Finite Group Databasebeta

← #28488092

Subgroups — #28488092

Order 15104.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21423782C211yesmaximal, 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
11C2×#14273637—21no—
12C2×C8×Dic59—21no—
13#16837248C2211yescharacteristic
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—
41#14273637—24no—
42C8×Dic59—24no—
43C59⋊#95—21no—
44C59⋊#95D412yes—
45C2×#41×C59—21no—
46C2×SD16×C59—21no—
47C2×SD16×C59—21no—
48C22×C8×C59D411yescharacteristic
49C22×C8×C59D411yescharacteristic
50C4⋊Q8×C59C2×C411yescharacteristic
51C4⋊Q8×C59C2×C411yescharacteristic
52C4⋊Q8×C59C2×C411yescharacteristic
53#127×C59C2×C411yescharacteristic
54#125×C59—21no—
55#124×C59—21no—
56C22⋊Q8×C59—21no—
57C22⋊Q8×C59—21no—
58#121×C59—21no—
59#121×C59—21no—
60C4×Q8×C59—21no—
61C4×Q8×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64#117×C59—21no—
65C2×C42×C59C2311yescharacteristic
66#106×C59—21no—
67#106×C59—21no—
68#103×C59—21no—
69#103×C59—21no—
70#102×C59—21no—
71#102×C59—21no—
72C4×C8×C59C2×C411yescharacteristic
73C4×C8×C59C2×C411yescharacteristic
74C4×C8×C59D411yescharacteristic
75C4×C8×C59D411yescharacteristic
76#41×C59—82no—
77C2×Q8×C59—21no—
78C2×Q8×C59—21no—
79C2×Q8×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59#4111yescharacteristic
88C22×Dic59—22no—
89SD16×C59—42no—
90SD16×C59—42no—
91C2×C8×C59—22no—
92C2×C8×C59—22no—
93C2×C8×C59C22⋊C411yescharacteristic
94C2×C8×C59C22⋊C411yescharacteristic
95C2×C8×C59C22⋊C411yescharacteristic
96C2×C8×C59C22⋊C411yescharacteristic
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C42×C59—41no—
109C42×C59C22×C411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59#4111yescharacteristic
113C4×Dic59—22no—
114C4×Dic59—22no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C23×C59—41no—
121C23×C59#12111yescharacteristic
122Q8×C59—41no—
123Q8×C59—41no—
124Q8×C59—81no—
125Q8×C59—81no—
126D4×C59—41no—
127D4×C59—41no—
128D4×C59—81no—
129D4×C59—81no—
130D4×C59—82no—
131C2×C4×C59—22no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—82no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59C4×D411yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12411yescharacteristic
144C2×C4×C59#12711yescharacteristic
145C2×Dic59—24no—
146C2×Dic59—42no—
147C2×Dic59—42no—
148C8×C59—21no—
149C8×C59—21no—
150C8×C59#10412yes—
151C8×C59#10412yes—
152C59⋊C8—24no—
153C59⋊C8—81no—
154C59⋊C8—81no—
155#9761—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59#39011yescharacteristic
162C22×C59#47112yes—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—81no—
169C4×C59—81no—
170C4×C59—81no—
171C4×C59C2×#10411yescharacteristic
172C4×C59C2×#10411yescharacteristic
173Dic59—44no—
174#3167D5911yescharacteristic
175#1726—591no—
176#1568—591no—
177#1531—591no—
178#1530—591no—
179#1506—591no—
180#1505—591no—
181C2×C59—22no—
182C2×C59—161no—
183C2×C59#196211yescharacteristic
184C2×C59#197611yescharacteristic
185C2×C59#197711yescharacteristic
186#533C2×D5911yescharacteristic
187#532C2×D5911yescharacteristic
188#498Dic5911yescharacteristic, complemented
189#492Dic5911yescharacteristic, complemented
190#487Dic5911yescharacteristic, complemented
191#486Dic5911yescharacteristic, complemented
192#465—21no—
193#465—21no—
194C4×SD16—21no—
195C4×SD16—21no—
196#432—591no—
197#432—591no—
198C2×#97—591no—
199C2×#97—1181no—
200C2×C4×C8—1181no—
201C2×C4×C8C2×D5911yescharacteristic
202#376—591no—
203#335—1181no—
204#334—1181no—
205#330—1182no—
206#329—1182no—
207#328—1182no—
208#327—1182no—
209C59#976111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×SD16—21no—
212C2×SD16—21no—
213C22×C8—1181no—
214C22×C8C22⋊D5911yescharacteristic
215C22×C8C22⋊D5911yescharacteristic
216C4⋊Q8C2×Dic5911yescharacteristic
217C4⋊Q8C2×Dic5911yescharacteristic
218C4⋊Q8C2×Dic5911yescharacteristic
219#127C2×Dic5911yescharacteristic
220#125—21no—
221#124—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224#121—21no—
225#121—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230#117—21no—
231C2×C42—1181no—
232C2×C42C22×D5911yescharacteristic
233#106—21no—
234#106—21no—
235C4⋊C8—1181no—
236C4⋊C8—1181no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C22⋊C8—1181no—
242C22⋊C8—1181no—
243#97—592no—
244#97—1184no—
245#97—1181no—
246#97—1181no—
247#97—1181no—
248C4×C8—1184no—
249C4×C8C2×Dic5911yescharacteristic
250C4×C8C2×Dic5911yescharacteristic
251C4×C8C22⋊D5911yescharacteristic
252C4×C8C22⋊D5911yescharacteristic
253#95—592no—
254#95—1181no—
255#41—82no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—41no—
264C22×C4—1182no—
265C22×C4D4×D5911yescharacteristic
266C22×C4#1174568711yescharacteristic
267C22×C4C2×C22⋊D5911yescharacteristic
268SD16—42no—
269SD16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—1182no—
273C2×C8—1182no—
274C2×C8—1182no—
275C2×C8—2361no—
276C2×C8—2361no—
277C2×C8C22⋊Dic5911yescharacteristic
278C2×C8C22⋊Dic5911yescharacteristic
279C2×C8C22⋊Dic5911yescharacteristic
280C2×C8C22⋊Dic5911yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42—41no—
293C42—1182no—
294C42—1182no—
295C42D4×D5911yescharacteristic
296C42#1174568711yescharacteristic
297C42C22×Dic5911yescharacteristic
298C42C2×C22⋊D5911yescharacteristic
299C23—41no—
300C23#1427369811yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—1184no—
319C2×C4—2362no—
320C2×C4—2362no—
321C2×C4C2×C22⋊Dic5911yescharacteristic
322C2×C4D4×Dic5911yescharacteristic
323C2×C4D4×Dic5911yescharacteristic
324C2×C4#1427369811yescharacteristic
325C2×C4#1427371411yescharacteristic
326C2×C4#1427372311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1184no—
330C8—4721no—
331C8—4721no—
332C8#1427365412yes—
333C8#1427365412yes—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1683730511yescharacteristic
340C22#1683766212yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—2364no—
350C4C2×#1427365411yescharacteristic
351C4C2×#1427365411yescharacteristic
352C2—22no—
353C2—161no—
354C2#2142492511yescharacteristic
355C2#2142499511yescharacteristic
356C2#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#9761
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. #21423782C2semidirect
2. #21422859C2semidirect
3. #21422711C2semidirect
5. #21422608C2semidirect
22. C59×#498C4semidirect
23. C59×#492C4semidirect
24. C59×#487C4semidirect
25. C59×#486C4semidirect
188. #498Dic59semidirect
189. #492Dic59semidirect
190. #487Dic59semidirect
191. #486Dic59semidirect
209. C59#9761semidirect

Maximal subgroups

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

subgroupindex
#214237822
#214228592
#214227112
#214227072
#214226082
#214226042
C59×#31672
#976159