Finite Group Databasebeta

← #28485437

Subgroups — #28485437

Order 15104.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435224C211yesmaximal, characteristic
2#21424928C211yesmaximal, characteristic
3#21422781C211yesmaximal, characteristic, complemented
4#21422655C211yesmaximal, characteristic
5#21421954C211yesmaximal, characteristic, complemented
6#21421925C211yesmaximal, characteristic, complemented
7C59×#2331C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837798C412yescomplemented
10#16837750C412yescomplemented
11#16837720—22no—
12#16837689—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×#14273655—21no—
17C2×#14273655—21no—
18#16837148—22no—
19#16837135—22no—
20C2×C4⋊Q8×C59C2211yescharacteristic
21C59×#391—22no—
22C59×#389—21no—
23C59×#387—22no—
24C59×#382C2211yescharacteristic
25C4×C4⋊C4×C59—21no—
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#14273658—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—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59—22no—
56C2×C4⋊C4×C59D412yes—
57C2×C42×C59—22no—
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61#95×C59—22no—
62#95×C59—22no—
63#95×C59—21no—
64#95×C59—21no—
65#95×C59—21no—
66#95×C59D412yes—
67C2×Q8×C59—22no—
68C2×Q8×C59—22no—
69C2×Q8×C59—82no—
70C22×C4×C59—21no—
71C22×C4×C59—22no—
72C22×C4×C59—21no—
73C22×C4×C59—22no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C22⋊D59—41no—
78#11745685—41no—
79#11745685—41no—
80#11745685—41no—
81#11745685—41no—
82C2×D236—21no—
83C2×D236—21no—
84C2×D236—21no—
85C2×C4×D59—41no—
86C2×Dic118—21no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—82no—
93C4⋊C4×C59C22⋊C414yes—
94C42×C59—21no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59—82no—
98C42×C59SD1612yes—
99C42×C59C22×C411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59#4111yescharacteristic
102C42×C59#4111yescharacteristic
103(C2×D59)⋊C4—41no—
104(C2×D59)⋊C4—41no—
105C4⋊Dic59—21no—
106C4⋊Dic59—21no—
107#11745664—41no—
108#11745664—41no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C23×C59#12011yescharacteristic
113C22×D59—41no—
114Q8×C59—82no—
115Q8×C59—82no—
116C2×C4×C59—24no—
117C2×C4×C59—22no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59C2×C22⋊C411yescharacteristic
131C2×C4×C59C4×D411yescharacteristic
132C2×C4×C59C4×D411yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12311yescharacteristic
136C2×C4×C59C2×SD1611yescharacteristic
137C2×C4×C59#13511yescharacteristic
138C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C59#13711yescharacteristic
140C22⋊D59—81no—
141C22⋊D59—81no—
142C2×Dic59—41no—
143D236—42no—
144D236—41no—
145D236—41no—
146D236—81no—
147C4×D59—81no—
148C4×D59—81no—
149Dic118—41no—
150Dic118—41no—
151C59⋊C8—42no—
152C59⋊C8—81no—
153C59⋊C8—81no—
154#9302—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59#38611yescharacteristic
158C22×C59#44811yescharacteristic
159C22×C59#44911yescharacteristic
160C2×D59—81no—
161C2×D59—81no—
162C2×D59—81no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—42no—
168C4×C59—44no—
169C4×C59—82no—
170C4×C59—81no—
171C4×C59C4×SD1611yescharacteristic
172C4×C59#44011yescharacteristic
173C4×C59#46011yescharacteristic
174C4×C59#46411yescharacteristic
175Dic59—81no—
176#3227—591no—
177#2331D5911yescharacteristic, complemented
178#1963—591no—
179#1549—591no—
180#1517—591no—
181#1286—591no—
182#1274—591no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59#188011yescharacteristic
186C2×C59#190411yescharacteristic
187C2×C59#191211yescharacteristic
188D59—161no—
189#532—591no—
190C2×C4⋊Q8C2×D5911yescharacteristic
191#498—592no—
192#486—592no—
193#481—1182no—
194#476—1182no—
195#431—591no—
196#431—591no—
197#423—591no—
198C2×#104—1181no—
199C2×#104—1181no—
200#391—22no—
201#389—21no—
202#387—22no—
203#382C2×D5911yescharacteristic
204C4×C4⋊C4—21no—
205#376C2×D5911yescharacteristic
206#332—1182no—
207#327—1182no—
208C59#930211yescharacteristic, complemented, Sylow
209C2×#41—1181no—
210C22×Q8—21no—
211C2×M16—1181no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—591no—
215C4⋊Q8C4×D5912yes—
216C4⋊Q8D23612yes—
217#127—591no—
218#124—1181no—
219#121—1181no—
220#121—1181no—
221C4×D4—1181no—
222C4×D4—1181no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D5912yes—
227C2×C42—22no—
228C2×C42C22⋊D5911yescharacteristic
229C2×C42C22⋊D5911yescharacteristic
230C2×C42C22×D5911yescharacteristic
231#106—1184no—
232C4⋊C8—1181no—
233C4⋊C8—1181no—
234C4⋊C8—1181no—
235C4⋊C8—1181no—
236#104—2362no—
237#104—2362no—
238#102—1184no—
239C22⋊C8—1181no—
240C22⋊C8—1181no—
241C4×C8—592no—
242C4×C8—1181no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C22⋊D5912yes—
249#41—2361no—
250#41—2361no—
251#41—2361no—
252#41—2361no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—1181no—
257C2×D4—1181no—
258C2×D4—1181no—
259C2×D4—1181no—
260C2×D4—2361no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—2361no—
266C22×C4C2×C22⋊D5911yescharacteristic
267C22×C4C2×C22⋊D5911yescharacteristic
268C22×C4C2×C22⋊D5911yescharacteristic
269M16—2361no—
270M16—2361no—
271C2×C8—1182no—
272C2×C8—2361no—
273C2×C8—2361no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—1181no—
281C4⋊C4—1181no—
282C4⋊C4(C2×D59)⋊C414yes—
283C22⋊C4—2361no—
284C22⋊C4—2361no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42#1174567112yes—
290C42C2×C4×D5911yescharacteristic
291C42C2×D23611yescharacteristic
292C42#1174568511yescharacteristic
293C42#1174568511yescharacteristic
294C23—2361no—
295C23#1427369611yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—2361no—
299Q8—2361no—
300D4—4721no—
301D4—4721no—
302D4—4721no—
303D4—2362no—
304D4—2361no—
305D4—2361no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—2361no—
323C2×C4C2×(C2×D59)⋊C411yescharacteristic
324C2×C4C4×C22⋊D5911yescharacteristic
325C2×C4C4×C22⋊D5911yescharacteristic
326C2×C4#1427370111yescharacteristic
327C2×C4#1427370111yescharacteristic
328C2×C4#1427371011yescharacteristic
329C2×C4C2×#1174567111yescharacteristic
330C2×C4#1427374711yescharacteristic
331C2×C4#1427375211yescharacteristic
332C2×C4#1427375811yescharacteristic
333C8—4721no—
334C8—4721no—
335C8—2362no—
336C22—21no—
337C22—21no—
338C22—4721no—
339C22—4721no—
340C22—4721no—
341C22#1683728111yescharacteristic
342C22#1683754111yescharacteristic
343C22#1683754711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—4721no—
353C4C4×#1174567111yescharacteristic
354C4#1683750011yescharacteristic
355C4#1683760211yescharacteristic
356C4#1683762411yescharacteristic
357C2—21no—
358C2—21no—
359C2—9441no—
360C2#2142445111yescharacteristic
361C2#2142458311yescharacteristic
362C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2331C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2331D59
hypocenterC59#9302
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #21422781C2semidirect
5. #21421954C2semidirect
6. #21421925C2semidirect
7. C59×#2331C2semidirect
9. #16837798C4semidirect
10. #16837750C4semidirect
177. #2331D59semidirect
208. C59#9302semidirect

Maximal subgroups

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

subgroupindex
#214352242
#214249282
#214227812
#214226552
#214219542
#214219252
C59×#23312
#930259