Finite Group Databasebeta

← #28485423

Subgroups — #28485423

Order 15104.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435217C211yesmaximal, characteristic
2#21424928C211yesmaximal, characteristic
3#21422768C211yesmaximal, characteristic, complemented
4#21422655C211yesmaximal, characteristic
5#21421954C211yesmaximal, characteristic, complemented
6#21421925C211yesmaximal, characteristic, complemented
7C59×#2327C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837805C412yescomplemented
10#16837750C412yescomplemented
11#16837715—22no—
12#16837679—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×#14273655—21no—
17C2×#14273655—21no—
18#16837148—22no—
19#16837135—22no—
20C2×C4⋊Q8×C59C2211yescharacteristic
21C2×C4×Q8×C59—21no—
22C59×#391—21no—
23C59×#387—22no—
24C59×#387—22no—
25C59×#382C2211yescharacteristic
26C59×#376C2211yescharacteristic
27C2×#11745685—21no—
28C2×#11745664—21no—
29#14273727C2×C411yescharacteristic
30#14273724C2×C411yescharacteristic
31#14273715—21no—
32#14273701—21no—
33#14273701—21no—
34C4×D236—21no—
35C4×D236—21no—
36#14273660—24no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41#14273655—42no—
42#14273655—42no—
43#14273647—24no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8C2×C412yes—
48C22×Q8×C59—21no—
49C4⋊Q8×C59—21no—
50C4⋊Q8×C59—21no—
51C4⋊Q8×C59C2×C412yes—
52C4⋊Q8×C59D412yes—
53C4×Q8×C59—42no—
54C4×Q8×C59—42no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—22no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59D412yes—
59C2×C42×C59—22no—
60C2×C42×C59D411yescharacteristic
61C2×C42×C59D411yescharacteristic
62C2×C42×C59C2311yescharacteristic
63#95×C59—24no—
64#95×C59—21no—
65#95×C59—21no—
66#95×C59—21no—
67#95×C59D412yes—
68C2×Q8×C59—22no—
69C2×Q8×C59—22no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C2×Q8×C59—41no—
73C2×Q8×C59—41no—
74C22×C4×C59—21no—
75C22×C4×C59—22no—
76C22×C4×C59—21no—
77C22×C4×C59—22no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C2×C22⋊D59—41no—
82#11745685—41no—
83#11745685—41no—
84#11745685—41no—
85#11745685—41no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236—21no—
89C2×C4×D59—41no—
90C2×Dic118—21no—
91C4⋊C4×C59—22no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—22no—
94C4⋊C4×C59—44no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59C22⋊C414yes—
98C42×C59—21no—
99C42×C59—21no—
100C42×C59—21no—
101C42×C59—44no—
102C42×C59D812yes—
103C42×C59C22×C411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C42×C59#4111yescharacteristic
107(C2×D59)⋊C4—41no—
108(C2×D59)⋊C4—41no—
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111#11745664—41no—
112#11745664—41no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C23×C59#12011yescharacteristic
117C22×D59—41no—
118Q8×C59—42no—
119Q8×C59—42no—
120Q8×C59—42no—
121Q8×C59—42no—
122C2×C4×C59—24no—
123C2×C4×C59—22no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—42no—
129C2×C4×C59—42no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—42no—
135C2×C4×C59—41no—
136C2×C4×C59C2×C22⋊C411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12311yescharacteristic
142C2×C4×C59C2×D811yescharacteristic
143C2×C4×C59#13511yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C22⋊D59—81no—
147C22⋊D59—81no—
148C2×Dic59—41no—
149D236—42no—
150D236—41no—
151D236—41no—
152D236—81no—
153C4×D59—81no—
154C4×D59—81no—
155Dic118—41no—
156Dic118—41no—
157C59⋊C8—42no—
158C59⋊C8—81no—
159C59⋊C8—81no—
160#9300—591nomaximal, Sylow
161C22×C59—21no—
162C22×C59—21no—
163C22×C59#38611yescharacteristic
164C22×C59#44711yescharacteristic
165C22×C59#44911yescharacteristic
166C2×D59—81no—
167C2×D59—81no—
168C2×D59—81no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—42no—
174C4×C59—44no—
175C4×C59—82no—
176C4×C59—81no—
177C4×C59C4×D811yescharacteristic
178C4×C59#44211yescharacteristic
179C4×C59#45911yescharacteristic
180C4×C59#46311yescharacteristic
181Dic59—81no—
182#3226—591no—
183#2327D5911yescharacteristic, complemented
184#1963—591no—
185#1546—591no—
186#1517—591no—
187#1286—591no—
188#1274—591no—
189C2×C59—21no—
190C2×C59—21no—
191C2×C59#187911yescharacteristic
192C2×C59#190511yescharacteristic
193C2×C59#191211yescharacteristic
194D59—161no—
195#532—591no—
196C2×C4⋊Q8C2×D5911yescharacteristic
197C2×C4×Q8—21no—
198#500—592no—
199#486—592no—
200#480—1182no—
201#474—1182no—
202#431—591no—
203#431—591no—
204#423—591no—
205C2×#104—1181no—
206C2×#104—1181no—
207#391—21no—
208#387—22no—
209#387—22no—
210#382C2×D5911yescharacteristic
211#376C2×D5911yescharacteristic
212#332—1182no—
213#327—1182no—
214C59#930011yescharacteristic, complemented, Sylow
215C2×#41—1181no—
216C22×Q8—21no—
217C2×M16—1181no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—591no—
221C4⋊Q8C4×D5912yes—
222C4⋊Q8D23612yes—
223#127—591no—
224#124—1181no—
225#121—1181no—
226#121—1181no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×D4—1181no—
230C4×D4—1181no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D5912yes—
235C2×C42—22no—
236C2×C42C22⋊D5911yescharacteristic
237C2×C42C22⋊D5911yescharacteristic
238C2×C42C22×D5911yescharacteristic
239#107—1184no—
240C4⋊C8—1181no—
241C4⋊C8—1181no—
242C4⋊C8—1181no—
243C4⋊C8—1181no—
244#104—2362no—
245#104—2362no—
246#102—1184no—
247C22⋊C8—1181no—
248C22⋊C8—1181no—
249C4×C8—592no—
250C4×C8—1181no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C22⋊D5912yes—
256#41—2361no—
257#41—2361no—
258#41—2361no—
259#41—2361no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—1181no—
267C2×D4—1181no—
268C2×D4—1181no—
269C2×D4—1181no—
270C2×D4—2361no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—2361no—
276C22×C4C2×C22⋊D5911yescharacteristic
277C22×C4C2×C22⋊D5911yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279M16—2361no—
280M16—2361no—
281C2×C8—1182no—
282C2×C8—2361no—
283C2×C8—2361no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—1181no—
291C4⋊C4—1181no—
292C4⋊C4(C2×D59)⋊C414yes—
293C22⋊C4—2361no—
294C22⋊C4—2361no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—44no—
299C42#1174566912yes—
300C42C2×C4×D5911yescharacteristic
301C42C2×D23611yescharacteristic
302C42#1174568511yescharacteristic
303C42#1174568511yescharacteristic
304C23—2361no—
305C23#1427369611yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310Q8—2361no—
311Q8—2361no—
312D4—4721no—
313D4—4721no—
314D4—4721no—
315D4—2362no—
316D4—2361no—
317D4—2361no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—4721no—
333C2×C4—4721no—
334C2×C4—2361no—
335C2×C4C2×(C2×D59)⋊C411yescharacteristic
336C2×C4C4×C22⋊D5911yescharacteristic
337C2×C4C4×C22⋊D5911yescharacteristic
338C2×C4#1427370111yescharacteristic
339C2×C4#1427370111yescharacteristic
340C2×C4#1427371011yescharacteristic
341C2×C4C2×#1174566911yescharacteristic
342C2×C4#1427374711yescharacteristic
343C2×C4#1427375211yescharacteristic
344C2×C4#1427375311yescharacteristic
345C8—4721no—
346C8—4721no—
347C8—2362no—
348C22—21no—
349C22—21no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22#1683728111yescharacteristic
354C22#1683753611yescharacteristic
355C22#1683754711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—4721no—
365C4C4×#1174566911yescharacteristic
366C4#1683751211yescharacteristic
367C4#1683759711yescharacteristic
368C4#1683761711yescharacteristic
369C2—21no—
370C2—21no—
371C2—9441no—
372C2#2142444411yescharacteristic
373C2#2142459011yescharacteristic
374C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2327C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2327D59
hypocenterC59#9300
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
3. #21422768C2semidirect
5. #21421954C2semidirect
6. #21421925C2semidirect
7. C59×#2327C2semidirect
9. #16837805C4semidirect
10. #16837750C4semidirect
183. #2327D59semidirect
214. C59#9300semidirect

Maximal subgroups

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

subgroupindex
#214352172
#214249282
#214227682
#214226552
#214219542
#214219252
C59×#23272
#930059