Group #25011250 — subgroups

Subgroups — #25011250

Order 11008.

306 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 99 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	#19136870	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136845	C₂	1	1	yes	maximal, characteristic
3	#19136819	C₂	1	2	yes	maximal, complemented
4	#19134187	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₈₈	—	2	2	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583276	C₂²	1	2	yes	—
9	#14583272	—	2	2	no	—
10	#14583246	—	2	2	no	—
11	#14582725	—	2	2	no	—
12	#14582701	—	2	4	no	—
13	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₈²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#10621873	—	4	4	no	—
17	C₂×D₃₄₄	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	#13063325	—	2	2	no	—
20	#13063322	—	2	2	no	—
21	D₆₈₈	—	4	4	no	—
22	#13063259	—	2	2	no	—
23	#13063259	—	2	2	no	—
24	#13063247	—	4	4	no	—
25	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
27	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×D₁₇₂	—	4	2	no	—
31	C₂×D₁₇₂	—	4	2	no	—
32	C₂×D₁₇₂	—	4	2	no	—
33	C₂×Dic₈₆	—	4	2	no	—
34	C₂×C₈×C₄₃	D₈	1	2	yes	—
35	C₂×C₈×C₄₃	D₈	1	2	yes	—
36	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₃	—	4	2	no	—
40	C₄⋊Dic₄₃	—	4	2	no	—
41	D₃₄₄	—	4	4	no	—
42	D₃₄₄	—	4	4	no	—
43	#10621873	—	8	8	no	—
44	C₁₆×C₄₃	—	2	2	no	—
45	C₁₆×C₄₃	SD₁₆	1	4	yes	complemented
46	C₂²×D₄₃	—	8	2	no	—
47	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₄₃	—	8	2	no	—
51	D₁₇₂	—	8	2	no	—
52	D₁₇₂	—	8	2	no	—
53	D₁₇₂	—	8	4	no	—
54	D₁₇₂	—	8	4	no	—
55	Dic₈₆	—	8	4	no	—
56	C₈×C₄₃	—	2	2	no	—
57	C₈×C₄₃	D₁₆	1	4	yes	complemented
58	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
59	C₈×C₄₃	#135	1	2	yes	—
60	#12503	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	#493	1	1	yes	characteristic
62	C₂×D₄₃	—	16	2	no	—
63	C₂×D₄₃	—	16	2	no	—
64	C₂×D₄₃	—	16	2	no	—
65	C₄×C₄₃	#492	1	1	yes	characteristic
66	C₄×C₄₃	#495	1	1	yes	characteristic
67	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
68	C₄×C₄₃	#508	1	2	yes	—
69	Dic₄₃	—	16	2	no	—
70	#2252	—	43	1	no	—
71	#2246	—	43	1	no	—
72	#2240	—	43	2	no	—
73	#1706	—	43	2	no	—
74	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
75	C₂×C₄₃	#1706	1	1	yes	characteristic
76	C₂×C₄₃	#2246	1	1	yes	characteristic
77	C₂×C₄₃	#2249	1	1	yes	characteristic
78	D₄₃	—	32	2	no	—
79	C₂×D₁₆	—	86	2	no	—
80	#500	—	43	2	no	—
81	#493	—	43	2	no	—
82	#492	—	86	2	no	—
83	#486	—	86	2	no	—
84	#366	—	86	2	no	—
85	#357	—	86	4	no	—
86	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
87	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
88	C₈²	C₂×D₄₃	1	1	yes	characteristic
89	C₄₃	#12503	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	172	4	no	—
91	C₂×D₈	—	86	2	no	—
92	C₂×D₈	—	86	2	no	—
93	C₄⋊Q₈	—	86	2	no	—
94	#127	—	86	2	no	—
95	D₁₆	—	172	4	no	—
96	C₂×C₁₆	D₁₇₂	1	2	yes	—
97	C₂×C₁₆	D₁₇₂	1	2	yes	—
98	#107	—	86	2	no	—
99	#107	—	86	2	no	—
100	#102	—	172	4	no	—
101	C₄×C₈	D₁₇₂	1	1	yes	characteristic
102	C₄×C₈	D₁₇₂	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×Q₈	—	172	2	no	—
105	C₂×D₄	—	172	2	no	—
106	C₂×D₄	—	172	2	no	—
107	C₂×D₄	—	172	2	no	—
108	SD₁₆	—	344	8	no	—
109	D₈	—	172	4	no	—
110	D₈	—	172	4	no	—
111	C₂×C₈	D₃₄₄	1	2	yes	—
112	C₂×C₈	D₃₄₄	1	2	yes	—
113	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
114	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
115	C₄⋊C₄	—	172	2	no	—
116	C₄⋊C₄	—	172	2	no	—
117	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	#10621873	1	4	yes	complemented
120	C₂³	—	344	2	no	—
121	Q₈	—	344	4	no	—
122	D₄	—	344	4	no	—
123	D₄	—	344	4	no	—
124	D₄	—	344	2	no	—
125	D₄	—	344	2	no	—
126	C₂×C₄	—	344	2	no	—
127	C₂×C₄	#13063322	1	1	yes	characteristic
128	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
129	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	D₆₈₈	1	4	yes	complemented
132	C₈	C₂×#10621873	1	2	yes	—
133	C₈	#13063347	1	2	yes	—
134	C₂²	—	688	2	no	—
135	C₂²	—	688	2	no	—
136	C₂²	—	688	2	no	—
137	C₂²	#14583276	1	1	yes	characteristic
138	C₄	—	688	2	no	—
139	C₄	#14583272	1	1	yes	characteristic
140	C₄	#14583283	1	1	yes	characteristic
141	C₄	C₂×D₆₈₈	1	2	yes	—
142	C₄	#14583331	1	2	yes	—
143	C₂	—	1376	2	no	—
144	C₂	#19134187	1	1	yes	characteristic
145	C₂	#19136845	1	1	yes	characteristic
146	C₂	#19136856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136870

C₂

yes

maximal, characteristic, complemented

#19136845

C₂

yes

maximal, characteristic

#19136819

C₂

yes

maximal, complemented

#19134187

C₂

yes

maximal, complemented

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₆₈₈

—

#14583304

C₂²

yes

—

#14583276

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×#10621873

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063325

—

#13063322

—

D₆₈₈

—

#13063259

—

#13063259

—

#13063247

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

complemented

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

D₁₆

yes

complemented

C₈×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×D₁₆

yes

—

C₄×C₄₃

#508

yes

—

Dic₄₃

—

—

—

—

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×D₁₆

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12503

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

172

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

#107

—

100

#102

—

172

—

101

C₄×C₈

D₁₇₂

yes

characteristic

102

C₄×C₈

D₁₇₂

yes

characteristic

103

C₄×C₈

C₂²×D₄₃

yes

characteristic

104

C₂×Q₈

—

172

—

105

C₂×D₄

—

172

—

106

C₂×D₄

—

172

—

107

C₂×D₄

—

172

—

108

SD₁₆

—

344

—

109

D₈

—

172

—

110

D₈

—

172

—

111

C₂×C₈

D₃₄₄

yes

—

112

C₂×C₈

D₃₄₄

yes

—

113

C₂×C₈

C₂×D₁₇₂

yes

characteristic

114

C₂×C₈

C₂×D₁₇₂

yes

characteristic

115

C₄⋊C₄

—

172

—

116

C₄⋊C₄

—

172

—

117

C₄²

C₂×D₁₇₂

yes

characteristic

118

C₁₆

—

119

C₁₆

#10621873

yes

complemented

120

C₂³

—

344

—

121

Q₈

—

344

—

122

D₄

—

344

—

123

D₄

—

344

—

124

D₄

—

344

—

125

D₄

—

344

—

126

C₂×C₄

—

344

—

127

C₂×C₄

#13063322

yes

characteristic

128

C₂×C₄

C₂×D₃₄₄

yes

characteristic

129

C₂×C₄

C₂×D₃₄₄

yes

characteristic

130

C₈

—

131

C₈

D₆₈₈

yes

complemented

132

C₈

C₂×#10621873

yes

—

133

C₈

#13063347

yes

—

134

C₂²

—

688

—

135

C₂²

—

688

—

136

C₂²

—

688

—

137

C₂²

#14583276

yes

characteristic

138

C₄

—

688

—

139

C₄

#14583272

yes

characteristic

140

C₄

#14583283

yes

characteristic

141

C₄

C₂×D₆₈₈

yes

—

142

C₄

#14583331

yes

—

143

C₂

—

1376

—

144

C₂

#19134187

yes

characteristic

145

C₂

#19136845

yes

characteristic

146

C₂

#19136856

yes

characteristic

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#493
prime radical	#19136845	C₂
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12503
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₈×C₁₆×C₄₃

C₂

C₂²×C₄₃

C₂

C₈×C₁₆

D₄₃

hypocenter

C₄₃

#12503

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #19136870	C₂	semidirect
3. #19136819	C₂	semidirect
4. #19134187	C₂	semidirect
5. C₈×C₁₆×C₄₃	C₂	semidirect
45. C₁₆×C₄₃	SD₁₆	semidirect
57. C₈×C₄₃	D₁₆	semidirect
74. C₈×C₁₆	D₄₃	semidirect
89. C₄₃	#12503	semidirect
119. C₁₆	#10621873	semidirect
131. C₈	D₆₈₈	semidirect

H ≅ G/N

type

1. #19136870

C₂

semidirect

3. #19136819

C₂

semidirect

4. #19134187

C₂

semidirect

5. C₈×C₁₆×C₄₃

C₂

semidirect

45. C₁₆×C₄₃

SD₁₆

semidirect

57. C₈×C₄₃

D₁₆

semidirect

74. C₈×C₁₆

D₄₃

semidirect

89. C₄₃

#12503

semidirect

119. C₁₆

#10621873

semidirect

131. C₈

D₆₈₈

semidirect

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

subgroup	index
#19136870	2
#19136845	2
#19136819	2
#19134187	2
C₈×C₁₆×C₄₃	2
#12503	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011250

Conjugacy classes of subgroups

Complements

Maximal subgroups