Finite Group Databasebeta

← #28484535

Subgroups — #28484535

Order 15104.

436 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433185C211yesmaximal, characteristic
2#21423926C211yesmaximal, characteristic
3#21422711C211yesmaximal, characteristic
4#21422595C211yesmaximal, characteristic
5#21421994C211yesmaximal, characteristic
6#21421953C211yesmaximal, characteristic
7C59×#3113C211yesmaximal, characteristic
8#16837533—22no—
9#16837533—21no—
10Q8×C59⋊C8—21no—
11#16837459—21no—
12#16837459C2211yescharacteristic
13#16837455—21no—
14#16837455C2211yescharacteristic
15C2×#14273651—21no—
16C2×#14273638C2211yescharacteristic
17C4×C4⋊Dic59C2211yescharacteristic
18#16837164—21no—
19#16837144—22no—
20#16837139—21no—
21#16837139—21no—
22C59×#533C2211yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#483—21no—
25C59×#475C411yescharacteristic, complemented
26C59×#475C411yescharacteristic, complemented
27C59×#462C411yescharacteristic, complemented
28C59×#462C411yescharacteristic, complemented
29C59×#451—21no—
30C59×#441—21no—
31C59×#441—21no—
32C59×#432C2211yescharacteristic
33C22×C59⋊C8—21no—
34C2×C4⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40#14273651—24no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)D411yescharacteristic
43C22⋊(C59⋊C8)D411yescharacteristic
44#14273638—22no—
45#14273638—21no—
46#14273638—21no—
47#14273638D412yes—
48C4×C59⋊C8—22no—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C22×Q8×C59D411yescharacteristic
52C2×Q16×C59—21no—
53C2×Q16×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×C59—21no—
62C4×Q8×C59—21no—
63C4×Q8×C59C2×C411yescharacteristic
64C4×Q8×C59C2×C411yescharacteristic
65C4×Q8×C59D411yescharacteristic
66C4×Q8×C59D411yescharacteristic
67#117×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#106×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#103×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—22no—
84C2×Q8×C59—21no—
85C2×Q8×C59—21no—
86C2×Q8×C59—41no—
87C2×Q8×C59C22⋊C411yescharacteristic
88C2×Q8×C59C22⋊C411yescharacteristic
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×Dic59—41no—
95Q16×C59—42no—
96Q16×C59—42no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C4⋊C4×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—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59C22⋊C411yescharacteristic
111C4⋊C4×C59C22⋊C411yescharacteristic
112C22⋊C4×C59—41no—
113C42×C59—21no—
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—
124C4×Dic59—81no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C23×C59#12111yescharacteristic
132Q8×C59—21no—
133Q8×C59—21no—
134Q8×C59—21no—
135Q8×C59—42no—
136Q8×C59—41no—
137Q8×C59—41no—
138Q8×C59—81no—
139Q8×C59#10412yes—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
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—41no—
166C59⋊C8—41no—
167C59⋊C8—42no—
168C59⋊C8—42no—
169#9166—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59#38611yescharacteristic
173C22×C59#47111yescharacteristic
174C22×C59#47311yescharacteristic
175C4×C59—21no—
176C4×C59—22no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59C2×#10411yescharacteristic
187C4×C59#42111yescharacteristic
188Dic59—82no—
189#3113D5911yescharacteristic
190#2984—591no—
191#1763—591no—
192#1531—591no—
193#1503—591no—
194#1302—591no—
195#1285—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59#187111yescharacteristic
199C2×C59#190511yescharacteristic
200C2×C59#190811yescharacteristic
201#533C2×D5911yescharacteristic
202C2×C4×Q8C2×D5911yescharacteristic
203#483—21no—
204#475Dic5911yescharacteristic, complemented
205#475Dic5911yescharacteristic, complemented
206#462Dic5911yescharacteristic, complemented
207#462Dic5911yescharacteristic, complemented
208#451—21no—
209#446—1182no—
210#446—1181no—
211C8×Q8—1181no—
212#441—21no—
213#441—21no—
214#432—591no—
215#432—1181no—
216#432C2×D5911yescharacteristic
217#431—591no—
218#431—1181no—
219C2×#103—1181no—
220C2×#97—591no—
221C4×C4⋊C4—591no—
222#340—1181no—
223#330—1182no—
224#328—1181no—
225#328—1181no—
226C59#916611yescharacteristic, complemented, Sylow
227C22×Q8C22⋊D5911yescharacteristic
228C2×Q16—21no—
229C2×Q16—21no—
230C22×C8—1181no—
231C4⋊Q8C2×Dic5911yescharacteristic
232C4⋊Q8C2×Dic5911yescharacteristic
233#125—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C2×Dic5911yescharacteristic
241C4×Q8C2×Dic5911yescharacteristic
242C4×Q8C22⋊D5911yescharacteristic
243C4×Q8C22⋊D5911yescharacteristic
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4C22⋊D5911yescharacteristic
248C2×C42—21no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#106—21no—
252#106—21no—
253C4⋊C8—1182no—
254C4⋊C8—1182no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257C4⋊C8C2×Dic5911yescharacteristic
258C4⋊C8C2×Dic5911yescharacteristic
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263#103—1184no—
264C22⋊C8—21no—
265C22⋊C8—591no—
266C22⋊C8—591no—
267C22⋊C8—1181no—
268#97—21no—
269#97—592no—
270#97—1182no—
271#97—1181no—
272#97—1181no—
273C4×C8—1182no—
274C4×C8—1181no—
275#95—1181no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8C22⋊Dic5911yescharacteristic
282C2×Q8C22⋊Dic5911yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—2361no—
286C22×C4D4×D5911yescharacteristic
287C22×C4#1174568711yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289Q16—42no—
290Q16—42no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—1182no—
294C2×C8—1181no—
295C2×C8—1182no—
296C2×C8—1181no—
297C2×C8—2361no—
298C2×C8—2361no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—1182no—
311C4⋊C4—2361no—
312C4⋊C4C22⋊Dic5911yescharacteristic
313C4⋊C4C22⋊Dic5911yescharacteristic
314C22⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—4721no—
321C42D4×D5911yescharacteristic
322C42#1174568711yescharacteristic
323C42C22×Dic5911yescharacteristic
324C42C2×C22⋊D5911yescharacteristic
325C23#1427369911yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8#1427365412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—4721no—
349C2×C4—4721no—
350C2×C4—2362no—
351C2×C4C2×C22⋊Dic5911yescharacteristic
352C2×C4D4×Dic5911yescharacteristic
353C2×C4D4×Dic5911yescharacteristic
354C2×C4#1427369511yescharacteristic
355C2×C4#1427369911yescharacteristic
356C2×C4#1427370811yescharacteristic
357C8—41no—
358C8—41no—
359C8—2361no—
360C8—2362no—
361C8—2361no—
362C8—2362no—
363C22—21no—
364C22—21no—
365C22#1683728211yescharacteristic
366C22#1683766711yescharacteristic
367C22#1683767511yescharacteristic
368C4—21no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—4722no—
380C4C2×#1427365411yescharacteristic
381C4#1683741911yescharacteristic
382C2—21no—
383C2—21no—
384C2#2142439911yescharacteristic
385C2#2142459211yescharacteristic
386C2#2142461311yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3113C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#3113D59
hypocenterC59#9166
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
25. C59×#475C4semidirect
26. C59×#475C4semidirect
27. C59×#462C4semidirect
28. C59×#462C4semidirect
204. #475Dic59semidirect
205. #475Dic59semidirect
206. #462Dic59semidirect
207. #462Dic59semidirect
226. C59#9166semidirect

Maximal subgroups

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

subgroupindex
#214331852
#214239262
#214227112
#214225952
#214219942
#214219532
C59×#31132
#916659