Finite Group Databasebeta

← #28481851

Subgroups — #28481851

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C59⋊C8C211yesmaximal, characteristic
2#21424116C211yesmaximal, characteristic
3C4×#14273658C211yesmaximal, characteristic, complemented
4#21422851C211yesmaximal, characteristic, complemented
5#21421999C211yesmaximal, characteristic, complemented
6#21421997C211yesmaximal, characteristic, complemented
7C59×#1806C211yesmaximal, characteristic
8Q8×C59⋊C8C411yescharacteristic, complemented
9Q8×C59⋊C8C411yescharacteristic, complemented
10D4×C59⋊C8—21no—
11D4×C59⋊C8—21no—
12D4×C59⋊C8C411yescharacteristic, complemented
13D4×C59⋊C8C411yescharacteristic, complemented
14#16837455—21no—
15#16837455C2211yescharacteristic
16C2×#14273658—21no—
17#16837426C2211yescharacteristic
18C2×#14273654—21no—
19C2×C4×C59⋊C8—21no—
20C2×C4×C59⋊C8C2211yescharacteristic
21C4×C4⋊Dic59C2211yescharacteristic
22#16837164—21no—
23#16837161—21no—
24#16837152—21no—
25#16837152—21no—
26C4×#41×C59C2211yescharacteristic
27C59×#423C2211yescharacteristic
28C2×#104×C59—21no—
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#339—21no—
31C2×#11745664—21no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34C22×C59⋊C8D411yescharacteristic
35C22×C59⋊C8D411yescharacteristic
36C2×C4⋊Dic59—21no—
37C2×C4×Dic59—21no—
38#14273658—24no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)C2×C411yescharacteristic
44C4⋊(C59⋊C8)C2×C411yescharacteristic
45#14273654—42no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C4×C59⋊C8C2×C411yescharacteristic
53C4×C59⋊C8C2×C411yescharacteristic
54C4×C59⋊C8D411yescharacteristic
55C4×C59⋊C8D411yescharacteristic
56C59⋊#95—21no—
57C2×#41×C59D411yescharacteristic
58C2×M16×C59—21no—
59C4×Q8×C59C2×C411yescharacteristic
60C4×Q8×C59Q811yescharacteristic
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59C2×C411yescharacteristic
64C4×D4×C59Q811yescharacteristic
65#117×C59—21no—
66#117×C59D411yescharacteristic
67C2×C4⋊C4×C59—21no—
68C2×C42×C59—21no—
69C2×C42×C59—21no—
70C2×C42×C59C2311yescharacteristic
71C4⋊C8×C59—21no—
72C4⋊C8×C59—21no—
73#104×C59—42no—
74#95×C59—21no—
75#41×C59—21no—
76#41×C59—21no—
77#41×C59SD1612yes—
78#41×C59SD1612yes—
79C2×Q8×C59C4⋊C411yescharacteristic
80C2×D4×C59—21no—
81C2×D4×C59C4⋊C411yescharacteristic
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—41no—
90M16×C59—41no—
91M16×C59—41no—
92C2×C8×C59—41no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59C4⋊C411yescharacteristic
98C4⋊C4×C59C4⋊C411yescharacteristic
99C22⋊C4×C59—21no—
100C22⋊C4×C59—21no—
101C22⋊C4×C59—21no—
102C42×C59—21no—
103C42×C59—21no—
104C42×C59—21no—
105C42×C59—41no—
106C42×C59—41no—
107C42×C59C22×C411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59C2×Q811yescharacteristic
110C42×C59#4111yescharacteristic
111C4⋊Dic59—22no—
112C4⋊Dic59—41no—
113C4×Dic59—41no—
114C4×Dic59—41no—
115#11745664—41no—
116#11745664—41no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—42no—
122C2×C59⋊C8—42no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8C22⋊C411yescharacteristic
125C2×C59⋊C8C22⋊C411yescharacteristic
126C2×C59⋊C8C22⋊C411yescharacteristic
127C2×C59⋊C8C22⋊C411yescharacteristic
128C23×C59—21no—
129C23×C59#12111yescharacteristic
130Q8×C59—21no—
131Q8×C59#10612yescomplemented
132D4×C59—22no—
133D4×C59—22no—
134D4×C59—21no—
135D4×C59#10612yescomplemented
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×C4×C59C2×C22⋊C411yescharacteristic
151C2×C4×C59C2×C4⋊C411yescharacteristic
152C2×C4×C59#11711yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59C22⋊Q811yescharacteristic
155C2×C4×C59C22⋊Q811yescharacteristic
156C2×C4×C59C2×SD1611yescharacteristic
157C2×C4×C59C2×SD1611yescharacteristic
158C2×C4×C59#13511yescharacteristic
159C2×C4×C59#13511yescharacteristic
160C2×Dic59—42no—
161C2×Dic59—81no—
162C2×Dic59—81no—
163C8×C59—81no—
164C59⋊C8—21no—
165C59⋊C8—21no—
166C59⋊C8—21no—
167C59⋊C8—41no—
168C59⋊C8—41no—
169C59⋊C8—81no—
170C59⋊C8#10412yescomplemented
171#8666—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—22no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59#38011yescharacteristic
178C22×C59#46511yescharacteristic
179C22×C59#46511yescharacteristic
180C4×C59—21no—
181C4×C59—22no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—41no—
185C4×C59—41no—
186C4×C59—82no—
187C4×C59C2×#10411yescharacteristic
188C4×C59#42111yescharacteristic
189C4×C59C2×#10611yescharacteristic
190C4×C59#42711yescharacteristic
191C4×C59#46511yescharacteristic
192C4×C59#46511yescharacteristic
193Dic59—82no—
194C8×#41—591no—
195#1806—591no—
196#1806D5911yescharacteristic
197C4×#106—591no—
198#1566—591no—
199#1303—591no—
200#1303—591no—
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—42no—
204C2×C59#180611yescharacteristic
205C2×C59#180711yescharacteristic
206C2×C59#181411yescharacteristic
207C4×#41C2×D5911yescharacteristic
208C8×Q8—591no—
209C8×Q8—591no—
210C8×D4—591no—
211C8×D4—591no—
212C8×D4—1181no—
213C8×D4—1181no—
214#431—591no—
215#431—1181no—
216C2×#106—1181no—
217#423—591no—
218#423C2×D5911yescharacteristic
219C2×#104—21no—
220C2×#104—1181no—
221C2×C4×C8—591no—
222C2×C4×C8—1181no—
223C4×C4⋊C4—591no—
224C4×C4⋊C4C2×D5911yescharacteristic
225#340—1181no—
226#339—21no—
227#339—1181no—
228#334—1181no—
229#334—1181no—
230C59#866611yescharacteristic, complemented, Sylow
231C2×#41C22⋊D5911yescharacteristic
232C2×M16—21no—
233C2×M16—1181no—
234C22×C8—591no—
235C22×C8—591no—
236C22×C8—1181no—
237C22×C8—1181no—
238C4×Q8Dic11811yescharacteristic
239C4×Q8C4×D5911yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4Dic11811yescharacteristic
243C4×D4C4×D5911yescharacteristic
244#117—21no—
245#117C22⋊D5911yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1181no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—1181no—
251C2×C42C22×D5911yescharacteristic
252#106—1184no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—591no—
256C4⋊C8—591no—
257C4⋊C8—1181no—
258C4⋊C8—1181no—
259C4⋊C8—1181no—
260C4⋊C8—1181no—
261#104—42no—
262#104—2362no—
263C22⋊C8—1181no—
264C22⋊C8—1181no—
265C22⋊C8—1181no—
266C4×C8—591no—
267C4×C8—591no—
268C4×C8—591no—
269C4×C8—591no—
270C4×C8—1181no—
271C4×C8—1181no—
272C4×C8—1181no—
273#95—21no—
274#95—1181no—
275#41—21no—
276#41—21no—
277#41#1174567012yes—
278#41#1174567112yes—
279C2×Q8Dic59⋊C411yescharacteristic
280C2×D4—21no—
281C2×D4Dic59⋊C411yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—2361no—
287C22×C4D4×D5911yescharacteristic
288C22×C4#1174568711yescharacteristic
289C22×C4C2×C22⋊D5911yescharacteristic
290M16—41no—
291M16—41no—
292M16—2361no—
293M16—2361no—
294C2×C8—41no—
295C2×C8—591no—
296C2×C8—591no—
297C2×C8—591no—
298C2×C8—591no—
299C2×C8—1182no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—2362no—
303C2×C8—2361no—
304C2×C8—2362no—
305C2×C8—2361no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—1182no—
311C4⋊C4—2361no—
312C4⋊C4Dic59⋊C411yescharacteristic
313C4⋊C4Dic59⋊C411yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—2361no—
323C42—2361no—
324C42C2×Dic11811yescharacteristic
325C42C2×C4×D5911yescharacteristic
326C42D4×D5911yescharacteristic
327C42#1174568711yescharacteristic
328C23—21no—
329C23#1427369911yescharacteristic
330Q8—21no—
331Q8#1427365812yescomplemented
332D4—22no—
333D4—22no—
334D4—21no—
335D4#1427365812yescomplemented
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—4721no—
351C2×C4—4721no—
352C2×C4—2362no—
353C2×C4C22⋊C4×D5911yescharacteristic
354C2×C4C2×Dic59⋊C411yescharacteristic
355C2×C4#1427368311yescharacteristic
356C2×C4#1427369911yescharacteristic
357C2×C4#1427370411yescharacteristic
358C2×C4#1427370411yescharacteristic
359C2×C4C2×#1174567111yescharacteristic
360C2×C4C2×#1174567011yescharacteristic
361C2×C4#1427374711yescharacteristic
362C2×C4#1427374711yescharacteristic
363C8—81no—
364C8—592no—
365C8—1181no—
366C8—1181no—
367C8—1181no—
368C8—4721no—
369C8—2361no—
370C8—2361no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22#1683726011yescharacteristic
377C22#1683762911yescharacteristic
378C22#1683763311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—4722no—
387C4#104×D5911yescharacteristic
388C4#1683741711yescharacteristic
389C4C2×#1427365811yescharacteristic
390C4#1683743811yescharacteristic
391C4#1683762911yescharacteristic
392C4#1683763311yescharacteristic
393C2—21no—
394C2—21no—
395C2—42no—
396C2#2142411411yescharacteristic
397C2#2142411911yescharacteristic
398C2#2142414811yescharacteristic
subgroupG/H
centerC2×C4#14273699
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1806C2
socleC22×C59#380
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#1806D59
hypocenterC59#8666
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. C4×#14273658C2semidirect
4. #21422851C2semidirect
5. #21421999C2semidirect
6. #21421997C2semidirect
8. Q8×C59⋊C8C4semidirect
9. Q8×C59⋊C8C4semidirect
12. D4×C59⋊C8C4semidirect
13. D4×C59⋊C8C4semidirect
131. Q8×C59#106semidirect
135. D4×C59#106semidirect
170. C59⋊C8#104semidirect
230. C59#8666semidirect
331. Q8#14273658semidirect
335. D4#14273658semidirect

Maximal subgroups

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

subgroupindex
#41×C59⋊C82
#214241162
C4×#142736582
#214228512
#214219992
#214219972
C59×#18062
#866659