Group #16928126 — subgroups

Subgroups — #16928126

Order 4352.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#14528956	C₂	1	1	yes	maximal, characteristic
2	#14527542	C₂	1	4	yes	maximal
3	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	—	2	2	no	—
6	#11762833	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	#11762561	—	2	8	no	—
9	C₈×#3622	C₄	1	1	yes	characteristic
10	C₈×#3622	C₄	1	2	yes	—
11	C₈×#3622	C₄	1	1	yes	characteristic
12	C₂×C₁₇×#108	—	2	2	no	—
13	C₁₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₇	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×#63223	—	2	2	no	—
19	C₂²×#3622	Q₈	1	4	yes	—
20	C₁₇×#108	—	4	4	no	—
21	C₁₇×C₄⋊C₈	—	2	4	no	—
22	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	Q₈	1	2	yes	—
25	#10587490	—	4	4	no	—
26	C₄⋊#3622	—	2	4	no	—
27	C₄×#3622	C₂×C₄	1	2	yes	—
28	C₄×#3622	C₂×C₄	1	2	yes	—
29	C₄×#3622	D₄	1	4	yes	—
30	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
32	M₁₆×C₁₇	—	4	4	no	—
33	C₂×C₈×C₁₇	—	4	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₇	D₈	1	4	yes	—
37	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
38	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂×D₄	1	2	yes	—
40	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	#63223	—	4	4	no	—
42	C₂×#3622	—	2	8	no	—
43	C₂×#3622	—	4	2	no	—
44	C₂×#3622	C₄⋊C₄	1	4	yes	—
45	C₂×#3622	C₄⋊C₄	1	4	yes	—
46	#8823	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	4	no	—
49	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
55	C₈×C₁₇	—	8	2	no	—
56	C₈×C₁₇	#107	1	4	yes	—
57	C₈×C₁₇	#107	1	4	yes	—
58	#3622	—	2	4	no	—
59	#3622	—	8	2	no	—
60	#3622	#108	1	8	yes	complemented
61	#1848	—	17	1	no	—
62	#1848	D₁₇	1	1	yes	characteristic
63	#1567	—	17	4	no	—
64	C₂×C₈²	—	17	1	no	—
65	C₂²×C₁₇	—	2	2	no	—
66	C₂²×C₁₇	#384	1	1	yes	characteristic
67	C₂²×C₁₇	#500	1	2	yes	—
68	C₄×C₁₇	—	2	2	no	—
69	C₄×C₁₇	C₂×#107	1	2	yes	—
70	C₄×C₁₇	C₂×#108	1	4	yes	—
71	C₄×C₁₇	#493	1	1	yes	characteristic
72	C₄×C₁₇	#494	1	1	yes	characteristic
73	C₂×#108	—	2	2	no	—
74	C₂×#108	—	34	2	no	—
75	#423	—	17	2	no	—
76	#423	C₂×D₁₇	1	2	yes	—
77	C₂×C₄×C₈	—	17	2	no	—
78	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
79	#335	—	34	8	no	—
80	C₈²	—	17	1	no	—
81	C₈²	—	17	2	no	—
82	C₈²	—	17	1	no	—
83	C₂×C₁₇	—	2	2	no	—
84	C₂×C₁₇	#1845	1	1	yes	characteristic
85	C₂×C₁₇	#1848	1	2	yes	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	34	2	no	—
88	C₂²×C₈	—	17	4	no	—
89	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
90	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
91	#108	—	4	4	no	—
92	#108	—	68	4	no	—
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	34	4	no	—
95	C₄×C₈	—	17	2	no	—
96	C₄×C₈	—	17	2	no	—
97	C₄×C₈	—	17	4	no	—
98	C₄×C₈	Dic₃₄	1	2	yes	—
99	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
101	C₁₇	#8823	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₄	Q₈×D₁₇	1	2	yes	—
103	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
104	M₁₆	—	4	4	no	—
105	M₁₆	—	68	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	17	4	no	—
108	C₂×C₈	—	17	4	no	—
109	C₂×C₈	—	34	8	no	—
110	C₂×C₈	—	68	2	no	—
111	C₂×C₈	#63225	1	2	yes	—
112	C₂×C₈	#63225	1	2	yes	—
113	C₂×C₈	#63228	1	4	yes	—
114	C₂×C₈	#63231	1	4	yes	—
115	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
117	C₄²	D₄×D₁₇	1	2	yes	—
118	C₂³	#10587630	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	#10587574	1	2	yes	—
121	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
122	C₂×C₄	C₂×#63225	1	1	yes	characteristic
123	C₂×C₄	C₂×#63228	1	2	yes	—
124	C₂×C₄	#10587626	1	1	yes	characteristic
125	C₂×C₄	C₂×#63231	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	17	8	no	—
128	C₈	—	34	4	no	—
129	C₈	—	136	2	no	—
130	C₈	#10587478	1	4	yes	—
131	C₈	#10587478	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#11762678	1	1	yes	characteristic
134	C₂²	#11763212	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#10587478	1	2	yes	—
137	C₄	#108×D₁₇	1	4	yes	—
138	C₄	#11763185	1	1	yes	characteristic
139	C₄	#11763188	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#14528943	1	1	yes	characteristic
142	C₂	#14528954	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal, characteristic

#14527542

C₂

yes

maximal

C₂×C₈×#3622

C₂

yes

maximal, characteristic

C₁₇×#1848

C₂

yes

maximal, characteristic

C₂×#10587490

—

#11762833

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

—

#11762561

—

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₄

yes

—

C₈×#3622

C₄

yes

characteristic

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂²×#3622

Q₈

yes

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

Q₈

yes

—

#10587490

—

C₄⋊#3622

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

#8823

—

maximal, Sylow

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

#107

yes

—

C₈×C₁₇

#107

yes

—

—

—

yes

—

D₁₇

yes

characteristic

#1567

—

C₂×C₈²

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#500

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

C₂×#108

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

—

C₂×D₁₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#335

—

C₈²

—

C₈²

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1845

yes

characteristic

C₂×C₁₇

#1848

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₄×D₁₇

yes

characteristic

100

C₄×C₈

C₄×D₁₇

yes

characteristic

101

C₁₇

#8823

yes

characteristic, complemented, Sylow

102

C₂²×C₄

Q₈×D₁₇

yes

—

103

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#63225

yes

—

112

C₂×C₈

#63225

yes

—

113

C₂×C₈

#63228

yes

—

114

C₂×C₈

#63231

yes

—

115

C₄²

C₂×Dic₃₄

yes

characteristic

116

C₄²

C₂×C₄×D₁₇

yes

characteristic

117

C₄²

D₄×D₁₇

yes

—

118

C₂³

#10587630

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

#10587574

yes

—

121

C₂×C₄

C₄⋊C₄×D₁₇

yes

—

122

C₂×C₄

C₂×#63225

yes

characteristic

123

C₂×C₄

C₂×#63228

yes

—

124

C₂×C₄

#10587626

yes

characteristic

125

C₂×C₄

C₂×#63231

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

136

—

130

C₈

#10587478

yes

—

131

C₈

#10587478

yes

—

132

C₂²

—

133

C₂²

#11762678

yes

characteristic

134

C₂²

#11763212

yes

—

135

C₄

—

136

C₄

C₂×#10587478

yes

—

137

C₄

#108×D₁₇

yes

—

138

C₄

#11763185

yes

characteristic

139

C₄

#11763188

yes

characteristic

140

C₂

—

141

C₂

#14528943

yes

characteristic

142

C₂

#14528954

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#8823
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1848

C₂

socle

C₂²×C₁₇

#384

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
60. #3622	#108	semidirect
101. C₁₇	#8823	semidirect

H ≅ G/N

type

60. #3622

#108

semidirect

101. C₁₇

#8823

semidirect

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

subgroup	index
#14528956	2
#14527542	2
C₂×C₈×#3622	2
C₁₇×#1848	2
#8823	17

subgroup

index

#14528956

#14527542

C₂×C₈×#3622

C₁₇×#1848

#8823

Subgroups — #16928126

Conjugacy classes of subgroups

Complements

Maximal subgroups