Finite Group Databasebeta

← #16943981

Subgroups — #16943981

Order 4352.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#14533971C211yesmaximal, characteristic
2#14527108C211yesmaximal, characteristic
3#14526729C212yesmaximal
4#14526727C212yesmaximal
5C17×#1467C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic
7#11762862—21no—
8#11762742—21no—
9#11762742—21no—
10#11762716C411yescharacteristic
11#11762716C411yescharacteristic
12#11762693—21no—
13#11762693—21no—
14#11762678C2211yescharacteristic
15#11762674C2211yescharacteristic
16C4×C4⋊Dic17C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#432—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C22×Q8×C17C2×C411yescharacteristic
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—21no—
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41#10587556—41no—
42#10587556—41no—
43C17×C4⋊C8—41no—
44C17×C22⋊C8—22no—
45C17×#97—41no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C2×Q8×C17—22no—
57C2×Q8×C17—22no—
58C2×Q8×C17—41no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C2×C8×C17—42no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C42×C17—21no—
75C42×C17—21no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C4⋊Dic17—42no—
81C4⋊Dic17—41no—
82C4⋊Dic17—41no—
83#63225—42no—
84#63225—42no—
85#63225—42no—
86#63225—41no—
87#63225—41no—
88C4×Dic17—22no—
89C4×Dic17—22no—
90C2×#3622—42no—
91#11571—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93Q8×C17—44no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17#9911yescharacteristic
109C2×C4×C17#9911yescharacteristic
110C2×C4×C17#10112yes—
111C2×C4×C17#13112yes—
112C2×Dic17—22no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C8×C17—82no—
124#3622—82no—
125#2565—171no—
126#1467D1711yescharacteristic
127#1461—171no—
128#1325—172no—
129#1324—172no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17#40811yescharacteristic
133C22×C17C2×#9911yescharacteristic
134C22×C17C2×#10111yescharacteristic
135C4×C17—21no—
136C4×C17—21no—
137C4×C17—22no—
138C4×C17—22no—
139C4×C17—41no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—41no—
143Dic17—42no—
144Dic17—81no—
145Dic17—81no—
146Dic17—81no—
147C2×#125—171no—
148C2×C4×Q8C2×D1711yescharacteristic
149#432—21no—
150#431—341no—
151#398—341no—
152#398—341no—
153#391—171no—
154#391—171no—
155#387—341no—
156#387—341no—
157#384—171no—
158#382—171no—
159C4×C4⋊C4—171no—
160#324—21no—
161#324—172no—
162#324—341no—
163#324C2×D1712yes—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17#147211yescharacteristic
167C2×C17#212611yescharacteristic
168C2×C17#213311yescharacteristic
169C22×Q8C4×D1711yescharacteristic
170#125—681no—
171#125—681no—
172C4×Q8—41no—
173C4×Q8—41no—
174C2×C4⋊C4—171no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—341no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—341no—
179C2×C4⋊C4—341no—
180C2×C4⋊C4—341no—
181C2×C4⋊C4D6811yescharacteristic
182C2×C4⋊C4C22⋊D1711yescharacteristic
183C2×C4⋊C4C22×D1711yescharacteristic
184C2×C42—171no—
185C2×C42—171no—
186C2×C42C4×D1711yescharacteristic
187C2×C42D6811yescharacteristic
188C2×C42C22⋊D1711yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—681no—
191C22⋊C8—22no—
192C22⋊C8—342no—
193#97—41no—
194C4×C8—681no—
195#95—341no—
196#95—341no—
197#95—341no—
198#95—341no—
199#95—341no—
200#95—341no—
201#95—341no—
202C17#1157111yescharacteristic, complemented, Sylow
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C22×C4—341no—
207C22×C4—341no—
208C22×C4—341no—
209C22×C4—341no—
210C22×C4#6322711yescharacteristic
211C22×C4#6322711yescharacteristic
212C22×C4#6322711yescharacteristic
213C22×C4#6322711yescharacteristic
214C22×C4C2×C4×D1711yescharacteristic
215C22×C4C2×D6811yescharacteristic
216C22×C4C2×C22⋊D1711yescharacteristic
217C2×C8—42no—
218C2×C8—682no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—681no—
223C4⋊C4—681no—
224C4⋊C4—682no—
225C4⋊C4—682no—
226C4⋊C4—682no—
227C4⋊C4—682no—
228C4⋊C4—681no—
229C4⋊C4—681no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—342no—
237C42—342no—
238C23C2×#6322711yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—342no—
255C2×C4—681no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4—681no—
259C2×C4—681no—
260C2×C4—681no—
261C2×C4—682no—
262C2×C4—681no—
263C2×C4—681no—
264C2×C4—681no—
265C2×C4#1058747711yescharacteristic
266C2×C4#1058747711yescharacteristic
267C2×C4#1058749512yes—
268C2×C4#1058759312yes—
269C8—82no—
270C8—1362no—
271C22—21no—
272C22—21no—
273C22#1176277711yescharacteristic
274C22C2×#1058747711yescharacteristic
275C22C2×#1058749511yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—682no—
285C4—1361no—
286C4—1361no—
287C4—1361no—
288C2—21no—
289C2—21no—
290C2#1452714911yescharacteristic
291C2#1453044711yescharacteristic
292C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1467C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1467D17
hypocenterC17#11571
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
202. C17#11571semidirect

Maximal subgroups

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

subgroupindex
#145339712
#145271082
#145267292
#145267272
C17×#14672
#1157117