Finite Group Databasebeta

← #28477373

Subgroups — #28477373

Order 15104.

450 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 95 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#21432405C211yesmaximal, characteristic
2#21424163C211yesmaximal, characteristic
3#21424075C211yesmaximal, characteristic
4#21421920C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421918C211yesmaximal, characteristic, complemented
7C59×#1271C211yesmaximal, characteristic, complemented
8C2×#14273718C411yescharacteristic
9C2×C42×D59C2211yescharacteristic
10C2×Dic59⋊C8—21no—
11C2×(C2×D59)⋊C8—21no—
12C2×(C2×D59)⋊C8—21no—
13#16837300C2211yescharacteristic
14#16837288—21no—
15#16837284—21no—
16#16837276—21no—
17#16837273—21no—
18#16837267C411yescharacteristic
19#16837259C411yescharacteristic
20#16837259C411yescharacteristic
21#16837157C2211yescharacteristic
22#16837157C2211yescharacteristic
23#16837156—21no—
24C59×#389C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×C4×D59C2×C411yescharacteristic
28C22×C4×D59D411yescharacteristic
29C22×C4×D59D411yescharacteristic
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273718—42no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×C4×Dic59C2×C411yescharacteristic
41C2×C4×Dic59D411yescharacteristic
42C2×C4×Dic59D411yescharacteristic
43C42×D59—42no—
44Dic59⋊C8—42no—
45(C2×D59)⋊C8—42no—
46(C2×D59)⋊C8—42no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×C8×C59—21no—
53C22×C8×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C4⋊C4×C59C2×C411yescharacteristic
57C2×C42×C59C2311yescharacteristic
58#95×C59—21no—
59C23×D59C22⋊C411yescharacteristic
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—21no—
66C22×Dic59—21no—
67C22×Dic59C22⋊C411yescharacteristic
68C22×Dic59C22⋊C411yescharacteristic
69C22×Dic59C22⋊C411yescharacteristic
70C2×C4×D59—22no—
71C2×C4×D59—21no—
72C2×C4×D59—21no—
73C2×C4×D59—21no—
74C2×C4×D59—21no—
75C2×C4×D59—21no—
76C2×C4×D59—21no—
77C2×C4×D59—22no—
78C2×C4×D59—41no—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C2×C8×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C42×C59—41no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96C4⋊Dic59—21no—
97C4⋊Dic59—21no—
98C4⋊Dic59—42no—
99C4⋊Dic59—41no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102Dic59⋊C4—42no—
103C4×Dic59—22no—
104C4×Dic59—22no—
105C4×Dic59—41no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C23×C59C2×C22⋊C411yescharacteristic
113C22×D59—21no—
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—21no—
117C22×D59—21no—
118C22×D59—21no—
119C22×D59#10112yes—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
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#13111yescharacteristic
133C2×C4×C59#13111yescharacteristic
134C2×C4×C59#13511yescharacteristic
135C2×C4×C59#13511yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C2×Dic59—21no—
139C2×Dic59—21no—
140C2×Dic59—21no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C2×Dic59—21no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59#9912yes—
153C4×D59—42no—
154C4×D59—42no—
155C4×D59—42no—
156C4×D59—42no—
157C4×D59—42no—
158C4×D59—42no—
159C8×C59—81no—
160C8×C59—81no—
161C59⋊C8—81no—
162C59⋊C8—81no—
163#7731—591nomaximal, Sylow
164C22×C59#40811yescharacteristic
165C22×C59C2×#9911yescharacteristic
166C22×C59C2×#10111yescharacteristic
167C22×C59#41611yescharacteristic
168C22×C59#41711yescharacteristic
169C22×C59#42111yescharacteristic
170C22×C59#42111yescharacteristic
171C2×D59—22no—
172C2×D59—22no—
173C2×D59—22no—
174C2×D59—41no—
175C2×D59—41no—
176C2×D59—41no—
177C2×D59—41no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—81no—
185C4×C59—81no—
186Dic59—22no—
187Dic59—41no—
188Dic59—41no—
189Dic59—81no—
190Dic59—81no—
191#2851—591no—
192#1817—591no—
193#1797—591no—
194#1271—591no—
195#1271—591no—
196#1271—591no—
197#1271D5911yescharacteristic, complemented
198C2×C59#147211yescharacteristic
199C2×C59#148211yescharacteristic
200C2×C59#149511yescharacteristic
201C2×C59#151311yescharacteristic
202C2×C59#151611yescharacteristic
203C2×C59#153411yescharacteristic
204C2×C59#155211yescharacteristic
205D59—42no—
206C2×#125—591no—
207C22×C42—591no—
208C2×C4⋊C8—1181no—
209C2×C22⋊C8—1181no—
210C2×C22⋊C8—1181no—
211#389—591no—
212#389C2×D5911yescharacteristic
213#387—1181no—
214#387—1181no—
215#385—1181no—
216#385—1181no—
217#382—591no—
218#380—591no—
219#380—591no—
220#336—591no—
221#336—591no—
222#336—1181no—
223#336C2×D5911yescharacteristic
224#336C2×D5911yescharacteristic
225C59#773111yescharacteristic, complemented, Sylow
226C23×C4—591no—
227C23×C4—591no—
228C23×C4—591no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—1181no—
232C22×C8—1181no—
233#125—2362no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—591no—
236C2×C4⋊C4—591no—
237C2×C4⋊C4—1181no—
238C2×C4⋊C4—1181no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4C4×D5911yescharacteristic
241C2×C4⋊C4C4×D5911yescharacteristic
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C42—591no—
245C2×C42—591no—
246C2×C42—591no—
247C2×C42—2362no—
248C2×C42C22×D5911yescharacteristic
249C4⋊C8—2362no—
250C22⋊C8—2362no—
251C22⋊C8—2362no—
252#95—21no—
253#95—1181no—
254#95—1181no—
255#95—1181no—
256#95—1181no—
257#95—1181no—
258C24—591no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—591no—
262C22×C4—591no—
263C22×C4—591no—
264C22×C4—1182no—
265C22×C4—1182no—
266C22×C4—1181no—
267C22×C4—1181no—
268C22×C4—1181no—
269C22×C4—1181no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4—2361no—
275C22×C4—2361no—
276C22×C4—2361no—
277C22×C4—2361no—
278C22×C4C2×C4×D5911yescharacteristic
279C22×C4D4×D5911yescharacteristic
280C22×C4D4×D5911yescharacteristic
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8—2361no—
291C2×C8—2361no—
292C2×C8—2361no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1181no—
299C4⋊C4—1181no—
300C4⋊C4—2361no—
301C4⋊C4—2362no—
302C4⋊C4—2362no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C22⋊C4—2362no—
306C22⋊C4—2362no—
307C42—41no—
308C42—1182no—
309C42—1182no—
310C42—2361no—
311C23—592no—
312C23—1181no—
313C23—1181no—
314C23—1181no—
315C23—1181no—
316C23—1181no—
317C23—1181no—
318C23C22⋊C4×D5911yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—592no—
332C2×C4—1181no—
333C2×C4—1181no—
334C2×C4—1181no—
335C2×C4—1181no—
336C2×C4—1181no—
337C2×C4—1181no—
338C2×C4—1181no—
339C2×C4—1181no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4—2362no—
345C2×C4—2362no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4#1427373311yescharacteristic
353C2×C4#1427373511yescharacteristic
354C2×C4#1427374611yescharacteristic
355C2×C4#1427374811yescharacteristic
356C2×C4#1427375411yescharacteristic
357C2×C4#1427375911yescharacteristic
358C8—81no—
359C8—81no—
360C8—4721no—
361C8—4721no—
362C22—1182no—
363C22—1182no—
364C22—1182no—
365C22—2361no—
366C22—2361no—
367C22—2361no—
368C22—2361no—
369C22#1683736711yescharacteristic
370C22#99×D5911yescharacteristic
371C22#101×D5911yescharacteristic
372C22#1683739511yescharacteristic
373C22#1683740211yescharacteristic
374C22#1683741711yescharacteristic
375C22#1683741711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—1182no—
385C4—4721no—
386C4—4721no—
387C4—2361no—
388C4—2361no—
389C2—2362no—
390C2#2142245811yescharacteristic
391C2#2142250111yescharacteristic
392C2#2142256011yescharacteristic
393C2#2142263811yescharacteristic
394C2#2142264911yescharacteristic
395C2#2142272111yescharacteristic
396C2#2142279311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1271C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1271D59
hypocenterC59#7731
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21421920C2semidirect
5. #21421919C2semidirect
6. #21421918C2semidirect
7. C59×#1271C2semidirect
197. #1271D59semidirect
225. C59#7731semidirect

Maximal subgroups

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

subgroupindex
#214324052
#214241632
#214240752
#214219202
#214219192
#214219182
C59×#12712
#773159