Finite Group Databasebeta

← #16947166

Subgroups — #16947166

Order 4352.

380 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14536777C211yesmaximal, characteristic
2#14527758C212yesmaximal
3#14526813C212yesmaximal
4#14526718C211yesmaximal, characteristic
5C17×#2028C211yesmaximal, characteristic
6#11762964—22no—
7#11762730—22no—
8#11762729C2212yes—
9#11762716—21no—
10#11762715—21no—
11#11762713—22no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#419—21no—
16C17×#391C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×Q8×C17D411yescharacteristic
20C22×Dic34—21no—
21C22×Dic34D412yes—
22C2×M16×C17—21no—
23C17×C4⋊Q8—41no—
24C17×#125—41no—
25C17×C22⋊Q8—22no—
26C17×#117—21no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×#63231—42no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×#63225—22no—
33C2×C4×Dic17—21no—
34C17×#103—41no—
35C17×#103—41no—
36C17×#95—22no—
37#10587502—22no—
38#10587502—22no—
39#10587502—22no—
40#10587502D412yes—
41C22⋊#3622—22no—
42#10587481—42no—
43C2×Q8×C17—21no—
44C2×Q8×C17—22no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17Q1612yescomplemented
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17C2×D412yes—
51C22×C4×C17C2×D411yescharacteristic
52C22×Dic17—22no—
53C22×Dic17—22no—
54C2×Dic34—24no—
55C2×Dic34—42no—
56C2×Dic34—82no—
57C2×Dic34—81no—
58C2×Dic34—81no—
59M16×C17—41no—
60M16×C17—41no—
61C2×C8×C17—41no—
62C4⋊C4×C17—22no—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C22⋊C4×C17—41no—
68C42×C17—41no—
69C42×C17—41no—
70#63231—84no—
71C4⋊Dic17—81no—
72C4⋊Dic17—81no—
73#63225—82no—
74C4×Dic17—81no—
75C2×#3622—42no—
76#12224—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—42no—
79Q8×C17—81no—
80Q8×C17—82no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—41no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—42no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17C2×Q1611yescharacteristic
94C2×C4×C17#13711yescharacteristic
95C2×Dic17—24no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101Dic34—44no—
102Dic34—84no—
103Dic34—82no—
104C8×C17—81no—
105#3622—82no—
106#2811—171no—
107#2028D1711yescharacteristic
108#1605—172no—
109#1354—172no—
110#1319—171no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17#45111yescharacteristic
114C22×C17#45511yescharacteristic
115C22×C17B(4,𝔽2)11yescharacteristic
116C4×C17—21no—
117C4×C17—21no—
118C4×C17—42no—
119C4×C17—42no—
120C4×C17—42no—
121C4×C17—81no—
122C4×C17—81no—
123Dic17—44no—
124Dic17—82no—
125#544C2×D1711yescharacteristic
126#451—342no—
127#419—21no—
128#395—172no—
129#395—342no—
130#391—342no—
131#391—341no—
132#391—341no—
133#391C2×D1711yescharacteristic
134#343C2×D1711yescharacteristic
135#339—21no—
136#328—342no—
137#324—172no—
138C2×C17—21no—
139C2×C17—21no—
140C2×C17#160211yescharacteristic
141C2×C17#219911yescharacteristic
142C2×C17#220511yescharacteristic
143C22×Q8—172no—
144C22×Q8—341no—
145C22×Q8C22⋊D1711yescharacteristic
146C2×Q16—682no—
147C2×M16—21no—
148C4⋊Q8—41no—
149#125—41no—
150C22⋊Q8—22no—
151#117—21no—
152C2×C4⋊C4—341no—
153C2×C4⋊C4—341no—
154C2×C4⋊C4—342no—
155C2×C4⋊C4C22×D1711yescharacteristic
156C2×C42—341no—
157C2×C42C22⋊D1711yescharacteristic
158#103—41no—
159#103—41no—
160#103—682no—
161C22⋊C8—342no—
162#95—22no—
163#95—172no—
164#95—342no—
165#95—342no—
166#95—342no—
167C17#1222411yescharacteristic, complemented, Sylow
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—344no—
173C2×Q8—682no—
174C2×Q8—1362no—
175C2×Q8—1361no—
176C2×Q8—1361no—
177C2×Q8#6323112yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4D4×D1712yes—
183C22×C4C2×C22⋊D1711yescharacteristic
184Q16—1364no—
185M16—41no—
186M16—41no—
187C2×C8—41no—
188C2×C8—682no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—1362no—
195C4⋊C4—1361no—
196C4⋊C4—1361no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—1361no—
201C23#1058762311yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—82no—
205Q8—684no—
206Q8—1364no—
207Q8—1362no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—344no—
221C2×C4—682no—
222C2×C4—682no—
223C2×C4—682no—
224C2×C4—682no—
225C2×C4—682no—
226C2×C4C2×#6323111yescharacteristic
227C2×C4#1058763711yescharacteristic
228C8—81no—
229C8—1362no—
230C22—21no—
231C22—21no—
232C22#1176296411yescharacteristic
233C22#1176298611yescharacteristic
234C22#1176299511yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—684no—
243C4—1362no—
244C2—21no—
245C2—21no—
246C2#1452774011yescharacteristic
247C2#1453078411yescharacteristic
248C2#1453081311yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2028C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2028D17
hypocenterC17#12224
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
47. C2×Q8×C17Q16semidirect
167. C17#12224semidirect
177. C2×Q8#63231semidirect

Maximal subgroups

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

subgroupindex
#145367772
#145277582
#145268132
#145267182
C17×#20282
#1222417