Finite Group Databasebeta

← #28488119

Subgroups — #28488119

Order 15104.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21423847C211yesmaximal, characteristic, complemented
2#21422859C211yesmaximal, characteristic, complemented
3#21422711C211yesmaximal, characteristic, complemented
4#21422707C211yesmaximal, characteristic
5#21422608C211yesmaximal, characteristic, complemented
6#21422604C211yesmaximal, characteristic
7C59×#3167C211yesmaximal, characteristic
8#16837459C2211yescharacteristic
9#16837459C2211yescharacteristic
10C2×#14273638C2211yescharacteristic
11#16837248C2211yescharacteristic
12#16837156—21no—
13#16837156—21no—
14#16837155—21no—
15#16837153—21no—
16#16837144—22no—
17#16837142—22no—
18#16837139—22no—
19#16837136—22no—
20C59×#533C2211yescharacteristic
21C59×#532C2211yescharacteristic
22C59×#498C411yescharacteristic, complemented
23C59×#492C411yescharacteristic, complemented
24C59×#487C411yescharacteristic, complemented
25C59×#486C411yescharacteristic, complemented
26C59×#465—21no—
27C59×#465—21no—
28C4×SD16×C59—21no—
29C4×SD16×C59—21no—
30C2×C4×C8×C59C2211yescharacteristic
31C22×C59⋊C8—21no—
32C2×C4×Dic59—21no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)—21no—
35C22⋊(C59⋊C8)—21no—
36C22⋊(C59⋊C8)—21no—
37#14273638—21no—
38#14273638—21no—
39#14273638—21no—
40#14273638D412yes—
41C59⋊#95—21no—
42C59⋊#95D412yes—
43C2×#41×C59—21no—
44C2×SD16×C59—21no—
45C2×SD16×C59—21no—
46C22×C8×C59D411yescharacteristic
47C22×C8×C59D411yescharacteristic
48C4⋊Q8×C59C2×C411yescharacteristic
49C4⋊Q8×C59C2×C411yescharacteristic
50C4⋊Q8×C59C2×C411yescharacteristic
51#127×C59C2×C411yescharacteristic
52#125×C59—21no—
53#124×C59—21no—
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56#121×C59—21no—
57#121×C59—21no—
58C4×Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62#117×C59—21no—
63C2×C42×C59C2311yescharacteristic
64#106×C59—21no—
65#106×C59—21no—
66#103×C59—21no—
67#103×C59—21no—
68#102×C59—21no—
69#102×C59—21no—
70C4×C8×C59C2×C411yescharacteristic
71C4×C8×C59C2×C411yescharacteristic
72C4×C8×C59D411yescharacteristic
73C4×C8×C59D411yescharacteristic
74#41×C59—82no—
75C2×Q8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×D4×C59—21no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59#4111yescharacteristic
86C22×Dic59—22no—
87SD16×C59—42no—
88SD16×C59—42no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59C22⋊C411yescharacteristic
92C2×C8×C59C22⋊C411yescharacteristic
93C2×C8×C59C22⋊C411yescharacteristic
94C2×C8×C59C22⋊C411yescharacteristic
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C42×C59—41no—
107C42×C59C22×C411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59#4111yescharacteristic
111C4×Dic59—41no—
112C4×Dic59—41no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C23×C59—41no—
119C23×C59#12111yescharacteristic
120Q8×C59—41no—
121Q8×C59—41no—
122Q8×C59—81no—
123Q8×C59—81no—
124D4×C59—41no—
125D4×C59—41no—
126D4×C59—81no—
127D4×C59—81no—
128D4×C59—82no—
129C2×C4×C59—22no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—82no—
137C2×C4×C59C2×C22⋊C411yescharacteristic
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12411yescharacteristic
142C2×C4×C59#12711yescharacteristic
143C2×Dic59—42no—
144C2×Dic59—42no—
145C2×Dic59—42no—
146C8×C59—21no—
147C8×C59—21no—
148C8×C59—21no—
149C8×C59—21no—
150C59⋊C8—42no—
151C59⋊C8—81no—
152C59⋊C8—81no—
153#9766—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59#39011yescharacteristic
160C22×C59#47112yes—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59—81no—
169C4×C59#42111yescharacteristic
170C4×C59#42111yescharacteristic
171Dic59—82no—
172#3167D5911yescharacteristic
173#1746—591no—
174#1568—591no—
175#1531—591no—
176#1530—591no—
177#1506—591no—
178#1505—591no—
179C2×C59—22no—
180C2×C59—161no—
181C2×C59#196311yescharacteristic
182C2×C59#197611yescharacteristic
183C2×C59#197711yescharacteristic
184#533C2×D5911yescharacteristic
185#532C2×D5911yescharacteristic
186#498Dic5911yescharacteristic, complemented
187#492Dic5911yescharacteristic, complemented
188#487Dic5911yescharacteristic, complemented
189#486Dic5911yescharacteristic, complemented
190#465—21no—
191#465—21no—
192C4×SD16—21no—
193C4×SD16—21no—
194#432—591no—
195#432—591no—
196C2×#97—591no—
197C2×C4×C8C2×D5911yescharacteristic
198#376—591no—
199#336—1181no—
200#336—1181no—
201#335—1181no—
202#334—1181no—
203#330—1182no—
204#329—1182no—
205#328—1182no—
206#327—1182no—
207C59#976611yescharacteristic, complemented, Sylow
208C2×#41—21no—
209C2×SD16—21no—
210C2×SD16—21no—
211C22×C8—1181no—
212C22×C8C22⋊D5911yescharacteristic
213C22×C8C22⋊D5911yescharacteristic
214C4⋊Q8C2×Dic5911yescharacteristic
215C4⋊Q8C2×Dic5911yescharacteristic
216C4⋊Q8C2×Dic5911yescharacteristic
217#127C2×Dic5911yescharacteristic
218#125—21no—
219#124—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C42—1181no—
230C2×C42C22×D5911yescharacteristic
231#106—21no—
232#106—21no—
233C4⋊C8—1181no—
234C4⋊C8—1181no—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—1181no—
240C22⋊C8—1181no—
241#97—592no—
242#97—1181no—
243#97—1181no—
244#97—1181no—
245C4×C8C2×Dic5911yescharacteristic
246C4×C8C2×Dic5911yescharacteristic
247C4×C8C22⋊D5911yescharacteristic
248C4×C8C22⋊D5911yescharacteristic
249#95—592no—
250#95—1181no—
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—1182no—
261C22×C4D4×D5911yescharacteristic
262C22×C4#1174568711yescharacteristic
263C22×C4C2×C22⋊D5911yescharacteristic
264SD16—42no—
265SD16—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—1182no—
269C2×C8—2361no—
270C2×C8—2361no—
271C2×C8—2361no—
272C2×C8—2361no—
273C2×C8C22⋊Dic5911yescharacteristic
274C2×C8C22⋊Dic5911yescharacteristic
275C2×C8C22⋊Dic5911yescharacteristic
276C2×C8C22⋊Dic5911yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—2361no—
290C42—2361no—
291C42D4×D5911yescharacteristic
292C42#1174568711yescharacteristic
293C42C22×Dic5911yescharacteristic
294C42C2×C22⋊D5911yescharacteristic
295C23—41no—
296C23#1427369811yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—81no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4—2362no—
315C2×C4—2362no—
316C2×C4—2362no—
317C2×C4C2×C22⋊Dic5911yescharacteristic
318C2×C4D4×Dic5911yescharacteristic
319C2×C4D4×Dic5911yescharacteristic
320C2×C4#1427369811yescharacteristic
321C2×C4#1427371411yescharacteristic
322C2×C4#1427372311yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—4721no—
328C8—4721no—
329C8—2362no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#1683730511yescharacteristic
336C22#1683766212yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—4722no—
346C4#1683741911yescharacteristic
347C4#1683741911yescharacteristic
348C2—22no—
349C2—161no—
350C2#2142492911yescharacteristic
351C2#2142499511yescharacteristic
352C2#2142500111yescharacteristic
subgroupG/H
centerC22#16837305
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3167C2
socleC22×C59#390
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3167D59
hypocenterC59#9766
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21423847C2semidirect
2. #21422859C2semidirect
3. #21422711C2semidirect
5. #21422608C2semidirect
22. C59×#498C4semidirect
23. C59×#492C4semidirect
24. C59×#487C4semidirect
25. C59×#486C4semidirect
186. #498Dic59semidirect
187. #492Dic59semidirect
188. #487Dic59semidirect
189. #486Dic59semidirect
207. C59#9766semidirect

Maximal subgroups

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

subgroupindex
#214238472
#214228592
#214227112
#214227072
#214226082
#214226042
C59×#31672
#976659