Group #25011366 — subgroups

Subgroups — #25011366

Order 11008.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19136848	C₂	1	2	yes	maximal
2	#19136823	C₂	1	2	yes	maximal
3	#19134213	C₂	1	1	yes	maximal, characteristic
4	#19134192	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	—	2	4	no	—
7	#14583304	—	2	1	no	—
8	#14583304	—	2	1	no	—
9	#14583304	C₂²	1	1	yes	characteristic
10	#14583297	—	2	1	no	—
11	#14583279	C₂²	1	2	yes	—
12	#14583279	C₂²	1	1	yes	characteristic
13	#14583250	—	2	1	no	—
14	#14582704	—	2	4	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	#13063325	—	2	1	no	—
21	#13063325	—	2	1	no	—
22	#13063325	—	2	2	no	—
23	Dic₃₄₄	—	4	8	no	—
24	#13063259	—	2	2	no	—
25	#13063259	—	4	2	no	—
26	#13063259	—	4	2	no	—
27	#13063257	—	4	2	no	—
28	#13063250	—	4	2	no	—
29	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₈₆	—	4	1	no	—
34	C₂×Dic₈₆	—	4	2	no	—
35	C₂×Dic₈₆	—	4	1	no	—
36	C₂×C₈×C₄₃	D₈	1	2	yes	—
37	C₂×C₈×C₄₃	D₈	1	2	yes	—
38	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
39	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₄₃	—	4	1	no	—
41	C₄⋊Dic₄₃	—	4	1	no	—
42	C₄⋊Dic₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	4	1	no	—
44	C₄⋊Dic₄₃	—	4	2	no	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	Dic₁₇₂	—	4	4	no	—
47	Dic₁₇₂	—	4	4	no	—
48	Dic₁₇₂	—	4	4	no	—
49	C₁₆×C₄₃	—	2	4	no	—
50	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₄₃	—	8	1	no	—
54	C₂×Dic₄₃	—	8	2	no	—
55	C₂×Dic₄₃	—	8	1	no	—
56	Dic₈₆	—	8	2	no	—
57	Dic₈₆	—	8	2	no	—
58	Dic₈₆	—	8	2	no	—
59	Dic₈₆	—	8	2	no	—
60	C₈×C₄₃	—	2	2	no	—
61	C₈×C₄₃	Q₃₂	1	4	yes	complemented
62	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
63	#12528	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	#493	1	1	yes	characteristic
65	C₄×C₄₃	#496	1	2	yes	—
66	C₄×C₄₃	C₂×Q₃₂	1	2	yes	—
67	C₄×C₄₃	#510	1	2	yes	—
68	Dic₄₃	—	16	1	no	—
69	Dic₄₃	—	16	2	no	—
70	Dic₄₃	—	16	1	no	—
71	#2247	—	43	2	no	—
72	#2241	—	43	2	no	—
73	#1713	—	43	1	no	—
74	#1707	—	43	1	no	—
75	#1312	D₄₃	1	1	yes	characteristic
76	C₂×C₄₃	#1721	1	1	yes	characteristic
77	C₂×C₄₃	#2247	1	1	yes	characteristic
78	C₂×C₄₃	#2251	1	1	yes	characteristic
79	C₂×Q₃₂	—	86	4	no	—
80	#500	—	43	1	no	—
81	#500	—	86	1	no	—
82	#500	—	86	1	no	—
83	#498	—	86	1	no	—
84	#494	—	43	1	no	—
85	#494	—	43	2	no	—
86	#487	—	86	1	no	—
87	#358	—	86	4	no	—
88	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
89	C₈²	C₂×D₄₃	1	1	yes	characteristic
90	C₄₃	#12528	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	86	2	no	—
92	C₂×Q₁₆	—	86	2	no	—
93	C₂×Q₁₆	—	86	2	no	—
94	C₄⋊Q₈	—	86	1	no	—
95	C₄⋊Q₈	—	86	1	no	—
96	C₄⋊Q₈	—	86	2	no	—
97	Q₃₂	—	172	8	no	—
98	C₂×C₁₆	D₁₇₂	1	4	yes	—
99	#107	—	86	2	no	—
100	#107	—	172	2	no	—
101	#107	—	172	2	no	—
102	#106	—	172	2	no	—
103	#103	—	172	2	no	—
104	C₄×C₈	D₁₇₂	1	1	yes	characteristic
105	C₄×C₈	D₁₇₂	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
107	C₂×Q₈	—	172	1	no	—
108	C₂×Q₈	—	172	2	no	—
109	C₂×Q₈	—	172	1	no	—
110	Q₁₆	—	172	4	no	—
111	Q₁₆	—	172	4	no	—
112	Q₁₆	—	172	4	no	—
113	C₂×C₈	D₃₄₄	1	2	yes	—
114	C₂×C₈	D₃₄₄	1	2	yes	—
115	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
116	C₄⋊C₄	—	172	1	no	—
117	C₄⋊C₄	—	172	1	no	—
118	C₄⋊C₄	—	172	1	no	—
119	C₄⋊C₄	—	172	1	no	—
120	C₄⋊C₄	—	172	2	no	—
121	C₄⋊C₄	—	172	2	no	—
122	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	344	2	no	—
125	Q₈	—	344	2	no	—
126	Q₈	—	344	2	no	—
127	Q₈	—	344	2	no	—
128	C₂×C₄	—	344	1	no	—
129	C₂×C₄	—	344	2	no	—
130	C₂×C₄	—	344	1	no	—
131	C₂×C₄	#13063322	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₃₄₄	1	4	yes	complemented
136	C₈	#13063353	1	4	yes	—
137	C₂²	#14583276	1	1	yes	characteristic
138	C₄	—	688	1	no	—
139	C₄	—	688	2	no	—
140	C₄	—	688	1	no	—
141	C₄	#14583288	1	2	yes	—
142	C₄	C₂×Dic₃₄₄	1	2	yes	—
143	C₄	#14583342	1	2	yes	—
144	C₂	#19134261	1	1	yes	characteristic
145	C₂	#19136848	1	1	yes	characteristic
146	C₂	#19136866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136848

C₂

yes

maximal

#19136823

C₂

yes

maximal

#19134213

C₂

yes

maximal, characteristic

#19134192

C₂

yes

maximal, characteristic

C₄₃×#1312

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

#14583304

—

#14583304

—

#14583304

C₂²

yes

characteristic

#14583297

—

#14583279

C₂²

yes

—

#14583279

C₂²

yes

characteristic

#14583250

—

#14582704

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

#13063325

—

#13063325

—

Dic₃₄₄

—

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

Q₃₂

yes

complemented

C₈×C₄₃

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

C₂×Q₃₂

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12528

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

100

#107

—

172

—

101

#107

—

172

—

102

#106

—

172

—

103

#103

—

172

—

104

C₄×C₈

D₁₇₂

yes

characteristic

105

C₄×C₈

D₁₇₂

yes

characteristic

106

C₄×C₈

C₂²×D₄₃

yes

characteristic

107

C₂×Q₈

—

172

—

108

C₂×Q₈

—

172

—

109

C₂×Q₈

—

172

—

110

Q₁₆

—

172

—

111

Q₁₆

—

172

—

112

Q₁₆

—

172

—

113

C₂×C₈

D₃₄₄

yes

—

114

C₂×C₈

D₃₄₄

yes

—

115

C₂×C₈

C₂×D₁₇₂

yes

—

116

C₄⋊C₄

—

172

—

117

C₄⋊C₄

—

172

—

118

C₄⋊C₄

—

172

—

119

C₄⋊C₄

—

172

—

120

C₄⋊C₄

—

172

—

121

C₄⋊C₄

—

172

—

122

C₄²

C₂×D₁₇₂

yes

characteristic

123

C₁₆

—

124

Q₈

—

344

—

125

Q₈

—

344

—

126

Q₈

—

344

—

127

Q₈

—

344

—

128

C₂×C₄

—

344

—

129

C₂×C₄

—

344

—

130

C₂×C₄

—

344

—

131

C₂×C₄

#13063322

yes

characteristic

132

C₂×C₄

C₂×D₃₄₄

yes

characteristic

133

C₂×C₄

C₂×D₃₄₄

yes

characteristic

134

C₈

—

135

C₈

Dic₃₄₄

yes

complemented

136

C₈

#13063353

yes

—

137

C₂²

#14583276

yes

characteristic

138

C₄

—

688

—

139

C₄

—

688

—

140

C₄

—

688

—

141

C₄

#14583288

yes

—

142

C₄

C₂×Dic₃₄₄

yes

—

143

C₄

#14583342

yes

—

144

C₂

#19134261

yes

characteristic

145

C₂

#19136848

yes

characteristic

146

C₂

#19136866

yes

characteristic

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1312	C₂
socle, prime radical	C₂²×C₄₃	#493
hypercenter	#1312	D₄₃
hypocenter	C₄₃	#12528
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1312

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
61. C₈×C₄₃	Q₃₂	semidirect
90. C₄₃	#12528	semidirect
135. C₈	Dic₃₄₄	semidirect

H ≅ G/N

type

61. C₈×C₄₃

Q₃₂

semidirect

90. C₄₃

#12528

semidirect

135. C₈

Dic₃₄₄

semidirect

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

subgroup	index
#19136848	2
#19136823	2
#19134213	2
#19134192	2
C₄₃×#1312	2
#12528	43

subgroup

index

C₄₃×#1312

Subgroups — #25011366

Conjugacy classes of subgroups

Complements

Maximal subgroups