×C₄	#13094031	1	4	yes	—
112	C₈	—	4	4	no	—
113	C₈	—	94	8	no	—
114	C₈	—	188	4	no	—
115	C₂²	#15688969	1	1	yes	characteristic
116	C₂²	M₁₆×Dic₄₇	1	2	yes	—
117	C₂²	#15689078	1	2	yes	—
118	C₂²	#15689079	1	2	yes	—
119	C₄	—	4	4	no	—
120	C₄	#15689178	1	4	yes	—
121	C₄	#15689182	1	4	yes	—
122	C₄	#15689184	1	4	yes	—
123	C₂	#20241683	1	4	yes	—
124	C₂	#20243136	1	1	yes	characteristic
125	C₂	#20243139	1	1	yes	characteristic
126	C₂	#20243140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243134

C₂

yes

maximal, characteristic

C₄×#13093935

C₂

yes

maximal, characteristic

C₄₇⋊#1273

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×#13093935

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₄×M₁₆

yes

—

C₂²×C₄₇

#405

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

—

C₄×C₄₇

#432

yes

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#433

yes

—

#1744

—

#1744

D₄₇

yes

characteristic

C₄×#97

—

#1273

—

C₂×C₄₇

#1453

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

Dic₄₇

yes

—

C₄₇

#8134

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₄×Dic₄₇

yes

—

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

#10625350

yes

—

102

C₂³

C₂×C₄×Dic₄₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#13093981

yes

—

107

C₂×C₄

#13093981

yes

—

108

C₂×C₄

#13093983

yes

—

109

C₂×C₄

M₁₆×D₄₇

yes

—

110

C₂×C₄

#13094030

yes

—

111

C₂×C₄

#13094031

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

188

—

115

C₂²

#15688969

yes

characteristic

116

C₂²

M₁₆×Dic₄₇

yes

—

117

C₂²

#15689078

yes

—

118

C₂²

#15689079

yes

—

119

C₄

—

120

C₄

#15689178

yes

—

121

C₄

#15689182

yes

—

122

C₄

#15689184

yes

—

123

C₂

#20241683

yes

—

124

C₂

#20243136

yes

characteristic

125

C₂

#20243139

yes

characteristic

126

C₂

#20243140

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1744	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8134
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂×C₄²

Fitting

C₄₇×#1744

C₂

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
75. C₄₇	#8134	semidirect

H ≅ G/N

type

75. C₄₇

#8134

semidirect

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

subgroup	index
#20243134	2
C₄×#13093935	2
C₄₇⋊#1273	2
C₄₇×#1744	2
#8134	47

subgroup

index

#20243134

C₄×#13093935

C₄₇⋊#1273

C₄₇×#1744

#8134

Subgroups — #26106710

Conjugacy classes of subgroups

Complements

Maximal subgroups