Group #25011283 — subgroups

Subgroups — #25011283

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	#19136819	C₂	1	4	yes	maximal, complemented
2	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134210	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583304	—	2	1	no	—
6	#14583304	C₂²	1	2	yes	—
7	#14583279	—	2	1	no	—
8	#14583276	—	2	1	no	—
9	#14583276	—	2	1	no	—
10	#14583276	C₂²	1	2	yes	—
11	#14582701	—	2	8	no	—
12	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
13	C₈²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₇₂	—	4	2	no	—
15	C₂×D₃₄₄	—	2	4	no	—
16	C₂×D₃₄₄	—	4	2	no	—
17	C₂×D₃₄₄	—	4	2	no	—
18	#13063325	—	2	2	no	—
19	#13063322	—	2	2	no	—
20	#13063259	—	2	4	no	—
21	#13063259	—	4	2	no	—
22	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
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×D₁₇₂	—	4	2	no	—
27	C₂×D₁₇₂	—	4	2	no	—
28	C₂×D₁₇₂	—	4	2	no	—
29	C₂×Dic₈₆	—	4	2	no	—
30	C₂×C₈×C₄₃	D₈	1	2	yes	—
31	C₂×C₈×C₄₃	D₈	1	2	yes	—
32	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
33	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₄₃	—	4	2	no	—
35	C₄⋊Dic₄₃	—	4	2	no	—
36	Dic₁₇₂	—	8	4	no	—
37	D₃₄₄	—	4	8	no	—
38	D₃₄₄	—	8	4	no	—
39	D₃₄₄	—	8	4	no	—
40	C₁₆×C₄₃	—	2	4	no	—
41	C₂²×D₄₃	—	8	2	no	—
42	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₄₃	—	8	2	no	—
46	D₁₇₂	—	8	4	no	—
47	D₁₇₂	—	8	4	no	—
48	D₁₇₂	—	8	4	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	#493	1	1	yes	characteristic
55	C₂×D₄₃	—	16	2	no	—
56	C₂×D₄₃	—	16	2	no	—
57	C₂×D₄₃	—	16	2	no	—
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	—	43	1	no	—
65	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
66	C₂×C₄₃	#1709	1	1	yes	characteristic
67	C₂×C₄₃	#2246	1	1	yes	characteristic
68	C₂×C₄₃	#2248	1	1	yes	characteristic
69	D₄₃	—	32	2	no	—
70	#500	—	43	2	no	—
71	#500	—	86	1	no	—
72	#494	—	86	1	no	—
73	#493	—	43	2	no	—
74	#493	—	86	1	no	—
75	#493	—	86	1	no	—
76	#357	—	86	8	no	—
77	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
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	—	86	4	no	—
82	C₂×D₈	—	172	2	no	—
83	C₂×D₈	—	172	2	no	—
84	C₄⋊Q₈	—	86	2	no	—
85	#127	—	86	2	no	—
86	C₂×C₁₆	D₁₇₂	1	4	yes	—
87	#107	—	86	4	no	—
88	#107	—	172	2	no	—
89	C₄×C₈	D₁₇₂	1	1	yes	characteristic
90	C₄×C₈	D₁₇₂	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
92	C₂×Q₈	—	172	2	no	—
93	C₂×D₄	—	172	2	no	—
94	C₂×D₄	—	172	2	no	—
95	C₂×D₄	—	172	2	no	—
96	Q₁₆	—	344	4	no	—
97	D₈	—	344	4	no	—
98	D₈	—	344	4	no	—
99	D₈	—	172	8	no	—
100	C₂×C₈	D₃₄₄	1	2	yes	—
101	C₂×C₈	D₃₄₄	1	2	yes	—
102	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
103	C₄⋊C₄	—	172	2	no	—
104	C₄⋊C₄	—	172	2	no	—
105	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
106	C₁₆	—	2	4	no	—
107	C₂³	—	344	2	no	—
108	Q₈	—	344	4	no	—
109	D₄	—	344	4	no	—
110	D₄	—	344	4	no	—
111	D₄	—	344	4	no	—
112	C₂×C₄	—	344	2	no	—
113	C₂×C₄	#13063322	1	1	yes	characteristic
114	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
116	C₈	D₆₈₈	1	4	yes	—
117	C₈	#13063270	1	4	yes	—
118	C₈	#13063347	1	4	yes	—
119	C₂²	—	688	2	no	—
120	C₂²	—	688	2	no	—
121	C₂²	—	688	2	no	—
122	C₂²	#14583276	1	1	yes	characteristic
123	C₄	—	688	2	no	—
124	C₄	#14583283	1	2	yes	—
125	C₄	C₂×D₆₈₈	1	2	yes	—
126	C₄	C₂×#13063270	1	2	yes	—
127	C₂	—	1376	2	no	—
128	C₂	#19134201	1	1	yes	characteristic
129	C₂	#19136845	1	1	yes	characteristic
130	C₂	#19136851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136819

C₂

yes

maximal, complemented

#19134213

C₂

yes

maximal, characteristic, complemented

#19134210

C₂

yes

maximal, characteristic

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

#14583304

—

#14583304

C₂²

yes

—

—

—

—

C₂²

yes

—

#14582701

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

—

—

—

—

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

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

—

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

—

Dic₈₆

—

C₈×C₄₃

D₁₆

yes

—

C₈×C₄₃

SD₃₂

yes

—

C₈×C₄₃

#135

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#495

yes

—

C₄×C₄₃

C₂×D₁₆

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

Dic₄₃

—

#2240

—

#1713

—

#1712

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

172

—

C₂×D₈

—

C₂×D₈

—

172

—

C₂×D₈

—

172

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₁₇₂

yes

—

#107

—

#107

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

Q₁₆

—

344

—

D₈

—

344

—

D₈

—

344

—

D₈

—

172

—

100

C₂×C₈

D₃₄₄

yes

—

101

C₂×C₈

D₃₄₄

yes

—

102

C₂×C₈

C₂×D₁₇₂

yes

—

103

C₄⋊C₄

—

172

—

104

C₄⋊C₄

—

172

—

105

C₄²

C₂×D₁₇₂

yes

characteristic

106

C₁₆

—

107

C₂³

—

344

—

108

Q₈

—

344

—

109

D₄

—

344

—

110

D₄

—

344

—

111

D₄

—

344

—

112

C₂×C₄

—

344

—

113

C₂×C₄

#13063322

yes

characteristic

114

C₂×C₄

C₂×D₃₄₄

yes

characteristic

115

C₂×C₄

C₂×D₃₄₄

yes

characteristic

116

C₈

D₆₈₈

yes

—

117

C₈

#13063270

yes

—

118

C₈

#13063347

yes

—

119

C₂²

—

688

—

120

C₂²

—

688

—

121

C₂²

—

688

—

122

C₂²

#14583276

yes

characteristic

123

C₄

—

688

—

124

C₄

#14583283

yes

—

125

C₄

C₂×D₆₈₈

yes

—

126

C₄

C₂×#13063270

yes

—

127

C₂

—

1376

—

128

C₂

#19134201

yes

characteristic

129

C₂

#19136845

yes

characteristic

130

C₂

#19136851

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	#19134210	C₂
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12509
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

#12509

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #19136819	C₂	semidirect
2. #19134213	C₂	semidirect
4. C₈×C₁₆×C₄₃	C₂	semidirect
65. C₈×C₁₆	D₄₃	semidirect
79. C₄₃	#12509	semidirect

H ≅ G/N

type

1. #19136819

C₂

semidirect

2. #19134213

C₂

semidirect

4. C₈×C₁₆×C₄₃

C₂

semidirect

65. C₈×C₁₆

D₄₃

semidirect

79. C₄₃

#12509

semidirect

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

subgroup	index
#19136819	2
#19134213	2
#19134210	2
C₈×C₁₆×C₄₃	2
#12509	43

subgroup

index

C₈×C₁₆×C₄₃

Subgroups — #25011283

Conjugacy classes of subgroups

Complements

Maximal subgroups