Finite Group Databasebeta

← #28501641

Subgroups — #28501641

Order 15104.

350 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425441C211yesmaximal, characteristic
2#21423103C211yesmaximal, characteristic
3#21423069C211yesmaximal, characteristic
4#21422126C211yesmaximal, characteristic
5#21422102C211yesmaximal, characteristic
6#21422042C211yesmaximal, characteristic
7C59×#2792C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837732—21no—
10#16837558—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#398—21no—
19C59×#398C2211yescharacteristic
20C59×#395—21no—
21C59×#395C2211yescharacteristic
22C59×#389—21no—
23C59×#389—21no—
24C59×#387—21no—
25C59×#387—21no—
26C59×#384C2211yescharacteristic
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#14273658—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×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C42×C59—21no—
53C2×C42×C59D411yescharacteristic
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—41no—
63C2×Q8×C59—81no—
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—24no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—81no—
82C4⋊C4×C59—81no—
83C4⋊C4×C59—81no—
84C4⋊C4×C59—81no—
85C4⋊C4×C59—81no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—81no—
89C22⋊Dic59—41no—
90#11745672—82no—
91C4⋊Dic59—21no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97C4×Dic59—41no—
98#11745664—22no—
99#11745664—41no—
100C2×C59⋊C8—41no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C23×C59#12011yescharacteristic
104Q8×C59—42no—
105Q8×C59—81no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
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×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59#13511yescharacteristic
131C2×C4×C59#13711yescharacteristic
132C2×Dic59—41no—
133C2×Dic59—41no—
134Dic118—81no—
135Dic118—81no—
136C59⋊C8—42no—
137C59⋊C8—81no—
138C59⋊C8—81no—
139#12240—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59#45211yescharacteristic
143C22×C59#45611yescharacteristic
144C22×C59B(4,𝔽2)11yescharacteristic
145C4×C59—21no—
146C4×C59—21no—
147C4×C59—42no—
148C4×C59—42no—
149C4×C59—42no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—81no—
153C4×C59—81no—
154C4×C59—81no—
155Dic59—81no—
156Dic59—81no—
157#2792D5911yescharacteristic
158#2060—591no—
159#1611—591no—
160#1605—591no—
161#1354—591no—
162#1346—591no—
163#1325—591no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59#160311yescharacteristic
167C2×C59#220011yescharacteristic
168C2×C59#220511yescharacteristic
169#544—591no—
170#483—1181no—
171#451—1181no—
172#428—1181no—
173#398—21no—
174#398C2×D5911yescharacteristic
175#395—21no—
176#395C2×D5911yescharacteristic
177#389—21no—
178#389—21no—
179#387—21no—
180#387—21no—
181#384C2×D5911yescharacteristic
182#343—591no—
183#339—1181no—
184#328—1181no—
185#328—1181no—
186#324—591no—
187#324—591no—
188C59#1224011yescharacteristic, complemented, Sylow
189C22×Q8C22⋊D5911yescharacteristic
190C2×Q16—2361no—
191C2×M16—1181no—
192C4⋊Q8—2361no—
193#125—2361no—
194C22⋊Q8—1181no—
195C22⋊Q8—1181no—
196#117—1181no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D5911yescharacteristic
205C2×C4⋊C4C22⋊D5911yescharacteristic
206C2×C4⋊C4C22×D5911yescharacteristic
207C2×C42—21no—
208C2×C42C22⋊D5911yescharacteristic
209#107—2361no—
210#106—2361no—
211#106—2361no—
212#103—2361no—
213#103—2361no—
214C22⋊C8—1181no—
215C22⋊C8—1181no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C22⋊D5911yescharacteristic
223#95C22⋊D5911yescharacteristic
224C2×Q8—41no—
225C2×Q8—81no—
226C2×Q8—1181no—
227C2×Q8—1181no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4C2×C22⋊D5911yescharacteristic
235C22×C4C2×C22⋊D5911yescharacteristic
236C22×C4C2×C22⋊D5911yescharacteristic
237Q16—4722no—
238M16—1182no—
239M16—2361no—
240C2×C8—2361no—
241C2×C8—2361no—
242C2×C8—2361no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—1181no—
255C4⋊C4—1181no—
256C4⋊C4—1181no—
257C4⋊C4—1181no—
258C4⋊C4—2361no—
259C4⋊C4—2361no—
260C22⋊C4—2361no—
261C42—21no—
262C42—21no—
263C42—81no—
264C42—2361no—
265C23#1427369611yescharacteristic
266Q8—42no—
267Q8—81no—
268Q8—4721no—
269Q8—4721no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4#1427374711yescharacteristic
297C2×C4#1427375711yescharacteristic
298C8—4721no—
299C8—4721no—
300C8—2362no—
301C22—21no—
302C22—21no—
303C22#1683756411yescharacteristic
304C22#1683758511yescharacteristic
305C22#1683758911yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—4721no—
317C4—4721no—
318C2—21no—
319C2—21no—
320C2#2142305911yescharacteristic
321C2#2142610311yescharacteristic
322C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2792C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2792D59
hypocenterC59#12240
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2792

Complements

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

NH ≅ G/Ntype
188. C59#12240semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214231032
#214230692
#214221262
#214221022
#214220422
C59×#27922
#1224059