Group #25011342 — subgroups

Subgroups — #25011342

Order 11008.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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	#19136875	C₂	1	2	yes	maximal
2	#19136868	C₂	1	2	yes	maximal
3	#19134205	C₂	1	1	yes	maximal, characteristic
4	#19132512	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1713	C₂	1	1	yes	maximal, characteristic
6	#14583304	C₂²	1	1	yes	characteristic
7	#14583304	C₂²	1	1	yes	characteristic
8	#14583250	—	2	1	no	—
9	#14583250	—	2	1	no	—
10	#14582726	—	2	4	no	—
11	#14582724	—	2	4	no	—
12	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
13	C₄₃×#500	—	2	1	no	—
14	C₄₃×#500	C₂²	1	2	yes	—
15	C₄₃×#494	—	2	1	no	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	#13063325	—	2	1	no	—
18	#13063325	—	2	1	no	—
19	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
20	#13063259	—	2	2	no	—
21	#13063259	—	2	2	no	—
22	#13063250	—	4	2	no	—
23	#13063250	—	4	2	no	—
24	C₂×Q₁₆×C₄₃	—	4	2	no	—
25	C₄⋊Q₈×C₄₃	—	2	2	no	—
26	#107×C₄₃	—	2	2	no	—
27	#107×C₄₃	—	2	2	no	—
28	#107×C₄₃	—	4	2	no	—
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×Q₈×C₄₃	—	4	2	no	—
33	C₂×Dic₈₆	—	4	1	no	—
34	C₂×Dic₈₆	—	4	1	no	—
35	Q₁₆×C₄₃	—	8	4	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	C₄⋊C₄×C₄₃	—	4	2	no	—
40	C₄⋊C₄×C₄₃	—	4	2	no	—
41	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	4	1	no	—
44	C₄⋊Dic₄₃	—	4	1	no	—
45	C₄⋊Dic₄₃	—	4	1	no	—
46	C₄₃⋊C₁₆	Q₁₆	1	8	yes	complemented
47	Q₈×C₄₃	—	8	4	no	—
48	C₂×C₄×C₄₃	—	8	2	no	—
49	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₄₃	—	8	1	no	—
53	C₂×Dic₄₃	—	8	1	no	—
54	Dic₈₆	—	8	2	no	—
55	Dic₈₆	—	8	2	no	—
56	C₈×C₄₃	—	2	2	no	—
57	C₈×C₄₃	—	2	2	no	—
58	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
59	#12524	—	43	1	no	maximal, Sylow
60	C₂²×C₄₃	#493	1	1	yes	characteristic
61	C₄×C₄₃	—	16	2	no	—
62	C₄×C₄₃	#494	1	2	yes	—
63	C₄×C₄₃	#509	1	2	yes	—
64	C₄×C₄₃	#510	1	2	yes	—
65	Dic₄₃	—	16	1	no	—
66	Dic₄₃	—	16	1	no	—
67	#2254	—	43	2	no	—
68	#2252	—	43	2	no	—
69	#1713	D₄₃	1	1	yes	characteristic
70	#1710	—	43	1	no	—
71	#1311	—	43	1	no	—
72	C₂×C₄₃	#1713	1	1	yes	characteristic
73	C₂×C₄₃	#2250	1	1	yes	characteristic
74	C₂×C₄₃	#2251	1	1	yes	characteristic
75	#500	—	2	1	no	—
76	#500	—	43	1	no	—
77	#500	—	43	1	no	—
78	#500	C₂×D₄₃	1	2	yes	—
79	#494	—	2	1	no	—
80	#487	—	86	1	no	—
81	#487	—	86	1	no	—
82	#367	—	86	4	no	—
83	#366	—	86	4	no	—
84	C₄×C₁₆	—	43	2	no	—
85	C₈²	C₂×D₄₃	1	1	yes	characteristic
86	C₄₃	#12524	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	4	2	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄⋊Q₈	—	86	1	no	—
90	C₄⋊Q₈	—	86	1	no	—
91	C₂×C₁₆	—	43	4	no	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	4	2	no	—
95	#107	—	86	2	no	—
96	#107	—	86	2	no	—
97	#103	—	172	2	no	—
98	#103	—	172	2	no	—
99	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	172	1	no	—
104	C₂×Q₈	—	172	1	no	—
105	Q₁₆	—	8	4	no	—
106	C₂×C₈	#10621882	1	2	yes	—
107	C₂×C₈	#10621882	1	2	yes	—
108	C₂×C₈	D₄×D₄₃	1	2	yes	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	172	1	no	—
112	C₄⋊C₄	—	172	1	no	—
113	C₄⋊C₄	—	172	1	no	—
114	C₄⋊C₄	—	172	1	no	—
115	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
116	C₁₆	—	43	8	no	—
117	Q₈	—	8	4	no	—
118	Q₈	—	344	2	no	—
119	Q₈	—	344	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	344	1	no	—
122	C₂×C₄	—	344	1	no	—
123	C₂×C₄	#13063321	1	1	yes	characteristic
124	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
125	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₄₃	1	4	yes	—
129	C₂²	#14583275	1	1	yes	characteristic
130	C₄	—	16	2	no	—
131	C₄	—	688	1	no	—
132	C₄	—	688	1	no	—
133	C₄	#14583277	1	2	yes	—
134	C₄	#14583335	1	2	yes	—
135	C₄	#14583340	1	2	yes	—
136	C₂	#19134211	1	1	yes	characteristic
137	C₂	#19136859	1	1	yes	characteristic
138	C₂	#19136864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136875

C₂

yes

maximal

#19136868

C₂

yes

maximal

#19134205

C₂

yes

maximal, characteristic

#19132512

C₂

yes

maximal, characteristic

C₄₃×#1713

C₂

yes

maximal, characteristic

#14583304

C₂²

yes

characteristic

#14583304

C₂²

yes

—

—

—

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#500

—

C₄₃×#500

C₂²

yes

—

C₄₃×#494

—

C₈²×C₄₃

C₂²

yes

characteristic

#13063325

—

#13063325

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄₃⋊C₁₆

Q₁₆

yes

complemented

Q₈×C₄₃

—

C₂×C₄×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

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₁₆

yes

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

#494

yes

—

C₄×C₄₃

#509

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

Dic₄₃

—

#2254

—

#2252

—

#1713

D₄₃

yes

characteristic

#1710

—

#1311

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

C₂×D₄₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12524

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

—

—

172

—

#103

—

172

—

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

100

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

101

C₄×C₈

C₂²×D₄₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

172

—

104

C₂×Q₈

—

172

—

105

Q₁₆

—

106

C₂×C₈

#10621882

yes

—

107

C₂×C₈

#10621882

yes

—

108

C₂×C₈

D₄×D₄₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

172

—

112

C₄⋊C₄

—

172

—

113

C₄⋊C₄

—

172

—

114

C₄⋊C₄

—

172

—

115

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

116

C₁₆

—

117

Q₈

—

118

Q₈

—

344

—

119

Q₈

—

344

—

120

C₂×C₄

—

121

C₂×C₄

—

344

—

122

C₂×C₄

—

344

—

123

C₂×C₄

#13063321

yes

characteristic

124

C₂×C₄

C₂×#10621882

yes

characteristic

125

C₂×C₄

C₂×#10621882

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₄₃

yes

—

129

C₂²

#14583275

yes

characteristic

130

C₄

—

131

C₄

—

688

—

132

C₄

—

688

—

133

C₄

#14583277

yes

—

134

C₄

#14583335

yes

—

135

C₄

#14583340

yes

—

136

C₂

#19134211

yes

characteristic

137

C₂

#19136859

yes

characteristic

138

C₂

#19136864

yes

characteristic

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1713	C₂
socle, prime radical	C₂²×C₄₃	#493
hypercenter	#1713	D₄₃
hypocenter	C₄₃	#12524
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1713

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
46. C₄₃⋊C₁₆	Q₁₆	semidirect
86. C₄₃	#12524	semidirect

H ≅ G/N

type

46. C₄₃⋊C₁₆

Q₁₆

semidirect

86. C₄₃

#12524

semidirect

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

subgroup	index
#19136875	2
#19136868	2
#19134205	2
#19132512	2
C₄₃×#1713	2
#12524	43

subgroup

index

C₄₃×#1713

Subgroups — #25011342

Conjugacy classes of subgroups

Complements

Maximal subgroups