Finite Group Databasebeta

← #17959074

Subgroups — #17959074

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#14543768C214yesmaximal, complemented
4#14534551C211yesmaximal, characteristic
5C2×D17⋊M16—21no—
6#11763825—21no—
7#11763689—21no—
8#11763689C2212yes—
9#11763655C2212yes—
10#11763653—21no—
11C2×#10587632—21no—
12C2×#10587558C2211yescharacteristic
13#11762740—21no—
14#11762740C2212yes—
15#11762735—21no—
16#11762735C412yes—
17#11762716—21no—
18#11762675C411yescharacteristic
19C2×C17×C4⋊Q8C411yescharacteristic
20C22×Q8×C17—21no—
21C22×C4×D17D411yescharacteristic
22C2×#63247—22no—
23D17⋊M16—24no—
24D17⋊M16—42no—
25C2×#63243—41no—
26C2×#63242—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C42×C17C2×C411yescharacteristic
32#10587632—42no—
33#10587632—42no—
34C2×#63227—22no—
35C2×#63227C2×C411yescharacteristic
36C2×#63227D411yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×#63225—22no—
39C2×#63225D411yescharacteristic
40C2×#63225C2311yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42#10587558—41no—
43#10587558—41no—
44C22⋊#3624—22no—
45C4⋊#3624—41no—
46C4×#3624—41no—
47#10587502—21no—
48#10587502—22no—
49#10587502—22no—
50#10587502C2×C412yes—
51C23×D17C22⋊C411yescharacteristic
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C22×C4×C17—22no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—22no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C2×C4×D17—21no—
65C2×C4×D17—21no—
66C2×C4×D17—22no—
67C2×C4×D17D812yes—
68C2×C4×D17SD1612yes—
69#63247—24no—
70#63247—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73#63243—44no—
74#63242—44no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C42×C17—21no—
78C42×C17—21no—
79#63227—41no—
80#63227—41no—
81#63227—44no—
82C4⋊Dic17—42no—
83#63225—41no—
84#63225—41no—
85#63225—42no—
86#63225—42no—
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—
96Q8×C17—82no—
97Q8×C17—82no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17#9911yescharacteristic
106C2×C4×C17#9911yescharacteristic
107C2×C4×C17#10212yes—
108C2×C4×C17#10412yes—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17C2×D811yescharacteristic
119C2×Dic17C2×SD1611yescharacteristic
120C4×D17—22no—
121C4×D17—22no—
122C4×D17—22no—
123C4×D17—22no—
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#59511yescharacteristic
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—171no—
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—21no—
172C23×C4—171no—
173C2×M16—342no—
174C2×M16—344no—
175C2×M16—682no—
176C2×M16—681no—
177C22×C8—681no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
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—342no—
188C2×C4⋊C4—341no—
189C2×C4⋊C4C2×#59512yes—
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—172no—
199#95—342no—
200#95—342no—
201#95—341no—
202C17#1170011yescharacteristic, complemented, Sylow
203C24—171no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—41no—
208C22×C4—22no—
209C22×C4—171no—
210C22×C4—172no—
211C22×C4—172no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—342no—
215C22×C4—341no—
216C22×C4—341no—
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—42no—
229C4⋊C4—681no—
230C4⋊C4—681no—
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—82no—
248Q8—82no—
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#1058754712yes—
272C2×C4#1058754812yes—
273C2×C4#1058755111yescharacteristic
274C2×C4#1058755111yescharacteristic
275C8—684no—
276C8—1362no—
277C22—21no—
278C22—21no—
279C22—342no—
280C22—682no—
281C22—681no—
282C22—681no—
283C22—1361no—
284C22C2×#1058755111yescharacteristic
285C22C2×#1058754711yescharacteristic
286C22C2×#1058754811yescharacteristic
287C4—41no—
288C4—41no—
289C4—82no—
290C4—341no—
291C4—341no—
292C4—681no—
293C4—681no—
294C4—1362no—
295C4#1176366012yes—
296C4#1176366012yes—
297C2—21no—
298C2—21no—
299C2—682no—
300C2C2×#1176366011yescharacteristic
301C2#1454504611yescharacteristic
302C2#1454508611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC2×#63225C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11700
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543768C2semidirect
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
#145437682
#145345512
#1170017