Finite Group Databasebeta

← #28484486

Subgroups — #28484486

Order 15104.

452 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433110C211yesmaximal, characteristic
2#21423906C211yesmaximal, characteristic
3#21422691C211yesmaximal, characteristic, complemented
4#21422608C211yesmaximal, characteristic, complemented
5#21421988C211yesmaximal, characteristic, complemented
6#21421967C211yesmaximal, characteristic, complemented
7C59×#3116C211yesmaximal, characteristic
8#16837533—21no—
9#16837474—22no—
10D4×C59⋊C8—21no—
11#16837459—21no—
12#16837459C2211yescharacteristic
13C2×#14273654—21no—
14C4×#11745664C2211yescharacteristic
15C4×#11745664C2211yescharacteristic
16C2×#14273638—21no—
17#16837248C2211yescharacteristic
18#16837161—21no—
19#16837144—21no—
20#16837144—21no—
21#16837139—22no—
22C59×#533C2211yescharacteristic
23C4×#41×C59C2211yescharacteristic
24C59×#485—21no—
25C59×#477C411yescharacteristic, complemented
26C59×#476C411yescharacteristic, complemented
27C59×#462C411yescharacteristic, complemented
28C59×#461C411yescharacteristic, complemented
29C59×#452—21no—
30C59×#441—21no—
31C59×#440—21no—
32C59×#432C2211yescharacteristic
33C2×#11745664—21no—
34C2×#11745664—21no—
35C22×C59⋊C8—22no—
36C2×C4×Dic59—21no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—21no—
39#14273654—24no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)D412yes—
42#14273638—22no—
43#14273638—21no—
44#14273638—21no—
45#14273638D411yescharacteristic
46#14273638D411yescharacteristic
47C4×C59⋊C8—21no—
48C4×C59⋊C8—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C2×#41×C59D411yescharacteristic
52C2×Q16×C59—21no—
53C2×SD16×C59—21no—
54C4⋊Q8×C59C2×C411yescharacteristic
55C4⋊Q8×C59C2×C411yescharacteristic
56#125×C59—21no—
57C22⋊Q8×C59—21no—
58C22⋊Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×Q8×C59—21no—
61C4×Q8×C59C2×C411yescharacteristic
62C4×Q8×C59D411yescharacteristic
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59C2×C411yescharacteristic
66C4×D4×C59D411yescharacteristic
67#117×C59—21no—
68#117×C59—21no—
69#117×C59D411yescharacteristic
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#107×C59—21no—
73#106×C59—21no—
74C4⋊C8×C59C2×C411yescharacteristic
75C4⋊C8×C59C2×C411yescharacteristic
76#103×C59—21no—
77#103×C59—21no—
78#103×C59—21no—
79#102×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82#41×C59—22no—
83#41×C59—41no—
84C2×Q8×C59—21no—
85C2×Q8×C59—21no—
86C2×Q8×C59C22⋊C411yescharacteristic
87C2×D4×C59—21no—
88C2×D4×C59C22⋊C411yescharacteristic
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C22×Dic59—41no—
96Q16×C59—42no—
97SD16×C59—42no—
98C2×C8×C59—21no—
99C2×C8×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59C22⋊C411yescharacteristic
109C4⋊C4×C59C22⋊C411yescharacteristic
110C22⋊C4×C59—21no—
111C22⋊C4×C59—21no—
112C22⋊C4×C59—21no—
113C22⋊C4×C59—41no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—21no—
117C42×C59—41no—
118C42×C59C22×C411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59#4111yescharacteristic
122C4×Dic59—22no—
123C4×Dic59—41no—
124#11745664—22no—
125#11745664—42no—
126#11745664—41no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—44no—
131C2×C59⋊C8—41no—
132C23×C59—21no—
133C23×C59#12111yescharacteristic
134Q8×C59—21no—
135Q8×C59—21no—
136Q8×C59—41no—
137Q8×C59—41no—
138Q8×C59—81no—
139D4×C59—21no—
140D4×C59—42no—
141D4×C59#10412yes—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—81no—
154C2×C4×C59C2×C22⋊C411yescharacteristic
155C2×C4×C59C4×D411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59#12011yescharacteristic
158C2×C4×C59#12111yescharacteristic
159C2×C4×C59#12311yescharacteristic
160C2×Dic59—42no—
161C2×Dic59—81no—
162C2×Dic59—81no—
163C8×C59—41no—
164C8×C59—41no—
165C59⋊C8—22no—
166C59⋊C8—42no—
167C59⋊C8—41no—
168C59⋊C8—42no—
169#9159—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—22no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59#38611yescharacteristic
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—41no—
185C4×C59—81no—
186C4×C59—81no—
187C4×C59C2×#10411yescharacteristic
188C4×C59#42111yescharacteristic
189Dic59—82no—
190#3116D5911yescharacteristic
191#2970—591no—
192#1759—591no—
193#1526—591no—
194#1506—591no—
195#1300—591no—
196#1291—591no—
197C2×C59—41no—
198C2×C59—42no—
199C2×C59#187111yescharacteristic
200C2×C59#190211yescharacteristic
201C2×C59#190611yescharacteristic
202#533C2×D5911yescharacteristic
203C4×#41C2×D5911yescharacteristic
204#485—21no—
205#477Dic5911yescharacteristic, complemented
206#476Dic5911yescharacteristic, complemented
207#462Dic5911yescharacteristic, complemented
208#461Dic5911yescharacteristic, complemented
209#452—21no—
210#446—1181no—
211#441—21no—
212#440—21no—
213#435—1182no—
214C8×D4—1181no—
215#432—591no—
216#432—1181no—
217#432C2×D5911yescharacteristic
218C2×#104—1181no—
219C4×M16—591no—
220C4×M16—591no—
221C2×#97—1181no—
222#376—591no—
223#339—1181no—
224#330—1181no—
225#330—1181no—
226#328—1182no—
227C59#915911yescharacteristic, complemented, Sylow
228C2×#41C22⋊D5911yescharacteristic
229C2×Q16—21no—
230C2×SD16—21no—
231C2×M16—1181no—
232C2×M16—1181no—
233C22×C8—1182no—
234C4⋊Q8C2×Dic5911yescharacteristic
235C4⋊Q8C2×Dic5911yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×Dic5911yescharacteristic
242C4×Q8C22⋊D5911yescharacteristic
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×Dic5911yescharacteristic
246C4×D4C22⋊D5911yescharacteristic
247#117—21no—
248#117—21no—
249#117C22⋊D5911yescharacteristic
250C2×C42—21no—
251C2×C42—1181no—
252C2×C42C22×D5911yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—1182no—
256C4⋊C8—1181no—
257C4⋊C8C2×Dic5911yescharacteristic
258C4⋊C8C2×Dic5911yescharacteristic
259#104—1184no—
260#103—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264C22⋊C8—21no—
265C22⋊C8—592no—
266C22⋊C8—1182no—
267#97—21no—
268#97—591no—
269#97—591no—
270#97—1182no—
271#97—1181no—
272#97—1181no—
273C4×C8—1181no—
274C4×C8—1181no—
275#95—1181no—
276#95—1181no—
277#41—22no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22⋊Dic5911yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊Dic5911yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—2361no—
288C22×C4D4×D5911yescharacteristic
289C22×C4#1174568711yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—1182no—
294M16—2362no—
295M16—2361no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1182no—
301C2×C8—2364no—
302C2×C8—2361no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊Dic5911yescharacteristic
312C4⋊C4C22⋊Dic5911yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—41no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—1182no—
322C42—2361no—
323C42D4×D5911yescharacteristic
324C42#1174568711yescharacteristic
325C42C22×Dic5911yescharacteristic
326C42C2×C22⋊D5911yescharacteristic
327C23—21no—
328C23#1427369911yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—41no—
332Q8—41no—
333Q8—81no—
334D4—21no—
335D4—42no—
336D4#1427365412yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—4721no—
350C2×C4—4721no—
351C2×C4—2362no—
352C2×C4C2×C22⋊Dic5911yescharacteristic
353C2×C4D4×Dic5911yescharacteristic
354C2×C4D4×Dic5911yescharacteristic
355C2×C4#1427369511yescharacteristic
356C2×C4#1427369911yescharacteristic
357C2×C4#1427370811yescharacteristic
358C8—41no—
359C8—41no—
360C8—1182no—
361C8—2362no—
362C8—2361no—
363C8—2362no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22#1683728211yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—4722no—
382C4C2×#1427365411yescharacteristic
383C4#1683741911yescharacteristic
384C2—41no—
385C2—42no—
386C2#2142439911yescharacteristic
387C2#2142457111yescharacteristic
388C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3116C2
socleC22×C59#386
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#3116D59
hypocenterC59#9159
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #21422691C2semidirect
4. #21422608C2semidirect
5. #21421988C2semidirect
6. #21421967C2semidirect
25. C59×#477C4semidirect
26. C59×#476C4semidirect
27. C59×#462C4semidirect
28. C59×#461C4semidirect
205. #477Dic59semidirect
206. #476Dic59semidirect
207. #462Dic59semidirect
208. #461Dic59semidirect
227. C59#9159semidirect

Maximal subgroups

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

subgroupindex
#214331102
#214239062
#214226912
#214226082
#214219882
#214219672
C59×#31162
#915959