Finite Group Databasebeta

← #17956573

Subgroups — #17956573

Order 4352.

548 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 102 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#14544641C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14537093C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×#10587543C2212yes—
8C2×(C2×#595)⋊C4—21no—
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⋊C4×C17—21no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42#10587576—42no—
43C42×D17—42no—
44(C2×#595)⋊C4—22no—
45(C2×#595)⋊C4—42no—
46(C2×#595)⋊C4—42no—
47(C2×#595)⋊C4Q814yes—
48#10587544—42no—
49#10587543—24no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C23×D17C2×D411yescharacteristic
57C22×#595—24no—
58C22×#595—42no—
59C22×#595—42no—
60C22×#595—42no—
61C22×#595—42no—
62C22×#595—42no—
63C22×#595—42no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17C2×Q812yes—
84C2×C4×D17#4112yes—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—42no—
91C42×C17—41no—
92#63227—42no—
93#63227—41no—
94#63227—41no—
95C4⋊Dic17—42no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98#63225—42no—
99#63225—42no—
100C4×Dic17—22no—
101C4×Dic17—22no—
102C4×Dic17—41no—
103#9913—171nomaximal, Sylow
104C23×C17C2×C22⋊C411yescharacteristic
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17C22⋊Q812yes—
110C22×D17#13512yes—
111C22×D17AGL(1,ℤ/8)11yescharacteristic
112C22×D17#13711yescharacteristic
113C2×#595—44no—
114C2×#595—44no—
115C2×#595—44no—
116C2×#595—82no—
117C2×#595—82no—
118C2×#595—82no—
119C2×#595—82no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17C4×Q812yes—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17#9912yes—
149C2×Dic17#12411yescharacteristic
150C2×Dic17C4⋊Q811yescharacteristic
151C4×D17—42no—
152C4×D17—42no—
153C4×D17—44no—
154C4×D17—42no—
155C4×D17—42no—
156#3624—82no—
157#2851—171no—
158#1797—171no—
159#1434—171no—
160#1294—172no—
161#1294—172no—
162C22×C17#39111yescharacteristic
163C22×C17C2×#9911yescharacteristic
164C22×C17#41011yescharacteristic
165C22×C17#41611yescharacteristic
166C22×C17#41711yescharacteristic
167C22×C17#42111yescharacteristic
168C22×C17#42111yescharacteristic
169C2×D17—22no—
170C2×D17—22no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—22no—
176C2×D17—22no—
177C2×D17#47812yes—
178C2×D17#47912yes—
179#595—84no—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—81no—
184C4×C17—81no—
185Dic17—21no—
186Dic17—21no—
187Dic17—22no—
188Dic17—41no—
189Dic17—81no—
190Dic17—81no—
191C2×#125—171no—
192C22×C42—171no—
193C2×C4⋊C8—341no—
194C2×C22⋊C8—172no—
195#389—171no—
196#389#59511yescharacteristic, complemented
197#387—341no—
198#387—341no—
199#385—341no—
200#385—341no—
201#382—171no—
202#380—171no—
203#380—171no—
204C2×#95—172no—
205C2×#95—341no—
206C2×#95—341no—
207#336—341no—
208C2×C17#151211yescharacteristic
209C2×C17#151311yescharacteristic
210C2×C17#151611yescharacteristic
211C2×C17#152111yescharacteristic
212C2×C17#198911yescharacteristic
213C2×C17#199511yescharacteristic
214C2×C17#199611yescharacteristic
215D17—22no—
216D17—22no—
217C23×C4—171no—
218C23×C4—171no—
219C23×C4—171no—
220C23×C4—342no—
221C22×C8—342no—
222#125—682no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4C2×#59511yescharacteristic
230C2×C4⋊C4C2×#59511yescharacteristic
231C2×C22⋊C4—341no—
232C2×C22⋊C4—341no—
233C2×C42—171no—
234C2×C42—171no—
235C2×C42—171no—
236C2×C42—682no—
237C2×C42C2×#59511yescharacteristic
238C4⋊C8—682no—
239C22⋊C8—344no—
240#95—21no—
241#95—174no—
242#95—341no—
243#95—341no—
244#95—341no—
245#95—341no—
246#95—342no—
247#95—341no—
248#95—682no—
249#95—682no—
250C17#991311yescharacteristic, complemented, Sylow
251C24—171no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—171no—
259C22×C4—342no—
260C22×C4—341no—
261C22×C4—342no—
262C22×C4—341no—
263C22×C4—344no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4—681no—
271C22×C4—682no—
272C22×C4—682no—
273C22×C4—682no—
274C22×C4—682no—
275C22×C4—682no—
276C22×C4—682no—
277C22×C4C22⋊#59511yescharacteristic
278C22×C4C22⋊#59511yescharacteristic
279C22×C4C22×#59511yescharacteristic
280C2×C8—682no—
281C2×C8—682no—
282C2×C8—682no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—682no—
289C4⋊C4—682no—
290C4⋊C4—682no—
291C22⋊C4—682no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C42—41no—
295C42—342no—
296C42—342no—
297C42—681no—
298C23—172no—
299C23—172no—
300C23—171no—
301C23—171no—
302C23—341no—
303C23—341no—
304C23—341no—
305C23—341no—
306C23C2×C22⋊#59511yescharacteristic
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—171no—
322C2×C4—171no—
323C2×C4—172no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—682no—
332C2×C4—682no—
333C2×C4—684no—
334C2×C4—682no—
335C2×C4—682no—
336C2×C4—684no—
337C2×C4—684no—
338C2×C4—684no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—1362no—
346C2×C4—1362no—
347C2×C4—1362no—
348C2×C4—1362no—
349C2×C4Q8×#59512yes—
350C8—1362no—
351C22—172no—
352C22—172no—
353C22—341no—
354C22—341no—
355C22—341no—
356C22—341no—
357C22—342no—
358C22—342no—
359C22—342no—
360C22—342no—
361C22(C2×#595)⋊Q811yescharacteristic
362C22D17⋊#9911yescharacteristic
363C22#1176378811yescharacteristic
364C22#1176379911yescharacteristic
365C22#1176380011yescharacteristic
366C22#1176380711yescharacteristic
367C22#1176380811yescharacteristic
368C4—22no—
369C4—41no—
370C4—41no—
371C4—81no—
372C4—81no—
373C4—341no—
374C4—341no—
375C4—342no—
376C4—681no—
377C4—1364no—
378C4—1361no—
379C4—1361no—
380C2—342no—
381C2—342no—
382C2#1454419511yescharacteristic
383C2#1454420211yescharacteristic
384C2#1454420911yescharacteristic
385C2#1454422811yescharacteristic
386C2#1454496411yescharacteristic
387C2#1454497211yescharacteristic
388C2#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#9913
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
196. #389#595semidirect
250. C17#9913semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145439602
#145436572
#145370932
#991317