Group #25011282 — subgroups

Subgroups — #25011282

Order 11008.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#19136818	C₂	1	4	yes	maximal, complemented
2	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1712	C₂	1	1	yes	maximal, characteristic
5	#14583304	—	2	1	no	—
6	#14583304	C₂²	1	2	yes	—
7	#14583279	—	2	1	no	—
8	#14582700	—	2	8	no	—
9	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	C₄₃×#493	—	2	1	no	—
11	C₄₃×#493	—	2	1	no	—
12	C₄₃×#493	C₂²	1	2	yes	—
13	C₈²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₇₂	—	4	2	no	—
15	#13063325	—	2	2	no	—
16	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
17	#13063259	—	2	4	no	—
18	#13063259	—	4	2	no	—
19	C₂×D₈×C₄₃	—	2	4	no	—
20	C₂×D₈×C₄₃	—	4	2	no	—
21	C₂×D₈×C₄₃	—	4	2	no	—
22	#127×C₄₃	—	2	2	no	—
23	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₄₃	—	4	2	no	—
27	C₂×D₄×C₄₃	—	4	2	no	—
28	C₂×D₄×C₄₃	—	4	2	no	—
29	C₂×Dic₈₆	—	4	2	no	—
30	D₈×C₄₃	—	4	8	no	—
31	D₈×C₄₃	—	8	4	no	—
32	D₈×C₄₃	—	8	4	no	—
33	C₂×C₈×C₄₃	D₈	1	2	yes	—
34	C₂×C₈×C₄₃	D₈	1	2	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	C₄⋊Dic₄₃	—	4	2	no	—
39	Dic₁₇₂	—	8	4	no	—
40	C₄₃⋊C₁₆	—	2	4	no	—
41	C₂³×C₄₃	—	8	2	no	—
42	D₄×C₄₃	—	8	4	no	—
43	D₄×C₄₃	—	8	4	no	—
44	D₄×C₄₃	—	8	4	no	—
45	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₄₃	—	8	2	no	—
49	Dic₈₆	—	8	4	no	—
50	C₈×C₄₃	D₁₆	1	4	yes	—
51	C₈×C₄₃	SD₃₂	1	4	yes	—
52	C₈×C₄₃	#135	1	4	yes	—
53	#12509	—	43	1	no	maximal, Sylow
54	C₂²×C₄₃	—	16	2	no	—
55	C₂²×C₄₃	—	16	2	no	—
56	C₂²×C₄₃	—	16	2	no	—
57	C₂²×C₄₃	#493	1	1	yes	characteristic
58	C₄×C₄₃	#495	1	2	yes	—
59	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
60	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
61	Dic₄₃	—	16	2	no	—
62	#2240	—	43	4	no	—
63	#1713	—	43	1	no	—
64	#1712	D₄₃	1	1	yes	characteristic
65	C₈×C₁₆	—	43	1	no	—
66	C₂×C₄₃	—	32	2	no	—
67	C₂×C₄₃	#1709	1	1	yes	characteristic
68	C₂×C₄₃	#2246	1	1	yes	characteristic
69	C₂×C₄₃	#2248	1	1	yes	characteristic
70	#500	—	43	2	no	—
71	#500	—	86	1	no	—
72	#494	—	86	1	no	—
73	#493	—	2	1	no	—
74	#493	—	2	1	no	—
75	#493	C₂×D₄₃	1	2	yes	—
76	#357	—	86	8	no	—
77	C₄×C₁₆	—	43	2	no	—
78	C₈²	C₂×D₄₃	1	1	yes	characteristic
79	C₄₃	#12509	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	172	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	86	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	43	4	no	—
87	#107	—	86	4	no	—
88	#107	—	172	2	no	—
89	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
92	C₂×Q₈	—	172	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	344	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#10621882	1	2	yes	—
101	C₂×C₈	#10621882	1	2	yes	—
102	C₂×C₈	D₄×D₄₃	1	2	yes	—
103	C₄⋊C₄	—	172	2	no	—
104	C₄⋊C₄	—	172	2	no	—
105	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
106	C₁₆	—	86	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	344	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	344	2	no	—
113	C₂×C₄	#13063321	1	1	yes	characteristic
114	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
115	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
116	C₈	#13063266	1	4	yes	—
117	C₈	#13063269	1	4	yes	—
118	C₈	#13063344	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#14583275	1	1	yes	characteristic
123	C₄	—	688	2	no	—
124	C₄	#14583282	1	2	yes	—
125	C₄	C₂×#13063266	1	2	yes	—
126	C₄	C₂×#13063269	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#19134200	1	1	yes	characteristic
129	C₂	#19136844	1	1	yes	characteristic
130	C₂	#19136850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136818

C₂

yes

maximal, complemented

#19134213

C₂

yes

maximal, characteristic, complemented

C₈×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₃×#1712

C₂

yes

maximal, characteristic

#14583304

—

#14583304

C₂²

yes

—

#14583279

—

#14582700

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#493

—

C₄₃×#493

—

C₄₃×#493

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

#13063325

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063259

—

#13063259

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

#127×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×Dic₈₆

—

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

—

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

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

D₁₆

yes

—

C₈×C₄₃

SD₃₂

yes

—

C₈×C₄₃

#135

yes

—

#12509

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#493

yes

characteristic

C₄×C₄₃

#495

yes

—

C₄×C₄₃

C₂×D₁₆

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

Dic₄₃

—

#2240

—

#1713

—

#1712

D₄₃

yes

characteristic

C₈×C₁₆

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

—

—

C₂×D₄₃

yes

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

172

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

172

—

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

344

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#10621882

yes

—

101

C₂×C₈

#10621882

yes

—

102

C₂×C₈

D₄×D₄₃

yes

—

103

C₄⋊C₄

—

172

—

104

C₄⋊C₄

—

172

—

105

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

106

C₁₆

—

107

C₂³

—

108

Q₈

—

344

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

344

—

113

C₂×C₄

#13063321

yes

characteristic

114

C₂×C₄

C₂×#10621882

yes

characteristic

115

C₂×C₄

C₂×#10621882

yes

characteristic

116

C₈

#13063266

yes

—

117

C₈

#13063269

yes

—

118

C₈

#13063344

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14583275

yes

characteristic

123

C₄

—

688

—

124

C₄

#14583282

yes

—

125

C₄

C₂×#13063266

yes

—

126

C₄

C₂×#13063269

yes

—

127

C₂

—

128

C₂

#19134200

yes

characteristic

129

C₂

#19136844

yes

characteristic

130

C₂

#19136850

yes

characteristic

	subgroup	G/H
center	C₂²	#14583275
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting, prime radical	C₄₃×#1712	C₂
socle	C₂²×C₄₃	#493
hypercenter	#1712	D₄₃
hypocenter	C₄₃	#12509
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#14583275

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting, prime radical

C₄₃×#1712

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #19136818	C₂	semidirect
2. #19134213	C₂	semidirect
3. C₈×C₄₃⋊C₁₆	C₂	semidirect
79. C₄₃	#12509	semidirect

H ≅ G/N

type

1. #19136818

C₂

semidirect

2. #19134213

C₂

semidirect

3. C₈×C₄₃⋊C₁₆

C₂

semidirect

79. C₄₃

#12509

semidirect

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

subgroup	index
#19136818	2
#19134213	2
C₈×C₄₃⋊C₁₆	2
C₄₃×#1712	2
#12509	43

subgroup

index

#19136818

#19134213

C₈×C₄₃⋊C₁₆

C₄₃×#1712

#12509

Subgroups — #25011282

Conjugacy classes of subgroups

Complements

Maximal subgroups