Finite Group Databasebeta

← #28480868

Subgroups — #28480868

Order 15104.

718 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 131 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#21424173C211yesmaximal, characteristic, complemented
2#21424167C211yesmaximal, characteristic, complemented
3#21422652C212yesmaximal, complemented
4#21422652C212yesmaximal, complemented
5C59×#3493C211yesmaximal, characteristic
6C2×C22⋊(C59⋊C8)C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2211yescharacteristic
8#16837265C2212yes—
9#16837169—22no—
10#16837157—22no—
11#16837139—22no—
12#16837139—22no—
13#16837136—22no—
14#16837136—22no—
15C59×#556—22no—
16C59×#555—22no—
17C59×#553—22no—
18C59×#552—22no—
19C59×#544C411yescharacteristic, complemented
20C59×#538—22no—
21C59×#537C411yescharacteristic, complemented
22C59×#529—22no—
23C59×#525C2212yes—
24C59×#520C412yescomplemented
25C2×#117×C59C2211yescharacteristic
26C22×C59⋊C8—21no—
27C22×C59⋊C8—21no—
28C2×C22⋊Dic59—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C59⋊#95—22no—
32C59⋊#95—22no—
33C2×#41×C59C2×C412yes—
34C23×C4×C59C2311yescharacteristic
35C4⋊Q8×C59—22no—
36#126×C59—24no—
37#126×C59—24no—
38#125×C59—22no—
39#125×C59—22no—
40#124×C59—22no—
41#124×C59—24no—
42#123×C59—24no—
43#123×C59—22no—
44C22⋊Q8×C59—22no—
45C22⋊Q8×C59—22no—
46C22⋊Q8×C59D412yes—
47C22⋊Q8×C59D412yes—
48#121×C59—24no—
49#121×C59—22no—
50#121×C59—22no—
51#121×C59D412yes—
52#121×C59D412yes—
53#120×C59—22no—
54C4×Q8×C59—22no—
55C4×Q8×C59—22no—
56C4×D4×C59—22no—
57C4×D4×C59—22no—
58C4×D4×C59—24no—
59C4×D4×C59—22no—
60C4×D4×C59—22no—
61#117×C59—24no—
62#117×C59—22no—
63#117×C59—22no—
64#117×C59—22no—
65#117×C59C2×C412yes—
66C2×C4⋊C4×C59—22no—
67C2×C4⋊C4×C59C2×C411yescharacteristic
68C2×C4⋊C4×C59C2×C411yescharacteristic
69C2×C22⋊C4×C59D412yes—
70C2×C42×C59D412yes—
71C24×C59C2×D411yescharacteristic
72#41×C59—44no—
73C2×Q8×C59—22no—
74C2×Q8×C59C22⋊C412yes—
75C2×D4×C59—22no—
76C2×D4×C59—22no—
77C2×D4×C59—22no—
78C2×D4×C59—22no—
79C2×D4×C59C22⋊C412yes—
80C22×C4×C59—24no—
81C22×C4×C59—22no—
82C22×C4×C59—42no—
83C22×C4×C59C22⋊C412yes—
84C22×C4×C59C22⋊C412yes—
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D412yes—
88C22×C4×C59C2×D412yes—
89C22×Dic59—42no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—24no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—22no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—22no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59C22⋊C412yes—
99C4⋊C4×C59C22⋊C412yes—
100C22⋊C4×C59—24no—
101C22⋊C4×C59—24no—
102C22⋊C4×C59—22no—
103C22⋊C4×C59—24no—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—22no—
108C42×C59—24no—
109C42×C59—22no—
110C42×C59—22no—
111C42×C59—22no—
112C22⋊Dic59—44no—
113C2×C59⋊C8—42no—
114C2×C59⋊C8—42no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C23×C59—22no—
118C23×C59—22no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59C2×C22⋊C411yescharacteristic
122C23×C59#12012yes—
123Q8×C59—42no—
124Q8×C59—82no—
125D4×C59—42no—
126D4×C59—44no—
127D4×C59—82no—
128D4×C59—82no—
129D4×C59—82no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—44no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—84no—
143C2×C4×C59C2×C22⋊C412yes—
144C2×C4×C59#12011yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×Dic59—82no—
147C2×Dic59—82no—
148C2×Dic59—82no—
149C59⋊C8—81no—
150C59⋊C8—81no—
151#8463—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59—42no—
158C22×C59—42no—
159C22×C59—42no—
160C22×C59—82no—
161C22×C59#37911yescharacteristic
162C22×C59#41012yes—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—22no—
166C4×C59—42no—
167C4×C59—42no—
168C4×C59—42no—
169C4×C59—42no—
170C4×C59—42no—
171Dic59—162no—
172#3493D5911yescharacteristic
173#1820—591no—
174#1818—591no—
175#1516—592no—
176#1516—592no—
177C2×C59—22no—
178C2×C59—42no—
179C2×C59—82no—
180C2×C59#178211yescharacteristic
181C2×C59#179411yescharacteristic
182C2×C59#179411yescharacteristic
183#556—22no—
184#555—22no—
185#553—22no—
186#552—22no—
187#544Dic5911yescharacteristic, complemented
188#538—22no—
189#537Dic5911yescharacteristic, complemented
190#529—22no—
191#525C2×D5912yes—
192#520Dic5912yescomplemented
193C2×#117C2×D5911yescharacteristic
194C2×C22⋊C8—591no—
195C2×C22⋊C8—591no—
196#381—592no—
197#342—1182no—
198#336—1182no—
199#328—1182no—
200#328—1182no—
201#327—1182no—
202#327—1182no—
203C59#846311yescharacteristic, complemented, Sylow
204C2×#41C2×Dic5912yes—
205C23×C4C22×D5911yescharacteristic
206C22×C8—1181no—
207C22×C8—1181no—
208C4⋊Q8—22no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125—22no—
213#124—22no—
214#124—24no—
215#123—24no—
216#123—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8C22⋊D5912yes—
220C22⋊Q8C22⋊D5912yes—
221#121—24no—
222#121—22no—
223#121—22no—
224#121C22⋊D5912yes—
225#121C22⋊D5912yes—
226#120—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234#117—24no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117C2×Dic5912yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×Dic5911yescharacteristic
241C2×C4⋊C4C2×Dic5911yescharacteristic
242C2×C22⋊C4—1182no—
243C2×C22⋊C4C22⋊D5912yes—
244C2×C42C22⋊D5912yes—
245C22⋊C8—1182no—
246C22⋊C8—1182no—
247#95—1182no—
248#95—1182no—
249C24C2×C22⋊D5911yescharacteristic
250#41—44no—
251C2×Q8—22no—
252C2×Q8C22⋊Dic5912yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊Dic5912yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—2362no—
262C22×C4C22⋊Dic5912yes—
263C22×C4C22⋊Dic5912yes—
264C22×C4C22×Dic5911yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266C22×C4C2×C22⋊D5912yes—
267C22×C4C2×C22⋊D5912yes—
268C2×C8—2362no—
269C2×C8—2362no—
270C2×C8—2361no—
271C2×C8—2361no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C22⋊Dic5912yes—
281C4⋊C4C22⋊Dic5912yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—2364no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic5911yescharacteristic
300C23#1427369612yes—
301Q8—42no—
302Q8—82no—
303D4—42no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—4722no—
322C2×C4—4722no—
323C2×C4—4722no—
324C2×C4C2×C22⋊Dic5912yes—
325C2×C4#1427369611yescharacteristic
326C2×C4#1427369611yescharacteristic
327C8—4721no—
328C8—4721no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22#1683725811yescharacteristic
339C22#1683737612yes—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—9442no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2#2142401411yescharacteristic
353C2#2142406011yescharacteristic
354C2#2142406011yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#3493C2
socleC22×C59#379
cosocleC23×C4×C59C23
hypercenter#3493D59
hypocenterC59#8463
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
1. #21424173C2semidirect
2. #21424167C2semidirect
3. #21422652C2semidirect
4. #21422652C2semidirect
19. C59×#544C4semidirect
21. C59×#537C4semidirect
24. C59×#520C4semidirect
187. #544Dic59semidirect
189. #537Dic59semidirect
192. #520Dic59semidirect
203. C59#8463semidirect

Maximal subgroups

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

subgroupindex
#214241732
#214241672
#214226522
C59×#34932
#846359