Finite Group Databasebeta

← #28507390

Subgroups — #28507390

Order 15104.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273670C211yesmaximal, characteristic, complemented
2C2×#16837640C211yesmaximal, characteristic, complemented
3C2×#16837639C211yesmaximal, characteristic
4C2×#16837204C211yesmaximal, characteristic, complemented
5C2×(C2×D59)⋊C16C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#358C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C22×C8×D59C2211yescharacteristic
10C2×#14273703—21no—
11C2×#14273702—21no—
12C2×#14273670—22no—
13C2×#14273670—22no—
14C2×#14273670—22no—
15C22×C59⋊C16C2211yescharacteristic
16#16837640—22no—
17#16837640—22no—
18#16837639—22no—
19#16837639—22no—
20C2×#14273661C2211yescharacteristic
21C2×#14273651—21no—
22C2×#14273647—21no—
23#16837204—22no—
24(C2×D59)⋊C16—22no—
25C22×Q16×C59C2211yescharacteristic
26C22×C16×C59C2211yescharacteristic
27C2×#107×C59C2211yescharacteristic
28C59×#358—22no—
29C22×D236—21no—
30C22×D236—41no—
31C22×C4×D59D411yescharacteristic
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472D412yes—
38C22×C59⋊C8D411yescharacteristic
39C2×C8×D59—21no—
40C2×C8×D59—21no—
41C2×C8×D59—21no—
42C2×C8×D59—21no—
43C2×C8×D59—21no—
44C2×C8×D59—21no—
45#14273703—24no—
46#14273703—42no—
47#14273702—24no—
48#14273702—42no—
49C2×C4⋊Dic59—21no—
50C2×Dic59⋊C4—41no—
51C2×(C2×D59)⋊C4—41no—
52C2×(C2×D59)⋊C4—41no—
53#14273670—28no—
54C2×C59⋊C16—21no—
55C2×C59⋊C16—21no—
56C2×C59⋊C16D412yescomplemented
57#14273661—21no—
58#14273661D412yescomplemented
59#14273651—24no—
60#14273647—24no—
61C22×Q8×C59—21no—
62C2×Q16×C59—21no—
63C2×Q16×C59—21no—
64C2×Q16×C59—21no—
65C2×Q16×C59—21no—
66C2×Q16×C59—21no—
67C2×Q16×C59D412yescomplemented
68C22×C8×C59C2311yescharacteristic
69C2×C4⋊C4×C59—21no—
70C2×C16×C59—21no—
71C2×C16×C59—21no—
72C2×C16×C59—21no—
73#107×C59—21no—
74#107×C59D412yes—
75C23×D59—41no—
76C23×D59D811yescharacteristic
77C2×Q8×C59—22no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×Q8×C59—41no—
82C2×Q8×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C2×D411yescharacteristic
86C22×Dic59—41no—
87C22×Dic59D811yescharacteristic
88C2×D236—22no—
89C2×D236—41no—
90C2×D236—41no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—44no—
94C2×D236—41no—
95C2×D236—82no—
96C2×D236—82no—
97C2×C4×D59—21no—
98C2×C4×D59—21no—
99C2×C4×D59—21no—
100C2×C4×D59—21no—
101C2×C4×D59—21no—
102C2×C4×D59—21no—
103Q16×C59—22no—
104Q16×C59—22no—
105Q16×C59—41no—
106Q16×C59—41no—
107C2×C8×C59C2×D411yescharacteristic
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59#4111yescharacteristic
113C4⋊C4×C59—22no—
114C4⋊C4×C59—41no—
115(C2×D59)⋊C4—44no—
116(C2×D59)⋊C4—44no—
117C4⋊Dic59—22no—
118C4⋊Dic59—41no—
119Dic59⋊C4—44no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123D472—22no—
124D472—22no—
125D472—41no—
126D472—41no—
127C8×D59—41no—
128C8×D59—41no—
129C8×D59—41no—
130C8×D59—41no—
131C16×C59—41no—
132C59⋊C16—22no—
133C23×C59C2×D811yescharacteristic
134C22×D59—21no—
135C22×D59—21no—
136C22×D59—21no—
137C22×D59—21no—
138C22×D59—21no—
139C22×D59—21no—
140C22×D59—42no—
141C22×D59—81no—
142C22×D59—81no—
143C22×D59—81no—
144C22×D59—81no—
145C22×D59—81no—
146C22×D59—81no—
147C22×D59SD3212yescomplemented
148Q8×C59—42no—
149Q8×C59—42no—
150Q8×C59—81no—
151Q8×C59—81no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59—81no—
155C2×C4×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59#12011yescharacteristic
159C2×C4×C59#12011yescharacteristic
160C2×C4×C59#12111yescharacteristic
161C2×C4×C59#13511yescharacteristic
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
164C2×Dic59—21no—
165C2×Dic59—21no—
166C2×Dic59—21no—
167C2×Dic59—42no—
168C2×Dic59—81no—
169C2×Dic59—81no—
170D236—42no—
171D236—42no—
172D236—88no—
173D236—81no—
174D236—81no—
175C4×D59—41no—
176C4×D59—41no—
177C4×D59—41no—
178C4×D59—41no—
179C8×C59#12011yescharacteristic
180C8×C59#12111yescharacteristic
181C8×C59#12111yescharacteristic
182C8×C59#12711yescharacteristic
183C59⋊C8—41no—
184#13305—591nomaximal, Sylow
185C22×C59#44711yescharacteristic
186C22×C59#44711yescharacteristic
187C22×C59#46611yescharacteristic
188C22×C59C2×SD3211yescharacteristic
189C22×C59#50811yescharacteristic
190C22×C59#50911yescharacteristic
191C22×C59#51011yescharacteristic
192C2×D59—22no—
193C2×D59—22no—
194C2×D59—22no—
195C2×D59—41no—
196C2×D59—41no—
197C2×D59—41no—
198C2×D59—41no—
199C2×D59—82no—
200C2×D59—82no—
201C2×D59—82no—
202C2×D59—161no—
203C2×D59—161no—
204C2×D59—161no—
205C2×D59—161no—
206C4×C59—82no—
207C4×C59—82no—
208C4×C59#39211yescharacteristic
209C4×C59#44911yescharacteristic
210C4×C59#44911yescharacteristic
211C4×C59#46911yescharacteristic
212Dic59—41no—
213Dic59—82no—
214C22×SD32—591no—
215C2×#467—591no—
216C2×#466—591no—
217C2×#358D5911yescharacteristic, complemented
218C2×#357—591no—
219C2×C22⋊C16—591no—
220#1380—591no—
221C2×C59#199911yescharacteristic
222C2×C59#218511yescharacteristic
223C2×C59#218611yescharacteristic
224C2×C59#218711yescharacteristic
225C2×C59#218911yescharacteristic
226C2×C59#221711yescharacteristic
227C2×C59#222011yescharacteristic
228D59—42no—
229D59—162no—
230C22×Q16C2×D5911yescharacteristic
231C22×D8—591no—
232C23×C8—591no—
233C2×C22⋊Q8—1181no—
234C2×#121—1181no—
235C2×SD32—1182no—
236C2×SD32—1182no—
237C2×SD32—1182no—
238C22×C16—591no—
239C22×C16C2×D5911yescharacteristic
240#467—1182no—
241#467—1182no—
242#466—1182no—
243#466—1182no—
244C2×#107—591no—
245C2×#107C2×D5911yescharacteristic
246C2×#103—1181no—
247C2×#102—1181no—
248#358—22no—
249#357—1182no—
250C22⋊C16—1182no—
251C59#1330511yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×D4—1181no—
254C22×D4—2361no—
255C23×C4—591no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16D23612yescomplemented
262C2×D8—592no—
263C2×D8—1181no—
264C2×D8—1181no—
265C2×D8—1181no—
266C2×D8—1181no—
267C2×D8—1181no—
268C22×C8—591no—
269C22×C8—1181no—
270C22×C8—1181no—
271C22×C8—1181no—
272C22×C8—1181no—
273C22×C8—1181no—
274C22×C8—1181no—
275C22×C8C22×D5911yescharacteristic
276C22⋊Q8—1184no—
277C22⋊Q8—2362no—
278#121—1184no—
279#121—2362no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—1181no—
282C2×C4⋊C4—2361no—
283C2×C22⋊C4—2361no—
284C2×C22⋊C4—2361no—
285SD32—1188no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—592no—
290C2×C16—1181no—
291C2×C16—1181no—
292#107—21no—
293#107—592no—
294#107—1181no—
295#107C22⋊D5912yes—
296#103—1184no—
297#102—1184no—
298C24—591no—
299C24—2361no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—1182no—
307C2×D4—4722no—
308C2×D4—4722no—
309C2×D4—2364no—
310C2×D4—2361no—
311C2×D4—2361no—
312C2×D4—2361no—
313C2×D4—2361no—
314C2×D4—2361no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—591no—
318C22×C4—1181no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4—1181no—
324C22×C4—2361no—
325C22×C4D4×D5911yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—1182no—
331D8—1182no—
332D8—2361no—
333D8—2361no—
334C2×C8—1181no—
335C2×C8—1181no—
336C2×C8—1181no—
337C2×C8—2361no—
338C2×C8—2361no—
339C2×C8—2361no—
340C2×C8—2361no—
341C2×C8C2×D23611yescharacteristic
342C2×C8#1174568511yescharacteristic
343C2×C8D4×D5911yescharacteristic
344C2×C8D4×D5911yescharacteristic
345C2×C8D4×D5911yescharacteristic
346C2×C8C2×C22⋊D5911yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—1182no—
350C4⋊C4—2364no—
351C4⋊C4—2361no—
352C22⋊C4—2364no—
353C22⋊C4—2364no—
354C16—41no—
355C16—1182no—
356C23—592no—
357C23—1181no—
358C23—1181no—
359C23—1181no—
360C23—1181no—
361C23—1181no—
362C23—1181no—
363C23—4721no—
364C23—4721no—
365C23—4721no—
366C23—4721no—
367C23—4721no—
368C23—4721no—
369C23—2362no—
370C23D8×D5911yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—4728no—
376D4—4721no—
377D4—4721no—
378D4—2362no—
379D4—2362no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—1181no—
387C2×C4—1181no—
388C2×C4—1181no—
389C2×C4—4721no—
390C2×C4—4721no—
391C2×C4—2362no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4#1427369511yescharacteristic
397C2×C4#1427369711yescharacteristic
398C2×C4#1427370011yescharacteristic
399C2×C4#1427374811yescharacteristic
400C2×C4#1427375111yescharacteristic
401C2×C4#1427375411yescharacteristic
402C8—2361no—
403C8#1427369711yescharacteristic
404C8#1427370011yescharacteristic
405C8#1427370111yescharacteristic
406C8#1427372311yescharacteristic
407C22—1182no—
408C22—1182no—
409C22—1182no—
410C22—9441no—
411C22—9441no—
412C22—9441no—
413C22—9441no—
414C22—4722no—
415C22—4722no—
416C22—4722no—
417C22—2361no—
418C22—2361no—
419C22—2361no—
420C22—2361no—
421C22#1683753411yescharacteristic
422C22#1683753711yescharacteristic
423C22#1683763711yescharacteristic
424C22SD32×D5911yescharacteristic
425C22#1683783511yescharacteristic
426C22#1683783911yescharacteristic
427C22#1683784411yescharacteristic
428C4—82no—
429C4—82no—
430C4—4722no—
431C4—2361no—
432C4#1683731111yescharacteristic
433C4#1683754611yescharacteristic
434C4#1683754811yescharacteristic
435C4#1683765411yescharacteristic
436C2—9442no—
437C2—2362no—
438C2#2142509311yescharacteristic
439C2#2142601411yescharacteristic
440C2#2142602211yescharacteristic
441C2#2142602311yescharacteristic
442C2#2142603511yescharacteristic
443C2#2142619511yescharacteristic
444C2#2142621211yescharacteristic
subgroupG/H
centerC23D8×D59
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#358C2
socleC23×C59C2×D8
prime radicalC2×#16837639C2
hypercenterC2×#358D59
hypocenterC59#13305
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C22×#14273670C2semidirect
2. C2×#16837640C2semidirect
4. C2×#16837204C2semidirect
5. C2×(C2×D59)⋊C16C2semidirect
6. #21422187C2semidirect
7. C2×C59×#358C2semidirect
56. C2×C59⋊C16D4semidirect
58. #14273661D4semidirect
67. C2×Q16×C59D4semidirect
147. C22×D59SD32semidirect
217. C2×#358D59semidirect
251. C59#13305semidirect
261. C2×Q16D236semidirect

Maximal subgroups

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

subgroupindex
C22×#142736702
C2×#168376402
C2×#168376392
C2×#168372042
C2×(C2×D59)⋊C162
#214221872
C2×C59×#3582
#1330559