Finite Group Databasebeta

← #17959054

Subgroups — #17959054

Order 4352.

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

normal subgroups only

This group has 468 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#14544483C211yesmaximal, characteristic
2#14544017C211yesmaximal, characteristic
3#14543764C214yesmaximal
4#14534551C211yesmaximal, characteristic
5C2×#10587677—21no—
6#11763825—21no—
7#11763687—21no—
8#11763687C2212yes—
9#11763655—21no—
10#11763653C2212yes—
11C2×#10587632—21no—
12C2×#10587558C2211yescharacteristic
13#11762740—21no—
14#11762740C412yescomplemented
15#11762735—21no—
16#11762735C2212yes—
17#11762716—21no—
18#11762675C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×C4×D17D411yescharacteristic
22C2×C22⋊#595—22no—
23#10587677—24no—
24#10587677—42no—
25C2×C4⋊#595—41no—
26C2×C4×#595—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×#63227D411yescharacteristic
36C2×#63227C2311yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×#63225—22no—
39C2×#63225C2×C411yescharacteristic
40C2×#63225D411yescharacteristic
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×D17C2×D411yescharacteristic
52C22×#595—42no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17—42no—
56C2×Q8×C17—42no—
57C22×C4×C17—22no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C2×C4×D17—21no—
66C2×C4×D17—21no—
67C2×C4×D17—22no—
68C2×C4×D17D812yes—
69C2×C4×D17SD1612yes—
70C22⋊#595—24no—
71C22⋊#595—42no—
72C2×#3624—42no—
73C4⋊#595—44no—
74C4×#595—44no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C42×C17—21no—
78C42×C17—21no—
79#63227—41no—
80#63227—41no—
81#63227—42no—
82#63227—42no—
83C4⋊Dic17—42no—
84#63225—41no—
85#63225—41no—
86#63225—44no—
87C4×Dic17—21no—
88C4×Dic17—21no—
89#11694—171nomaximal, Sylow
90C23×C17C2×C22⋊C411yescharacteristic
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—41no—
94C22×D17—41no—
95C22×D17C2×D811yescharacteristic
96C22×D17C2×SD1611yescharacteristic
97C2×#595—44no—
98C2×#595—82no—
99C2×#595—82no—
100Q8×C17—82no—
101Q8×C17—82no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17#9911yescharacteristic
110C2×C4×C17#9911yescharacteristic
111C2×C4×C17#10212yes—
112C2×C4×C17#10412yes—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17#10212yes—
123C4×D17—22no—
124C4×D17—22no—
125C4×D17—22no—
126C4×D17—22no—
127#3624—82no—
128#2621—171no—
129#1740—171no—
130#1458—171no—
131#1344—174no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17C2×#9911yescharacteristic
135C22×C17C2×#10211yescharacteristic
136C22×C17C2×#10411yescharacteristic
137C2×D17—21no—
138C2×D17—21no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17—41no—
143C2×D17—81no—
144#595—84no—
145C4×C17—41no—
146C4×C17—41no—
147C4×C17—82no—
148C4×C17#32712yes—
149C4×C17#32712yes—
150Dic17—22no—
151Dic17—41no—
152Dic17—41no—
153Dic17—82no—
154C2×C4⋊Q8#59511yescharacteristic, complemented
155C2×C22⋊Q8—341no—
156C2×#117—171no—
157C2×#117—341no—
158#431—341no—
159#398—172no—
160#398—341no—
161#397—172no—
162#397—341no—
163#391—341no—
164#382—171no—
165#342—172no—
166#342—341no—
167#324—341no—
168#323—172no—
169C2×C17—21no—
170C2×C17—21no—
171C2×C17C2×#32711yescharacteristic
172C2×C17#212311yescharacteristic
173C2×C17#212911yescharacteristic
174D17—41no—
175D17—41no—
176C22×Q8—21no—
177C23×C4—171no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C22⋊Q8—682no—
181C22⋊Q8—682no—
182#117—344no—
183#117—681no—
184#117—681no—
185#117—682no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—342no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—681no—
192C2×C4⋊C4C2×#59512yes—
193C2×C22⋊C4—171no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—342no—
197C2×C42—171no—
198C2×C42—681no—
199C2×C42C2×#59511yescharacteristic
200C4⋊C8—681no—
201C22⋊C8—342no—
202C4×C8—681no—
203#95—172no—
204#95—342no—
205#95—342no—
206#95—341no—
207C17#1169411yescharacteristic, complemented, Sylow
208C24—171no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C22×C4—22no—
214C22×C4—171no—
215C22×C4—172no—
216C22×C4—172no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—342no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—342no—
223C22×C4—682no—
224C22×C4C22⋊#59511yescharacteristic
225C22×C4C22⋊#59511yescharacteristic
226C22×C4C22×#59511yescharacteristic
227C2×C8—682no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—681no—
231C4⋊C4—681no—
232C4⋊C4—684no—
233C4⋊C4—684no—
234C4⋊C4—682no—
235C22⋊C4—344no—
236C22⋊C4—681no—
237C22⋊C4—681no—
238C22⋊C4—682no—
239C22⋊C4—682no—
240C22⋊C4—682no—
241C42—21no—
242C42—21no—
243C42—341no—
244C42—341no—
245C42—684no—
246C23—171no—
247C23—171no—
248C23—341no—
249C23—341no—
250C23—681no—
251C23—681no—
252C23C2×C22⋊#59511yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—172no—
263C2×C4—341no—
264C2×C4—341no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—682no—
274C2×C4—682no—
275C2×C4—684no—
276C2×C4—682no—
277C2×C4—1362no—
278C2×C4—1362no—
279C2×C4#1058754712yes—
280C2×C4#1058754812yes—
281C2×C4#1058755111yescharacteristic
282C2×C4#1058755111yescharacteristic
283C8—1362no—
284C22—21no—
285C22—21no—
286C22—341no—
287C22—341no—
288C22—681no—
289C22—681no—
290C22—681no—
291C22—681no—
292C22—1361no—
293C22C2×#1058755111yescharacteristic
294C22C2×#1058754711yescharacteristic
295C22C2×#1058754811yescharacteristic
296C4—41no—
297C4—41no—
298C4—82no—
299C4—342no—
300C4—681no—
301C4—681no—
302C4—1362no—
303C4—1364no—
304C4#1176366012yes—
305C4#1176366012yes—
306C2—21no—
307C2—21no—
308C2—681no—
309C2—681no—
310C2C2×#1176366011yescharacteristic
311C2#1454504611yescharacteristic
312C2#1454506611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×D17C2×D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11694
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
14. #11762740C4semidirect
18. #11762675C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
154. C2×C4⋊Q8#595semidirect
207. C17#11694semidirect

Maximal subgroups

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

subgroupindex
#145444832
#145440172
#145437642
#145345512
#1169417