Finite Group Databasebeta

← #17958661

Subgroups — #17958661

Order 4352.

554 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 74 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#14544043C211yesmaximal, characteristic
2#14544019C211yesmaximal, characteristic
3#14543729C214yesmaximal, complemented
4#14532143C211yesmaximal, characteristic
5#11763830—21no—
6#11763777—21no—
7#11763655—21no—
8#11763653—21no—
9#11763652C2212yes—
10#11763652C2212yes—
11C2×#10587632—21no—
12C2×C4×C22⋊D17C2211yescharacteristic
13#11762728—21no—
14#11762720—21no—
15#11762716—21no—
16#11762688C412yes—
17#11762682C2212yes—
18#11762669C411yescharacteristic
19C23×Q8×C17C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—21no—
22C22×Q8×C17—41no—
23C22×Q8×C17—41no—
24C22×Q8×C17—44no—
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C412yes—
27C22×C22⋊D17C2×C411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C2×#63242—41no—
30C2×C22⋊Dic17C2×C412yes—
31C2×C22⋊Dic17C2311yescharacteristic
32#10587632—44no—
33C4×C22⋊D17—41no—
34C4×C22⋊D17—41no—
35C2×#63227—22no—
36C2×#63227D411yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×#63225—22no—
39C2×#63225D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C22⋊#3624—22no—
42C22⋊#3624—22no—
43#10587543—41no—
44#10587542—41no—
45C4⋊#3624—41no—
46#10587502—21no—
47#10587502—22no—
48#10587502—22no—
49C24×C17C22×C411yescharacteristic
50C23×D17C22⋊C411yescharacteristic
51C2×Q8×C17—24no—
52C2×Q8×C17—42no—
53C2×Q8×C17—44no—
54C2×Q8×C17—42no—
55C2×Q8×C17—44no—
56C2×Q8×C17—48no—
57C2×Q8×C17—44no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C22×C4×C17—22no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—44no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C2×C22⋊D17—22no—
69C2×C22⋊D17—22no—
70C2×C22⋊D17—41no—
71C22×Dic17—22no—
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80#63242—82no—
81C22⋊Dic17—24no—
82C22⋊Dic17—41no—
83#63227—41no—
84#63227—44no—
85C4⋊Dic17—42no—
86#63225—41no—
87#63225—44no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90#11575—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17C2×C22⋊C411yescharacteristic
96C23×C17#13112yes—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17#10112yes—
102Q8×C17—44no—
103Q8×C17—48no—
104Q8×C17—44no—
105C2×C4×C17—22no—
106C2×C4×C17—24no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—44no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—44no—
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#9911yescharacteristic
117C22⋊D17—44no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C4×D17—42no—
129C4×D17—81no—
130#3624—82no—
131#3624—82no—
132#2409—171no—
133#1466—171no—
134#1459—171no—
135#1326—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17#40811yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10111yescharacteristic
146C2×D17—22no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—42no—
150C2×D17—81no—
151C4×C17—21no—
152C4×C17—44no—
153C4×C17—41no—
154C4×C17AΣL(1,𝔽16)12yes—
155Dic17—41no—
156Dic17—41no—
157Dic17—41no—
158Dic17—82no—
159C23×Q8#59511yescharacteristic
160C2×C22⋊Q8—341no—
161C2×C4×D4—171no—
162#432—341no—
163C2×C22⋊C8—341no—
164#395—341no—
165#393—341no—
166#391—341no—
167#386—172no—
168#385—172no—
169#380—171no—
170#324—341no—
171#323—172no—
172#323—172no—
173#323—341no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—42no—
177C2×C17#147211yescharacteristic
178C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
179C2×C17#212311yescharacteristic
180D17—42no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—44no—
186C22×D4—171no—
187C23×C4—171no—
188C23×C4C2×#59512yes—
189C23×C4C2×#59511yescharacteristic
190C22×C8—681no—
191C22⋊Q8—684no—
192C4×D4—681no—
193C4×D4—681no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—341no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4—171no—
199C2×C22⋊C4—171no—
200C2×C22⋊C4—342no—
201C2×C42—171no—
202C4⋊C8—681no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205C22⋊C8—681no—
206#97—681no—
207#95—342no—
208#95—342no—
209#95—341no—
210C17#1157511yescharacteristic, complemented, Sylow
211C24—171no—
212C24C22×#59511yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—342no—
223C2×D4—342no—
224C2×D4—681no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—341no—
235C22×C4—341no—
236C22×C4—342no—
237C22×C4—682no—
238C22×C4C22⋊#59511yescharacteristic
239C22×C4C22⋊#59511yescharacteristic
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—1362no—
243C4⋊C4—681no—
244C4⋊C4—684no—
245C4⋊C4—682no—
246C22⋊C4—344no—
247C22⋊C4—681no—
248C22⋊C4—684no—
249C22⋊C4—681no—
250C42—341no—
251C42—341no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—172no—
257C23—341no—
258C23—341no—
259C23—681no—
260C23—681no—
261C23#1058768012yes—
262C23C2×C22⋊#59511yescharacteristic
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—684no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—682no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—1361no—
289C2×C4#1058755111yescharacteristic
290C2×C4#1058755111yescharacteristic
291C8—1362no—
292C8—1362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—342no—
301C22—682no—
302C22—681no—
303C22—681no—
304C22—1361no—
305C22#1176378411yescharacteristic
306C22C2×#1058755111yescharacteristic
307C22D17⋊#10111yescharacteristic
308C4—21no—
309C4—44no—
310C4—41no—
311C4—681no—
312C4—681no—
313C4—681no—
314C4—1362no—
315C4#1176370912yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—682no—
320C2#1454406411yescharacteristic
321C2C2×#1176370911yescharacteristic
322C2#1454504611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC22×C22⋊D17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenterC23×Q8#595
hypocenterC17#11575
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #14543729C2semidirect
210. C17#11575semidirect

Maximal subgroups

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

subgroupindex
#145440432
#145440192
#145437292
#145321432
#1157517