Finite Group Databasebeta

← #28484631

Subgroups — #28484631

Order 15104.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433568C211yesmaximal, characteristic
2#21422655C211yesmaximal, characteristic
3#21422590C211yesmaximal, characteristic
4#21422491C211yesmaximal, characteristic, complemented
5#21421999C211yesmaximal, characteristic, complemented
6#21421954C211yesmaximal, characteristic, complemented
7C59×#2351C211yesmaximal, characteristic, complemented
8C2×#14273758—21no—
9#16837952C2211yescharacteristic
10#16837624—22no—
11#16837612C412yescomplemented
12#16837602C412yescomplemented
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16#16837401—21no—
17#16837135—22no—
18#16837135—21no—
19#16837135—21no—
20Q8⋊(C59⋊C8)—22no—
21C2×C4×Q8×C59—21no—
22C2×C4×Q8×C59C2211yescharacteristic
23C59×#389—21no—
24C59×#389—22no—
25C59×#382—21no—
26C59×#382C2211yescharacteristic
27C4×C4⋊C4×C59—21no—
28C4×C4⋊C4×C59C2211yescharacteristic
29C2×#11745685—21no—
30#14273758—42no—
31#14273758—42no—
32C2×#11745672—22no—
33C2×#11745671—22no—
34C2×#11745664—21no—
35#14273727C2×C411yescharacteristic
36#14273724C2×C411yescharacteristic
37#14273715—21no—
38#14273701—21no—
39#14273701—21no—
40C4×D236—21no—
41C4×D236—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)C2×C412yes—
46#14273650—22no—
47#14273647—22no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C22×Q8×C59D411yescharacteristic
53C4×Q8×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—42no—
56C4×Q8×C59—42no—
57C4×Q8×C59C2×C412yes—
58C4×Q8×C59D412yes—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—22no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C42×C59—21no—
66C2×C42×C59—22no—
67C2×C42×C59—21no—
68C2×C42×C59C2311yescharacteristic
69#95×C59—22no—
70#95×C59—21no—
71#95×C59D411yescharacteristic
72#95×C59D411yescharacteristic
73#95×C59D412yes—
74C2×Q8×C59—21no—
75C2×Q8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59C22⋊C412yes—
78C2×Q8×C59D812yes—
79C2×Q8×C59SD1612yes—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C2×C22⋊D59—41no—
89#11745685—41no—
90#11745685—41no—
91#11745685—41no—
92#11745685—41no—
93C2×D236—21no—
94C2×D236—21no—
95C2×D236—21no—
96C2×C4×D59—41no—
97C2×Dic118—21no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—42no—
102C4⋊C4×C59—44no—
103C4⋊C4×C59—44no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59C22⋊C412yes—
107C42×C59—22no—
108C42×C59—22no—
109C42×C59—44no—
110C42×C59—41no—
111C42×C59—41no—
112C42×C59C22×C411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59#4111yescharacteristic
115C42×C59#4111yescharacteristic
116#11745672—44no—
117#11745671—44no—
118(C2×D59)⋊C4—41no—
119(C2×D59)⋊C4—41no—
120C4⋊Dic59—21no—
121C4⋊Dic59—21no—
122#11745664—41no—
123#11745664—41no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—41no—
127C23×C59#12011yescharacteristic
128C22×D59—41no—
129Q8×C59—22no—
130Q8×C59—22no—
131Q8×C59—22no—
132Q8×C59#10214yes—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—41no—
146C2×C4×C59—42no—
147C2×C4×C59—42no—
148C2×C4×C59—41no—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59C4×D411yescharacteristic
152C2×C4×C59#12111yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12311yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59C2×SD1611yescharacteristic
157C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C59#13711yescharacteristic
159C22⋊D59—81no—
160C22⋊D59—81no—
161C2×Dic59—41no—
162D236—41no—
163D236—41no—
164D236—81no—
165D236—81no—
166C4×D59—81no—
167C4×D59—81no—
168Dic118—41no—
169Dic118—41no—
170C59⋊C8—42no—
171C59⋊C8—81no—
172C59⋊C8—81no—
173#9180—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59#38611yescharacteristic
177C22×C59#44711yescharacteristic
178C22×C59#44811yescharacteristic
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—24no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—42no—
188C4×C59—42no—
189C4×C59—82no—
190C4×C59—81no—
191C4×C59C2×#10211yescharacteristic
192C4×C59#41711yescharacteristic
193C4×C59#48011yescharacteristic
194C4×C59#48111yescharacteristic
195Dic59—81no—
196#3037—591no—
197#2351D5911yescharacteristic, complemented
198#1517—591no—
199#1502—591no—
200#1479—591no—
201#1303—591no—
202#1286—591no—
203C2×C59—21no—
204C2×C59—21no—
205C2×C59#187411yescharacteristic
206C2×C59#190511yescharacteristic
207C2×C59#190611yescharacteristic
208D59—161no—
209C2×#137—1181no—
210#532—591no—
211C2×C4×Q8—21no—
212C2×C4×Q8C2×D5911yescharacteristic
213#464—1182no—
214#462—592no—
215#460—592no—
216#431—591no—
217#431—591no—
218#423—591no—
219#417—1181no—
220#389—21no—
221#389—22no—
222#382—21no—
223#382C2×D5911yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2×D5911yescharacteristic
226#327—1181no—
227#327—1181no—
228#327—1182no—
229Q8⋊C8—1182no—
230C59#918011yescharacteristic, complemented, Sylow
231C2×#41—1181no—
232C22×Q8C22⋊D5911yescharacteristic
233#137—2362no—
234#137—2362no—
235C2×Q16—1182no—
236C2×SD16—1182no—
237C2×M16—1181no—
238C4⋊Q8—591no—
239#127—591no—
240#124—1181no—
241#121—1181no—
242#121—1181no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8C4×D5912yes—
248C4×Q8D23612yes—
249C4×D4—1181no—
250C4×D4—1181no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D5911yescharacteristic
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C22×D5911yescharacteristic
261C4⋊C8—592no—
262C4⋊C8—1181no—
263C4⋊C8—1181no—
264C4⋊C8—1181no—
265#103—1182no—
266#102—1182no—
267C22⋊C8—1181no—
268C22⋊C8—1181no—
269C4×C8—1181no—
270C4×C8—1181no—
271#95—22no—
272#95—21no—
273#95C22⋊D5911yescharacteristic
274#95C22⋊D5911yescharacteristic
275#95C22⋊D5912yes—
276#41—2361no—
277#41—2361no—
278#41—2361no—
279#41—2361no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—1181no—
284C2×Q8(C2×D59)⋊C412yes—
285C2×Q8#1174566912yes—
286C2×Q8#1174567112yes—
287C2×D4—1181no—
288C2×D4—1181no—
289C2×D4—1181no—
290C2×D4—2361no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—2361no—
297C22×C4C2×C22⋊D5911yescharacteristic
298C22×C4C2×C22⋊D5911yescharacteristic
299C22×C4C2×C22⋊D5911yescharacteristic
300Q16—2364no—
301SD16—2364no—
302M16—2361no—
303M16—2361no—
304C2×C8—1182no—
305C2×C8—2361no—
306C2×C8—2361no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1181no—
316C4⋊C4—1181no—
317C4⋊C4(C2×D59)⋊C412yes—
318C22⋊C4—2361no—
319C22⋊C4—2361no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C2×C4×D5911yescharacteristic
326C42C2×D23611yescharacteristic
327C42#1174568511yescharacteristic
328C42#1174568511yescharacteristic
329C23—2361no—
330C23#1427369611yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—2361no—
335Q8—2361no—
336Q8#1427364714yes—
337D4—4721no—
338D4—4721no—
339D4—4721no—
340D4—4721no—
341D4—2361no—
342D4—2361no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—4721no—
360C2×C4—4721no—
361C2×C4—2361no—
362C2×C4C2×(C2×D59)⋊C411yescharacteristic
363C2×C4C4×C22⋊D5911yescharacteristic
364C2×C4C4×C22⋊D5911yescharacteristic
365C2×C4#1427370111yescharacteristic
366C2×C4#1427370111yescharacteristic
367C2×C4#1427371011yescharacteristic
368C2×C4C2×#1174566911yescharacteristic
369C2×C4C2×#1174567111yescharacteristic
370C2×C4#1427375311yescharacteristic
371C2×C4#1427375811yescharacteristic
372C8—4721no—
373C8—4721no—
374C8—2362no—
375C22—21no—
376C22—21no—
377C22—4721no—
378C22—4721no—
379C22—4721no—
380C22#1683728111yescharacteristic
381C22#1683753611yescharacteristic
382C22#1683754111yescharacteristic
383C4—24no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—4721no—
393C4C2×#1427364711yescharacteristic
394C4#1683740111yescharacteristic
395C4#1683771511yescharacteristic
396C4#1683772011yescharacteristic
397C2—21no—
398C2—21no—
399C2—9441no—
400C2#2142441311yescharacteristic
401C2#2142459011yescharacteristic
402C2#2142459711yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2351C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2351D59
hypocenterC59#9180
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #21422491C2semidirect
5. #21421999C2semidirect
6. #21421954C2semidirect
7. C59×#2351C2semidirect
11. #16837612C4semidirect
12. #16837602C4semidirect
197. #2351D59semidirect
230. C59#9180semidirect

Maximal subgroups

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

subgroupindex
#214335682
#214226552
#214225902
#214224912
#214219992
#214219542
C59×#23512
#918059