Finite Group Databasebeta

← #28498075

Subgroups — #28498075

Order 15104.

574 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 87 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#21426689C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic, complemented
3#21422029C212yesmaximal, complemented
4#21422027C212yesmaximal, complemented
5C59×#1463C211yesmaximal, characteristic, complemented
6C22×#11745685C411yescharacteristic
7C2×#14273686C411yescharacteristic
8#16837366—21no—
9#16837307C411yescharacteristic
10#16837307C2211yescharacteristic
11#16837303C411yescharacteristic
12#16837303C2211yescharacteristic
13#16837281—21no—
14#16837281—21no—
15#16837259—21no—
16#16837259—21no—
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C2×#11745685—22no—
24C2×#11745685—22no—
25C2×#11745685—41no—
26C2×#11745685—41no—
27C2×#11745685—42no—
28C22×Dic118C2×C411yescharacteristic
29C22×D236C2×C411yescharacteristic
30C22×C22⋊D59—21no—
31C22×C4×D59—21no—
32C2×#11745664—41no—
33#14273686—42no—
34C2×C4⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×C4×Dic59C2×C411yescharacteristic
41C2×C4×Dic59C2×C411yescharacteristic
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—41no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C23×C4×C59C2×C411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C2×M16×C59—41no—
52C2×C4⋊C4×C59—41no—
53C2×C4⋊C4×C59—41no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C4⋊C4×C59C2311yescharacteristic
58C22⋊C8×C59—22no—
59C22⋊C8×C59—41no—
60C24×C59C22⋊C411yescharacteristic
61C23×D59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—22no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C2×C22⋊D59—24no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59—41no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86#11745685—44no—
87#11745685—44no—
88#11745685—44no—
89#11745685—44no—
90C2×D236—21no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236M1612yescomplemented
94C2×C4×D59—24no—
95C2×C4×D59—42no—
96C2×C4×D59—42no—
97C2×Dic118—21no—
98C2×Dic118—41no—
99C2×Dic118—41no—
100C2×Dic118M1612yescomplemented
101M16×C59—82no—
102C2×C8×C59—42no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C22⋊Dic59—42no—
108(C2×D59)⋊C4—81no—
109(C2×D59)⋊C4—81no—
110(C2×D59)⋊C4—81no—
111(C2×D59)⋊C4—81no—
112C4⋊Dic59—42no—
113C4×Dic59—41no—
114C4×Dic59—41no—
115#11745664—82no—
116C2×C59⋊C8—42no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59#9911yescharacteristic
122C23×C59#9911yescharacteristic
123C23×C59C2×C22⋊C411yescharacteristic
124C22×D59—22no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—81no—
150C2×C4×C59—81no—
151C2×C4×C59#10012yes—
152C2×C4×C59C2×M1611yescharacteristic
153C2×C4×C59C2×M1611yescharacteristic
154C22⋊D59—44no—
155C22⋊D59—44no—
156C2×Dic59—22no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165D236—42no—
166D236—42no—
167C4×D59—48no—
168Dic118—42no—
169Dic118—42no—
170C8×C59—82no—
171C59⋊C8—82no—
172#11508—591nomaximal, Sylow
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—42no—
182C22×C59—42no—
183C22×C59#40711yescharacteristic
184C22×C59C2×#9911yescharacteristic
185C22×C59C2×#10011yescharacteristic
186C2×D59—42no—
187C2×D59—42no—
188C2×D59—42no—
189C2×D59—81no—
190C2×D59—81no—
191C2×D59—81no—
192C2×D59—81no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—22no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200Dic59—42no—
201Dic59—81no—
202Dic59—81no—
203#2311—591no—
204#1463—591no—
205#1463D5911yescharacteristic, complemented
206#1319—592no—
207#1318—592no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—41no—
211C2×C59—41no—
212C2×C59#146211yescharacteristic
213C2×C59#212711yescharacteristic
214C2×C59#213311yescharacteristic
215D59—82no—
216C22×#41—591no—
217C2×#117—591no—
218C22×C4⋊C4C2×D5911yescharacteristic
219#407—21no—
220#407—1181no—
221#391—591no—
222#391—591no—
223#390—591no—
224#390—591no—
225#386—1181no—
226#386—1181no—
227#380—1181no—
228#380—1181no—
229#324—21no—
230#324—592no—
231#324—1181no—
232#324C2×D5912yes—
233C59#1150811yescharacteristic, complemented, Sylow
234C2×#41—1182no—
235C2×#41—1182no—
236C2×#41—2361no—
237C2×#41—2361no—
238C2×#41—2362no—
239C22×Q8—591no—
240C22×D4—591no—
241C22×D4—1181no—
242C23×C4—1181no—
243C23×C4C4×D5911yescharacteristic
244C23×C4D23611yescharacteristic
245C23×C4C22⋊D5911yescharacteristic
246C2×M16—41no—
247C2×M16—2361no—
248#117—2362no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4C4×D5911yescharacteristic
253C2×C4⋊C4D23611yescharacteristic
254C2×C4⋊C4C22⋊D5911yescharacteristic
255C2×C4⋊C4C22×D5911yescharacteristic
256C2×C22⋊C4—1181no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4—1181no—
261C2×C42—591no—
262C2×C42—591no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—1182no—
266C22⋊C8—2361no—
267#95—1181no—
268#95—1181no—
269#95—1181no—
270#95—1181no—
271C24—1181no—
272C24(C2×D59)⋊C411yescharacteristic
273#41—2364no—
274#41—2364no—
275#41—2364no—
276#41—2364no—
277C2×Q8—592no—
278C2×Q8—1181no—
279C2×Q8—2361no—
280C2×Q8—2361no—
281C2×D4—592no—
282C2×D4—1184no—
283C2×D4—1181no—
284C2×D4—2361no—
285C2×D4—2361no—
286C2×D4—2362no—
287C2×D4—2361no—
288C2×D4—2361no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—1184no—
301C22×C4—1181no—
302C22×C4—1181no—
303C22×C4—1181no—
304C22×C4—2362no—
305C22×C4—2362no—
306C22×C4(C2×D59)⋊C411yescharacteristic
307C22×C4(C2×D59)⋊C411yescharacteristic
308C22×C4(C2×D59)⋊C411yescharacteristic
309C22×C4C2×C4×D5911yescharacteristic
310C22×C4C2×D23611yescharacteristic
311C22×C4C2×C22⋊D5911yescharacteristic
312M16—82no—
313M16—4722no—
314C2×C8—42no—
315C2×C8—2362no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—2362no—
321C22⋊C4—4721no—
322C22⋊C4—4721no—
323C22⋊C4—4721no—
324C22⋊C4—4721no—
325C22⋊C4—2362no—
326C42—2361no—
327C42—2361no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—1182no—
333C23—2361no—
334C23—2361no—
335C23—2361no—
336C23—2361no—
337C23—2361no—
338C23—2361no—
339C23#1427364111yescharacteristic
340C23#1427364111yescharacteristic
341C23C2×(C2×D59)⋊C411yescharacteristic
342Q8—2362no—
343Q8—2362no—
344D4—2362no—
345D4—2362no—
346D4—2364no—
347D4—2364no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1182no—
369C2×C4—2368no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4#1427364312yes—
379C2×C4M16×D5911yescharacteristic
380C2×C4M16×D5911yescharacteristic
381C8—82no—
382C8—4722no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—4721no—
394C22—4721no—
395C22—4721no—
396C22—4721no—
397C22—2362no—
398C22—2362no—
399C22—2362no—
400C22#1683736511yescharacteristic
401C22C2×#1427364111yescharacteristic
402C22C2×#1427364311yescharacteristic
403C4—22no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4721no—
411C4—4721no—
412C4—2362no—
413C2—21no—
414C2—21no—
415C2—41no—
416C2—41no—
417C2—4722no—
418C2#2142242211yescharacteristic
419C2#2142576311yescharacteristic
420C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1463C2
socleC22×C59C2×#99
prime radicalC22×#11745685C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1463D59
hypocenterC59#11508
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #21422426C2semidirect
3. #21422029C2semidirect
4. #21422027C2semidirect
5. C59×#1463C2semidirect
93. C2×D236M16semidirect
100. C2×Dic118M16semidirect
205. #1463D59semidirect
233. C59#11508semidirect

Maximal subgroups

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

subgroupindex
#214266892
#214224262
#214220292
#214220272
C59×#14632
#1150859