Group #26106719 — subgroups

Subgroups — #26106719

Order 12032.

354 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 195 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	#20241671	C₂	1	2	yes	maximal
2	#20241219	C₂	1	2	yes	maximal
3	#20241217	C₂	1	2	yes	maximal
4	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
5	#15689176	C₂²	1	4	yes	—
6	#15688844	C₄	1	4	yes	complemented
7	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
8	C₂×#97×C₄₇	C₄	1	2	yes	—
9	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#376	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
14	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
15	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
21	#97×C₄₇	—	2	4	no	—
22	#97×C₄₇	—	2	2	no	—
23	C₄×C₈×C₄₇	C₈	1	8	yes	complemented
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	#95×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	—	2	2	no	—
28	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	#41	1	2	yes	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄²	1	2	yes	—
35	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
38	C₄²×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	C₂×C₈	1	4	yes	—
40	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
41	C₄²×C₄₇	#41	1	2	yes	—
42	C₂×C₄₇⋊C₈	—	2	8	no	—
43	C₂³×C₄₇	#117	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#117	1	4	yes	—
49	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
52	C₈×C₄₇	—	2	4	no	—
53	C₈×C₄₇	C₄×C₈	1	2	yes	—
54	C₈×C₄₇	C₄×C₈	1	2	yes	—
55	C₈×C₄₇	#97	1	4	yes	—
56	C₄₇⋊C₈	—	4	8	no	—
57	#8137	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	#376	1	1	yes	characteristic
60	C₂²×C₄₇	#431	1	2	yes	—
61	C₄×C₄₇	—	2	4	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
65	C₄×C₄₇	#431	1	2	yes	—
66	#1726	D₄₇	1	1	yes	characteristic
67	#1450	—	47	2	no	—
68	#1275	—	47	2	no	—
69	#1274	—	47	2	no	—
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	#1744	1	1	yes	characteristic
72	C₂×C₄₇	#1747	1	2	yes	—
73	#431	—	47	4	no	—
74	C₂×#97	Dic₄₇	1	2	yes	—
75	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
76	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
78	#376	C₂×D₄₇	1	1	yes	characteristic
79	#322	—	47	4	no	—
80	C₈²	—	47	4	no	—
81	C₄₇	#8137	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
83	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
85	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
86	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	94	4	no	—
88	C₂²⋊C₈	—	94	4	no	—
89	#97	—	2	4	no	—
90	#97	—	2	2	no	—
91	C₄×C₈	—	47	8	no	—
92	C₄×C₈	C₄₇⋊C₈	1	8	yes	complemented
93	C₄×C₈	C₄×D₄₇	1	2	yes	—
94	C₄×C₈	C₄×D₄₇	1	2	yes	—
95	#95	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	#10625350	1	2	yes	—
98	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	94	8	no	—
102	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
103	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
104	C₂×C₈	#10625329	1	4	yes	—
105	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
107	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
108	C₄²	—	2	2	no	—
109	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
110	C₄²	C₂×C₄×D₄₇	1	2	yes	—
111	C₄²	#10625350	1	2	yes	—
112	C₂³	#13093983	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
117	C₂×C₄	#13093981	1	4	yes	—
118	C₂×C₄	#13093983	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
120	C₂×C₄	C₂×#10625329	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	188	8	no	—
123	C₈	C₄×C₄₇⋊C₈	1	2	yes	—
124	C₈	C₄×C₄₇⋊C₈	1	2	yes	—
125	C₈	#13093935	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#15688969	1	1	yes	characteristic
128	C₂²	#15689176	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
132	C₄	C₂×#13093935	1	1	yes	characteristic
133	C₄	#15689176	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	#20243134	1	1	yes	characteristic
136	C₂	#20243143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20241671

C₂

yes

maximal

#20241219

C₂

yes

maximal

#20241217

C₂

yes

maximal

C₄₇×#1726

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15688844

C₄

yes

complemented

C₈×C₄₇⋊C₈

C₄

yes

complemented

C₂×#97×C₄₇

C₄

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#376

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

complemented

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

—

C₈×C₄₇

C₄×C₈

yes

—

C₈×C₄₇

#97

yes

—

C₄₇⋊C₈

—

#8137

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#376

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄×C₈

yes

characteristic

C₄×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

#431

yes

—

#1726

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1744

yes

characteristic

C₂×C₄₇

#1747

yes

—

#431

—

C₂×#97

Dic₄₇

yes

—

C₂×#97

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

C₂×D₄₇

yes

characteristic

#322

—

C₈²

—

C₄₇

#8137

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

complemented

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₄×D₄₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#10625350

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

103

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

104

C₂×C₈

#10625329

yes

—

105

C₂×C₈

C₄×Dic₄₇

yes

characteristic

106

C₂×C₈

C₄×Dic₄₇

yes

—

107

C₂×C₈

C₄×Dic₄₇

yes

characteristic

108

C₄²

—

109

C₄²

C₂×C₄₇⋊C₈

yes

—

110

C₄²

C₂×C₄×D₄₇

yes

—

111

C₄²

#10625350

yes

—

112

C₂³

#13093983

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

117

C₂×C₄

#13093981

yes

—

118

C₂×C₄

#13093983

yes

characteristic

119

C₂×C₄

C₂²×C₄₇⋊C₈

yes

—

120

C₂×C₄

C₂×#10625329

yes

—

121

C₈

—

122

C₈

—

188

—

123

C₈

C₄×C₄₇⋊C₈

yes

—

124

C₈

C₄×C₄₇⋊C₈

yes

—

125

C₈

#13093935

yes

—

126

C₂²

—

127

C₂²

#15688969

yes

characteristic

128

C₂²

#15689176

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×C₄₇⋊C₈

yes

characteristic

132

C₄

C₂×#13093935

yes

characteristic

133

C₄

#15689176

yes

—

134

C₂

—

135

C₂

#20243134

yes

characteristic

136

C₂

#20243143

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1726	C₂
socle	C₂²×C₄₇	#376
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#8137
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₄×Dic₄₇

commutator

C₄×C₄₇

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1726

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₈ ◁ #1726

N	H ≅ G/N	type
6. #15688844	C₄	semidirect
7. C₈×C₄₇⋊C₈	C₄	semidirect
23. C₄×C₈×C₄₇	C₈	semidirect
81. C₄₇	#8137	semidirect
92. C₄×C₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

6. #15688844

C₄

semidirect

7. C₈×C₄₇⋊C₈

C₄

semidirect

23. C₄×C₈×C₄₇

C₈

semidirect

81. C₄₇

#8137

semidirect

92. C₄×C₈

C₄₇⋊C₈

semidirect

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

subgroup	index
#20241671	2
#20241219	2
#20241217	2
C₄₇×#1726	2
#8137	47

subgroup

index

C₄₇×#1726

Subgroups — #26106719

Conjugacy classes of subgroups

Complements

Maximal subgroups