Group #25011331 — subgroups

Subgroups — #25011331

Order 11008.

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

normal subgroups only

This group has 316 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	#19136850	C₂	1	2	yes	maximal
2	#19136847	C₂	1	2	yes	maximal, complemented
3	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063271	—	2	4	no	—
7	C₂×#13063269	—	2	4	no	—
8	#14583304	—	2	2	no	—
9	#14583279	C₂²	1	2	yes	—
10	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
11	C₄₃×#494	C₂²	1	1	yes	characteristic
12	C₄₃×#493	C₂²	1	1	yes	characteristic
13	C₄₃×#486	—	2	2	no	—
14	C₈²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₂	—	2	2	no	—
16	C₂×Dic₁₇₂	—	2	2	no	—
17	#13063325	—	2	2	no	—
18	#13063271	—	4	8	no	—
19	#13063269	—	4	8	no	—
20	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
21	#13063259	—	4	4	no	—
22	C₂×Q₁₆×C₄₃	—	2	2	no	—
23	C₂×D₈×C₄₃	—	2	2	no	—
24	C₄⋊Q₈×C₄₃	—	2	1	no	—
25	#127×C₄₃	—	2	1	no	—
26	#102×C₄₃	—	4	4	no	—
27	C₄×C₈×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₃	—	4	1	no	—
30	C₂×D₄×C₄₃	—	4	1	no	—
31	C₂×D₄×C₄₃	—	4	2	no	—
32	C₂×Dic₈₆	—	4	2	no	—
33	Q₁₆×C₄₃	—	4	4	no	—
34	D₈×C₄₃	—	4	4	no	—
35	C₂×C₈×C₄₃	D₈	1	4	yes	—
36	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₄₃	—	4	2	no	—
38	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₃	—	4	4	no	—
40	Dic₁₇₂	—	4	4	no	—
41	Dic₁₇₂	—	4	4	no	—
42	C₄₃⋊C₁₆	D₈	1	8	yes	—
43	C₂³×C₄₃	—	8	1	no	—
44	Q₈×C₄₃	—	8	2	no	—
45	D₄×C₄₃	—	8	2	no	—
46	D₄×C₄₃	—	8	4	no	—
47	C₂×C₄×C₄₃	—	8	1	no	—
48	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
50	C₂×Dic₄₃	—	8	2	no	—
51	Dic₈₆	—	8	2	no	—
52	Dic₈₆	—	8	2	no	—
53	C₈×C₄₃	—	2	4	no	—
54	C₈×C₄₃	C₂×D₈	1	4	yes	—
55	#12520	—	43	1	no	maximal, Sylow
56	C₂²×C₄₃	—	16	1	no	—
57	C₂²×C₄₃	—	16	2	no	—
58	C₂²×C₄₃	#493	1	1	yes	characteristic
59	C₄×C₄₃	—	16	1	no	—
60	C₄×C₄₃	#493	1	2	yes	—
61	C₄×C₄₃	#510	1	4	yes	—
62	Dic₄₃	—	16	2	no	—
63	#2248	—	43	2	no	—
64	#2247	—	43	2	no	—
65	#1713	—	43	1	no	—
66	#1709	D₄₃	1	1	yes	characteristic
67	#1311	—	43	1	no	—
68	C₂×C₄₃	—	32	1	no	—
69	C₂×C₄₃	#1712	1	1	yes	characteristic
70	C₂×C₄₃	#2251	1	2	yes	—
71	C₂×Q₃₂	—	86	4	no	—
72	C₂×SD₃₂	—	86	4	no	—
73	#500	—	86	2	no	—
74	#494	—	43	2	no	—
75	#494	C₂×D₄₃	1	1	yes	characteristic
76	#493	C₂×D₄₃	1	1	yes	characteristic
77	#486	—	2	2	no	—
78	C₄×C₁₆	—	43	2	no	—
79	C₈²	C₂×D₄₃	1	1	yes	characteristic
80	C₄₃	#12520	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	86	2	no	—
83	C₂×Q₁₆	—	86	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	86	2	no	—
87	#127	—	2	1	no	—
88	Q₃₂	—	172	8	no	—
89	SD₃₂	—	172	8	no	—
90	C₂×C₁₆	—	43	4	no	—
91	#107	—	172	4	no	—
92	#102	—	4	4	no	—
93	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
94	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	172	2	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	172	4	no	—
101	Q₁₆	—	172	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#10621882	1	4	yes	—
104	C₂×C₈	D₄×D₄₃	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	172	4	no	—
107	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
108	C₁₆	—	43	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	344	2	no	—
112	Q₈	—	344	2	no	—
113	D₄	—	8	4	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	344	2	no	—
117	C₂×C₄	#13063321	1	1	yes	characteristic
118	C₂×C₄	C₂×#10621882	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₄₃	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#14583275	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	688	2	no	—
126	C₄	#14583274	1	2	yes	—
127	C₄	#14583339	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#19134209	1	1	yes	characteristic
130	C₂	#19136867	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136850

C₂

yes

maximal

#19136847

C₂

yes

maximal, complemented

#19134213

C₂

yes

maximal, characteristic, complemented

#19132512

C₂

yes

maximal, characteristic, complemented

C₄₃×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#494

C₂²

yes

characteristic

C₄₃×#493

C₂²

yes

characteristic

C₄₃×#486

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

#13063271

—

#13063269

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063259

—

C₂×Q₁₆×C₄₃

—

C₂×D₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

#127×C₄₃

—

#102×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₄₃⋊C₁₆

D₈

yes

—

C₂³×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₈

yes

—

#12520

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#493

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

#510

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

#1311

—

C₂×C₄₃

—

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2251

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₄₃

yes

characteristic

#493

C₂×D₄₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12520

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

172

—

SD₃₂

—

172

—

C₂×C₁₆

—

#107

—

172

—

#102

—

C₄×C₈

C₂²⋊D₄₃

yes

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

172

—

101

Q₁₆

—

172

—

102

D₈

—

103

C₂×C₈

#10621882

yes

—

104

C₂×C₈

D₄×D₄₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

172

—

107

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

344

—

112

Q₈

—

344

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

344

—

117

C₂×C₄

#13063321

yes

characteristic

118

C₂×C₄

C₂×#10621882

yes

—

119

C₈

—

120

C₈

D₈×D₄₃

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#14583275

yes

characteristic

124

C₄

—

125

C₄

—

688

—

126

C₄

#14583274

yes

—

127

C₄

#14583339

yes

—

128

C₂

—

129

C₂

#19134209

yes

characteristic

130

C₂

#19136867

yes

—

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1709	C₂
socle	C₂²×C₄₃	#493
prime radical	C₄₃×#493	C₂²
hypercenter	#1709	D₄₃
hypocenter	C₄₃	#12520
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1709

C₂

socle

C₂²×C₄₃

#493

prime radical

C₄₃×#493

C₂²

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #19136847	C₂	semidirect
3. #19134213	C₂	semidirect
4. #19132512	C₂	semidirect
80. C₄₃	#12520	semidirect

H ≅ G/N

type

2. #19136847

C₂

semidirect

3. #19134213

C₂

semidirect

4. #19132512

C₂

semidirect

80. C₄₃

#12520

semidirect

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

subgroup	index
#19136850	2
#19136847	2
#19134213	2
#19132512	2
C₄₃×#1709	2
#12520	43

subgroup

index

C₄₃×#1709

Subgroups — #25011331

Conjugacy classes of subgroups

Complements

Maximal subgroups