Finite Group Databasebeta

← #28485719

Subgroups — #28485719

Order 15104.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427262C211yesmaximal, characteristic
2#21425082C211yesmaximal, characteristic
3#21422629C211yesmaximal, characteristic
4#21422616C211yesmaximal, characteristic
5#21422003C211yesmaximal, characteristic
6#21421950C211yesmaximal, characteristic
7C59×#3195C211yesmaximal, characteristic
8#16837366C2211yescharacteristic
9#16837309—21no—
10#16837301—21no—
11#16837295—21no—
12#16837295—21no—
13#16837288—21no—
14#16837288—21no—
15#16837265C2211yescharacteristic
16#16837262C2211yescharacteristic
17C4×C22⋊Dic59C2211yescharacteristic
18#16837169—21no—
19#16837169—21no—
20#16837139—21no—
21#16837139—21no—
22#16837139—22no—
23C2×C22⋊Q8×C59C2211yescharacteristic
24C2×C22⋊Q8×C59C2211yescharacteristic
25C59×#483—21no—
26C59×#483C412yescomplemented
27C59×#470—22no—
28C59×#451—21no—
29C59×#451C412yescomplemented
30C59×#407C2211yescharacteristic
31C2×#11745664—21no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59D411yescharacteristic
38C2×C22⋊Dic59D411yescharacteristic
39C2×C4×Dic59—21no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59—21no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95D411yescharacteristic
52C59⋊#95D411yescharacteristic
53C22×Q8×C59C2×C411yescharacteristic
54C22×Q8×C59D411yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C2×Q16×C59—22no—
57C2×M16×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59C2×C412yes—
65C22⋊Q8×C59D412yes—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59C2×C411yescharacteristic
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72#106×C59—22no—
73#103×C59—22no—
74#103×C59—22no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59C2×C412yes—
77C24×C59C2×D411yescharacteristic
78C2×Q8×C59—21no—
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—81no—
82C2×Q8×C59—81no—
83C2×Q8×C59C22⋊C412yes—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—81no—
89C22×C4×C59C22×C411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—21no—
99C22×Dic59—41no—
100Q16×C59—82no—
101M16×C59—81no—
102C2×C8×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59C22⋊C412yes—
111C22⋊C4×C59—22no—
112C22⋊C4×C59—22no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C22⋊Dic59—22no—
116C22⋊Dic59—41no—
117C22⋊Dic59—41no—
118C22⋊Dic59—41no—
119C22⋊Dic59—81no—
120C4⋊Dic59—81no—
121C4⋊Dic59—81no—
122C4⋊Dic59—81no—
123C4×Dic59—81no—
124C4×Dic59—81no—
125C4×Dic59—81no—
126#11745664—81no—
127C2×C59⋊C8—41no—
128C23×C59—21no—
129C23×C59—21no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59C4×D411yescharacteristic
133C23×C59#12011yescharacteristic
134C23×C59#12111yescharacteristic
135Q8×C59—42no—
136Q8×C59—42no—
137Q8×C59—81no—
138Q8×C59—81no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59#12311yescharacteristic
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—42no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—81no—
173C2×Dic59—81no—
174C8×C59—42no—
175C59⋊C8—81no—
176#9342—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59#38611yescharacteristic
188C22×C59#45511yescharacteristic
189C22×C59#45611yescharacteristic
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—42no—
195C4×C59—42no—
196C4×C59—81no—
197C4×C59—81no—
198Dic59—81no—
199Dic59—81no—
200Dic59—81no—
201Dic59—81no—
202Dic59—82no—
203#3195D5911yescharacteristic
204#2378—591no—
205#1996—591no—
206#1511—591no—
207#1508—591no—
208#1305—591no—
209#1284—591no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—41no—
213C2×C59—41no—
214C2×C59#188211yescharacteristic
215C2×C59#190511yescharacteristic
216C2×C59#191311yescharacteristic
217C2×C22⋊Q8C2×D5911yescharacteristic
218C2×C22⋊Q8C2×D5911yescharacteristic
219#483—21no—
220#483Dic5912yescomplemented
221#470—22no—
222#451—21no—
223#451Dic5912yescomplemented
224#407—591no—
225#407C2×D5911yescharacteristic
226#391—1181no—
227#389—1181no—
228#388—1181no—
229#388—1181no—
230#387—1181no—
231#387—1181no—
232#381—591no—
233#380—591no—
234C4×C22⋊C4—591no—
235#342—1181no—
236#342—1181no—
237#328—1181no—
238#328—1181no—
239#328—1182no—
240C59#934211yescharacteristic, complemented, Sylow
241C22×Q8C2×Dic5911yescharacteristic
242C22×Q8C22⋊D5911yescharacteristic
243C23×C4C22×D5911yescharacteristic
244C2×Q16—22no—
245C2×M16—21no—
246C2×M16—1181no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8C2×Dic5912yes—
254C22⋊Q8C22⋊D5912yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—1181no—
258C2×C4⋊C4—1181no—
259C2×C4⋊C4—1181no—
260C2×C4⋊C4C2×Dic5911yescharacteristic
261C2×C4⋊C4C22⋊D5911yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—591no—
265C2×C22⋊C4—591no—
266C2×C22⋊C4—1181no—
267C2×C22⋊C4—1181no—
268C2×C42—1181no—
269C2×C42—1181no—
270C2×C42—1181no—
271#106—22no—
272#103—22no—
273#103—22no—
274C22⋊C8—21no—
275C22⋊C8—1181no—
276C22⋊C8—1181no—
277C22⋊C8C2×Dic5912yes—
278#95—591no—
279#95—591no—
280#95—1181no—
281#95—1181no—
282#95—1181no—
283#95—1181no—
284#95—1181no—
285#95—1181no—
286#95—1181no—
287C24D4×D5911yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—81no—
292C2×Q8—81no—
293C2×Q8C22⋊Dic5912yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—81no—
299C22×C4—1181no—
300C22×C4—1181no—
301C22×C4—1181no—
302C22×C4—1181no—
303C22×C4—2361no—
304C22×C4D4×D5911yescharacteristic
305C22×C4#1174568711yescharacteristic
306C22×C4#1174568711yescharacteristic
307C22×C4C22×Dic5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310Q16—82no—
311M16—81no—
312M16—4721no—
313C2×C8—22no—
314C2×C8—2361no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—4721no—
323C4⋊C4—4721no—
324C4⋊C4—4721no—
325C4⋊C4C22⋊Dic5912yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1182no—
331C22⋊C4—4721no—
332C22⋊C4—2361no—
333C22⋊C4—2361no—
334C22⋊C4—2361no—
335C42—4721no—
336C42—4721no—
337C42—4721no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23D4×Dic5911yescharacteristic
343C23#1427369511yescharacteristic
344C23#1427369911yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—81no—
348Q8—81no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—4721no—
366C2×C4—4721no—
367C2×C4—2361no—
368C2×C4—2361no—
369C2×C4—2362no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4—2361no—
379C2×C4—2361no—
380C2×C4C2×C22⋊Dic5911yescharacteristic
381C2×C4D4×Dic5911yescharacteristic
382C2×C4#1427369911yescharacteristic
383C2×C4#1427370811yescharacteristic
384C8—42no—
385C8—4721no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#1683728211yescharacteristic
397C22#1683757811yescharacteristic
398C22#1683758311yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—4721no—
408C4—4721no—
409C4—4721no—
410C4—4721no—
411C4—4722no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#2142446711yescharacteristic
417C2#2142459211yescharacteristic
418C2#2142463911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#3195C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#3195D59
hypocenterC59#9342
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
26. C59×#483C4semidirect
29. C59×#451C4semidirect
220. #483Dic59semidirect
223. #451Dic59semidirect
240. C59#9342semidirect

Maximal subgroups

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

subgroupindex
#214272622
#214250822
#214226292
#214226162
#214220032
#214219502
C59×#31952
#934259