Group #26102875 — subgroups

Subgroups — #26102875

Order 12032.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20243241	C₂	1	2	yes	maximal
2	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×#13093935	C₄	1	4	yes	—
5	#15688878	—	2	2	no	—
6	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
7	C₄³×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
11	#13093935	—	4	4	no	—
12	C₂²×C₈×C₄₇	—	2	2	no	—
13	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	D₄	1	2	yes	—
16	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₄₇	—	4	2	no	—
19	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	—	4	2	no	—
25	C₂×C₈×C₄₇	—	4	4	no	—
26	C₄²×C₄₇	—	2	1	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄²×C₄₇	—	2	1	no	—
29	C₄²×C₄₇	—	2	2	no	—
30	C₄²×C₄₇	—	2	4	no	—
31	C₄²×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	—	4	1	no	—
33	C₂×C₄₇⋊C₈	—	2	8	no	—
34	C₂×C₄₇⋊C₈	—	4	4	no	—
35	C₂×C₄₇⋊C₈	—	4	4	no	—
36	C₂³×C₄₇	#95	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	1	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	#100	1	2	yes	—
47	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#101	1	2	yes	—
49	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#104	1	4	yes	—
51	C₂×C₄×C₄₇	#108	1	2	yes	—
52	C₈×C₄₇	—	8	2	no	—
53	C₄₇⋊C₈	—	4	8	no	—
54	#7042	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂²×C₄₇	#339	1	2	yes	—
57	C₂²×C₄₇	#341	1	1	yes	characteristic
58	C₂²×C₄₇	#343	1	2	yes	—
59	C₂²×C₄₇	#343	1	1	yes	characteristic
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	—	4	2	no	—
64	C₄×C₄₇	—	4	1	no	—
65	C₄×C₄₇	#329	1	2	yes	—
66	C₄×C₄₇	#330	1	2	yes	—
67	#1831	D₄₇	1	1	yes	characteristic
68	#1767	—	47	2	no	—
69	C₂×C₄₇	#1285	1	2	yes	—
70	C₂×C₄₇	#1290	1	1	yes	characteristic
71	C₂×C₄₇	#1295	1	1	yes	characteristic
72	C₂×C₄₇	#1299	1	2	yes	—
73	C₂×C₄₇	#1309	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	94	2	no	—
75	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
76	C₂×#97	—	47	4	no	—
77	C₄³	C₂×D₄₇	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#336	—	94	2	no	—
80	C₄₇	#7042	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	94	4	no	—
83	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
84	C₂×C₄²	C₄×D₄₇	1	2	yes	—
85	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	94	8	no	—
90	#97	—	188	4	no	—
91	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
92	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
94	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
95	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	94	8	no	—
99	C₂×C₈	—	188	4	no	—
100	C₂×C₈	—	188	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	C₄₇⋊#95	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	#13093940	1	2	yes	—
119	C₂×C₄	#13093941	1	1	yes	characteristic
120	C₂×C₄	#13093942	1	2	yes	—
121	C₂×C₄	#13093943	1	1	yes	characteristic
122	C₂×C₄	#13093951	1	4	yes	—
123	C₂×C₄	#13093960	1	2	yes	—
124	C₈	—	8	2	no	—
125	C₈	—	188	8	no	—
126	C₂²	#15688880	1	1	yes	characteristic
127	C₂²	#15688882	1	2	yes	—
128	C₂²	#15688888	1	1	yes	characteristic
129	C₂²	#15688891	1	2	yes	—
130	C₂²	#15688892	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	1	no	—
136	C₄	#15688863	1	2	yes	—
137	C₄	#15688865	1	2	yes	—
138	C₂	#20241245	1	2	yes	—
139	C₂	#20241258	1	1	yes	characteristic
140	C₂	#20241269	1	1	yes	characteristic
141	C₂	#20241279	1	2	yes	—
142	C₂	C₄₇⋊#1309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243241

C₂

yes

maximal

C₄₇×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×#13093935

C₄

yes

—

#15688878

—

C₂×C₄⋊C₈×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#100

yes

—

C₂×C₄×C₄₇

#100

yes

characteristic

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#101

yes

characteristic

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#341

yes

characteristic

C₂²×C₄₇

#343

yes

—

C₂²×C₄₇

#343

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#329

yes

—

C₄×C₄₇

#330

yes

—

#1831

D₄₇

yes

characteristic

#1767

—

C₂×C₄₇

#1285

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1309

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₄₇

yes

—

C₂×#97

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

—

C₄₇

#7042

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₉₄

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

188

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂²×C₄

Dic₄₇⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

100

C₂×C₈

—

188

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

C₄₇⋊#95

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#13093940

yes

—

119

C₂×C₄

#13093941

yes

characteristic

120

C₂×C₄

#13093942

yes

—

121

C₂×C₄

#13093943

yes

characteristic

122

C₂×C₄

#13093951

yes

—

123

C₂×C₄

#13093960

yes

—

124

C₈

—

125

C₈

—

188

—

126

C₂²

#15688880

yes

characteristic

127

C₂²

#15688882

yes

—

128

C₂²

#15688888

yes

characteristic

129

C₂²

#15688891

yes

—

130

C₂²

#15688892

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#15688863

yes

—

137

C₄

#15688865

yes

—

138

C₂

#20241245

yes

—

139

C₂

#20241258

yes

characteristic

140

C₂

#20241269

yes

characteristic

141

C₂

#20241279

yes

—

142

C₂

C₄₇⋊#1309

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#7042
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1831

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
80. C₄₇	#7042	semidirect

H ≅ G/N

type

80. C₄₇

#7042

semidirect

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

subgroup	index
#20243241	2
C₄₇×#1831	2
#7042	47

subgroup

index

#20243241

C₄₇×#1831

#7042

Subgroups — #26102875

Conjugacy classes of subgroups

Complements

Maximal subgroups