Finite Group Databasebeta

← #28477668

Subgroups — #28477668

Order 15104.

478 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 81 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#21431055C211yesmaximal, characteristic
2#21425426C211yesmaximal, characteristic, complemented
3#21425421C211yesmaximal, characteristic, complemented
4#21421920C211yesmaximal, characteristic, complemented
5#21421918C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2013C211yesmaximal, characteristic
8C2×#14273720—21no—
9C2×#14273660—21no—
10C2×#14273658—21no—
11#16837338—21no—
12#16837325—21no—
13#16837316—21no—
14#16837300C2211yescharacteristic
15#16837300C2211yescharacteristic
16#16837297—21no—
17#16837295—21no—
18#16837164—21no—
19#16837164—21no—
20#16837157C2211yescharacteristic
21#16837157C2211yescharacteristic
22C22×SD16×C59—21no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×#103×C59—21no—
25C59×#390C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×C59⋊C8—21no—
28C22×C59⋊C8—21no—
29#14273720—42no—
30#14273720—42no—
31C2×C4⋊Dic59D411yescharacteristic
32C2×C4⋊Dic59D411yescharacteristic
33C2×C4⋊Dic59D411yescharacteristic
34C2×C4⋊Dic59D411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C4×Dic59—21no—
44#14273660—42no—
45#14273658—42no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C22×Q8×C59—21no—
50C22×D4×C59D411yescharacteristic
51C2×SD16×C59—42no—
52C2×SD16×C59—42no—
53C2×SD16×C59—42no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—42no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C42×C59C2311yescharacteristic
62#103×C59—42no—
63C24×C59—21no—
64C2×Q8×C59—22no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×D4×C59—21no—
71C2×D4×C59—21no—
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C22×C4×C59—21no—
75C22×C4×C59—41no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83SD16×C59—48no—
84C2×C8×C59—22no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C42×C59—41no—
97C22⋊Dic59—42no—
98C22⋊Dic59—42no—
99C22⋊Dic59—81no—
100C22⋊Dic59—81no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59—21no—
104C4⋊Dic59—41no—
105C4⋊Dic59—41no—
106C4⋊Dic59SD1612yescomplemented
107Dic59⋊C4—42no—
108Dic59⋊C4—41no—
109Dic59⋊C4—41no—
110Dic59⋊C4—41no—
111Dic59⋊C4—41no—
112Dic59⋊C4—81no—
113C4×Dic59—42no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59#12011yescharacteristic
128Q8×C59—42no—
129Q8×C59—42no—
130Q8×C59—81no—
131Q8×C59—81no—
132D4×C59—42no—
133D4×C59—81no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59C2×SD1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59#13511yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C59#13711yescharacteristic
152C2×Dic59—22no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C8×C59—42no—
165C59⋊C8—42no—
166C59⋊C8—42no—
167#7780—591nomaximal, Sylow
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59#44911yescharacteristic
176C22×C59#45011yescharacteristic
177C22×C59#45211yescharacteristic
178C22×C59#45311yescharacteristic
179C22×C59#45611yescharacteristic
180C22×C59#45611yescharacteristic
181C22×C59B(4,𝔽2)11yescharacteristic
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—82no—
189C4×C59—81no—
190Dic59—42no—
191Dic59—81no—
192Dic59—81no—
193Dic59—81no—
194#2727—591no—
195#2058—591no—
196#2057—591no—
197#2013D5911yescharacteristic
198#1271—591no—
199#1271—591no—
200#1270—591no—
201C2×C59—161no—
202C2×C59#160011yescharacteristic
203C2×C59#160311yescharacteristic
204C2×C59#160411yescharacteristic
205C2×C59#163111yescharacteristic
206C2×C59#165011yescharacteristic
207C2×C59#165411yescharacteristic
208C2×C59#166011yescharacteristic
209C22×SD16—21no—
210C2×C4⋊Q8C2×D5911yescharacteristic
211C2×#126—1181no—
212C2×#107—1181no—
213C2×#106—1181no—
214C2×#103—21no—
215#399—1181no—
216#396—1181no—
217#394—1181no—
218#390C2×D5911yescharacteristic
219#389—591no—
220#389—591no—
221#389—1181no—
222#388—1181no—
223#340—1181no—
224#340—1181no—
225#336—591no—
226#336—591no—
227#336C2×D5911yescharacteristic
228C59#778011yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×D4C22⋊D5911yescharacteristic
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8—21no—
235C22×C8—1181no—
236C22×C8—1181no—
237C4⋊Q8—42no—
238#126—2362no—
239#126—2362no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—591no—
243C2×C4⋊C4—591no—
244C2×C4⋊C4—591no—
245C2×C4⋊C4—591no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—1181no—
249C2×C4⋊C4—1181no—
250C2×C4⋊C4C22⋊D5911yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1181no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C42—1181no—
258C2×C42C22×D5911yescharacteristic
259#107—2362no—
260#106—2362no—
261#103—42no—
262#95—1181no—
263#95—1181no—
264#95—1181no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—1181no—
281C22×C4—1181no—
282C22×C4D4×D5911yescharacteristic
283C22×C4D4×D5911yescharacteristic
284C22×C4C2×C22⋊D5911yescharacteristic
285SD16—48no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1182no—
290C2×C8—1182no—
291C2×C8—2361no—
292C2×C8—2361no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—592no—
301C4⋊C4—1181no—
302C4⋊C4—1181no—
303C4⋊C4—1181no—
304C4⋊C4—4721no—
305C4⋊C4—2361no—
306C4⋊C4—2361no—
307C4⋊C4—2362no—
308C4⋊C4—2361no—
309C4⋊C4—2361no—
310C4⋊C4—2361no—
311C4⋊C4—2361no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—4721no—
317C22⋊C4—4721no—
318C22⋊C4—2362no—
319C22⋊C4—2362no—
320C42—41no—
321C42—2362no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1427369511yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—1182no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4SD16×D5911yescharacteristic
361C2×C4#1427374711yescharacteristic
362C2×C4#1427374811yescharacteristic
363C2×C4#1427375311yescharacteristic
364C2×C4#1427375411yescharacteristic
365C2×C4#1427375611yescharacteristic
366C8—42no—
367C8—2362no—
368C8—2362no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#1683754511yescharacteristic
377C22#1683755111yescharacteristic
378C22#1683756111yescharacteristic
379C22#1683756911yescharacteristic
380C22#1683758311yescharacteristic
381C22#1683758311yescharacteristic
382C22#1683758811yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—82no—
391C4—4721no—
392C4—4721no—
393C4—4721no—
394C4—2362no—
395C2—161no—
396C2#2142304011yescharacteristic
397C2#2142305611yescharacteristic
398C2#2142306311yescharacteristic
399C2#2142322011yescharacteristic
400C2#2142335011yescharacteristic
401C2#2142337811yescharacteristic
402C2#2142340511yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2013C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2013D59
hypocenterC59#7780
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #21425426C2semidirect
3. #21425421C2semidirect
4. #21421920C2semidirect
5. #21421918C2semidirect
106. C4⋊Dic59SD16semidirect
228. C59#7780semidirect

Maximal subgroups

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

subgroupindex
#214310552
#214254262
#214254212
#214219202
#214219182
#214219172
C59×#20132
#778059