Group #25011269 — subgroups

Subgroups — #25011269

Order 11008.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	#19136877	C₂	1	1	yes	maximal, characteristic
2	#19136822	C₂	1	2	yes	maximal
3	#19134189	C₂	1	2	yes	maximal
4	#19132511	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#14583304	C₂²	1	2	yes	—
7	#14583295	—	2	2	no	—
8	#14583251	—	2	2	no	—
9	#14582727	—	2	2	no	—
10	#14582703	—	2	4	no	—
11	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₄₃	—	2	2	no	—
14	C₄₃×#494	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	#13063325	—	2	2	no	—
17	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
18	#13063259	—	2	2	no	—
19	#13063259	—	2	2	no	—
20	#13063256	—	4	4	no	—
21	#13063248	—	4	4	no	—
22	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₄₃	—	2	2	no	—
24	C₂×Q₁₆×C₄₃	—	2	2	no	—
25	C₄⋊Q₈×C₄₃	—	2	2	no	—
26	Q₃₂×C₄₃	—	4	4	no	—
27	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₃	—	4	2	no	—
30	C₂×Dic₈₆	—	4	2	no	—
31	Q₁₆×C₄₃	—	4	4	no	—
32	Q₁₆×C₄₃	—	4	4	no	—
33	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₃	—	4	2	no	—
36	C₄⋊C₄×C₄₃	—	4	2	no	—
37	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₃	—	4	2	no	—
39	C₄⋊Dic₄₃	—	4	2	no	—
40	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
41	C₁₆×C₄₃	SD₁₆	1	4	yes	—
42	C₄₃⋊C₁₆	—	2	2	no	—
43	Q₈×C₄₃	—	8	2	no	—
44	Q₈×C₄₃	—	8	2	no	—
45	C₂×C₄×C₄₃	—	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	2	no	—
49	Dic₈₆	—	8	4	no	—
50	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
51	C₈×C₄₃	#135	1	2	yes	—
52	C₄₃⋊C₈	—	2	4	no	—
53	#12507	—	43	1	no	maximal, Sylow
54	C₂²×C₄₃	#493	1	1	yes	characteristic
55	C₄×C₄₃	—	16	2	no	—
56	C₄×C₄₃	#492	1	1	yes	characteristic
57	C₄×C₄₃	#495	1	1	yes	characteristic
58	C₄×C₄₃	#510	1	4	yes	—
59	Dic₄₃	—	16	2	no	—
60	#2254	—	43	1	no	—
61	#2247	D₄₃	1	1	yes	characteristic
62	#2241	—	43	2	no	—
63	#1707	—	43	2	no	—
64	#1311	—	43	1	no	—
65	C₂×C₄₃	#1706	1	1	yes	characteristic
66	C₂×C₄₃	#2251	1	2	yes	—
67	C₂×Q₃₂	—	2	2	no	—
68	#500	—	43	2	no	—
69	#498	—	86	2	no	—
70	#494	C₂×D₄₃	1	2	yes	—
71	#487	—	86	2	no	—
72	#367	—	86	2	no	—
73	#358	—	86	4	no	—
74	C₄×C₁₆	—	43	1	no	—
75	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
76	C₈²	—	43	1	no	—
77	C₄₃	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	86	2	no	—
82	Q₃₂	—	4	4	no	—
83	C₂×C₁₆	—	43	2	no	—
84	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
85	#107	—	86	2	no	—
86	#107	—	86	2	no	—
87	#106	—	172	4	no	—
88	#103	—	172	4	no	—
89	C₄×C₈	—	43	2	no	—
90	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	172	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	—	43	4	no	—
96	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	172	2	no	—
101	C₄⋊C₄	—	172	2	no	—
102	C₄²	D₄×D₄₃	1	1	yes	characteristic
103	C₁₆	—	86	2	no	—
104	C₁₆	#10621883	1	4	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	344	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	344	2	no	—
110	C₂×C₄	#13063321	1	1	yes	characteristic
111	C₂×C₄	D₈×D₄₃	1	2	yes	—
112	C₈	—	86	4	no	—
113	C₈	C₂×#10621883	1	2	yes	—
114	C₈	#13063344	1	2	yes	—
115	C₂²	#14583274	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	688	2	no	—
118	C₄	#14583271	1	1	yes	characteristic
119	C₄	#14583282	1	1	yes	characteristic
120	C₄	#14583341	1	4	yes	—
121	C₂	#19134184	1	1	yes	characteristic
122	C₂	#19136863	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136877

C₂

yes

maximal, characteristic

#19136822

C₂

yes

maximal

#19134189

C₂

yes

maximal

#19132511

C₂

yes

maximal, characteristic

C₄₃×#2247

C₂

yes

maximal, characteristic

#14583304

C₂²

yes

—

—

—

—

—

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₄₃

—

C₄₃×#494

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

#13063325

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

Q₃₂×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

D₈

yes

—

C₁₆×C₄₃

SD₁₆

yes

—

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#135

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

#2254

—

#2247

D₄₃

yes

—

—

—

C₂×C₄₃

#1706

yes

characteristic

C₂×C₄₃

#2251

yes

—

C₂×Q₃₂

—

#500

—

#498

—

#494

C₂×D₄₃

yes

—

#487

—

#367

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

—

C₄₃

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₃

yes

—

#107

—

#107

—

#106

—

172

—

#103

—

172

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

172

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₄₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

172

—

101

C₄⋊C₄

—

172

—

102

C₄²

D₄×D₄₃

yes

characteristic

103

C₁₆

—

104

C₁₆

#10621883

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

344

—

108

C₂×C₄

—

109

C₂×C₄

—

344

—

110

C₂×C₄

#13063321

yes

characteristic

111

C₂×C₄

D₈×D₄₃

yes

—

112

C₈

—

113

C₈

C₂×#10621883

yes

—

114

C₈

#13063344

yes

—

115

C₂²

#14583274

yes

characteristic

116

C₄

—

117

C₄

—

688

—

118

C₄

#14583271

yes

characteristic

119

C₄

#14583282

yes

characteristic

120

C₄

#14583341

yes

—

121

C₂

#19134184

yes

characteristic

122

C₂

#19136863

yes

—

	subgroup	G/H
center	C₂²	#14583274
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2247	C₂
socle, prime radical	C₂²×C₄₃	#493
hypercenter	#2247	D₄₃
hypocenter	C₄₃	#12507
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#14583274

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#2247

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
77. C₄₃	#12507	semidirect

H ≅ G/N

type

77. C₄₃

#12507

semidirect

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

subgroup	index
#19136877	2
#19136822	2
#19134189	2
#19132511	2
C₄₃×#2247	2
#12507	43

subgroup

index

C₄₃×#2247

Subgroups — #25011269

Conjugacy classes of subgroups

Complements

Maximal subgroups