Finite Group Databasebeta

← #17956562

Subgroups — #17956562

Order 4352.

640 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 110 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#14544639C211yesmaximal, characteristic
2C4×(C2×#595)⋊C4C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14537093C211yesmaximal, characteristic
6C22×C4×#595—21no—
7C2×C4⋊#3624—21no—
8C2×#10587543C2212yes—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12C2×#10587576C411yescharacteristic, complemented
13C2×C42×D17C2211yescharacteristic
14#11762707C411yescharacteristic, complemented
15#11762695—21no—
16#11762691—21no—
17#11762683—21no—
18#11762680—21no—
19#11762674C411yescharacteristic, complemented
20#11762666C2211yescharacteristic
21#11762666C2211yescharacteristic
22C17×#389C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×#3624—22no—
28C2×C4×#595—24no—
29C2×C4×#595—42no—
30C2×C4×#595—42no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×#63227—21no—
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45#10587576—42no—
46C42×D17—42no—
47(C2×#595)⋊C4—24no—
48(C2×#595)⋊C4—22no—
49(C2×#595)⋊C4Q814yes—
50#10587543—24no—
51C4⋊#3624—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C23×D17C2×D411yescharacteristic
59C22×#595—24no—
60C22×#595—24no—
61C22×#595—24no—
62C22×#595—42no—
63C22×#595—42no—
64C22×#595—42no—
65C22×#595—42no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C2×D411yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—22no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17C2×Q812yes—
86C2×C4×D17#4112yes—
87C2×#3624—42no—
88C2×#3624—42no—
89C2×#3624—42no—
90C4×#595—48no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—42no—
94C42×C17—41no—
95#63227—42no—
96#63227—41no—
97#63227—41no—
98C4⋊Dic17—42no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63225—42no—
102#63225—42no—
103C4×Dic17—22no—
104C4×Dic17—22no—
105C4×Dic17—41no—
106#9910—171nomaximal, Sylow
107C23×C17C2×C22⋊C411yescharacteristic
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17C22⋊Q812yes—
113C22×D17#13512yes—
114C22×D17AGL(1,ℤ/8)11yescharacteristic
115C22×D17#13711yescharacteristic
116C2×#595—28no—
117C2×#595—44no—
118C2×#595—44no—
119C2×#595—44no—
120C2×#595—44no—
121C2×#595—44no—
122C2×#595—44no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17#10412yes—
136C2×C4×C17#10412yes—
137C2×C4×C17C4×Q812yes—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17#9912yes—
152C2×Dic17#12411yescharacteristic
153C2×Dic17C4⋊Q811yescharacteristic
154C4×D17—24no—
155C4×D17—24no—
156C4×D17—42no—
157C4×D17—42no—
158C4×D17—44no—
159#3624—82no—
160#2851—171no—
161#1797—171no—
162C4×#95—171no—
163#1294—172no—
164#1294—172no—
165C22×C17#39111yescharacteristic
166C22×C17C2×#9911yescharacteristic
167C22×C17#41011yescharacteristic
168C22×C17#41611yescharacteristic
169C22×C17#41711yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C22×C17C2×#10411yescharacteristic
172C2×D17—22no—
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17—21no—
177C2×D17—21no—
178C2×D17—21no—
179C2×D17—21no—
180C2×D17#47812yes—
181C2×D17#47912yes—
182#595—48no—
183C4×C17—22no—
184C4×C17—22no—
185C4×C17—22no—
186C4×C17—81no—
187C4×C17—81no—
188Dic17—21no—
189Dic17—21no—
190Dic17—22no—
191Dic17—41no—
192Dic17—81no—
193Dic17—81no—
194C2×#125—171no—
195C22×C42—171no—
196C22×C42—341no—
197C2×C4⋊C8—341no—
198C2×C22⋊C8—172no—
199#389—171no—
200#389#59511yescharacteristic, complemented
201#387—341no—
202#387—341no—
203#385—341no—
204#385—341no—
205#382—171no—
206#380—171no—
207#380—171no—
208C2×#95—172no—
209C2×#95—341no—
210#336—341no—
211C2×C17#150011yescharacteristic
212C2×C17#150111yescharacteristic
213C2×C17#150411yescharacteristic
214C2×C17#150911yescharacteristic
215C2×C17#198911yescharacteristic
216C2×C17#199411yescharacteristic
217C2×C17#199611yescharacteristic
218D17—22no—
219D17—22no—
220C23×C4—171no—
221C23×C4—171no—
222C23×C4—171no—
223C23×C4—342no—
224C22×C8—342no—
225#125—682no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4C2×#59511yescharacteristic
233C2×C4⋊C4C2×#59511yescharacteristic
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C42—171no—
237C2×C42—171no—
238C2×C42—171no—
239C2×C42—344no—
240C2×C42—682no—
241C2×C42—682no—
242C2×C42—682no—
243C2×C42C2×#59511yescharacteristic
244C4⋊C8—682no—
245C22⋊C8—344no—
246#95—21no—
247#95—174no—
248#95—341no—
249#95—341no—
250#95—341no—
251#95—341no—
252#95—342no—
253#95—344no—
254#95—341no—
255C17#991011yescharacteristic, complemented, Sylow
256C24—171no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—172no—
262C22×C4—172no—
263C22×C4—171no—
264C22×C4—342no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—342no—
268C22×C4—342no—
269C22×C4—344no—
270C22×C4—344no—
271C22×C4—344no—
272C22×C4—342no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—681no—
276C22×C4—681no—
277C22×C4—681no—
278C22×C4—682no—
279C22×C4—682no—
280C22×C4—682no—
281C22×C4—682no—
282C22×C4C22⋊#59511yescharacteristic
283C22×C4C22⋊#59511yescharacteristic
284C22×C4C22×#59511yescharacteristic
285C2×C8—682no—
286C2×C8—682no—
287C2×C8—682no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—681no—
292C4⋊C4—681no—
293C4⋊C4—682no—
294C4⋊C4—682no—
295C4⋊C4—682no—
296C22⋊C4—682no—
297C22⋊C4—681no—
298C22⋊C4—681no—
299C42—41no—
300C42—342no—
301C42—342no—
302C42—681no—
303C42—688no—
304C23—172no—
305C23—172no—
306C23—171no—
307C23—171no—
308C23—341no—
309C23—341no—
310C23—341no—
311C23—341no—
312C23C2×C22⋊#59511yescharacteristic
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—171no—
326C2×C4—171no—
327C2×C4—172no—
328C2×C4—344no—
329C2×C4—344no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—348no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—682no—
339C2×C4—682no—
340C2×C4—684no—
341C2×C4—684no—
342C2×C4—684no—
343C2×C4—684no—
344C2×C4—684no—
345C2×C4—684no—
346C2×C4—684no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4#1058754812yes—
354C2×C4#1058755012yes—
355C2×C4Q8×#59512yes—
356C8—1362no—
357C22—172no—
358C22—172no—
359C22—342no—
360C22—342no—
361C22—341no—
362C22—341no—
363C22—341no—
364C22—341no—
365C22—342no—
366C22—342no—
367C22(C2×#595)⋊Q811yescharacteristic
368C22D17⋊#9911yescharacteristic
369C22#1176378811yescharacteristic
370C22#1176379911yescharacteristic
371C22#1176380011yescharacteristic
372C22C2×#1058754811yescharacteristic
373C22C2×#1058755011yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—81no—
378C4—81no—
379C4—342no—
380C4—341no—
381C4—341no—
382C4—688no—
383C4—681no—
384C4—1361no—
385C4—1361no—
386C2—342no—
387C2—342no—
388C2#1454415711yescharacteristic
389C2#1454416011yescharacteristic
390C2#1454417111yescharacteristic
391C2#1454418611yescharacteristic
392C2#1454496411yescharacteristic
393C2#1454496911yescharacteristic
394C2#1454497511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#389#595
hypocenterC17#9910
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
12. C2×#10587576C4semidirect
14. #11762707C4semidirect
19. #11762674C4semidirect
22. C17×#389C4semidirect
200. #389#595semidirect
255. C17#9910semidirect

Maximal subgroups

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

subgroupindex
#145446392
C4×(C2×#595)⋊C42
#145436572
#145370932
#991017