Finite Group Databasebeta

← #17959464

Subgroups — #17959464

Order 4352.

402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 64 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#14544682C211yesmaximal, characteristic
2#14544038C211yesmaximal, characteristic
3#14543780C212yesmaximal
4#14543779C212yesmaximal
5#14532055C211yesmaximal, characteristic
6#11763830—21no—
7#11763752—21no—
8#11763687—21no—
9#11763687C2212yes—
10#11763654—21no—
11#11763653C2212yes—
12C2×Q8×Dic17C411yescharacteristic, complemented
13C2×#10587578C2211yescharacteristic
14#11762748C411yescharacteristic, complemented
15#11762746C2211yescharacteristic
16#11762741—21no—
17#11762731—21no—
18#11762720C2211yescharacteristic
19C4⋊C4×Dic17—21no—
20#11762660C411yescharacteristic, complemented
21#11762656—21no—
22C17×#395C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C2×C22⋊#595—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28Q8×Dic17—41no—
29Q8×Dic17—41no—
30C2×#63227—21no—
31C2×#63227—21no—
32C2×#63227D411yescharacteristic
33C2×#63227C2311yescharacteristic
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17D411yescharacteristic
38C2×#63225—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43#10587578—41no—
44#10587578—41no—
45C22⋊#3624—22no—
46(C2×#595)⋊C4—41no—
47(C2×#595)⋊C4—41no—
48#10587542—41no—
49C4⋊#3624—41no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56C23×D17C2×D411yescharacteristic
57C22×#595—42no—
58C2×Q8×C17—42no—
59C2×Q8×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C2×C4×D17—22no—
71C2×C4×D17—21no—
72C2×C4×D17—21no—
73C2×C4×D17—41no—
74C22⋊#595—42no—
75C22⋊#595—42no—
76C2×#3624—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—42no—
79#63227—41no—
80#63227—41no—
81#63227—42no—
82#63227—42no—
83C4⋊Dic17—41no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86C4⋊Dic17—41no—
87#63225—42no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90C4×Dic17—21no—
91C4×Dic17—21no—
92C4×Dic17—42no—
93C4×Dic17—41no—
94C4×Dic17—41no—
95#11810—171nomaximal, Sylow
96C23×C17C2×C22⋊C411yescharacteristic
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17#13712yes—
102C2×#595—82no—
103C2×#595—82no—
104C2×#595—82no—
105Q8×C17—42no—
106Q8×C17—42no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17#9911yescharacteristic
133C2×Dic17#9911yescharacteristic
134C4×D17—42no—
135C4×D17—42no—
136#3624—82no—
137#2396—171no—
138#1820—171no—
139#1464—171no—
140#1352—172no—
141#1351—172no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17C2×#9911yescharacteristic
145C22×C17#41911yescharacteristic
146C22×C17#42111yescharacteristic
147C2×D17—22no—
148C2×D17—41no—
149C2×D17—41no—
150C2×D17—42no—
151C2×D17—81no—
152#595—162no—
153C4×C17—42no—
154C4×C17—41no—
155C4×C17—42no—
156C4×C17—41no—
157C4×C17—41no—
158Dic17—21no—
159Dic17—21no—
160Dic17—41no—
161Dic17—41no—
162Dic17—81no—
163Dic17—81no—
164C2×C4×Q8—171no—
165C2×#117—171no—
166#432—341no—
167#400—171no—
168#399—171no—
169#398—341no—
170#396—341no—
171#395#59511yescharacteristic, complemented
172#393—171no—
173#381—341no—
174C4×C4⋊C4—171no—
175C4×C4⋊C4—341no—
176C4×C22⋊C4—341no—
177#342—172no—
178#342—341no—
179#324—341no—
180#323—172no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17#151111yescharacteristic
184C2×C17#212011yescharacteristic
185C2×C17#212611yescharacteristic
186D17—42no—
187C22×Q8C2×#59511yescharacteristic
188C23×C4—171no—
189C4×Q8—681no—
190C4×Q8—681no—
191#117—681no—
192#117—681no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4C2×#59511yescharacteristic
200C2×C22⋊C4—171no—
201C2×C22⋊C4—171no—
202C2×C22⋊C4—341no—
203C2×C22⋊C4—341no—
204C2×C22⋊C4—342no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—341no—
208C2×C42—341no—
209C4⋊C8—681no—
210C22⋊C8—342no—
211#97—681no—
212#95—21no—
213#95—171no—
214#95—341no—
215#95—341no—
216#95—341no—
217#95—681no—
218#95—681no—
219#95C2×#59511yescharacteristic
220C17#1181011yescharacteristic, complemented, Sylow
221C24—171no—
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—342no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—681no—
235C22×C4—682no—
236C22×C4C22⋊#59511yescharacteristic
237C22×C4C22⋊#59511yescharacteristic
238C22×C4C22×#59511yescharacteristic
239C2×C8—682no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—682no—
245C4⋊C4—682no—
246C4⋊C4—682no—
247C22⋊C4—681no—
248C22⋊C4—681no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4—682no—
252C22⋊C4—682no—
253C42—341no—
254C42—341no—
255C42—341no—
256C42—341no—
257C42—681no—
258C42—681no—
259C42—682no—
260C23—172no—
261C23—341no—
262C23—341no—
263C23—681no—
264C23—681no—
265C23C2×C22⋊#59511yescharacteristic
266Q8—42no—
267Q8—42no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—171no—
282C2×C4—171no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4—1362no—
300C8—1362no—
301C22—21no—
302C22—21no—
303C22—342no—
304C22—682no—
305C22—681no—
306C22—681no—
307C22—1361no—
308C22D17⋊#9911yescharacteristic
309C22#1176380211yescharacteristic
310C22#1176380611yescharacteristic
311C4—42no—
312C4—41no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—2722no—
317C4—341no—
318C4—341no—
319C4—681no—
320C4—681no—
321C4—1361no—
322C4—1361no—
323C2—21no—
324C2—21no—
325C2—682no—
326C2#1454419311yescharacteristic
327C2#1454503611yescharacteristic
328C2#1454505811yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#395C4
socleC22×C17C2×#99
prime radicalC23×D17C2×D4
cosocleC2×#63227C23
hypercenter#395#595
hypocenterC17#11810
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
12. C2×Q8×Dic17C4semidirect
14. #11762748C4semidirect
20. #11762660C4semidirect
22. C17×#395C4semidirect
171. #395#595semidirect
220. C17#11810semidirect

Maximal subgroups

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

subgroupindex
#145446822
#145440382
#145437802
#145437792
#145320552
#1181017