Group #26109949 — subgroups

Subgroups — #26109949

Order 12032.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20243540	C₂	1	2	yes	maximal
2	#20241223	C₂	1	4	yes	maximal
3	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
8	C₄³×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#336	C₄	1	4	yes	—
10	C₂²×C₄₇⋊C₈	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
12	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
13	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
14	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
15	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₄₇	—	2	4	no	—
21	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	4	no	—
27	C₂×C₈×C₄₇	—	2	8	no	—
28	C₄²×C₄₇	—	2	4	no	—
29	C₄²×C₄₇	—	4	2	no	—
30	C₄²×C₄₇	D₈	1	2	yes	—
31	C₄²×C₄₇	SD₁₆	1	4	yes	—
32	C₄²×C₄₇	Q₁₆	1	2	yes	—
33	C₄²×C₄₇	#41	1	4	yes	—
34	C₂×C₄₇⋊C₈	—	4	4	no	—
35	C₂×C₄₇⋊C₈	—	4	4	no	—
36	C₂×C₄₇⋊C₈	—	4	4	no	—
37	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	#106	1	4	yes	—
44	C₂×C₄×C₄₇	#107	1	4	yes	—
45	C₂×C₄×C₄₇	#117	1	2	yes	—
46	C₂×C₄×C₄₇	#124	1	2	yes	—
47	C₂×C₄×C₄₇	#125	1	2	yes	—
48	C₂×C₄×C₄₇	#131	1	4	yes	—
49	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₈×C₄₇	—	4	4	no	—
53	C₄₇⋊C₈	—	8	4	no	—
54	#8787	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	#382	1	1	yes	characteristic
56	C₂²×C₄₇	#424	1	2	yes	—
57	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
58	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
59	C₂²×C₄₇	#430	1	2	yes	—
60	C₄×C₄₇	—	2	4	no	—
61	C₄×C₄₇	—	2	4	no	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	#433	1	4	yes	—
64	C₄×C₄₇	#486	1	2	yes	—
65	C₄×C₄₇	#487	1	2	yes	—
66	#1842	—	47	2	no	—
67	#1744	D₄₇	1	1	yes	characteristic
68	#1277	—	47	4	no	—
69	C₂×C₄₇	#1570	1	2	yes	—
70	C₂×C₄₇	#1571	1	2	yes	—
71	C₂×C₄₇	#1832	1	1	yes	characteristic
72	C₂×C₄₇	#1834	1	1	yes	characteristic
73	C₂×C₄₇	#1838	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	47	4	no	—
75	C₂×C₄⋊C₈	—	94	2	no	—
76	C₂×C₄⋊C₈	—	94	2	no	—
77	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
78	C₄³	C₂×D₄₇	1	1	yes	characteristic
79	#336	Dic₄₇	1	4	yes	—
80	C₄₇	#8787	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	94	4	no	—
82	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
83	C₂×C₄²	Dic₉₄	1	2	yes	—
84	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
85	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
87	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
88	C₄⋊C₈	—	94	8	no	—
89	C₄⋊C₈	—	188	4	no	—
90	C₄⋊C₈	—	188	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
94	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
96	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	—	188	4	no	—
100	C₂×C₈	—	188	4	no	—
101	C₂×C₈	—	188	4	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	#10625325	1	4	yes	—
105	C₄²	D₃₇₆	1	2	yes	—
106	C₄²	Dic₁₈₈	1	2	yes	—
107	C₄²	#10625350	1	4	yes	—
108	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#13093956	1	4	yes	—
115	C₂×C₄	#13093958	1	4	yes	—
116	C₂×C₄	#13093983	1	2	yes	—
117	C₂×C₄	#13094012	1	2	yes	—
118	C₂×C₄	#13094016	1	2	yes	—
119	C₂×C₄	#13094031	1	4	yes	—
120	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
121	C₂×C₄	C₂×#10625325	1	2	yes	—
122	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
123	C₈	—	4	4	no	—
124	C₈	—	376	4	no	—
125	C₂²	#15688989	1	1	yes	characteristic
126	C₂²	#15689150	1	2	yes	—
127	C₂²	C₂×#13093956	1	1	yes	characteristic
128	C₂²	C₂×#13093958	1	1	yes	characteristic
129	C₂²	#15689172	1	2	yes	—
130	C₄	—	2	2	no	—
131	C₄	—	2	4	no	—
132	C₄	—	2	4	no	—
133	C₄	#15689184	1	4	yes	—
134	C₄	#15689470	1	2	yes	—
135	C₄	#15689474	1	2	yes	—
136	C₂	#20242159	1	2	yes	—
137	C₂	#20242163	1	2	yes	—
138	C₂	#20243508	1	1	yes	characteristic
139	C₂	#20243516	1	1	yes	characteristic
140	C₂	#20243529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243540

C₂

yes

maximal

#20241223

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#106

yes

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#486

yes

—

C₄×C₄₇

#487

yes

—

#1842

—

#1744

D₄₇

yes

characteristic

#1277

—

C₂×C₄₇

#1570

yes

—

C₂×C₄₇

#1571

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

Dic₄₇

yes

—

C₄₇

#8787

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

Dic₉₄

yes

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

188

—

C₄⋊C₈

—

188

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₄₇

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

100

C₂×C₈

—

188

—

101

C₂×C₈

—

188

—

102

C₄²

—

103

C₄²

—

104

C₄²

#10625325

yes

—

105

C₄²

D₃₇₆

yes

—

106

C₄²

Dic₁₈₈

yes

—

107

C₄²

#10625350

yes

—

108

C₂³

C₂×C₄⋊Dic₄₇

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#13093956

yes

—

115

C₂×C₄

#13093958

yes

—

116

C₂×C₄

#13093983

yes

—

117

C₂×C₄

#13094012

yes

—

118

C₂×C₄

#13094016

yes

—

119

C₂×C₄

#13094031

yes

—

120

C₂×C₄

C₂×D₃₇₆

yes

characteristic

121

C₂×C₄

C₂×#10625325

yes

—

122

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

123

C₈

—

124

C₈

—

376

—

125

C₂²

#15688989

yes

characteristic

126

C₂²

#15689150

yes

—

127

C₂²

C₂×#13093956

yes

characteristic

128

C₂²

C₂×#13093958

yes

characteristic

129

C₂²

#15689172

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#15689184

yes

—

134

C₄

#15689470

yes

—

135

C₄

#15689474

yes

—

136

C₂

#20242159

yes

—

137

C₂

#20242163

yes

—

138

C₂

#20243508

yes

characteristic

139

C₂

#20243516

yes

characteristic

140

C₂

#20243529

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8787
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₈₈

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
80. C₄₇	#8787	semidirect

H ≅ G/N

type

80. C₄₇

#8787

semidirect

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

subgroup	index
#20243540	2
#20241223	2
C₄₇×#1744	2
#8787	47

subgroup

index

#20243540

#20241223

C₄₇×#1744

#8787

Subgroups — #26109949

Conjugacy classes of subgroups

Complements

Maximal subgroups