Finite Group Databasebeta

← #28501755

Subgroups — #28501755

Order 15104.

352 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425446C211yesmaximal, characteristic
2#21423137C211yesmaximal, characteristic
3#21423069C211yesmaximal, characteristic
4#21422126C211yesmaximal, characteristic
5#21422123C211yesmaximal, characteristic
6#21422040C211yesmaximal, characteristic
7C59×#2612C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837737—21no—
10#16837558—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400C2211yescharacteristic
19C59×#398—21no—
20C59×#395C2211yescharacteristic
21C59×#391—21no—
22C59×#389—21no—
23C59×#387—21no—
24C59×#387—21no—
25C59×#382C2211yescharacteristic
26C59×#376—21no—
27C2×#11745672—41no—
28C2×#11745664—21no—
29#14273726—41no—
30#14273718—41no—
31#14273706—21no—
32#14273706—21no—
33#14273686—21no—
34#14273660—41no—
35#14273660—41no—
36#14273658—41no—
37#14273651—41no—
38#14273650—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×Q8×C59D411yescharacteristic
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C42×C59—21no—
50C2×C42×C59—21no—
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59D411yescharacteristic
61#95×C59D411yescharacteristic
62C2×Q8×C59—42no—
63C2×Q8×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C2×Dic118—21no—
74C2×Dic118—21no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—81no—
82C42×C59—21no—
83C42×C59—42no—
84C42×C59—42no—
85C42×C59Q1612yescomplemented
86C22⋊Dic59—41no—
87#11745672—82no—
88C4⋊Dic59—21no—
89C4⋊Dic59—21no—
90C4⋊Dic59—21no—
91C4⋊Dic59—21no—
92Dic59⋊C4—41no—
93Dic59⋊C4—41no—
94C4×Dic59—41no—
95#11745664—22no—
96#11745664—41no—
97C2×C59⋊C8—41no—
98C2×C59⋊C8—41no—
99C2×C59⋊C8—41no—
100C23×C59#12011yescharacteristic
101Q8×C59—42no—
102Q8×C59—81no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59C2×Q1611yescharacteristic
128C2×C4×C59#13711yescharacteristic
129C2×Dic59—41no—
130C2×Dic59—41no—
131Dic118—81no—
132Dic118—81no—
133C59⋊C8—42no—
134C59⋊C8—81no—
135C59⋊C8—81no—
136#12258—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59#45111yescharacteristic
140C22×C59#45611yescharacteristic
141C22×C59B(4,𝔽2)11yescharacteristic
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—42no—
145C4×C59—42no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—42no—
149C4×C59—81no—
150C4×C59—81no—
151C4×C59—81no—
152Dic59—81no—
153Dic59—81no—
154#2612D5911yescharacteristic
155#2061—591no—
156#1617—591no—
157#1605—591no—
158#1354—591no—
159#1353—591no—
160#1324—591no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59#160511yescharacteristic
164C2×C59#219711yescharacteristic
165C2×C59#220411yescharacteristic
166#544—591no—
167#484—1181no—
168#451—1181no—
169#428—1181no—
170#400C2×D5911yescharacteristic
171#398—21no—
172#395C2×D5911yescharacteristic
173#391—21no—
174#389—21no—
175#387—21no—
176#387—21no—
177#382C2×D5911yescharacteristic
178#376—21no—
179#343—591no—
180#339—1181no—
181#328—1181no—
182#328—1181no—
183#324—591no—
184#324—591no—
185C59#1225811yescharacteristic, complemented, Sylow
186C22×Q8C22⋊D5911yescharacteristic
187C2×Q16—2361no—
188C2×M16—1181no—
189C4⋊Q8—2361no—
190#125—2361no—
191C22⋊Q8—1181no—
192C22⋊Q8—1181no—
193#117—1181no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D5911yescharacteristic
199C2×C4⋊C4C22⋊D5911yescharacteristic
200C2×C4⋊C4C22×D5911yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊D5911yescharacteristic
204#107—2361no—
205#107—2361no—
206#106—2361no—
207#103—2361no—
208#103—2361no—
209C22⋊C8—1181no—
210C22⋊C8—1181no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C22⋊D5911yescharacteristic
220#95C22⋊D5911yescharacteristic
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—1181no—
224C2×Q8—1181no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×C22⋊D5911yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233C22×C4C2×C22⋊D5911yescharacteristic
234Q16—4722no—
235M16—1182no—
236M16—2361no—
237C2×C8—2361no—
238C2×C8—2361no—
239C2×C8—2361no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—1181no—
248C4⋊C4—1181no—
249C4⋊C4—1181no—
250C4⋊C4—1181no—
251C4⋊C4—2361no—
252C4⋊C4—2361no—
253C22⋊C4—2361no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42—2361no—
258C42#1174567212yescomplemented
259C23#1427369611yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—4721no—
263Q8—4721no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4C2×#1174567211yescharacteristic
291C2×C4#1427375811yescharacteristic
292C8—4721no—
293C8—4721no—
294C8—2362no—
295C22—21no—
296C22—21no—
297C22#1683755811yescharacteristic
298C22#1683758511yescharacteristic
299C22#1683758911yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—4721no—
311C4—4721no—
312C2—21no—
313C2—21no—
314C2#2142307011yescharacteristic
315C2#2142608611yescharacteristic
316C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2612C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2612D59
hypocenterC59#12258
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2612

Complements

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

NH ≅ G/Ntype
85. C42×C59Q16semidirect
185. C59#12258semidirect
258. C42#11745672semidirect

Maximal subgroups

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

subgroupindex
#214254462
#214231372
#214230692
#214221262
#214221232
#214220402
C59×#26122
#1225859