Finite Group Databasebeta

← #28507389

Subgroups — #28507389

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×#16837638C211yesmaximal, characteristic
3C2×#16837206C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#467C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273701—21no—
10C2×#14273670—22no—
11C2×#14273670—22no—
12C2×#14273670—22no—
13C22×C59⋊C16C2211yescharacteristic
14C22×C59⋊C16C2211yescharacteristic
15#16837638—22no—
16#16837638—22no—
17C2×#14273661C2211yescharacteristic
18C2×#14273649—21no—
19#16837206—22no—
20#16837202—22no—
21C22⋊(C59⋊C16)—22no—
22C22×Q16×C59C2211yescharacteristic
23C23×C8×C59C2211yescharacteristic
24C2×C22⋊Q8×C59—21no—
25C59×#467—22no—
26C59×#467—22no—
27C2×#107×C59C2211yescharacteristic
28C2×#103×C59—21no—
29C22×D236—21no—
30C22×C22⋊D59—41no—
31C2×D472—21no—
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472D412yes—
37#14273701—24no—
38#14273701—42no—
39C2×C4⋊Dic59—21no—
40C2×(C2×D59)⋊C4—41no—
41#14273670—28no—
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16D412yescomplemented
48#14273661—21no—
49#14273661D412yes—
50#14273649—24no—
51C22×Q8×C59—21no—
52C23×C4×C59D411yescharacteristic
53C2×Q16×C59—21no—
54C2×Q16×C59—21no—
55C2×Q16×C59—21no—
56C2×Q16×C59—21no—
57C2×Q16×C59—21no—
58C2×Q16×C59D412yescomplemented
59C22×C8×C59—21no—
60C22×C8×C59—21no—
61C22×C8×C59—21no—
62C22×C8×C59—21no—
63C22×C8×C59—21no—
64C22×C8×C59—21no—
65C22×C8×C59D411yescharacteristic
66C22×C8×C59C2311yescharacteristic
67C22⋊Q8×C59—24no—
68C22⋊Q8×C59—42no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—41no—
71C2×C22⋊C4×C59—41no—
72#107×C59—21no—
73#107×C59D412yescomplemented
74#103×C59—24no—
75C24×C59D811yescharacteristic
76C23×D59—41no—
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—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59D811yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C2×C22⋊D59—44no—
94C2×C22⋊D59—82no—
95C2×C22⋊D59—82no—
96C22×Dic59—41no—
97C2×D236—22no—
98C2×D236—41no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236—41no—
102C2×D236—41no—
103Q16×C59—22no—
104Q16×C59—22no—
105Q16×C59—41no—
106Q16×C59—41no—
107C2×C8×C59—21no—
108C2×C8×C59—21no—
109C2×C8×C59—21no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C2×C8×C59—41no—
114C2×C8×C59C2×D411yescharacteristic
115C2×C8×C59C2×D411yescharacteristic
116C2×C8×C59C2×D411yescharacteristic
117C2×C8×C59C2×D411yescharacteristic
118C2×C8×C59C2×D411yescharacteristic
119C2×C8×C59#4111yescharacteristic
120C4⋊C4×C59—22no—
121C4⋊C4×C59—44no—
122C4⋊C4×C59—41no—
123C22⋊C4×C59—44no—
124(C2×D59)⋊C4—44no—
125C4⋊Dic59—22no—
126C4⋊Dic59—41no—
127D472—22no—
128D472—22no—
129D472—41no—
130D472—41no—
131C59⋊C16—22no—
132C59⋊C16—41no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59SD3212yescomplemented
140C23×C59C2×D811yescharacteristic
141C22×D59—42no—
142C22×D59—81no—
143C22×D59—81no—
144C22×D59—81no—
145C22×D59—81no—
146C22×D59—81no—
147C22×D59—81no—
148Q8×C59—42no—
149Q8×C59—42no—
150Q8×C59—81no—
151Q8×C59—81no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—42no—
160C2×C4×C59—42no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59#12011yescharacteristic
166C2×C4×C59#12011yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#13511yescharacteristic
169C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
171C22⋊D59—88no—
172C2×Dic59—42no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175D236—42no—
176D236—42no—
177D236—81no—
178D236—81no—
179C8×C59—41no—
180C8×C59#12011yescharacteristic
181C8×C59#12111yescharacteristic
182C8×C59#12111yescharacteristic
183C8×C59#12711yescharacteristic
184#13305—591nomaximal, Sylow
185C22×C59—22no—
186C22×C59—22no—
187C22×C59—22no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59#44711yescharacteristic
193C22×C59#44711yescharacteristic
194C22×C59#46611yescharacteristic
195C22×C59C2×SD3211yescharacteristic
196C22×C59#50811yescharacteristic
197C22×C59#50911yescharacteristic
198C22×C59#51011yescharacteristic
199C2×D59—82no—
200C2×D59—82no—
201C2×D59—82no—
202C2×D59—161no—
203C2×D59—161no—
204C2×D59—161no—
205C2×D59—161no—
206C4×C59—41no—
207C4×C59—82no—
208C4×C59—82no—
209C4×C59#39211yescharacteristic
210C4×C59#44911yescharacteristic
211C4×C59#44911yescharacteristic
212C4×C59#46911yescharacteristic
213Dic59—82no—
214C22×SD32—591no—
215C2×#467D5911yescharacteristic, complemented
216C2×#466—591no—
217C2×#358—591no—
218C2×#357—591no—
219C2×C22⋊C16—591no—
220#1380—591no—
221C2×C59—42no—
222C2×C59#199911yescharacteristic
223C2×C59#218511yescharacteristic
224C2×C59#218611yescharacteristic
225C2×C59#218711yescharacteristic
226C2×C59#218911yescharacteristic
227C2×C59#221711yescharacteristic
228C2×C59#222011yescharacteristic
229D59—162no—
230C22×Q16C2×D5911yescharacteristic
231C22×D8—591no—
232C23×C8C2×D5911yescharacteristic
233C2×C22⋊Q8—21no—
234C2×#121—1181no—
235C2×SD32—1182no—
236C2×SD32—1182no—
237C2×SD32—1182no—
238C22×C16—591no—
239C22×C16—591no—
240#467—22no—
241#467—22no—
242#466—1182no—
243#466—1182no—
244C2×#107—591no—
245C2×#107C2×D5911yescharacteristic
246C2×#103—21no—
247C2×#102—1181no—
248#358—1182no—
249#357—1182no—
250C22⋊C16—1182no—
251C59#1330511yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×D4—1181no—
254C22×D4—2361no—
255C23×C4C22⋊D5911yescharacteristic
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16C22⋊D5912yescomplemented
262C2×D8—592no—
263C2×D8—1181no—
264C2×D8—1181no—
265C2×D8—1181no—
266C2×D8—1181no—
267C2×D8—1181no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8C22⋊D5911yescharacteristic
275C22×C8C22×D5911yescharacteristic
276C22⋊Q8—24no—
277C22⋊Q8—42no—
278#121—1184no—
279#121—2362no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—1181no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—2361no—
285SD32—1188no—
286C2×C16—592no—
287C2×C16—1181no—
288C2×C16—1181no—
289C2×C16—1181no—
290C2×C16—1181no—
291C2×C16—1181no—
292#107—21no—
293#107—592no—
294#107—1181no—
295#107C22⋊D5912yescomplemented
296#103—24no—
297#102—1184no—
298C24—2361no—
299C24#1174566911yescharacteristic
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—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—2361no—
324C22×C4#1174566911yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—1182no—
331D8—1182no—
332D8—2361no—
333D8—2361no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8D4×D5911yescharacteristic
342C2×C8D4×D5911yescharacteristic
343C2×C8D4×D5911yescharacteristic
344C2×C8#1174568711yescharacteristic
345C2×C8C2×C22⋊D5911yescharacteristic
346C2×C8C2×C22⋊D5911yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—41no—
350C4⋊C4—1182no—
351C4⋊C4—2361no—
352C22⋊C4—44no—
353C22⋊C4—2364no—
354C16—1182no—
355C16—2361no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—4721no—
363C23—4721no—
364C23—4721no—
365C23—4721no—
366C23—4721no—
367C23—4721no—
368C23—2362no—
369C23#1427367012yescomplemented
370C23C2×#1174566911yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—4728no—
376D4—4721no—
377D4—4721no—
378D4—2362no—
379D4—2362no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4721no—
394C2×C4—4721no—
395C2×C4—2362no—
396C2×C4#1427369511yescharacteristic
397C2×C4#1427369611yescharacteristic
398C2×C4#1427369911yescharacteristic
399C2×C4#1427374711yescharacteristic
400C2×C4#1427375211yescharacteristic
401C2×C4#1427375311yescharacteristic
402C8—41no—
403C8#1427369511yescharacteristic
404C8#1427369811yescharacteristic
405C8#1427369911yescharacteristic
406C8#1427372311yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—9441no—
415C22—9441no—
416C22—9441no—
417C22—9441no—
418C22—4722no—
419C22—4722no—
420C22—4722no—
421C22#1683753511yescharacteristic
422C22#1683753611yescharacteristic
423C22#1683763611yescharacteristic
424C22C2×#1427367011yescharacteristic
425C22#1683783211yescharacteristic
426C22#1683783711yescharacteristic
427C22#1683784311yescharacteristic
428C4—41no—
429C4—82no—
430C4—82no—
431C4—4722no—
432C4#1683731011yescharacteristic
433C4#1683754511yescharacteristic
434C4#1683754711yescharacteristic
435C4#1683765311yescharacteristic
436C2—42no—
437C2—9442no—
438C2#2142509211yescharacteristic
439C2#2142601311yescharacteristic
440C2#2142601911yescharacteristic
441C2#2142602411yescharacteristic
442C2#2142603411yescharacteristic
443C2#2142619411yescharacteristic
444C2#2142621111yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#467C2
socleC23×C59C2×D8
prime radicalC2×#16837638C2
hypercenterC2×#467D59
hypocenterC59#13305
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C22×#14273670C2semidirect
3. C2×#16837206C2semidirect
4. C2×#16837202C2semidirect
5. C2×C22⋊(C59⋊C16)C2semidirect
6. #21422187C2semidirect
7. C2×C59×#467C2semidirect
47. C2×C59⋊C16D4semidirect
58. C2×Q16×C59D4semidirect
73. #107×C59D4semidirect
139. C23×C59SD32semidirect
215. C2×#467D59semidirect
251. C59#13305semidirect
261. C2×Q16C22⋊D59semidirect
295. #107C22⋊D59semidirect
369. C23#14273670semidirect

Maximal subgroups

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

subgroupindex
C22×#142736702
C2×#168376382
C2×#168372062
C2×#168372022
C2×C22⋊(C59⋊C16)2
#214221872
C2×C59×#4672
#1330559