Group #25011325 — subgroups

Subgroups — #25011325

Order 11008.

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

normal subgroups only

This group has 360 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	#19136851	C₂	1	2	yes	maximal, complemented
2	#19136845	C₂	1	2	yes	maximal, complemented
3	#19134210	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134201	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063270	—	2	4	no	—
7	C₂×D₆₈₈	—	2	4	no	—
8	#14583279	C₂²	1	1	yes	characteristic
9	#14583276	—	2	2	no	—
10	#14583276	C₂²	1	2	yes	—
11	#14583276	C₂²	1	1	yes	characteristic
12	#14583246	—	2	2	no	—
13	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₂	—	2	2	no	—
16	C₂×D₃₄₄	—	2	2	no	—
17	C₂×D₃₄₄	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	C₂×D₃₄₄	—	4	4	no	—
20	#13063325	—	2	1	no	—
21	#13063322	—	2	1	no	—
22	#13063322	—	2	2	no	—
23	#13063270	—	4	8	no	—
24	D₆₈₈	—	4	8	no	—
25	#13063247	—	4	4	no	—
26	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
27	C₄×C₈×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×D₁₇₂	—	4	1	no	—
30	C₂×D₁₇₂	—	4	2	no	—
31	C₂×D₁₇₂	—	4	2	no	—
32	C₂×D₁₇₂	—	4	4	no	—
33	C₂×Dic₈₆	—	4	1	no	—
34	C₂×C₈×C₄₃	D₈	1	4	yes	—
35	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
36	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₃	—	4	2	no	—
38	Dic₁₇₂	—	4	4	no	—
39	D₃₄₄	—	4	4	no	—
40	D₃₄₄	—	4	4	no	—
41	D₃₄₄	—	4	4	no	—
42	D₃₄₄	—	8	8	no	—
43	C₁₆×C₄₃	D₈	1	8	yes	complemented
44	C₂²×D₄₃	—	8	1	no	—
45	C₂²×D₄₃	—	8	2	no	—
46	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
48	C₂×Dic₄₃	—	8	1	no	—
49	D₁₇₂	—	8	2	no	—
50	D₁₇₂	—	8	2	no	—
51	D₁₇₂	—	8	2	no	—
52	D₁₇₂	—	8	8	no	—
53	D₁₇₂	—	8	4	no	—
54	Dic₈₆	—	8	2	no	—
55	C₈×C₄₃	—	2	4	no	—
56	C₈×C₄₃	C₂×D₈	1	4	yes	—
57	#12519	—	43	1	no	maximal, Sylow
58	C₂²×C₄₃	#493	1	1	yes	characteristic
59	C₂×D₄₃	—	16	1	no	—
60	C₂×D₄₃	—	16	2	no	—
61	C₂×D₄₃	—	16	4	no	—
62	C₂×D₄₃	—	16	2	no	—
63	C₄×C₄₃	#493	1	2	yes	—
64	C₄×C₄₃	#509	1	4	yes	—
65	Dic₄₃	—	16	1	no	—
66	#2248	—	43	2	no	—
67	#2246	—	43	2	no	—
68	#1712	—	43	1	no	—
69	#1709	—	43	1	no	—
70	#1311	D₄₃	1	1	yes	characteristic, complemented
71	C₂×C₄₃	#1712	1	1	yes	characteristic
72	C₂×C₄₃	#2250	1	2	yes	—
73	D₄₃	—	32	2	no	—
74	D₄₃	—	32	1	no	—
75	C₂×SD₃₂	—	86	4	no	—
76	C₂×D₁₆	—	86	4	no	—
77	#494	—	43	1	no	—
78	#493	—	43	1	no	—
79	#493	—	43	2	no	—
80	#493	—	86	2	no	—
81	#486	—	86	2	no	—
82	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
83	C₈²	C₂×D₄₃	1	1	yes	characteristic
84	C₄₃	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	86	2	no	—
86	C₂×D₈	—	86	2	no	—
87	C₂×D₈	—	86	2	no	—
88	C₂×D₈	—	86	2	no	—
89	C₂×D₈	—	172	4	no	—
90	C₄⋊Q₈	—	86	1	no	—
91	#127	—	86	1	no	—
92	#127	—	86	2	no	—
93	SD₃₂	—	172	8	no	—
94	D₁₆	—	172	8	no	—
95	C₂×C₁₆	D₁₇₂	1	4	yes	—
96	#102	—	172	4	no	—
97	C₄×C₈	D₁₇₂	1	2	yes	—
98	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×Q₈	—	172	1	no	—
100	C₂×D₄	—	172	1	no	—
101	C₂×D₄	—	172	2	no	—
102	C₂×D₄	—	172	2	no	—
103	C₂×D₄	—	172	4	no	—
104	Q₁₆	—	172	4	no	—
105	D₈	—	344	8	no	—
106	D₈	—	172	4	no	—
107	D₈	—	172	4	no	—
108	D₈	—	172	4	no	—
109	C₂×C₈	D₃₄₄	1	4	yes	—
110	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
111	C₄⋊C₄	—	172	2	no	—
112	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
113	C₁₆	D₃₄₄	1	8	yes	complemented
114	C₂³	—	344	1	no	—
115	C₂³	—	344	2	no	—
116	Q₈	—	344	2	no	—
117	D₄	—	344	8	no	—
118	D₄	—	344	4	no	—
119	D₄	—	344	2	no	—
120	D₄	—	344	2	no	—
121	D₄	—	344	2	no	—
122	C₂×C₄	—	344	1	no	—
123	C₂×C₄	#13063322	1	1	yes	characteristic
124	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₃₄₄	1	4	yes	—
127	C₂²	—	688	1	no	—
128	C₂²	—	688	2	no	—
129	C₂²	—	688	4	no	—
130	C₂²	—	688	2	no	—
131	C₂²	#14583276	1	1	yes	characteristic
132	C₄	—	688	1	no	—
133	C₄	#14583276	1	2	yes	—
134	C₄	#14583337	1	4	yes	—
135	C₂	—	1376	2	no	—
136	C₂	—	1376	1	no	—
137	C₂	#19134210	1	1	yes	characteristic
138	C₂	#19136861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136851

C₂

yes

maximal, complemented

#19136845

C₂

yes

maximal, complemented

#19134210

C₂

yes

maximal, characteristic, complemented

#19134201

C₂

yes

maximal, characteristic, complemented

C₄₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#13063270

—

C₂×D₆₈₈

—

#14583279

C₂²

yes

characteristic

#14583276

—

#14583276

C₂²

yes

—

#14583276

C₂²

yes

characteristic

#14583246

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

—

—

—

—

D₆₈₈

—

#13063247

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

D₈

yes

complemented

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₈

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

#509

yes

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2250

yes

—

D₄₃

—

D₄₃

—

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12519

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

172

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

172

—

D₁₆

—

172

—

C₂×C₁₆

D₁₇₂

yes

—

#102

—

172

—

C₄×C₈

D₁₇₂

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

100

C₂×D₄

—

172

—

101

C₂×D₄

—

172

—

102

C₂×D₄

—

172

—

103

C₂×D₄

—

172

—

104

Q₁₆

—

172

—

105

D₈

—

344

—

106

D₈

—

172

—

107

D₈

—

172

—

108

D₈

—

172

—

109

C₂×C₈

D₃₄₄

yes

—

110

C₂×C₈

C₂×D₁₇₂

yes

—

111

C₄⋊C₄

—

172

—

112

C₄²

C₂×D₁₇₂

yes

characteristic

113

C₁₆

D₃₄₄

yes

complemented

114

C₂³

—

344

—

115

C₂³

—

344

—

116

Q₈

—

344

—

117

D₄

—

344

—

118

D₄

—

344

—

119

D₄

—

344

—

120

D₄

—

344

—

121

D₄

—

344

—

122

C₂×C₄

—

344

—

123

C₂×C₄

#13063322

yes

characteristic

124

C₂×C₄

C₂×D₃₄₄

yes

—

125

C₈

—

126

C₈

C₂×D₃₄₄

yes

—

127

C₂²

—

688

—

128

C₂²

—

688

—

129

C₂²

—

688

—

130

C₂²

—

688

—

131

C₂²

#14583276

yes

characteristic

132

C₄

—

688

—

133

C₄

#14583276

yes

—

134

C₄

#14583337

yes

—

135

C₂

—

1376

—

136

C₂

—

1376

—

137

C₂

#19134210

yes

characteristic

138

C₂

#19136861

yes

—

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1311	C₂
socle	C₂²×C₄₃	#493
hypercenter	#1311	D₄₃
hypocenter	C₄₃	#12519
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1311

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #19136851	C₂	semidirect
2. #19136845	C₂	semidirect
3. #19134210	C₂	semidirect
4. #19134201	C₂	semidirect
5. C₄₃×#1311	C₂	semidirect
43. C₁₆×C₄₃	D₈	semidirect
70. #1311	D₄₃	semidirect
84. C₄₃	#12519	semidirect
113. C₁₆	D₃₄₄	semidirect

H ≅ G/N

type

1. #19136851

C₂

semidirect

2. #19136845

C₂

semidirect

3. #19134210

C₂

semidirect

4. #19134201

C₂

semidirect

5. C₄₃×#1311

C₂

semidirect

43. C₁₆×C₄₃

D₈

semidirect

70. #1311

D₄₃

semidirect

84. C₄₃

#12519

semidirect

113. C₁₆

D₃₄₄

semidirect

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

subgroup	index
#19136851	2
#19136845	2
#19134210	2
#19134201	2
C₄₃×#1311	2
#12519	43

subgroup

index

C₄₃×#1311

Subgroups — #25011325

Conjugacy classes of subgroups

Complements

Maximal subgroups