Group #25011405 — subgroups

Subgroups — #25011405

Order 11008.

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

normal subgroups only

This group has 250 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	#19136877	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	#14583304	—	2	1	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583304	C₂²	1	1	yes	characteristic
9	#14583297	—	2	1	no	—
10	#14583279	—	2	1	no	—
11	#14583279	C₂²	1	1	yes	characteristic
12	#14583250	—	2	1	no	—
13	#14582727	—	2	4	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	—	4	2	no	—
19	#13063325	—	2	1	no	—
20	#13063325	—	2	1	no	—
21	#13063325	—	2	2	no	—
22	#13063259	—	2	2	no	—
23	#13063259	—	2	2	no	—
24	#13063259	—	2	2	no	—
25	#13063259	—	4	2	no	—
26	#13063257	—	4	2	no	—
27	#13063250	—	4	2	no	—
28	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₈₆	—	4	1	no	—
33	C₂×Dic₈₆	—	4	1	no	—
34	C₂×Dic₈₆	—	4	2	no	—
35	C₂×C₈×C₄₃	D₈	1	2	yes	—
36	C₂×C₈×C₄₃	D₈	1	2	yes	—
37	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
38	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₃	—	4	2	no	—
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	Dic₁₇₂	—	4	4	no	—
46	Dic₁₇₂	—	8	4	no	—
47	C₁₆×C₄₃	—	2	4	no	—
48	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₄₃	—	8	1	no	—
52	C₂×Dic₄₃	—	8	2	no	—
53	C₂×Dic₄₃	—	8	1	no	—
54	Dic₈₆	—	8	2	no	—
55	Dic₈₆	—	8	4	no	—
56	Dic₈₆	—	8	2	no	—
57	C₈×C₄₃	—	2	2	no	—
58	C₈×C₄₃	SD₃₂	1	4	yes	—
59	C₈×C₄₃	#137	1	4	yes	—
60	#12535	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	#493	1	1	yes	characteristic
62	C₄×C₄₃	#497	1	2	yes	—
63	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
64	C₄×C₄₃	#510	1	2	yes	—
65	Dic₄₃	—	16	1	no	—
66	Dic₄₃	—	16	2	no	—
67	Dic₄₃	—	16	1	no	—
68	#2254	—	43	2	no	—
69	#2241	—	43	2	no	—
70	#1713	—	43	1	no	—
71	#1707	—	43	1	no	—
72	#1312	D₄₃	1	1	yes	characteristic
73	C₂×C₄₃	#1722	1	1	yes	characteristic
74	C₂×C₄₃	#2248	1	1	yes	characteristic
75	C₂×C₄₃	#2251	1	1	yes	characteristic
76	#500	—	43	1	no	—
77	#500	—	43	2	no	—
78	#500	—	86	1	no	—
79	#498	—	86	1	no	—
80	#494	—	43	1	no	—
81	#494	—	86	1	no	—
82	#487	—	86	1	no	—
83	#367	—	86	4	no	—
84	#358	—	86	4	no	—
85	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
86	C₈²	C₂×D₄₃	1	1	yes	characteristic
87	C₄₃	#12535	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	86	2	no	—
89	C₂×Q₁₆	—	172	2	no	—
90	C₄⋊Q₈	—	86	1	no	—
91	C₄⋊Q₈	—	86	1	no	—
92	C₄⋊Q₈	—	86	2	no	—
93	C₂×C₁₆	D₁₇₂	1	4	yes	—
94	#107	—	86	2	no	—
95	#107	—	86	2	no	—
96	#107	—	86	2	no	—
97	#107	—	172	2	no	—
98	#106	—	172	2	no	—
99	#103	—	172	2	no	—
100	C₄×C₈	D₁₇₂	1	1	yes	characteristic
101	C₄×C₈	D₁₇₂	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
103	C₂×Q₈	—	172	1	no	—
104	C₂×Q₈	—	172	1	no	—
105	C₂×Q₈	—	172	2	no	—
106	Q₁₆	—	344	4	no	—
107	Q₁₆	—	172	4	no	—
108	C₂×C₈	D₃₄₄	1	2	yes	—
109	C₂×C₈	D₃₄₄	1	2	yes	—
110	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
111	C₄⋊C₄	—	172	2	no	—
112	C₄⋊C₄	—	172	1	no	—
113	C₄⋊C₄	—	172	1	no	—
114	C₄⋊C₄	—	172	1	no	—
115	C₄⋊C₄	—	172	1	no	—
116	C₄⋊C₄	—	172	2	no	—
117	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	Q₈	—	344	4	no	—
120	Q₈	—	344	2	no	—
121	Q₈	—	344	2	no	—
122	C₂×C₄	—	344	1	no	—
123	C₂×C₄	—	344	2	no	—
124	C₂×C₄	—	344	1	no	—
125	C₂×C₄	#13063322	1	1	yes	characteristic
126	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
127	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	#13063270	1	4	yes	—
130	C₈	#13063358	1	4	yes	—
131	C₂²	#14583276	1	1	yes	characteristic
132	C₄	—	688	1	no	—
133	C₄	—	688	2	no	—
134	C₄	—	688	1	no	—
135	C₄	#14583293	1	2	yes	—
136	C₄	C₂×#13063270	1	2	yes	—
137	C₄	#14583342	1	2	yes	—
138	C₂	#19134266	1	1	yes	characteristic
139	C₂	#19136851	1	1	yes	characteristic
140	C₂	#19136866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136877

C₂

yes

maximal

#19136823

C₂

yes

maximal

#19134213

C₂

yes

maximal, characteristic

#19134192

C₂

yes

maximal, characteristic

C₄₃×#1312

C₂

yes

maximal, characteristic

#14583304

—

#14583304

C₂²

yes

—

#14583304

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

—

—

—

—

—

—

—

—

—

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

—

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

—

C₈×C₄₃

—

C₈×C₄₃

SD₃₂

yes

—

C₈×C₄₃

#137

yes

—

—

C₂²×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12535

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₁₇₂

yes

—

—

—

—

—

172

—

#106

—

172

—

#103

—

172

—

100

C₄×C₈

D₁₇₂

yes

characteristic

101

C₄×C₈

D₁₇₂

yes

characteristic

102

C₄×C₈

C₂²×D₄₃

yes

characteristic

103

C₂×Q₈

—

172

—

104

C₂×Q₈

—

172

—

105

C₂×Q₈

—

172

—

106

Q₁₆

—

344

—

107

Q₁₆

—

172

—

108

C₂×C₈

D₃₄₄

yes

—

109

C₂×C₈

D₃₄₄

yes

—

110

C₂×C₈

C₂×D₁₇₂

yes

—

111

C₄⋊C₄

—

172

—

112

C₄⋊C₄

—

172

—

113

C₄⋊C₄

—

172

—

114

C₄⋊C₄

—

172

—

115

C₄⋊C₄

—

172

—

116

C₄⋊C₄

—

172

—

117

C₄²

C₂×D₁₇₂

yes

characteristic

118

C₁₆

—

119

Q₈

—

344

—

120

Q₈

—

344

—

121

Q₈

—

344

—

122

C₂×C₄

—

344

—

123

C₂×C₄

—

344

—

124

C₂×C₄

—

344

—

125

C₂×C₄

#13063322

yes

characteristic

126

C₂×C₄

C₂×D₃₄₄

yes

characteristic

127

C₂×C₄

C₂×D₃₄₄

yes

characteristic

128

C₈

—

129

C₈

#13063270

yes

—

130

C₈

#13063358

yes

—

131

C₂²

#14583276

yes

characteristic

132

C₄

—

688

—

133

C₄

—

688

—

134

C₄

—

688

—

135

C₄

#14583293

yes

—

136

C₄

C₂×#13063270

yes

—

137

C₄

#14583342

yes

—

138

C₂

#19134266

yes

characteristic

139

C₂

#19136851

yes

characteristic

140

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	#12535
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
87. C₄₃	#12535	semidirect

H ≅ G/N

type

87. C₄₃

#12535

semidirect

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

subgroup	index
#19136877	2
#19136823	2
#19134213	2
#19134192	2
C₄₃×#1312	2
#12535	43

subgroup

index

C₄₃×#1312

Subgroups — #25011405

Conjugacy classes of subgroups

Complements

Maximal subgroups