Finite Group Databasebeta

← #28484528

Subgroups — #28484528

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433131C211yesmaximal, characteristic
2#21423921C211yesmaximal, characteristic
3#21422711C211yesmaximal, characteristic, complemented
4#21422595C211yesmaximal, characteristic, complemented
5#21421994C211yesmaximal, characteristic, complemented
6#21421953C211yesmaximal, characteristic, complemented
7C59×#3110C211yesmaximal, characteristic
8#16837480—21no—
9#16837474—22no—
10D4×C59⋊C8—21no—
11#16837459C2211yescharacteristic
12#16837455C2211yescharacteristic
13C2×#14273648—21no—
14#16837366—21no—
15C2×C22⋊(C59⋊C8)—21no—
16C2×#14273638C2211yescharacteristic
17C4×C4⋊Dic59C2211yescharacteristic
18#16837164—21no—
19#16837144—22no—
20#16837139—21no—
21#16837139—21no—
22C59×#533C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C59×#484—21no—
25C59×#474C411yescharacteristic, complemented
26C59×#474C411yescharacteristic, complemented
27C59×#461C411yescharacteristic, complemented
28C59×#461C411yescharacteristic, complemented
29C59×#450—21no—
30C59×#440—21no—
31C59×#440—21no—
32C59×#432C2211yescharacteristic
33C2×#11745664—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—22no—
36C2×C4⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40#14273648—24no—
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)D411yescharacteristic
45C22⋊(C59⋊C8)D411yescharacteristic
46#14273638—21no—
47#14273638—21no—
48#14273638D412yes—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C22×D4×C59D411yescharacteristic
52C23×C4×C59—21no—
53C2×SD16×C59—21no—
54C2×SD16×C59—21no—
55C4⋊Q8×C59C2×C411yescharacteristic
56C4⋊Q8×C59C2×C411yescharacteristic
57#125×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C4×Q8×C59—21no—
61C4×Q8×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59C2×C411yescharacteristic
65C4×D4×C59C2×C411yescharacteristic
66C4×D4×C59D411yescharacteristic
67C4×D4×C59D411yescharacteristic
68#117×C59—21no—
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#107×C59—21no—
73#107×C59—21no—
74C4⋊C8×C59C2×C411yescharacteristic
75C4⋊C8×C59C2×C411yescharacteristic
76#103×C59—21no—
77#103×C59—21no—
78#102×C59—21no—
79#102×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82C24×C59—21no—
83C2×Q8×C59—21no—
84C2×Q8×C59—21no—
85C2×D4×C59—22no—
86C2×D4×C59—21no—
87C2×D4×C59—41no—
88C2×D4×C59C22⋊C411yescharacteristic
89C2×D4×C59C22⋊C411yescharacteristic
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—82no—
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59#4111yescharacteristic
99C22×Dic59—41no—
100SD16×C59—42no—
101SD16×C59—42no—
102C2×C8×C59—21no—
103C2×C8×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59C22⋊C411yescharacteristic
112C4⋊C4×C59C22⋊C411yescharacteristic
113C22⋊C4×C59—21no—
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C22⋊C4×C59—41no—
118C42×C59—41no—
119C42×C59C22×C411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C4⋊Dic59—22no—
124C4⋊Dic59—41no—
125C4×Dic59—81no—
126#11745664—82no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8—82no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—42no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59—81no—
140C23×C59#12111yescharacteristic
141Q8×C59—41no—
142Q8×C59—41no—
143Q8×C59—81no—
144D4×C59—21no—
145D4×C59—21no—
146D4×C59—21no—
147D4×C59—42no—
148D4×C59#10412yes—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—41no—
156C2×C4×C59—42no—
157C2×C4×C59—42no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59C2×C22⋊C411yescharacteristic
164C2×C4×C59C4×D411yescharacteristic
165C2×C4×C59C4×D411yescharacteristic
166C2×C4×C59#12011yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12311yescharacteristic
169C2×Dic59—42no—
170C2×Dic59—81no—
171C2×Dic59—81no—
172C8×C59—41no—
173C8×C59—41no—
174C59⋊C8—42no—
175C59⋊C8—41no—
176C59⋊C8—42no—
177C59⋊C8—41no—
178#9165—591nomaximal, Sylow
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—22no—
182C22×C59—41no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—82no—
188C22×C59—82no—
189C22×C59#38611yescharacteristic
190C22×C59#47111yescharacteristic
191C22×C59#47311yescharacteristic
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—81no—
200C4×C59—81no—
201C4×C59C2×#10411yescharacteristic
202C4×C59#42111yescharacteristic
203Dic59—82no—
204#3110D5911yescharacteristic
205#2974—591no—
206#1762—591no—
207#1531—591no—
208#1503—591no—
209#1302—591no—
210#1285—591no—
211C2×C59—21no—
212C2×C59—21no—
213C2×C59—42no—
214C2×C59—81no—
215C2×C59#187111yescharacteristic
216C2×C59#190411yescharacteristic
217C2×C59#190811yescharacteristic
218#533C2×D5911yescharacteristic
219C2×C4×D4C2×D5911yescharacteristic
220#484—21no—
221#474Dic5911yescharacteristic, complemented
222#474Dic5911yescharacteristic, complemented
223#461Dic5911yescharacteristic, complemented
224#461Dic5911yescharacteristic, complemented
225#450—21no—
226#440—21no—
227#440—21no—
228#436—1181no—
229#435—1182no—
230C8×D4—1181no—
231#432—591no—
232#432C2×D5911yescharacteristic
233#431—591no—
234C2×#102—1181no—
235#407—1181no—
236C2×C22⋊C8—1181no—
237C2×#97—591no—
238C4×C4⋊C4—591no—
239#340—1181no—
240#330—1182no—
241#328—1181no—
242#328—1181no—
243C59#916511yescharacteristic, complemented, Sylow
244C22×D4C22⋊D5911yescharacteristic
245C23×C4—21no—
246C2×SD16—21no—
247C2×SD16—21no—
248C2×M16—1182no—
249C22×C8—1182no—
250C22×C8—1181no—
251C4⋊Q8C2×Dic5911yescharacteristic
252C4⋊Q8C2×Dic5911yescharacteristic
253#125—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic5911yescharacteristic
261C4×D4C2×Dic5911yescharacteristic
262C4×D4C22⋊D5911yescharacteristic
263C4×D4C22⋊D5911yescharacteristic
264#117—21no—
265C2×C4⋊C4—1181no—
266C2×C4⋊C4C22⋊D5911yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—1181no—
269C2×C42C22×D5911yescharacteristic
270#107—21no—
271#107—21no—
272C4⋊C8—1181no—
273C4⋊C8—1181no—
274C4⋊C8C2×Dic5911yescharacteristic
275C4⋊C8C2×Dic5911yescharacteristic
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280#102—1184no—
281C22⋊C8—21no—
282C22⋊C8—591no—
283C22⋊C8—591no—
284C22⋊C8—1182no—
285C22⋊C8—1182no—
286C22⋊C8—1181no—
287#97—21no—
288#97—592no—
289#97—1181no—
290#97—1181no—
291C4×C8—1181no—
292#95—1181no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4C22⋊Dic5911yescharacteristic
300C2×D4C22⋊Dic5911yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—2361no—
308C22×C4D4×D5911yescharacteristic
309C22×C4#1174568711yescharacteristic
310C22×C4C2×C22⋊D5911yescharacteristic
311SD16—42no—
312SD16—42no—
313M16—4722no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—1182no—
317C2×C8—1181no—
318C2×C8—1181no—
319C2×C8—1182no—
320C2×C8—4722no—
321C2×C8—2361no—
322C2×C8—2361no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1182no—
331C4⋊C4—2361no—
332C4⋊C4C22⋊Dic5911yescharacteristic
333C4⋊C4C22⋊Dic5911yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—41no—
339C42—41no—
340C42—4721no—
341C42D4×D5911yescharacteristic
342C42#1174568711yescharacteristic
343C42C22×Dic5911yescharacteristic
344C42C2×C22⋊D5911yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1427369911yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—81no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1427365412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—4721no—
375C2×C4—4721no—
376C2×C4—2362no—
377C2×C4C2×C22⋊Dic5911yescharacteristic
378C2×C4D4×Dic5911yescharacteristic
379C2×C4D4×Dic5911yescharacteristic
380C2×C4#1427369511yescharacteristic
381C2×C4#1427369911yescharacteristic
382C2×C4#1427370811yescharacteristic
383C8—41no—
384C8—41no—
385C8—2361no—
386C8—2362no—
387C8—2362no—
388C8—2361no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1683728211yescharacteristic
400C22#1683766711yescharacteristic
401C22#1683767511yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—4722no—
412C4C2×#1427365411yescharacteristic
413C4#1683741911yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2142439911yescharacteristic
419C2#2142458511yescharacteristic
420C2#2142461311yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3110C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#3110D59
hypocenterC59#9165
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #21422711C2semidirect
4. #21422595C2semidirect
5. #21421994C2semidirect
6. #21421953C2semidirect
25. C59×#474C4semidirect
26. C59×#474C4semidirect
27. C59×#461C4semidirect
28. C59×#461C4semidirect
221. #474Dic59semidirect
222. #474Dic59semidirect
223. #461Dic59semidirect
224. #461Dic59semidirect
243. C59#9165semidirect

Maximal subgroups

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

subgroupindex
#214331312
#214239212
#214227112
#214225952
#214219942
#214219532
C59×#31102
#916559