Finite Group Databasebeta

← #16945227

Subgroups — #16945227

Order 4352.

572 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 91 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#14528864C211yesmaximal, characteristic, complemented
2C2×#11762535C211yesmaximal, characteristic, complemented
3#14526814C212yesmaximal, complemented
4#14526787C212yesmaximal, complemented
5C17×#2406C211yesmaximal, characteristic
6#11762773—21no—
7C2×C22⋊#3622—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535—21no—
11#11762535C2212yes—
12C22×#41×C17C411yescharacteristic
13C2×C17×#123—21no—
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#395C411yescharacteristic
16C17×#395C2211yescharacteristic
17C17×#394C411yescharacteristic
18C17×#394C2211yescharacteristic
19C17×#386—21no—
20C17×#385—21no—
21C17×#385—21no—
22C17×#381C411yescharacteristic
23C2×#41×C17—41no—
24C2×#41×C17—41no—
25C2×#41×C17—44no—
26C22×Q8×C17C2×C411yescharacteristic
27C22×D4×C17—21no—
28C22×D4×C17C2×C411yescharacteristic
29C23×C4×C17—21no—
30C23×C4×C17C2×C411yescharacteristic
31C23×C4×C17D411yescharacteristic
32C23×C4×C17D411yescharacteristic
33C17×#123—44no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—41no—
37C2×C4⋊C4×C17—41no—
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C4⋊C4×C17D411yescharacteristic
41C2×C4⋊C4×C17C2311yescharacteristic
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—21no—
47C2×#63223—22no—
48C22×#3622—41no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95C2×C411yescharacteristic
53C17×#95C2×C411yescharacteristic
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C22⋊#3622—41no—
57C22⋊#3622—41no—
58C24×C17—21no—
59C24×C17C22⋊C411yescharacteristic
60#41×C17—44no—
61#41×C17—44no—
62#41×C17—44no—
63#41×C17—44no—
64C2×Q8×C17—42no—
65C2×Q8×C17—41no—
66C2×D4×C17—24no—
67C2×D4×C17—41no—
68C2×D4×C17—42no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C2×D4×C17—42no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—22no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—41no—
83C22×C4×C17—42no—
84C22×C4×C17—42no—
85C22×C4×C17—42no—
86C22×C4×C17—42no—
87C22×C4×C17—42no—
88C22×C4×C17C22⋊C411yescharacteristic
89C22×C4×C17C22⋊C411yescharacteristic
90C22×C4×C17C22⋊C411yescharacteristic
91C22×C4×C17C22×C411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—42no—
99C4⋊C4×C17—42no—
100C22⋊C4×C17—42no—
101C22⋊C4×C17—42no—
102C22⋊C4×C17—42no—
103C22⋊C4×C17—42no—
104C22⋊C4×C17—42no—
105#63223—42no—
106#63223—42no—
107C2×#3622—42no—
108C2×#3622—42no—
109C2×#3622—82no—
110#11848—171nomaximal, Sylow
111C23×C17—22no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—22no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17#9911yescharacteristic
123C23×C17#9911yescharacteristic
124C23×C17C2×C22⋊C411yescharacteristic
125Q8×C17—42no—
126Q8×C17—42no—
127D4×C17—44no—
128D4×C17—42no—
129D4×C17—42no—
130D4×C17—44no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—22no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—41no—
144C2×C4×C17—42no—
145C2×C4×C17—48no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—81no—
160C2×C4×C17#13512yes—
161#3622—82no—
162#3622—82no—
163#2406D1711yescharacteristic
164#1821—171no—
165C2×#324—171no—
166#1354—172no—
167#1345—172no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—42no—
178C22×C17—42no—
179C22×C17—42no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17—81no—
184C22×C17AΣL(1,𝔽16)12yes—
185C22×C17C2×#9911yescharacteristic
186C22×C17#41611yescharacteristic
187C22×C17#42111yescharacteristic
188C4×C17—22no—
189C4×C17—41no—
190C4×C17—41no—
191C4×C17—42no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17—41no—
195C4×C17—81no—
196C4×C17—81no—
197C22×#41Dic1711yescharacteristic
198C2×#123—21no—
199C22×C4⋊C4C2×D1711yescharacteristic
200#407—341no—
201C2×C22⋊C8—341no—
202#395Dic1711yescharacteristic
203#395C2×D1711yescharacteristic
204#394Dic1711yescharacteristic
205#394C2×D1711yescharacteristic
206#386—21no—
207#385—21no—
208#385—21no—
209#381Dic1711yescharacteristic
210#343—172no—
211#343—341no—
212#324—172no—
213#324—341no—
214C2×C17—21no—
215C2×C17—21no—
216C2×C17—22no—
217C2×C17—41no—
218C2×C17—82no—
219C2×C17#151611yescharacteristic
220C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
221C2×C17#213311yescharacteristic
222C2×#41—41no—
223C2×#41—41no—
224C2×#41—44no—
225C22×Q8C2×Dic1711yescharacteristic
226C22×D4—21no—
227C22×D4C2×Dic1711yescharacteristic
228C23×C4—21no—
229C23×C4C2×Dic1711yescharacteristic
230C23×C4C22⋊D1711yescharacteristic
231C23×C4C22⋊D1711yescharacteristic
232C2×M16—342no—
233C22×C8—681no—
234#123—44no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×Dic1711yescharacteristic
240C2×C4⋊C4C22⋊D1711yescharacteristic
241C2×C4⋊C4C22⋊D1711yescharacteristic
242C2×C4⋊C4C22×D1711yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C22⋊C8—342no—
249C22⋊C8—681no—
250C22⋊C8—681no—
251C22⋊C8—681no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C2×Dic1711yescharacteristic
256#95C2×Dic1711yescharacteristic
257C17#1184811yescharacteristic, complemented, Sylow
258C24—21no—
259C24C22⋊Dic1711yescharacteristic
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊Dic1711yescharacteristic
289C22×C4C22⋊Dic1711yescharacteristic
290C22×C4C22⋊Dic1711yescharacteristic
291C22×C4C22×Dic1711yescharacteristic
292C22×C4C2×C22⋊D1711yescharacteristic
293C22×C4C2×C22⋊D1711yescharacteristic
294M16—682no—
295M16—682no—
296C2×C8—682no—
297C2×C8—682no—
298C2×C8—1362no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#1058750511yescharacteristic
322C23#1058750511yescharacteristic
323C23C2×C22⋊Dic1711yescharacteristic
324Q8—42no—
325Q8—42no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4#1058763612yes—
360C8—1362no—
361C8—1362no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1176259612yes—
379C22C2×#1058750511yescharacteristic
380C22#1176280511yescharacteristic
381C22#1176282611yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—82no—
396C2#1452734011yescharacteristic
397C2C2×#1176259611yescharacteristic
398C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2406C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2406D17
hypocenterC17#11848
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. #14528864C2semidirect
2. C2×#11762535C2semidirect
3. #14526814C2semidirect
4. #14526787C2semidirect
257. C17#11848semidirect

Maximal subgroups

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

subgroupindex
#145288642
C2×#117625352
#145268142
#145267872
C17×#24062
#1184817