Finite Group Databasebeta

← #28483933

Subgroups — #28483933

Order 15104.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433045C211yesmaximal, characteristic
2#21425041C211yesmaximal, characteristic
3#21422595C211yesmaximal, characteristic
4#21422499C211yesmaximal, characteristic
5#21421936C211yesmaximal, characteristic
6#21421933C211yesmaximal, characteristic
7C59×#3170C211yesmaximal, characteristic
8#16837533—21no—
9#16837533—22no—
10Q8×C59⋊C8—21no—
11#16837455C2211yescharacteristic
12#16837429—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14#16837358—21no—
15C4×#11745664C2211yescharacteristic
16C4×C4⋊Dic59C2211yescharacteristic
17#16837161—21no—
18#16837161—21no—
19#16837139—22no—
20Q8⋊(C59⋊C8)—21no—
21Q8⋊(C59⋊C8)—21no—
22C59×#533C2211yescharacteristic
23C59×#533C2211yescharacteristic
24C59×#500C411yescharacteristic, complemented
25C59×#494C411yescharacteristic, complemented
26C59×#487C411yescharacteristic, complemented
27C59×#487C411yescharacteristic, complemented
28C59×#467—22no—
29C4×Q16×C59—22no—
30C2×C4×C8×C59C2211yescharacteristic
31C2×#11745664D411yescharacteristic
32C2×#11745664D411yescharacteristic
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—
40C4⋊(C59⋊C8)D411yescharacteristic
41C4⋊(C59⋊C8)D411yescharacteristic
42C22⋊(C59⋊C8)—21no—
43#14273638—22no—
44#14273638—21no—
45C4×C59⋊C8—22no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8—21no—
48C59⋊#95—21no—
49C2×Q16×C59—22no—
50C22×C8×C59—21no—
51C4⋊Q8×C59C2×C411yescharacteristic
52C4⋊Q8×C59C2×C411yescharacteristic
53C4⋊Q8×C59C2×C411yescharacteristic
54C4⋊Q8×C59C2×C411yescharacteristic
55#125×C59—21no—
56#125×C59—21no—
57C22⋊Q8×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59D412yes—
61C4×Q8×C59—21no—
62C4×Q8×C59—21no—
63C4×Q8×C59—21no—
64C4×Q8×C59D412yes—
65#117×C59—21no—
66#117×C59—21no—
67C2×C42×C59C2311yescharacteristic
68#107×C59—22no—
69#103×C59—22no—
70#103×C59—22no—
71C4×C8×C59—21no—
72C4×C8×C59C2×C411yescharacteristic
73C4×C8×C59C2×C411yescharacteristic
74C2×Q8×C59—21no—
75C2×Q8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59C22⋊C412yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×Dic59—41no—
82Q16×C59—44no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—42no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59C22⋊C412yes—
98C22⋊C4×C59—22no—
99C22⋊C4×C59—41no—
100C42×C59—22no—
101C42×C59—41no—
102C42×C59C22×C411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C4⋊Dic59—22no—
107C4⋊Dic59—41no—
108C4×Dic59—41no—
109C4×Dic59—41no—
110#11745664—21no—
111#11745664—21no—
112#11745664—21no—
113#11745664Q1612yescomplemented
114C2×C59⋊C8—21no—
115C2×C59⋊C8—21no—
116C2×C59⋊C8—21no—
117C2×C59⋊C8—21no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C2×C59⋊C8—41no—
121C23×C59#12111yescharacteristic
122Q8×C59—42no—
123Q8×C59—42no—
124Q8×C59—41no—
125Q8×C59—41no—
126Q8×C59—41no—
127Q8×C59—41no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59C2×C22⋊C411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12311yescharacteristic
141C2×C4×C59C2×Q1611yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59#13711yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C2×Dic59—42no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C8×C59—22no—
149C8×C59—41no—
150C59⋊C8—22no—
151C59⋊C8—41no—
152C59⋊C8—41no—
153C59⋊C8—41no—
154C59⋊C8—81no—
155#9056—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59#38611yescharacteristic
159C22×C59#46211yescharacteristic
160C22×C59#46411yescharacteristic
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—42no—
165C4×C59—42no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59C4×Q1611yescharacteristic
169C4×C59#44111yescharacteristic
170C4×C59#44311yescharacteristic
171C4×C59#44411yescharacteristic
172C4×C59#45111yescharacteristic
173C4×C59#45211yescharacteristic
174Dic59—82no—
175#3170D5911yescharacteristic
176#2962—591no—
177#1986—591no—
178#1503—591no—
179#1481—591no—
180#1279—591no—
181#1278—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59#187611yescharacteristic
185C2×C59#187811yescharacteristic
186C2×C59#188411yescharacteristic
187#533C2×D5911yescharacteristic
188#533C2×D5911yescharacteristic
189#500Dic5911yescharacteristic, complemented
190#494Dic5911yescharacteristic, complemented
191#487Dic5911yescharacteristic, complemented
192#487Dic5911yescharacteristic, complemented
193#467—22no—
194#446—1181no—
195#446—1182no—
196C8×Q8—1181no—
197C4×Q16—22no—
198#431—591no—
199#424—1181no—
200C2×C4⋊C8—591no—
201#405—1181no—
202C4×M16—591no—
203C2×C4×C8C2×D5911yescharacteristic
204C4×C4⋊C4—591no—
205#339—1181no—
206#339—1181no—
207#328—1182no—
208Q8⋊C8—1181no—
209Q8⋊C8—1181no—
210C59#905611yescharacteristic, complemented, Sylow
211C2×Q16—22no—
212C2×M16—591no—
213C2×M16—591no—
214C22×C8—21no—
215C22×C8—1181no—
216C4⋊Q8C2×Dic5911yescharacteristic
217C4⋊Q8C2×Dic5911yescharacteristic
218C4⋊Q8C2×Dic5911yescharacteristic
219C4⋊Q8C2×Dic5911yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D5912yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D5912yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—1181no—
233C2×C42—1181no—
234C2×C42C22×D5911yescharacteristic
235#107—22no—
236C4⋊C8—591no—
237C4⋊C8—591no—
238C4⋊C8—1182no—
239C4⋊C8—1182no—
240C4⋊C8—1181no—
241C4⋊C8—1181no—
242#103—22no—
243#103—22no—
244C22⋊C8—1181no—
245#97—1182no—
246#97—1181no—
247C4×C8—21no—
248C4×C8—1182no—
249C4×C8—1181no—
250C4×C8—1181no—
251C4×C8C2×Dic5911yescharacteristic
252C4×C8C2×Dic5911yescharacteristic
253#95—1181no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic5912yes—
258C22×C4—2361no—
259C22×C4D4×D5911yescharacteristic
260C22×C4#1174568711yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262Q16—44no—
263M16—592no—
264M16—1181no—
265M16—1181no—
266M16—1181no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—1181no—
271C2×C8—1181no—
272C2×C8—1181no—
273C2×C8—1181no—
274C2×C8—2361no—
275C2×C8—2361no—
276C2×C8—2361no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—1182no—
289C4⋊C4—2361no—
290C4⋊C4C22⋊Dic5912yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—2361no—
296C42—2361no—
297C42D4×D5911yescharacteristic
298C42#1174568711yescharacteristic
299C42C22×Dic5911yescharacteristic
300C42C2×C22⋊D5911yescharacteristic
301C23#1427369911yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4721no—
316C2×C4—4721no—
317C2×C4—2362no—
318C2×C4C2×C22⋊Dic5911yescharacteristic
319C2×C4D4×Dic5911yescharacteristic
320C2×C4D4×Dic5911yescharacteristic
321C2×C4#1427369511yescharacteristic
322C2×C4#1427369911yescharacteristic
323C2×C4#1427370811yescharacteristic
324C2×C4Q16×D5911yescharacteristic
325C2×C4#1427375011yescharacteristic
326C2×C4#1427375911yescharacteristic
327C2×C4#1427375911yescharacteristic
328C8—22no—
329C8—41no—
330C8—1182no—
331C8—4721no—
332C8—2361no—
333C8—2361no—
334C8—2361no—
335C22—21no—
336C22—21no—
337C22#1683728211yescharacteristic
338C22#1683761011yescharacteristic
339C22#1683762611yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—4722no—
348C4Q16×Dic5911yescharacteristic
349C4#1683750911yescharacteristic
350C4#1683751911yescharacteristic
351C4#1683752411yescharacteristic
352C4#1683755611yescharacteristic
353C4#1683756311yescharacteristic
354C2—21no—
355C2—21no—
356C2#2142442511yescharacteristic
357C2#2142443911yescharacteristic
358C2#2142448011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3170C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#3170D59
hypocenterC59#9056
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. C59×#500C4semidirect
25. C59×#494C4semidirect
26. C59×#487C4semidirect
27. C59×#487C4semidirect
113. #11745664Q16semidirect
189. #500Dic59semidirect
190. #494Dic59semidirect
191. #487Dic59semidirect
192. #487Dic59semidirect
210. C59#9056semidirect

Maximal subgroups

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

subgroupindex
#214330452
#214250412
#214225952
#214224992
#214219362
#214219332
C59×#31702
#905659