Group #16950745 — subgroups

Subgroups — #16950745

Order 4352.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14530550	C₂	1	1	yes	maximal, characteristic
2	#14530549	C₂	1	2	yes	maximal
3	#14530542	C₂	1	2	yes	maximal
4	#14528539	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	#11762764	C₂²	1	2	yes	—
8	#11762630	—	2	2	no	—
9	#11762629	—	2	4	no	—
10	#11762627	—	2	4	no	—
11	#11762567	C₄	1	1	yes	characteristic
12	#11762566	C₄	1	1	yes	characteristic
13	#11762565	C₄	1	2	yes	—
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#405	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊C₁₆	—	2	2	no	—
18	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
21	#10587612	C₂×C₄	1	2	yes	—
22	C₂×#63223	C₂×C₄	1	2	yes	—
23	C₁₇×M₃₂	—	2	2	no	—
24	C₂×C₁₆×C₁₇	—	2	1	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₁₇×#97	Q₈	1	2	yes	—
28	C₄×C₈×C₁₇	D₄	1	2	yes	—
29	#10587486	Q₈	1	4	yes	—
30	C₈×Dic₁₇	D₄	1	4	yes	—
31	#10587483	—	2	2	no	—
32	#10587483	—	2	2	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₇	—	2	1	no	—
43	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂²⋊Dic₁₇	—	2	2	no	—
45	C₄⋊Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	C₄⋊C₄	1	4	yes	—
47	#63223	—	2	2	no	—
48	#63223	—	2	2	no	—
49	C₂×#3622	C₄⋊C₄	1	4	yes	—
50	C₁₆×C₁₇	—	4	1	no	—
51	C₁₆×C₁₇	—	4	1	no	—
52	#63214	—	4	2	no	—
53	#12902	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#131	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#131	1	2	yes	—
60	C₂×Dic₁₇	—	2	4	no	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	#127	1	1	yes	characteristic
63	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
64	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
65	#3622	—	2	4	no	—
66	#2153	—	17	2	no	—
67	#2153	—	17	1	no	—
68	#2151	—	17	2	no	—
69	#2151	D₁₇	1	1	yes	characteristic
70	#1747	—	17	1	no	—
71	C₂²×C₁₇	—	4	1	no	—
72	C₂²×C₁₇	#424	1	2	yes	—
73	C₂²×C₁₇	#424	1	1	yes	characteristic
74	C₄×C₁₇	—	4	2	no	—
75	C₄×C₁₇	#384	1	1	yes	characteristic
76	C₄×C₁₇	#424	1	2	yes	—
77	C₄×C₁₇	#424	1	1	yes	characteristic
78	Dic₁₇	—	4	4	no	—
79	C₂×M₃₂	—	17	2	no	—
80	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
82	#405	—	17	2	no	—
83	#405	C₂×D₁₇	1	1	yes	characteristic
84	#364	—	34	4	no	—
85	#364	—	34	2	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	C₄⋊C₁₆	—	34	4	no	—
88	#338	—	17	1	no	—
89	H(3,ℤ/4)	—	17	1	no	—
90	H(3,ℤ/4)	—	17	2	no	—
91	C₂×C₁₇	—	2	2	no	—
92	C₂×C₁₇	—	2	1	no	—
93	C₂×C₁₇	#1843	1	1	yes	characteristic
94	C₂×M₁₆	—	17	2	no	—
95	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	#117	—	17	2	no	—
98	#117	C₄×D₁₇	1	1	yes	characteristic
99	M₃₂	—	2	2	no	—
100	M₃₂	—	34	2	no	—
101	M₃₂	—	34	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	34	2	no	—
106	#97	—	17	4	no	—
107	#97	Dic₃₄	1	2	yes	—
108	C₄×C₈	—	17	4	no	—
109	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
110	C₁₇	#12902	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	34	2	no	—
114	M₁₆	—	34	2	no	—
115	C₂×C₈	—	17	4	no	—
116	C₂×C₈	#63225	1	2	yes	—
117	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
118	C₂×C₈	D₄×D₁₇	1	2	yes	—
119	C₂×C₈	Q₈×D₁₇	1	2	yes	—
120	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	34	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	34	2	no	—
125	C₄²	—	17	4	no	—
126	C₄²	#63225	1	2	yes	—
127	C₁₆	—	4	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	—	68	2	no	—
130	C₂³	#10587587	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	34	4	no	—
133	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
134	C₂×C₄	#10587587	1	1	yes	characteristic
135	C₂×C₄	#10587593	1	2	yes	—
136	C₂×C₄	C₂×#63225	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	34	4	no	—
139	C₈	#10587574	1	2	yes	—
140	C₈	#10587626	1	1	yes	characteristic
141	C₈	#10587630	1	1	yes	characteristic
142	C₂²	—	4	1	no	—
143	C₂²	#11762834	1	2	yes	—
144	C₂²	#11762835	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	68	4	no	—
147	C₄	#11762678	1	1	yes	characteristic
148	C₄	#11762834	1	2	yes	—
149	C₄	#11762837	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#14528937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530550

C₂

yes

maximal, characteristic

#14530549

C₂

yes

maximal

#14530542

C₂

yes

maximal

#14528539

C₂

yes

maximal, characteristic

C₁₇×#2151

C₂

yes

maximal, characteristic

C₂×#10587483

C₂²

yes

—

#11762764

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#11762566

C₄

yes

characteristic

#11762565

C₄

yes

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#405

C₂²

yes

characteristic

C₁₇×C₄⋊C₁₆

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

C₂×C₄

yes

characteristic

#10587612

C₂×C₄

yes

—

C₂×#63223

C₂×C₄

yes

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₁₇×#97

Q₈

yes

—

C₄×C₈×C₁₇

D₄

yes

—

#10587486

Q₈

yes

—

C₈×Dic₁₇

D₄

yes

—

#10587483

—

#10587483

—

C₂×#63214

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

#63223

—

#63223

—

C₂×#3622

C₄⋊C₄

yes

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#127

yes

characteristic

C₈×C₁₇

C₄⋊Q₈

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

—

—

—

—

D₁₇

yes

characteristic

#1747

—

C₂²×C₁₇

—

C₂²×C₁₇

#424

yes

—

C₂²×C₁₇

#424

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#384

yes

characteristic

C₄×C₁₇

#424

yes

—

C₄×C₁₇

#424

yes

characteristic

Dic₁₇

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

#405

—

#405

C₂×D₁₇

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1843

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₄×D₁₇

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

#117

—

#117

C₄×D₁₇

yes

characteristic

M₃₂

—

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

#97

—

107

#97

Dic₃₄

yes

—

108

C₄×C₈

—

109

C₄×C₈

C₂²⋊D₁₇

yes

—

110

C₁₇

#12902

yes

characteristic, complemented, Sylow

111

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

#63225

yes

—

117

C₂×C₈

C₂×Dic₃₄

yes

characteristic

118

C₂×C₈

D₄×D₁₇

yes

—

119

C₂×C₈

Q₈×D₁₇

yes

—

120

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

#63225

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

130

C₂³

#10587587

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₁₇

yes

—

134

C₂×C₄

#10587587

yes

characteristic

135

C₂×C₄

#10587593

yes

—

136

C₂×C₄

C₂×#63225

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#10587574

yes

—

140

C₈

#10587626

yes

characteristic

141

C₈

#10587630

yes

characteristic

142

C₂²

—

143

C₂²

#11762834

yes

—

144

C₂²

#11762835

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

#11762678

yes

characteristic

148

C₄

#11762834

yes

—

149

C₄

#11762837

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#14528937

yes

characteristic

	subgroup	G/H
center	C₈	#10587626
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2151	C₂
socle	C₂×C₁₇	#1843
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#2151	D₁₇
hypocenter	C₁₇	#12902
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#10587626

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2151

C₂

C₂×C₁₇

C₂³×C₁₇

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
110. C₁₇	#12902	semidirect

H ≅ G/N

type

110. C₁₇

#12902

semidirect

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

subgroup	index
#14530550	2
#14530549	2
#14530542	2
#14528539	2
C₁₇×#2151	2
#12902	17

subgroup

index

C₁₇×#2151

Subgroups — #16950745

Conjugacy classes of subgroups

Complements

Maximal subgroups