Group #44506813 — subgroups

Subgroups — #44506813

Order 30848.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#29508096	C₂	1	1	yes	maximal, characteristic
2	#29508002	C₂	1	1	yes	maximal, characteristic
3	Q₈×#14515349	C₂	1	1	yes	maximal, characteristic
4	#29507469	C₂	1	1	yes	maximal, characteristic
5	#29507463	C₂	1	1	yes	maximal, characteristic
6	#29507462	C₂	1	1	yes	maximal, characteristic
7	#475×C₂₄₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16838760	—	2	1	no	—
9	C₂×#16838760	—	2	1	no	—
10	#21440642	C₂²	1	1	yes	characteristic
11	#21440573	—	2	1	no	—
12	C₄⋊#14515349	—	2	1	no	—
13	C₄⋊#14515349	C₂²	1	1	yes	characteristic
14	C₄⋊#14515349	C₂²	1	1	yes	characteristic
15	#21440566	—	2	1	no	—
16	C₄×#14515349	—	2	1	no	—
17	C₄×#14515349	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
20	#106×C₂₄₁	—	2	1	no	—
21	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
22	#103×C₂₄₁	—	2	1	no	—
23	C₂×Dic₄₈₂	—	2	1	no	—
24	#16838760	—	4	2	no	—
25	#16838760	—	4	2	no	—
26	C₂×#14515349	—	2	1	no	—
27	C₂×#14515349	—	2	1	no	—
28	C₂×#14515349	D₄	1	1	yes	characteristic
29	C₂×#14515349	D₄	1	1	yes	characteristic
30	C₄⋊Dic₂₄₁	—	2	1	no	—
31	C₄⋊Dic₂₄₁	—	2	1	no	—
32	C₂×Q₈×C₂₄₁	—	2	1	no	—
33	C₂×Q₈×C₂₄₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₄₁	—	2	1	no	—
35	C₄⋊C₄×C₂₄₁	—	2	1	no	—
36	C₄⋊C₄×C₂₄₁	—	2	1	no	—
37	C₄⋊C₄×C₂₄₁	—	2	1	no	—
38	C₄⋊C₄×C₂₄₁	D₄	1	1	yes	characteristic
39	C₄²×C₂₄₁	—	2	1	no	—
40	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
41	Dic₄₈₂	—	4	1	no	—
42	Dic₄₈₂	—	4	1	no	—
43	C₂×Dic₂₄₁	—	4	1	no	—
44	#14515349	—	2	1	no	—
45	#14515349	—	4	1	no	—
46	#14515349	—	4	1	no	—
47	#14515349	SD₁₆	1	2	yes	—
48	Q₈×C₂₄₁	—	2	1	no	—
49	Q₈×C₂₄₁	—	4	1	no	—
50	Q₈×C₂₄₁	—	4	1	no	—
51	Q₈×C₂₄₁	Q₁₆	1	2	yes	complemented
52	C₂×C₄×C₂₄₁	—	2	1	no	—
53	C₂×C₄×C₂₄₁	—	2	1	no	—
54	C₂×C₄×C₂₄₁	—	4	1	no	—
55	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
58	C₈×C₂₄₁	—	4	1	no	—
59	C₂²×C₂₄₁	#121	1	1	yes	characteristic
60	C₄×C₂₄₁	—	2	2	no	—
61	C₄×C₂₄₁	—	2	1	no	—
62	C₄×C₂₄₁	—	4	1	no	—
63	C₄×C₂₄₁	—	8	1	no	—
64	C₄×C₂₄₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₂₄₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₂₄₁	#135	1	1	yes	characteristic
67	C₄×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	Dic₂₄₁	—	8	1	no	—
69	C₂×C₂₄₁	#460	1	1	yes	characteristic
70	C₂×C₂₄₁	#467	1	1	yes	characteristic
71	C₂×C₂₄₁	#473	1	1	yes	characteristic
72	C₂₄₁	#1670	1	1	yes	characteristic, complemented, Sylow
73	#1670	—	241	1	no	maximal, Sylow
74	#494	—	241	1	no	—
75	#475	—	241	1	no	—
76	#475	D₂₄₁	1	1	yes	characteristic
77	C₈×Q₈	—	241	1	no	—
78	#335	—	241	1	no	—
79	#332	—	241	1	no	—
80	#332	—	241	1	no	—
81	C₂×Q₁₆	—	482	1	no	—
82	C₂×Q₁₆	—	482	1	no	—
83	C₄⋊Q₈	—	241	1	no	—
84	C₄⋊Q₈	C₂×D₂₄₁	1	1	yes	characteristic
85	C₄×Q₈	C₂×D₂₄₁	1	1	yes	characteristic
86	#106	—	2	1	no	—
87	#106	—	482	1	no	—
88	C₄⋊C₈	—	482	1	no	—
89	C₄⋊C₈	—	241	1	no	—
90	C₄⋊C₈	—	241	1	no	—
91	C₄⋊C₈	C₂×D₂₄₁	1	1	yes	characteristic
92	#103	—	2	1	no	—
93	#103	—	482	1	no	—
94	C₄×C₈	—	482	1	no	—
95	C₄×C₈	—	241	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	482	1	no	—
98	C₂×Q₈	C₂²⋊D₂₄₁	1	1	yes	characteristic
99	Q₁₆	—	964	2	no	—
100	Q₁₆	—	964	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	482	1	no	—
103	C₂×C₈	—	482	1	no	—
104	C₂×C₈	—	241	1	no	—
105	C₂×C₈	—	241	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	482	1	no	—
110	C₄⋊C₄	—	482	1	no	—
111	C₄⋊C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₂₄₁	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	964	1	no	—
118	Q₈	—	964	1	no	—
119	Q₈	#16838760	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	964	1	no	—
124	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
126	C₂×C₄	#16838769	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	964	1	no	—
129	C₈	—	964	1	no	—
130	C₈	—	482	1	no	—
131	C₈	—	241	2	no	—
132	C₂²	#21440614	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	1928	1	no	—
138	C₄	SD₁₆×D₂₄₁	1	1	yes	characteristic
139	C₄	C₂×#16838760	1	1	yes	characteristic
140	C₄	#21440662	1	1	yes	characteristic
141	C₄	#21440669	1	1	yes	characteristic
142	C₂	#29507915	1	1	yes	characteristic
143	C₂	#29507956	1	1	yes	characteristic
144	C₂	#29507990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29508096

C₂

yes

maximal, characteristic

#29508002

C₂

yes

maximal, characteristic

Q₈×#14515349

C₂

yes

maximal, characteristic

#29507469

C₂

yes

maximal, characteristic

#29507463

C₂

yes

maximal, characteristic

#29507462

C₂

yes

maximal, characteristic

#475×C₂₄₁

C₂

yes

maximal, characteristic

C₂×#16838760

—

C₂×#16838760

—

#21440642

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#14515349

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₂₄₁

C₂²

yes

characteristic

C₄×Q₈×C₂₄₁

C₂²

yes

characteristic

#106×C₂₄₁

—

C₄⋊C₈×C₂₄₁

C₂²

yes

characteristic

#103×C₂₄₁

—

C₂×Dic₄₈₂

—

—

—

—

—

D₄

yes

characteristic

C₂×#14515349

D₄

yes

characteristic

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₂×Q₈×C₂₄₁

—

C₂×Q₈×C₂₄₁

D₄

yes

characteristic

C₂×C₈×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

D₄

yes

characteristic

C₄²×C₂₄₁

—

C₄²×C₂₄₁

C₂³

yes

characteristic

Dic₄₈₂

—

Dic₄₈₂

—

C₂×Dic₂₄₁

—

—

—

—

SD₁₆

yes

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

Q₁₆

yes

complemented

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

characteristic

C₈×C₂₄₁

—

C₂²×C₂₄₁

#121

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂×SD₁₆

yes

characteristic

C₄×C₂₄₁

C₂×Q₁₆

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

Dic₂₄₁

—

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂₄₁

yes

characteristic, complemented, Sylow

#1670

—

241

maximal, Sylow

#494

—

241

—

#475

—

241

—

#475

D₂₄₁

yes

characteristic

C₈×Q₈

—

241

—

#335

—

241

—

#332

—

241

—

#332

—

241

—

C₂×Q₁₆

—

482

—

C₂×Q₁₆

—

482

—

C₄⋊Q₈

—

241

—

C₄⋊Q₈

C₂×D₂₄₁

yes

characteristic

C₄×Q₈

C₂×D₂₄₁

yes

characteristic

#106

—

#106

—

482

—

C₄⋊C₈

—

482

—

C₄⋊C₈

—

241

—

C₄⋊C₈

—

241

—

C₄⋊C₈

C₂×D₂₄₁

yes

characteristic

#103

—

#103

—

482

—

C₄×C₈

—

482

—

C₄×C₈

—

241

—

C₂×Q₈

—

C₂×Q₈

—

482

—

C₂×Q₈

C₂²⋊D₂₄₁

yes

characteristic

Q₁₆

—

964

—

100

Q₁₆

—

964

—

101

C₂×C₈

—

102

C₂×C₈

—

482

—

103

C₂×C₈

—

482

—

104

C₂×C₈

—

241

—

105

C₂×C₈

—

241

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

482

—

110

C₄⋊C₄

—

482

—

111

C₄⋊C₄

C₂²⋊D₂₄₁

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₂₄₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

964

—

118

Q₈

—

964

—

119

Q₈

#16838760

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

964

—

124

C₂×C₄

D₄×D₂₄₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

characteristic

126

C₂×C₄

#16838769

yes

characteristic

127

C₈

—

128

C₈

—

964

—

129

C₈

—

964

—

130

C₈

—

482

—

131

C₈

—

241

—

132

C₂²

#21440614

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

1928

—

138

C₄

SD₁₆×D₂₄₁

yes

characteristic

139

C₄

C₂×#16838760

yes

characteristic

140

C₄

#21440662

yes

characteristic

141

C₄

#21440669

yes

characteristic

142

C₂

#29507915

yes

characteristic

143

C₂

#29507956

yes

characteristic

144

C₂

#29507990

yes

characteristic

	subgroup	G/H
center	C₂²	#21440614
commutator, cosocle	C₄²×C₂₄₁	C₂³
Frattini	C₄²	C₂²×D₂₄₁
Fitting	#475×C₂₄₁	C₂
socle, prime radical	C₂²×C₂₄₁	#121
hypercenter	#475	D₂₄₁
hypocenter	C₂₄₁	#1670
LCS	G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#21440614

commutator, cosocle

C₄²×C₂₄₁

C₂³

Frattini

C₄²

C₂²×D₂₄₁

Fitting

#475×C₂₄₁

C₂

C₂²×C₂₄₁

D₂₄₁

C₂₄₁

G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
51. Q₈×C₂₄₁	Q₁₆	semidirect
72. C₂₄₁	#1670	semidirect
119. Q₈	#16838760	semidirect

H ≅ G/N

type

51. Q₈×C₂₄₁

Q₁₆

semidirect

72. C₂₄₁

#1670

semidirect

119. Q₈

#16838760

semidirect

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

subgroup	index
#29508096	2
#29508002	2
Q₈×#14515349	2
#29507469	2
#29507463	2
#29507462	2
#475×C₂₄₁	2
#1670	241

subgroup

index

#475×C₂₄₁

241

Subgroups — #44506813

Conjugacy classes of subgroups

Complements

Maximal subgroups