Finite Group Databasebeta

← #28488390

Subgroups — #28488390

Order 15104.

522 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837631C211yesmaximal, characteristic
2#21424134C211yesmaximal, characteristic, complemented
3#21422615C211yesmaximal, characteristic, complemented
4#21422611C211yesmaximal, characteristic
5#21422459C211yesmaximal, characteristic, complemented
6#21422455C211yesmaximal, characteristic
7C8×C22⋊C4×C59C211yesmaximal, characteristic, complemented
8C22×#11745660C411yescharacteristic, complemented
9C2×#14273706C2211yescharacteristic
10C2×#14273701C2211yescharacteristic
11#16837631—21no—
12#16837631—21no—
13#16837631—21no—
14#16837631—21no—
15C2×#14273663—21no—
16C2×#14273659C411yescharacteristic, complemented
17C2×#14273652C411yescharacteristic, complemented
18C2×#14273649C411yescharacteristic, complemented
19#16837366C2211yescharacteristic
20#16837366C2211yescharacteristic
21#16837167—21no—
22#16837138—22no—
23#16837135—22no—
24#16837127—22no—
25#16837125—22no—
26C23×C8×C59C2211yescharacteristic
27C2×C22⋊C8×C59C2211yescharacteristic
28C2×C4×C8×C59—21no—
29C4×C22⋊C4×C59C2211yescharacteristic
30C59×#336—21no—
31C22×Dic118C2×C411yescharacteristic
32C22×D236C2×C411yescharacteristic
33C22×C22⋊D59—21no—
34C2×#11745660—21no—
35C2×#11745660—21no—
36C2×#11745660—42no—
37C2×#11745664—21no—
38C2×#11745664—21no—
39#14273706—21no—
40#14273706—21no—
41#14273706—21no—
42#14273706—21no—
43#14273701—21no—
44#14273701—21no—
45#14273701—21no—
46#14273701—21no—
47C2×C4⋊Dic59C2×C411yescharacteristic
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×Dic59⋊C4—21no—
50C2×C22⋊Dic59—21no—
51C2×(C2×D59)⋊C4—21no—
52#14273663—42no—
53#14273659—21no—
54#14273659—21no—
55#14273652—21no—
56#14273652—21no—
57#14273649—21no—
58#14273649—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)—21no—
62C22⋊(C59⋊C8)—21no—
63C23×C4×C59C2311yescharacteristic
64C22×C8×C59—21no—
65C22×C8×C59—42no—
66C22×C8×C59C2×C411yescharacteristic
67C22×C8×C59C2×C411yescharacteristic
68C22×C8×C59D411yescharacteristic
69C22×C8×C59D411yescharacteristic
70C22×C8×C59D411yescharacteristic
71C22×C8×C59D411yescharacteristic
72C2×C22⋊C4×C59—21no—
73C2×C42×C59—21no—
74C22⋊C8×C59—21no—
75C22⋊C8×C59D412yes—
76C4×C8×C59—42no—
77#95×C59D412yes—
78C24×C59#4111yescharacteristic
79C23×D59—21no—
80C22×C4×C59—22no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C2×C22⋊D59—41no—
90C2×C22⋊D59—41no—
91C2×C22⋊D59—82no—
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—21no—
95C2×D236—21no—
96C2×D236—21no—
97C2×D236—41no—
98C2×D236—41no—
99C2×Dic118—21no—
100C2×Dic118—21no—
101C2×Dic118—41no—
102C2×Dic118—41no—
103C2×C8×C59—22no—
104C2×C8×C59—22no—
105C2×C8×C59—22no—
106C2×C8×C59—22no—
107C2×C8×C59—41no—
108C2×C8×C59—41no—
109C2×C8×C59—41no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C2×C8×C59C22⋊C411yescharacteristic
113C2×C8×C59C22⋊C411yescharacteristic
114C2×C8×C59C22⋊C411yescharacteristic
115C2×C8×C59C22⋊C411yescharacteristic
116C22⋊C4×C59—24no—
117C42×C59—41no—
118C42×C59—41no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121(C2×D59)⋊C4—41no—
122(C2×D59)⋊C4—41no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127Dic59⋊C4—41no—
128Dic59⋊C4—41no—
129#11745664—41no—
130#11745664—41no—
131#11745664—41no—
132#11745664—41no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135#11745660—42no—
136#11745660—42no—
137C23×C59—22no—
138C23×C59—22no—
139C23×C59—41no—
140C23×C59C4×D411yescharacteristic
141C23×C59#12111yescharacteristic
142C23×C59#12411yescharacteristic
143C22×D59—41no—
144C22×D59—42no—
145C22×D59—81no—
146C22×D59—81no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—24no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59C2×C22⋊C411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59#12111yescharacteristic
162C2×C4×C59#12711yescharacteristic
163C22⋊D59—82no—
164C22⋊D59—82no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—42no—
169C2×Dic59—81no—
170C2×Dic59—81no—
171D236—41no—
172D236—41no—
173D236—81no—
174Dic118—41no—
175Dic118—41no—
176Dic118—81no—
177C8×C59—21no—
178C8×C59—21no—
179C8×C59—21no—
180C8×C59—42no—
181C8×C59#9912yes—
182C59⋊C8—81no—
183C59⋊C8—81no—
184#9813—591nomaximal, Sylow
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—44no—
193C22×C59#39011yescharacteristic
194C22×C59#44312yes—
195C22×C59#47112yes—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—162no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—44no—
204C4×C59C2×#9911yescharacteristic
205C4×C59#41011yescharacteristic
206Dic59—81no—
207Dic59—81no—
208Dic59—81no—
209C2×#465—591no—
210#1810—591no—
211C8×C22⋊C4D5911yescharacteristic, complemented
212#1508—591no—
213#1507—591no—
214#1472—591no—
215#1471—591no—
216C2×C59—22no—
217C2×C59—22no—
218C2×C59—41no—
219C2×C59#196411yescharacteristic
220C2×C59#197411yescharacteristic
221C2×C59#197711yescharacteristic
222D59—161no—
223C22×SD16—591no—
224C23×C8C2×D5911yescharacteristic
225C2×C22⋊Q8—591no—
226C2×#121—591no—
227#465—1181no—
228#465—1181no—
229#465—1181no—
230#465—1181no—
231C2×#108—1181no—
232C2×#106—591no—
233C2×#103—591no—
234C2×#102—591no—
235#407—591no—
236#407—591no—
237C2×C22⋊C8C2×D5911yescharacteristic
238C2×C4×C8—21no—
239C4×C22⋊C4C2×D5911yescharacteristic
240#341—1181no—
241#336—21no—
242#328—1182no—
243#327—1182no—
244#324—1182no—
245#323—1182no—
246C59#981311yescharacteristic, complemented, Sylow
247C22×Q8—591no—
248C22×D4—591no—
249C22×D4—1181no—
250C23×C4C22×D5911yescharacteristic
251C2×SD16—1181no—
252C2×SD16—1181no—
253C2×SD16—2362no—
254C2×M16—1181no—
255C2×M16—1181no—
256C22×C8—21no—
257C22×C8—42no—
258C22×C8C4×D5911yescharacteristic
259C22×C8C4×D5911yescharacteristic
260C22×C8D23611yescharacteristic
261C22×C8D23611yescharacteristic
262C22×C8C22⋊D5911yescharacteristic
263C22×C8C22⋊D5911yescharacteristic
264C22⋊Q8—1181no—
265C22⋊Q8—1181no—
266C22⋊Q8—1181no—
267C22⋊Q8—1181no—
268#121—1181no—
269#121—1181no—
270#121—1181no—
271#121—1181no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—591no—
274C2×C4⋊C4—1181no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C42—21no—
279#108—2362no—
280#106—1181no—
281#106—1181no—
282#103—1181no—
283#103—1181no—
284#102—1181no—
285#102—1181no—
286C22⋊C8—21no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8—1181no—
290C22⋊C8—1181no—
291C22⋊C8D23612yes—
292C4×C8—42no—
293#95D23612yes—
294C24—1181no—
295C24#1174568511yescharacteristic
296C2×Q8—1181no—
297C2×Q8—1181no—
298C2×Q8—2361no—
299C2×Q8—2361no—
300C2×D4—1181no—
301C2×D4—1181no—
302C2×D4—4722no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2361no—
306C2×D4—2361no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1181no—
311C22×C4—1181no—
312C22×C4—1181no—
313C22×C4C2×C4×D5911yescharacteristic
314C22×C4C2×D23611yescharacteristic
315C22×C4C2×D23611yescharacteristic
316C22×C4C2×D23611yescharacteristic
317C22×C4#1174568511yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319SD16—2362no—
320SD16—2362no—
321M16—2361no—
322M16—2361no—
323M16—2361no—
324M16—2361no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—2361no—
335C2×C8—2361no—
336C2×C8(C2×D59)⋊C411yescharacteristic
337C2×C8(C2×D59)⋊C411yescharacteristic
338C2×C8(C2×D59)⋊C411yescharacteristic
339C2×C8(C2×D59)⋊C411yescharacteristic
340C4⋊C4—1181no—
341C4⋊C4—1181no—
342C4⋊C4—1181no—
343C4⋊C4—1181no—
344C4⋊C4—2361no—
345C4⋊C4—2361no—
346C22⋊C4—24no—
347C22⋊C4—2361no—
348C22⋊C4—2361no—
349C22⋊C4—2361no—
350C22⋊C4—2361no—
351C42—41no—
352C42—41no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—4721no—
357C23—4721no—
358C23—2362no—
359C23—2361no—
360C23C4×D23611yescharacteristic
361C23#1427370111yescharacteristic
362C23#1427371511yescharacteristic
363Q8—4721no—
364Q8—2361no—
365Q8—2361no—
366D4—4721no—
367D4—4722no—
368D4—4722no—
369D4—2361no—
370D4—2361no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—4721no—
384C2×C4—4721no—
385C2×C4—2362no—
386C2×C4—2361no—
387C2×C4—2361no—
388C2×C4—2361no—
389C2×C4C2×(C2×D59)⋊C411yescharacteristic
390C2×C4C4×D23611yescharacteristic
391C2×C4#1427370111yescharacteristic
392C2×C4#1427372411yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—42no—
397C8—4721no—
398C8—4721no—
399C8#1427364112yes—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—9442no—
409C22—4721no—
410C22—4721no—
411C22—4721no—
412C22#1683730411yescharacteristic
413C22#1683752012yes—
414C22#1683766512yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—44no—
419C4—4721no—
420C4—4721no—
421C4—4721no—
422C4C2×#1427364111yescharacteristic
423C4#1683737511yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—9441no—
428C2#2142493211yescharacteristic
429C2#2142498411yescharacteristic
430C2#2142499911yescharacteristic
subgroupG/H
centerC22#16837304
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC8×C22⋊C4×C59C2
socleC22×C59#390
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenterC8×C22⋊C4D59
hypocenterC59#9813
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #21424134C2semidirect
3. #21422615C2semidirect
5. #21422459C2semidirect
7. C8×C22⋊C4×C59C2semidirect
8. C22×#11745660C4semidirect
16. C2×#14273659C4semidirect
17. C2×#14273652C4semidirect
18. C2×#14273649C4semidirect
211. C8×C22⋊C4D59semidirect
246. C59#9813semidirect

Maximal subgroups

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

subgroupindex
C2×#168376312
#214241342
#214226152
#214226112
#214224592
#214224552
C8×C22⋊C4×C592
#981359