Finite Group Databasebeta

← #17956558

Subgroups — #17956558

Order 4352.

684 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 120 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
2C4×(C2×#595)⋊C4C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14537135C211yesmaximal, characteristic
6C22×C4×#595—21no—
7C2×#10587544—21no—
8C2×#10587543C2212yes—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12C2×#10587574C411yescharacteristic, complemented
13C2×#10587582—21no—
14C2×C42×D17C2211yescharacteristic
15#11762713—21no—
16#11762707C411yescharacteristic, complemented
17#11762704—21no—
18#11762678C411yescharacteristic, complemented
19#11762668—21no—
20#11762666C2211yescharacteristic
21#11762666C2211yescharacteristic
22C17×#389C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×Dic34—21no—
28C22×#3624—22no—
29C2×C4×#595—24no—
30C2×C4×#595—42no—
31C2×C4×#595—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47#10587582—44no—
48#10587574—42no—
49C42×D17—42no—
50(C2×#595)⋊C4—24no—
51(C2×#595)⋊C4—22no—
52(C2×#595)⋊C4Q814yescomplemented
53#10587544—42no—
54#10587543—24no—
55C17×#95—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59C23×D17C2×D411yescharacteristic
60C22×#595—24no—
61C22×#595—24no—
62C22×#595—24no—
63C22×#595—42no—
64C22×#595—42no—
65C22×#595—42no—
66C22×#595—42no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C22⋊C411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17SD1612yes—
85C2×C4×D17Q1612yes—
86C2×C4×D17C2×Q812yes—
87C2×C4×D17#4112yes—
88C2×Dic34—42no—
89C2×Dic34—42no—
90C2×Dic34—42no—
91C2×#3624—42no—
92C2×#3624—42no—
93C2×#3624—42no—
94C4×#595—48no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—42no—
98C42×C17—41no—
99#63227—42no—
100#63227—42no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103#63225—42no—
104#63225—42no—
105#63225—42no—
106#63225—42no—
107C4×Dic17—22no—
108C4×Dic17—22no—
109C4×Dic17—41no—
110#9908—171nomaximal, Sylow
111C23×C17C2×C22⋊C411yescharacteristic
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17C22⋊Q812yes—
117C22×D17C2×SD1611yescharacteristic
118C22×D17C2×Q1611yescharacteristic
119C22×D17AGL(1,ℤ/8)12yes—
120C2×#595—28no—
121C2×#595—44no—
122C2×#595—44no—
123C2×#595—44no—
124C2×#595—44no—
125C2×#595—44no—
126C2×#595—44no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17#10312yes—
141C2×C4×C17C4×Q812yes—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17#9912yes—
153C2×Dic17#10312yes—
154C2×Dic17#10412yes—
155C2×Dic17#10412yes—
156C2×Dic17#12411yescharacteristic
157C2×Dic17C4⋊Q811yescharacteristic
158C4×D17—24no—
159C4×D17—24no—
160C4×D17—42no—
161C4×D17—42no—
162C4×D17—44no—
163Dic34—84no—
164#3624—82no—
165#2857—171no—
166#1797—171no—
167C4×#95—171no—
168#1294—172no—
169#1294—172no—
170C22×C17#39111yescharacteristic
171C22×C17C2×#9911yescharacteristic
172C22×C17#41011yescharacteristic
173C22×C17C2×#10311yescharacteristic
174C22×C17#41811yescharacteristic
175C22×C17C2×#10411yescharacteristic
176C22×C17C2×#10411yescharacteristic
177C2×D17—22no—
178C2×D17—22no—
179C2×D17—21no—
180C2×D17—21no—
181C2×D17—21no—
182C2×D17—21no—
183C2×D17—22no—
184C2×D17—22no—
185C2×D17#47612yes—
186C2×D17#47712yes—
187#595—48no—
188C4×C17—22no—
189C4×C17—22no—
190C4×C17—41no—
191C4×C17—81no—
192C4×C17—81no—
193Dic17—21no—
194Dic17—21no—
195Dic17—22no—
196Dic17—81no—
197Dic17—81no—
198Dic17#32814yes—
199C2×C4⋊Q8—171no—
200C2×C22⋊Q8—341no—
201C22×C42—171no—
202C22×C42—341no—
203C2×C4⋊C8—341no—
204C2×C22⋊C8—172no—
205#391—341no—
206#389—171no—
207#389—341no—
208#389#59511yescharacteristic, complemented
209#384—171no—
210#380—171no—
211#380—171no—
212#380—341no—
213C2×#95—172no—
214C2×#95—341no—
215#336—341no—
216C2×C17C2×#32811yescharacteristic
217C2×C17#150311yescharacteristic
218C2×C17#150511yescharacteristic
219C2×C17#150711yescharacteristic
220C2×C17#198711yescharacteristic
221C2×C17#199411yescharacteristic
222C2×C17#199611yescharacteristic
223D17—22no—
224D17—22no—
225C22×Q8—341no—
226C23×C4—171no—
227C23×C4—171no—
228C23×C4—171no—
229C23×C4—342no—
230C22×C8—342no—
231C4⋊Q8—682no—
232C22⋊Q8—684no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4—341no—
240C2×C4⋊C4C2×#59511yescharacteristic
241C2×C4⋊C4C2×#59511yescharacteristic
242C2×C22⋊C4—341no—
243C2×C22⋊C4—341no—
244C2×C42—171no—
245C2×C42—171no—
246C2×C42—171no—
247C2×C42—344no—
248C2×C42—682no—
249C2×C42—682no—
250C2×C42—682no—
251C2×C42C2×#59511yescharacteristic
252C4⋊C8—682no—
253C22⋊C8—344no—
254#95—21no—
255#95—174no—
256#95—341no—
257#95—341no—
258#95—342no—
259#95—344no—
260#95—341no—
261C17#990811yescharacteristic, complemented, Sylow
262C24—171no—
263C2×Q8—682no—
264C2×Q8—682no—
265C2×Q8—682no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—172no—
272C22×C4—172no—
273C22×C4—172no—
274C22×C4—172no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—342no—
278C22×C4—342no—
279C22×C4—344no—
280C22×C4—344no—
281C22×C4—344no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4—681no—
287C22×C4—682no—
288C22×C4—682no—
289C22×C4—682no—
290C22×C4—682no—
291C22×C4C22⋊#59511yescharacteristic
292C22×C4C22⋊#59511yescharacteristic
293C22×C4C22×#59511yescharacteristic
294C2×C8—682no—
295C2×C8—682no—
296C2×C8—682no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—681no—
301C4⋊C4—681no—
302C4⋊C4—682no—
303C4⋊C4—682no—
304C4⋊C4—682no—
305C4⋊C4—682no—
306C22⋊C4—682no—
307C22⋊C4—682no—
308C42—41no—
309C42—342no—
310C42—342no—
311C42—681no—
312C42—688no—
313C23—172no—
314C23—171no—
315C23—171no—
316C23—172no—
317C23—341no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23C2×C22⋊#59511yescharacteristic
322Q8—1364no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—172no—
337C2×C4—172no—
338C2×C4—172no—
339C2×C4—171no—
340C2×C4—171no—
341C2×C4—172no—
342C2×C4—344no—
343C2×C4—344no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—348no—
349C2×C4—682no—
350C2×C4—682no—
351C2×C4—684no—
352C2×C4—684no—
353C2×C4—684no—
354C2×C4—684no—
355C2×C4—684no—
356C2×C4—684no—
357C2×C4—684no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4#1058754912yes—
365C2×C4Q8×#59512yes—
366C8—1362no—
367C22—172no—
368C22—172no—
369C22—341no—
370C22—341no—
371C22—341no—
372C22—341no—
373C22—342no—
374C22—342no—
375C22—342no—
376C22—342no—
377C22(C2×#595)⋊Q811yescharacteristic
378C22D17⋊#9911yescharacteristic
379C22#1176378811yescharacteristic
380C22C2×#1058754911yescharacteristic
381C22#1176380111yescharacteristic
382C22D17⋊#10411yescharacteristic
383C22D17⋊#10411yescharacteristic
384C4—22no—
385C4—22no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—174no—
390C4—342no—
391C4—341no—
392C4—341no—
393C4—688no—
394C4—1361no—
395C4—1361no—
396C2—342no—
397C2—342no—
398C2D17⋊#32811yescharacteristic
399C2#1454416611yescharacteristic
400C2#1454417411yescharacteristic
401C2#1454417911yescharacteristic
402C2#1454496211yescharacteristic
403C2#1454497011yescharacteristic
404C2#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#9908
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×#10587574C4semidirect
16. #11762707C4semidirect
18. #11762678C4semidirect
22. C17×#389C4semidirect
52. (C2×#595)⋊C4Q8semidirect
208. #389#595semidirect
261. C17#9908semidirect

Maximal subgroups

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

subgroupindex
#145446412
C4×(C2×#595)⋊C42
#145436572
#145371352
#990817