Finite Group Databasebeta

← #28501842

Subgroups — #28501842

Order 15104.

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

normal subgroups only

This group has 396 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#21425257C211yesmaximal, characteristic, complemented
2#21423137C211yesmaximal, characteristic, complemented
3#21423035C211yesmaximal, characteristic
4#21422123C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic
6#21422030C211yesmaximal, characteristic, complemented
7C59×#2572C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837737—21no—
10#16837553—21no—
11#16837411—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400C2211yescharacteristic
19C59×#394C2211yescharacteristic
20C59×#393—21no—
21C59×#391C2211yescharacteristic
22C59×#388—21no—
23C59×#386—21no—
24C59×#383—21no—
25C59×#381—21no—
26C4×C22⋊C4×C59—21no—
27C2×#11745670—41no—
28C2×#11745664—21no—
29#14273726—41no—
30#14273718—41no—
31#14273706—21no—
32#14273706—21no—
33#14273686—21no—
34#14273660—41no—
35#14273651—41no—
36#14273651—41no—
37#14273650—41no—
38#14273648—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×Q8×C59D411yescharacteristic
42C22×D4×C59D411yescharacteristic
43C23×C4×C59—21no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59C2311yescharacteristic
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C42×C59—21no—
52C2×C42×C59D411yescharacteristic
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
62C24×C59—21no—
63C2×Q8×C59—21no—
64C2×Q8×C59—21no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—42no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—42no—
75C22×C4×C59—42no—
76C22×C4×C59—81no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C2×Dic118—21no—
81C2×Dic118—21no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—81no—
89C22⋊C4×C59—81no—
90C42×C59—42no—
91C42×C59—41no—
92C22⋊Dic59—41no—
93#11745670—82no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59—21no—
97C4⋊Dic59—21no—
98Dic59⋊C4—41no—
99Dic59⋊C4—41no—
100C4×Dic59—41no—
101#11745664—41no—
102#11745664—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C2×C59⋊C8—41no—
106C23×C59—22no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59#12011yescharacteristic
114Q8×C59—42no—
115Q8×C59—81no—
116D4×C59—42no—
117D4×C59—81no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—22no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
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—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—84no—
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59#13711yescharacteristic
142C2×Dic59—41no—
143C2×Dic59—41no—
144Dic118—81no—
145Dic118—81no—
146C59⋊C8—81no—
147C59⋊C8—81no—
148C59⋊C8—81no—
149#12271—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59—42no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59#45211yescharacteristic
160C22×C59#45511yescharacteristic
161C22×C59B(4,𝔽2)11yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—42no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—81no—
168C4×C59—81no—
169C4×C59—81no—
170C4×C59—81no—
171Dic59—81no—
172Dic59—81no—
173#2572D5911yescharacteristic
174#2028—591no—
175#1617—591no—
176#1599—591no—
177#1353—591no—
178#1345—591no—
179#1319—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—82no—
183C2×C59#160611yescharacteristic
184C2×C59#219711yescharacteristic
185C2×C59#220411yescharacteristic
186#544—591no—
187#484—1181no—
188#450—1181no—
189#419—1181no—
190#400C2×D5911yescharacteristic
191#394C2×D5911yescharacteristic
192#393—21no—
193#391C2×D5911yescharacteristic
194#388—21no—
195#386—21no—
196#383—21no—
197#381—21no—
198C4×C22⋊C4—21no—
199#343—591no—
200#339—1181no—
201#328—1181no—
202#328—1181no—
203#324—591no—
204#324—591no—
205C59#1227111yescharacteristic, complemented, Sylow
206C22×Q8C22⋊D5911yescharacteristic
207C22×D4C22⋊D5911yescharacteristic
208C23×C4—21no—
209C2×SD16—2361no—
210C2×M16—1181no—
211C4⋊Q8—2361no—
212#125—2361no—
213C22⋊Q8—1181no—
214C22⋊Q8—1181no—
215#117—1181no—
216C2×C4⋊C4C22⋊D5911yescharacteristic
217C2×C4⋊C4C22×D5911yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42C22⋊D5911yescharacteristic
225#107—2361no—
226#103—2361no—
227#103—2361no—
228#103—2361no—
229#102—2361no—
230C22⋊C8—1181no—
231C22⋊C8—1181no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C22⋊D5911yescharacteristic
240#95C22⋊D5911yescharacteristic
241C24—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—1181no—
247C2×Q8—1181no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4C2×C22⋊D5911yescharacteristic
259C22×C4C2×C22⋊D5911yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261SD16—4722no—
262M16—2361no—
263M16—2361no—
264C2×C8—2361no—
265C2×C8—2361no—
266C2×C8—2361no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—1181no—
273C4⋊C4—2361no—
274C4⋊C4—2361no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—2361no—
282C42—41no—
283C42—42no—
284C42—2361no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23#1427369611yescharacteristic
293Q8—42no—
294Q8—81no—
295Q8—4721no—
296Q8—4721no—
297D4—42no—
298D4—81no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4#1427374711yescharacteristic
324C2×C4#1427375711yescharacteristic
325C8—4721no—
326C8—4721no—
327C8—4721no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22#1683756411yescharacteristic
338C22#1683758011yescharacteristic
339C22#1683758911yescharacteristic
340C4—21no—
341C4—21no—
342C4—42no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—4721no—
350C4—4721no—
351C2—21no—
352C2—21no—
353C2—82no—
354C2#2142307611yescharacteristic
355C2#2142608611yescharacteristic
356C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2572C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2572D59
hypocenterC59#12271
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2572

Complements

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

NH ≅ G/Ntype
1. #21425257C2semidirect
2. #21423137C2semidirect
4. #21422123C2semidirect
6. #21422030C2semidirect
205. C59#12271semidirect

Maximal subgroups

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

subgroupindex
#214252572
#214231372
#214230352
#214221232
#214220992
#214220302
C59×#25722
#1227159