Group #25011321 — subgroups

Subgroups — #25011321

Order 11008.

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

normal subgroups only

This group has 330 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	#19136852	C₂	1	2	yes	maximal, complemented
2	#19136845	C₂	1	1	yes	maximal, characteristic
3	#19134202	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063268	—	2	4	no	—
7	C₂×D₆₈₈	—	2	2	no	—
8	#14583276	C₂²	1	2	yes	—
9	#14583247	—	2	4	no	—
10	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₄₃	—	2	2	no	—
13	C₄₃×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×D₃₄₄	—	2	2	no	—
16	C₂×D₃₄₄	—	2	2	no	—
17	#13063322	—	2	2	no	—
18	#13063268	—	4	8	no	—
19	D₆₈₈	—	4	4	no	—
20	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
21	#13063245	—	4	8	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×D₁₇₂	—	4	2	no	—
31	C₂×D₁₇₂	—	4	4	no	—
32	Q₁₆×C₄₃	—	4	4	no	—
33	Q₁₆×C₄₃	—	4	4	no	—
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₃	—	4	4	no	—
37	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
39	D₃₄₄	—	4	4	no	—
40	D₃₄₄	—	4	4	no	—
41	C₁₆×C₄₃	D₈	1	4	yes	—
42	C₄₃⋊C₁₆	D₈	1	4	yes	—
43	C₂²×D₄₃	—	8	2	no	—
44	Q₈×C₄₃	—	8	2	no	—
45	Q₈×C₄₃	—	8	2	no	—
46	C₂×C₄×C₄₃	—	8	2	no	—
47	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
49	D₁₇₂	—	8	2	no	—
50	D₁₇₂	—	8	2	no	—
51	D₁₇₂	—	8	8	no	—
52	C₈×C₄₃	C₂×D₈	1	2	yes	—
53	C₈×C₄₃	C₂×D₈	1	2	yes	—
54	C₄₃⋊C₈	—	2	4	no	—
55	#12518	—	43	1	no	maximal, Sylow
56	C₂²×C₄₃	#493	1	1	yes	characteristic
57	C₂×D₄₃	—	16	2	no	—
58	C₂×D₄₃	—	16	4	no	—
59	C₄×C₄₃	—	16	2	no	—
60	C₄×C₄₃	#493	1	1	yes	characteristic
61	C₄×C₄₃	#493	1	1	yes	characteristic
62	C₄×C₄₃	#508	1	4	yes	—
63	#2248	—	43	2	no	—
64	#2247	D₄₃	1	1	yes	characteristic, complemented
65	#2246	—	43	1	no	—
66	#1709	—	43	2	no	—
67	C₈×C₁₆	—	43	1	no	—
68	C₂×C₄₃	#1712	1	1	yes	characteristic
69	C₂×C₄₃	#2249	1	2	yes	—
70	D₄₃	—	32	2	no	—
71	C₂×Q₃₂	—	2	2	no	—
72	C₂×SD₃₂	—	86	4	no	—
73	C₂×D₁₆	—	86	2	no	—
74	#494	C₂×D₄₃	1	2	yes	—
75	#493	—	43	2	no	—
76	#486	—	86	4	no	—
77	C₄×C₁₆	—	43	1	no	—
78	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
79	C₈²	—	43	1	no	—
80	C₄₃	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	86	2	no	—
84	C₂×D₈	—	86	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	86	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	172	8	no	—
89	D₁₆	—	172	4	no	—
90	C₂×C₁₆	—	43	2	no	—
91	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
92	#102	—	172	8	no	—
93	C₄×C₈	—	43	2	no	—
94	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	172	2	no	—
97	C₂×D₄	—	172	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	172	4	no	—
101	D₈	—	172	4	no	—
102	C₂×C₈	—	43	4	no	—
103	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₄₃	1	1	yes	characteristic
107	C₁₆	—	43	4	no	—
108	C₁₆	#10621882	1	4	yes	—
109	C₂³	—	344	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	344	8	no	—
113	D₄	—	344	2	no	—
114	D₄	—	344	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#13063321	1	1	yes	characteristic
117	C₂×C₄	D₈×D₄₃	1	2	yes	—
118	C₈	—	86	4	no	—
119	C₈	D₈×D₄₃	1	2	yes	—
120	C₈	C₂×#10621882	1	2	yes	—
121	C₂²	—	688	2	no	—
122	C₂²	—	688	4	no	—
123	C₂²	#14583274	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#14583274	1	1	yes	characteristic
126	C₄	#14583275	1	1	yes	characteristic
127	C₄	#14583332	1	4	yes	—
128	C₂	—	1376	2	no	—
129	C₂	#19134209	1	1	yes	characteristic
130	C₂	#19136857	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136852

C₂

yes

maximal, complemented

#19136845

C₂

yes

maximal, characteristic

#19134202

C₂

yes

maximal, complemented

C₁₆×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₃×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#13063268

—

C₂×D₆₈₈

—

#14583276

C₂²

yes

—

#14583247

—

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

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063322

—

#13063268

—

D₆₈₈

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

#13063245

—

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×D₁₇₂

—

C₂×D₁₇₂

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₈

yes

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

D₈

yes

—

C₄₃⋊C₁₆

D₈

yes

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

C₂×D₈

yes

—

C₈×C₄₃

C₂×D₈

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

#2248

—

#2247

D₄₃

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₂×C₄₃

#1712

yes

characteristic

C₂×C₄₃

#2249

yes

—

D₄₃

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₄₃

yes

—

#493

—

#486

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

—

C₄₃

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

172

—

D₁₆

—

172

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₃

yes

—

#102

—

172

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

Q₁₆

—

Q₁₆

—

100

D₈

—

172

—

101

D₈

—

172

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₄₃

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₄₃

yes

characteristic

107

C₁₆

—

108

C₁₆

#10621882

yes

—

109

C₂³

—

344

—

110

Q₈

—

111

Q₈

—

112

D₄

—

344

—

113

D₄

—

344

—

114

D₄

—

344

—

115

C₂×C₄

—

116

C₂×C₄

#13063321

yes

characteristic

117

C₂×C₄

D₈×D₄₃

yes

—

118

C₈

—

119

C₈

D₈×D₄₃

yes

—

120

C₈

C₂×#10621882

yes

—

121

C₂²

—

688

—

122

C₂²

—

688

—

123

C₂²

#14583274

yes

characteristic

124

C₄

—

125

C₄

#14583274

yes

characteristic

126

C₄

#14583275

yes

characteristic

127

C₄

#14583332

yes

—

128

C₂

—

1376

—

129

C₂

#19134209

yes

characteristic

130

C₂

#19136857

yes

—

	subgroup	G/H
center	C₂²	#14583274
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2247	C₂
socle	C₂²×C₄₃	#493
prime radical	#19136845	C₂
hypercenter	#2247	D₄₃
hypocenter	C₄₃	#12518
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

C₂

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #19136852	C₂	semidirect
3. #19134202	C₂	semidirect
4. C₁₆×C₄₃⋊C₈	C₂	semidirect
5. C₄₃×#2247	C₂	semidirect
64. #2247	D₄₃	semidirect
80. C₄₃	#12518	semidirect

H ≅ G/N

type

1. #19136852

C₂

semidirect

3. #19134202

C₂

semidirect

4. C₁₆×C₄₃⋊C₈

C₂

semidirect

5. C₄₃×#2247

C₂

semidirect

64. #2247

D₄₃

semidirect

80. C₄₃

#12518

semidirect

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

subgroup	index
#19136852	2
#19136845	2
#19134202	2
C₁₆×C₄₃⋊C₈	2
C₄₃×#2247	2
#12518	43

subgroup

index

#19136852

#19136845

#19134202

C₁₆×C₄₃⋊C₈

C₄₃×#2247

#12518

Subgroups — #25011321

Conjugacy classes of subgroups

Complements

Maximal subgroups