Group #16950706 — subgroups

Subgroups — #16950706

Order 4352.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526875	C₂	1	4	yes	maximal
4	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587488	—	2	1	no	—
8	#11762675	—	2	1	no	—
9	C₄×#63214	C₄	1	4	yes	complemented
10	C₂×C₁₇×#107	C₂²	1	2	yes	—
11	C₂×C₁₇×#106	—	2	1	no	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#382	—	2	1	no	—
14	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
17	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₁₇×#107	—	2	4	no	—
20	C₁₇×#106	—	4	2	no	—
21	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₇	Q₈	1	2	yes	—
23	C₁₇×#95	—	4	2	no	—
24	#10587502	—	4	2	no	—
25	#10587489	—	2	4	no	—
26	#10587488	—	4	2	no	—
27	C₂×#63214	—	2	2	no	—
28	C₂×#63214	—	2	2	no	—
29	C₂×#63214	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₇	—	4	2	no	—
31	C₂²×C₄×C₁₇	D₈	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₇	—	4	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	#41	1	2	yes	—
40	C₂×C₈×C₁₇	#41	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	4	2	no	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₇	Q₁₆	1	2	yes	—
45	C₄⋊Dic₁₇	—	4	2	no	—
46	C₄⋊Dic₁₇	—	4	2	no	—
47	#63214	—	2	4	no	—
48	#12889	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	4	no	—
51	C₂×C₄×C₁₇	—	8	2	no	—
52	C₂×C₄×C₁₇	—	8	4	no	—
53	C₂×C₄×C₁₇	#107	1	4	yes	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×Dic₁₇	—	8	2	no	—
61	C₂×Dic₁₇	—	8	4	no	—
62	C₈×C₁₇	#106	1	4	yes	—
63	C₈×C₁₇	#117	1	2	yes	—
64	C₈×C₁₇	#125	1	2	yes	—
65	C₂×C₄×C₁₆	—	17	1	no	—
66	#1846	—	17	1	no	—
67	#1846	D₁₇	1	1	yes	characteristic
68	#1380	—	17	4	no	—
69	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
70	C₂²×C₁₇	#486	1	1	yes	characteristic
71	C₂²×C₁₇	#486	1	1	yes	characteristic
72	C₂²×C₁₇	#508	1	4	yes	—
73	C₄×C₁₇	—	2	4	no	—
74	C₄×C₁₇	—	16	2	no	—
75	C₄×C₁₇	#382	1	1	yes	characteristic
76	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
77	C₄×C₁₇	#487	1	1	yes	characteristic
78	C₄×C₁₇	#487	1	1	yes	characteristic
79	Dic₁₇	—	16	2	no	—
80	C₂²×C₁₆	—	17	2	no	—
81	C₂×#107	—	17	2	no	—
82	C₂×#107	C₂×D₁₇	1	2	yes	—
83	C₂×#106	—	2	1	no	—
84	C₂×#106	—	34	1	no	—
85	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
86	#382	—	2	1	no	—
87	#382	—	34	1	no	—
88	C₄×C₁₆	—	17	4	no	—
89	C₂×C₁₇	#1834	1	1	yes	characteristic
90	C₂×C₁₇	#2158	1	2	yes	—
91	C₂×C₁₇	#2242	1	2	yes	—
92	C₂×C₁₇	#2242	1	2	yes	—
93	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	34	2	no	—
97	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
98	C₂×C₁₆	—	17	4	no	—
99	C₂×C₁₆	—	34	2	no	—
100	C₂×C₁₆	—	34	2	no	—
101	#107	—	2	4	no	—
102	#107	—	34	4	no	—
103	#106	—	4	2	no	—
104	#106	—	68	2	no	—
105	C₄×C₈	Dic₃₄	1	2	yes	—
106	C₄×C₈	C₄×D₁₇	1	2	yes	—
107	#95	—	4	2	no	—
108	#95	—	68	2	no	—
109	C₁₇	#12889	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	68	2	no	—
112	C₂²×C₄	#63228	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
114	C₂×C₈	#63225	1	2	yes	—
115	C₂×C₈	#63229	1	2	yes	—
116	C₂×C₈	#63230	1	2	yes	—
117	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
119	C₂×C₈	#63254	1	2	yes	—
120	C₂×C₈	#63256	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	68	2	no	—
124	C₄⋊C₄	—	68	2	no	—
125	C₄²	#63225	1	2	yes	—
126	C₄²	#63231	1	2	yes	—
127	C₁₆	—	34	4	no	—
128	C₂³	C₂×#63228	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	136	4	no	—
133	C₂×C₄	—	136	2	no	—
134	C₂×C₄	#10587478	1	4	yes	—
135	C₂×C₄	C₂×#63225	1	1	yes	characteristic
136	C₂×C₄	C₂×#63229	1	1	yes	characteristic
137	C₂×C₄	#10587622	1	1	yes	characteristic
138	C₂×C₄	C₂×#63230	1	1	yes	characteristic
139	C₂×C₄	C₂×#63231	1	1	yes	characteristic
140	C₂×C₄	#10587633	1	1	yes	characteristic
141	C₈	#10587479	1	4	yes	—
142	C₈	#10587576	1	2	yes	—
143	C₈	#10587578	1	2	yes	—
144	C₂²	C₂×#10587478	1	1	yes	characteristic
145	C₂²	#11763155	1	1	yes	characteristic
146	C₂²	#11763157	1	1	yes	characteristic
147	C₂²	#11763239	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	272	2	no	—
151	C₄	#11762674	1	1	yes	characteristic
152	C₄	C₂×#10587479	1	1	yes	characteristic
153	C₄	#11763159	1	1	yes	characteristic
154	C₄	#11763161	1	1	yes	characteristic
155	C₂	#14528911	1	1	yes	characteristic
156	C₂	#14530566	1	2	yes	—
157	C₂	#14531015	1	2	yes	—
158	C₂	#14531018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#63214

C₂

yes

maximal, characteristic

#14528948

C₂

yes

maximal, characteristic

#14526875

C₂

yes

maximal

C₁₇×#1846

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

C₂×#10587488

—

#11762675

—

C₄×#63214

C₄

yes

complemented

C₂×C₁₇×#107

C₂²

yes

—

C₂×C₁₇×#106

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#382

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₁₇×#107

—

C₁₇×#106

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

C₁₇×#95

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

Q₁₆

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

#12889

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#117

yes

—

C₈×C₁₇

#125

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₁₇

yes

characteristic

#1380

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₁₇

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#382

—

#382

—

C₄×C₁₆

—

C₂×C₁₇

#1834

yes

characteristic

C₂×C₁₇

#2158

yes

—

C₂×C₁₇

#2242

yes

—

C₂×C₁₇

#2242

yes

—

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

#107

—

102

#107

—

103

#106

—

104

#106

—

105

C₄×C₈

Dic₃₄

yes

—

106

C₄×C₈

C₄×D₁₇

yes

—

107

#95

—

108

#95

—

109

C₁₇

#12889

yes

characteristic, complemented, Sylow

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

#63228

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

114

C₂×C₈

#63225

yes

—

115

C₂×C₈

#63229

yes

—

116

C₂×C₈

#63230

yes

—

117

C₂×C₈

C₂×Dic₃₄

yes

characteristic

118

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

119

C₂×C₈

#63254

yes

—

120

C₂×C₈

#63256

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

#63225

yes

—

126

C₄²

#63231

yes

—

127

C₁₆

—

128

C₂³

C₂×#63228

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

136

—

133

C₂×C₄

—

136

—

134

C₂×C₄

#10587478

yes

—

135

C₂×C₄

C₂×#63225

yes

characteristic

136

C₂×C₄

C₂×#63229

yes

characteristic

137

C₂×C₄

#10587622

yes

characteristic

138

C₂×C₄

C₂×#63230

yes

characteristic

139

C₂×C₄

C₂×#63231

yes

characteristic

140

C₂×C₄

#10587633

yes

characteristic

141

C₈

#10587479

yes

—

142

C₈

#10587576

yes

—

143

C₈

#10587578

yes

—

144

C₂²

C₂×#10587478

yes

characteristic

145

C₂²

#11763155

yes

characteristic

146

C₂²

#11763157

yes

characteristic

147

C₂²

#11763239

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

272

—

151

C₄

#11762674

yes

characteristic

152

C₄

C₂×#10587479

yes

characteristic

153

C₄

#11763159

yes

characteristic

154

C₄

#11763161

yes

characteristic

155

C₂

#14528911

yes

characteristic

156

C₂

#14530566

yes

—

157

C₂

#14531015

yes

—

158

C₂

#14531018

yes

—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1846	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1846	D₁₇
hypocenter	C₁₇	#12889
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#63228

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1846

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₈

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×#63214	C₄	semidirect
109. C₁₇	#12889	semidirect

H ≅ G/N

type

9. C₄×#63214

C₄

semidirect

109. C₁₇

#12889

semidirect

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

subgroup	index
C₂×C₄×#63214	2
#14528948	2
#14526875	2
C₁₇×#1846	2
#12889	17

subgroup

index

C₂×C₄×#63214

#14528948

#14526875

C₁₇×#1846

#12889

Subgroups — #16950706

Conjugacy classes of subgroups

Complements

Maximal subgroups