Finite Group Databasebeta

← #17959076

Subgroups — #17959076

Order 4352.

452 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 88 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#14544489C211yesmaximal, characteristic
2#14544017C211yesmaximal, characteristic
3#14543769C214yesmaximal, complemented
4#14534553C211yesmaximal, characteristic
5C2×D17⋊M16—21no—
6#11763825—21no—
7#11763689—21no—
8#11763689C2212yes—
9#11763655C2212yes—
10#11763653—21no—
11C2×#10587555C411yescharacteristic
12C2×#10587583—21no—
13C2×#10587558C2211yescharacteristic
14#11762741—21no—
15#11762741C2212yes—
16#11762736—21no—
17#11762736C412yes—
18#11762714—21no—
19C17×#382C411yescharacteristic
20C22×C4×D17D411yescharacteristic
21C22×Dic34—21no—
22C2×#63247—22no—
23D17⋊M16—24no—
24D17⋊M16—42no—
25C2×#63243—41no—
26C2×#63242—41no—
27C2×C4⋊C4×C17—21no—
28C2×C42×C17C2×C411yescharacteristic
29C2×#63227—22no—
30C2×#63227C2×C411yescharacteristic
31C2×#63227D411yescharacteristic
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17C2×C412yes—
35C2×#63225D411yescharacteristic
36C2×#63225C2311yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38#10587583—42no—
39#10587583—42no—
40#10587558—41no—
41#10587558—41no—
42#10587555—41no—
43#10587555—41no—
44C22⋊#3624—22no—
45C4⋊#3624—41no—
46C4×#3624—41no—
47C17×#95—21no—
48C17×#95C2×C412yes—
49#10587502—22no—
50#10587502—22no—
51C23×D17C22⋊C411yescharacteristic
52C22×C4×C17—22no—
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22×C411yescharacteristic
56C22×Dic17—22no—
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C2×D411yescharacteristic
59C22×Dic17C2×D411yescharacteristic
60C2×C4×D17—21no—
61C2×C4×D17—21no—
62C2×C4×D17—22no—
63C2×C4×D17D812yes—
64C2×C4×D17SD1612yes—
65C2×Dic34—41no—
66C2×Dic34—41no—
67C2×Dic34—42no—
68C2×Dic34—42no—
69#63247—24no—
70#63247—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73#63243—44no—
74#63242—44no—
75C4⋊C4×C17—42no—
76C42×C17—21no—
77C42×C17—21no—
78#63227—41no—
79#63227—41no—
80#63227—44no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85#63225—41no—
86#63225—41no—
87C4×Dic17—21no—
88C4×Dic17—21no—
89#11700—171nomaximal, Sylow
90C23×C17C2×C22⋊C411yescharacteristic
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—41no—
94C22×D17—41no—
95C22×D17#10212yes—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17#9911yescharacteristic
104C2×C4×C17#9911yescharacteristic
105C2×C4×C17#10212yes—
106C2×C4×C17#10412yes—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—41no—
112C2×Dic17—41no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17C2×D811yescharacteristic
117C2×Dic17C2×SD1611yescharacteristic
118C4×D17—22no—
119C4×D17—22no—
120C4×D17—22no—
121C4×D17—22no—
122Dic34—82no—
123Dic34—82no—
124#3624—44no—
125#3624—82no—
126#2621—171no—
127#1742—171no—
128#1458—171no—
129#1346—174no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17C2×#9911yescharacteristic
133C22×C17C2×#10211yescharacteristic
134C22×C17C2×#10411yescharacteristic
135C2×D17—22no—
136C2×D17—41no—
137C2×D17—41no—
138C2×D17—42no—
139C2×D17—81no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—82no—
143C4×C17#32712yes—
144C4×C17#32712yes—
145Dic17—21no—
146Dic17—21no—
147Dic17—41no—
148Dic17—41no—
149Dic17—82no—
150C22×M16—341no—
151C2×C4⋊Q8—171no—
152C2×C22⋊Q8—341no—
153C2×#117—171no—
154#431—341no—
155#398—172no—
156#398—341no—
157#397—172no—
158#397—341no—
159#391—341no—
160#382#59511yescharacteristic
161#343—172no—
162#343—341no—
163#324—172no—
164#323—341no—
165C2×C17—21no—
166C2×C17—21no—
167C2×C17C2×#32711yescharacteristic
168C2×C17#212311yescharacteristic
169C2×C17#213511yescharacteristic
170D17—42no—
171C22×Q8—341no—
172C23×C4—171no—
173C2×M16—342no—
174C2×M16—344no—
175C2×M16—682no—
176C2×M16—681no—
177C22×C8—681no—
178C4⋊Q8—681no—
179C4⋊Q8—681no—
180C22⋊Q8—682no—
181C22⋊Q8—682no—
182#117—681no—
183#117—681no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—171no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—172no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4—341no—
190C2×C22⋊C4—171no—
191C2×C22⋊C4—171no—
192C2×C22⋊C4—342no—
193C2×C42—171no—
194C2×C42C2×#59511yescharacteristic
195C4⋊C8—681no—
196C22⋊C8—342no—
197C4×C8—681no—
198#95—21no—
199#95—342no—
200#95—342no—
201#95C2×#59512yes—
202C17#1170011yescharacteristic, complemented, Sylow
203C24—171no—
204C2×Q8—681no—
205C2×Q8—681no—
206C2×Q8—682no—
207C2×Q8—682no—
208C22×C4—22no—
209C22×C4—171no—
210C22×C4—172no—
211C22×C4—172no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—341no—
215C22×C4—341no—
216C22×C4—342no—
217C22×C4—342no—
218C22×C4C22⋊#59511yescharacteristic
219C22×C4C22⋊#59511yescharacteristic
220C22×C4C22×#59511yescharacteristic
221M16—344no—
222M16—682no—
223M16—684no—
224C2×C8—684no—
225C2×C8—682no—
226C2×C8—682no—
227C4⋊C4—42no—
228C4⋊C4—681no—
229C4⋊C4—681no—
230C4⋊C4—682no—
231C4⋊C4—682no—
232C4⋊C4—682no—
233C4⋊C4—682no—
234C22⋊C4—681no—
235C22⋊C4—681no—
236C22⋊C4—684no—
237C42—21no—
238C42—21no—
239C42—341no—
240C42—341no—
241C23—172no—
242C23—341no—
243C23—341no—
244C23—681no—
245C23—681no—
246C23C2×C22⋊#59511yescharacteristic
247Q8—1362no—
248Q8—1362no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—171no—
257C2×C4—171no—
258C2×C4—341no—
259C2×C4—341no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4#1058753911yescharacteristic
272C2×C4#1058753911yescharacteristic
273C2×C4#1058754712yes—
274C2×C4#1058755012yes—
275C8—684no—
276C8—1362no—
277C22—21no—
278C22—21no—
279C22—342no—
280C22—682no—
281C22—681no—
282C22—681no—
283C22—1361no—
284C22C2×#1058753911yescharacteristic
285C22C2×#1058754711yescharacteristic
286C22C2×#1058755011yescharacteristic
287C4—41no—
288C4—41no—
289C4—82no—
290C4—341no—
291C4—341no—
292C4—681no—
293C4—681no—
294C4—1362no—
295C4#1176366112yes—
296C4#1176366112yes—
297C2—21no—
298C2—21no—
299C2—682no—
300C2C2×#1176366111yescharacteristic
301C2#1454504811yescharacteristic
302C2#1454508811yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#382C4
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC2×#63225C23
hypercenter#382#595
hypocenterC17#11700
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #14543769C2semidirect
202. C17#11700semidirect

Maximal subgroups

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

subgroupindex
#145444892
#145440172
#145437692
#145345532
#1170017