Finite Group Databasebeta

← #16945213

Subgroups — #16945213

Order 4352.

506 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 83 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#14533942C211yesmaximal, characteristic
2#14528858C211yesmaximal, characteristic
3#14526806C212yesmaximal, complemented
4#14526785C212yesmaximal, complemented
5C2×C17×#323C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic, complemented
7#11762734C411yescharacteristic, complemented
8#11762734C2211yescharacteristic
9#11762721C411yescharacteristic, complemented
10#11762721C2211yescharacteristic
11#11762703—21no—
12#11762703—21no—
13#11762672—21no—
14#11762665—21no—
15#11762665—21no—
16C4×C22⋊Dic17C411yescharacteristic, complemented
17#11762576—21no—
18#11762576C2212yes—
19C22×C22⋊C4×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#323—21no—
22C17×#323C2212yes—
23C25×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×C8×C17—41no—
28C2×C22⋊C4×C17—41no—
29C2×C22⋊C4×C17—41no—
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—41no—
42C2×C22⋊Dic17—41no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44#10587610—41no—
45#10587610—41no—
46C2×#63225—21no—
47C2×C4×Dic17—21no—
48C17×C22⋊C8—22no—
49C17×C22⋊C8—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C24×C17—21no—
57C24×C17—21no—
58C24×C17—22no—
59C24×C17—41no—
60C24×C17—42no—
61C24×C17—81no—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22⋊C411yescharacteristic
64C24×C17C2×D411yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—42no—
70C22×C4×C17—81no—
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—42no—
80C2×Dic34—41no—
81C2×Dic34—81no—
82C2×C8×C17—42no—
83C2×C8×C17—82no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C22⋊Dic17—41no—
91C22⋊Dic17—41no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C22⋊Dic17—42no—
95C22⋊Dic17—81no—
96C22⋊Dic17—81no—
97C22⋊Dic17—82no—
98C22⋊Dic17—82no—
99C4⋊Dic17—41no—
100#63225—42no—
101C4×Dic17—81no—
102#11845—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—42no—
113C23×C17—42no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—42no—
119C23×C17—42no—
120C23×C17—81no—
121C23×C17—81no—
122C23×C17—84no—
123C23×C17—84no—
124C23×C17#9911yescharacteristic
125C23×C17#9911yescharacteristic
126C23×C17C2×C22⋊C411yescharacteristic
127C23×C17#13512yes—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17—81no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—82no—
153C2×Dic17—82no—
154C2×Dic17—82no—
155Dic34—82no—
156C8×C17—82no—
157#2561—171no—
158#1819—171no—
159C2×#323D1711yescharacteristic
160#1351—172no—
161#1344—172no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—22no—
169C22×C17—21no—
170C22×C17—42no—
171C22×C17—42no—
172C22×C17—42no—
173C22×C17—42no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—81no—
180C22×C17—84no—
181C22×C17—81no—
182C22×C17—84no—
183C22×C17#35412yes—
184C22×C17C2×#9911yescharacteristic
185C22×C17#41611yescharacteristic
186C22×C17#42111yescharacteristic
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—41no—
191Dic17—81no—
192Dic17—81no—
193Dic17—81no—
194Dic17—81no—
195Dic17—162no—
196C2×C22⋊Q8—171no—
197C22×C22⋊C4C2×D1711yescharacteristic
198C2×C22⋊C8—21no—
199#397—171no—
200#397—171no—
201#393—171no—
202#393—171no—
203#388—341no—
204#388—341no—
205#381—341no—
206#379—341no—
207#379—341no—
208C4×C22⋊C4—171no—
209#342—172no—
210#342—341no—
211#323—21no—
212#323C2×D1712yes—
213C2×C17—21no—
214C2×C17—21no—
215C2×C17—22no—
216C2×C17—42no—
217C2×C17—41no—
218C2×C17—81no—
219C2×C17#151611yescharacteristic
220C2×C17#212111yescharacteristic
221C2×C17C2×#35411yescharacteristic
222C25D6811yescharacteristic
223C22×Q8—171no—
224C23×C4C4×D1711yescharacteristic
225C23×C4C22⋊D1711yescharacteristic
226C22×C8—41no—
227C22⋊Q8—681no—
228C22⋊Q8—681no—
229C2×C4⋊C4—171no—
230C2×C4⋊C4—341no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C22⋊C4—341no—
237C2×C22⋊C4—342no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4—341no—
240C2×C22⋊C4—681no—
241C2×C22⋊C4—681no—
242C2×C22⋊C4C4×D1711yescharacteristic
243C2×C22⋊C4D6811yescharacteristic
244C2×C22⋊C4C22⋊D1711yescharacteristic
245C2×C22⋊C4C22×D1711yescharacteristic
246C2×C42—341no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249#95—171no—
250#95—171no—
251#95—341no—
252#95—341no—
253#95—341no—
254#95—341no—
255C17#1184511yescharacteristic, complemented, Sylow
256C24—21no—
257C24—21no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—81no—
262C24#6322711yescharacteristic
263C24#6322711yescharacteristic
264C24C2×D6811yescharacteristic
265C2×Q8—681no—
266C2×Q8—1361no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—81no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4—682no—
278C22×C4#6322711yescharacteristic
279C22×C4#6322711yescharacteristic
280C22×C4C2×C4×D1711yescharacteristic
281C22×C4C2×C22⋊D1711yescharacteristic
282C2×C8—42no—
283C2×C8—82no—
284C4⋊C4—681no—
285C4⋊C4—682no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—682no—
291C22⋊C4—682no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—682no—
295C22⋊C4—682no—
296C22⋊C4—682no—
297C22⋊C4—1361no—
298C22⋊C4—1361no—
299C22⋊C4—1362no—
300C22⋊C4—1362no—
301C42—1361no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23—84no—
323C23#1058747711yescharacteristic
324C23#1058747711yescharacteristic
325C23#1058759012yes—
326C23C2×#6322711yescharacteristic
327Q8—1362no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—1362no—
353C2×C4—1362no—
354C2×C4—1362no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22#1176260312yes—
378C22C2×#1058747711yescharacteristic
379C22#1176280611yescharacteristic
380C22#1176282511yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—2722no—
386C4—1361no—
387C4—1361no—
388C4—1361no—
389C4—1361no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2#1452734111yescharacteristic
397C2#1453043111yescharacteristic
398C2C2×#1176260311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC2×C17×#323C2
socleC22×C17C2×#99
prime radicalC25×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenterC2×#323D17
hypocenterC17#11845
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #14526806C2semidirect
4. #14526785C2semidirect
6. C2×#10587610C4semidirect
7. #11762734C4semidirect
9. #11762721C4semidirect
16. C4×C22⋊Dic17C4semidirect
255. C17#11845semidirect

Maximal subgroups

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

subgroupindex
#145339422
#145288582
#145268062
#145267852
C2×C17×#3232
#1184517