Finite Group Databasebeta

← #28477187

Subgroups — #28477187

Order 15104.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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#21423857C211yesmaximal, characteristic
3#336×D59C211yesmaximal, characteristic
4#21421920C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421918C211yesmaximal, characteristic, complemented
7C59×#1271C211yesmaximal, characteristic, complemented
8C22×C8×D59—21no—
9C2×#14273718C411yescharacteristic
10C2×C42×D59C2211yescharacteristic
11C2×(C2×D59)⋊C8—21no—
12C2×#14273637—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—
32C2×C8×D59—24no—
33C2×C8×D59—42no—
34C2×C8×D59—42no—
35#14273718—42no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59C2×C411yescharacteristic
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×Dic59⋊C4—21no—
40C2×Dic59⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×C4×Dic59C2×C411yescharacteristic
44C2×C4×Dic59D411yescharacteristic
45C2×C4×Dic59D411yescharacteristic
46C42×D59—42no—
47(C2×D59)⋊C8—24no—
48#14273637—42no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C22×C8×C59—21no—
55C22×C8×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C4⋊C4×C59C2×C411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60#95×C59—21no—
61C23×D59C22⋊C411yescharacteristic
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59C22⋊C411yescharacteristic
70C22×Dic59C22⋊C411yescharacteristic
71C22×Dic59C22⋊C411yescharacteristic
72C2×C4×D59—22no—
73C2×C4×D59—21no—
74C2×C4×D59—21no—
75C2×C4×D59—22no—
76C2×C4×D59—22no—
77C2×C4×D59—22no—
78C2×C4×D59—21no—
79C2×C4×D59—21no—
80C2×C4×D59—21no—
81C2×C4×D59—21no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C8×C59—22no—
85C2×C8×C59—22no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C42×C59—41no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—42no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100C4⋊Dic59—42no—
101C4⋊Dic59—41no—
102Dic59⋊C4—42no—
103Dic59⋊C4—41no—
104Dic59⋊C4—41no—
105C4×Dic59—22no—
106C4×Dic59—22no—
107C4×Dic59—41no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C8×D59—48no—
115C23×C59C2×C22⋊C411yescharacteristic
116C22×D59—21no—
117C22×D59—21no—
118C22×D59—21no—
119C22×D59—21no—
120C22×D59#10112yes—
121C22×D59#10412yes—
122C22×D59#10412yes—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
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#13111yescharacteristic
136C2×C4×C59#13111yescharacteristic
137C2×C4×C59#13511yescharacteristic
138C2×C4×C59#13511yescharacteristic
139C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C59#13711yescharacteristic
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C2×Dic59—21no—
146C2×Dic59—21no—
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59#9912yes—
156C4×D59—24no—
157C4×D59—24no—
158C4×D59—42no—
159C4×D59—42no—
160C4×D59—42no—
161C4×D59—42no—
162C8×C59—42no—
163C8×C59—42no—
164C59⋊C8—42no—
165C59⋊C8—42no—
166#7691—591nomaximal, Sylow
167C22×C59#40811yescharacteristic
168C22×C59C2×#9911yescharacteristic
169C22×C59C2×#10111yescharacteristic
170C22×C59#41611yescharacteristic
171C22×C59#41711yescharacteristic
172C22×C59C2×#10411yescharacteristic
173C22×C59C2×#10411yescharacteristic
174C2×D59—22no—
175C2×D59—22no—
176C2×D59—22no—
177C2×D59—22no—
178C2×D59—22no—
179C2×D59—22no—
180C2×D59#33014yes—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—81no—
188C4×C59—81no—
189Dic59—22no—
190Dic59—41no—
191Dic59—41no—
192Dic59—81no—
193Dic59—81no—
194#2851—591no—
195#1749—591no—
196C2×#336—591no—
197#1271—591no—
198#1271—591no—
199#1271—591no—
200#1271D5911yescharacteristic, complemented
201C2×C59#147211yescharacteristic
202C2×C59#148311yescharacteristic
203C2×C59#149411yescharacteristic
204C2×C59#150111yescharacteristic
205C2×C59#150411yescharacteristic
206C2×C59C2×#33011yescharacteristic
207C2×C59#155211yescharacteristic
208D59—24no—
209C23×C8—1181no—
210C2×#125—591no—
211C22×C42—591no—
212C2×C22⋊C8—1181no—
213C2×#97—1181no—
214#389—591no—
215#389C2×D5911yescharacteristic
216#387—1181no—
217#387—1181no—
218#385—1181no—
219#385—1181no—
220#382—591no—
221#380—591no—
222#380—591no—
223#336—591no—
224#336—591no—
225#336—1181no—
226#336C2×D5911yescharacteristic
227#336C2×D5911yescharacteristic
228C59#769111yescharacteristic, complemented, Sylow
229C23×C4—591no—
230C23×C4—591no—
231C23×C4—591no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—1184no—
235C22×C8—1181no—
236C22×C8—1181no—
237C22×C8—2362no—
238C22×C8—2362no—
239#125—2362no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—591no—
242C2×C4⋊C4—591no—
243C2×C4⋊C4—1181no—
244C2×C4⋊C4—1181no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4C4×D5911yescharacteristic
247C2×C4⋊C4C4×D5911yescharacteristic
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C42—591no—
251C2×C42—591no—
252C2×C42—591no—
253C2×C42—2362no—
254C2×C42C22×D5911yescharacteristic
255C22⋊C8—1184no—
256#97—2362no—
257#95—21no—
258#95—1181no—
259#95—1181no—
260#95—1181no—
261#95—1181no—
262#95—1181no—
263C24—591no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—591no—
267C22×C4—591no—
268C22×C4—591no—
269C22×C4—1182no—
270C22×C4—1182no—
271C22×C4—1182no—
272C22×C4—1182no—
273C22×C4—1181no—
274C22×C4—1181no—
275C22×C4—1181no—
276C22×C4—1181no—
277C22×C4—1181no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—1181no—
281C22×C4—2361no—
282C22×C4—2361no—
283C22×C4C2×C4×D5911yescharacteristic
284C22×C4D4×D5911yescharacteristic
285C22×C4D4×D5911yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—1182no—
293C2×C8—1182no—
294C2×C8—2368no—
295C2×C8—2361no—
296C2×C8—2361no—
297C2×C8—2361no—
298C2×C8—2361no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—1181no—
305C4⋊C4—1181no—
306C4⋊C4—2361no—
307C4⋊C4—2362no—
308C4⋊C4—2362no—
309C4⋊C4—2361no—
310C4⋊C4—2361no—
311C22⋊C4—2362no—
312C22⋊C4—2362no—
313C42—41no—
314C42—1182no—
315C42—1182no—
316C42—2361no—
317C23—592no—
318C23—592no—
319C23—592no—
320C23—1181no—
321C23—1181no—
322C23—1181no—
323C23—1181no—
324C23C22⋊C4×D5911yescharacteristic
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—592no—
338C2×C4—1184no—
339C2×C4—1184no—
340C2×C4—1181no—
341C2×C4—1181no—
342C2×C4—1181no—
343C2×C4—1181no—
344C2×C4—1181no—
345C2×C4—1181no—
346C2×C4—1181no—
347C2×C4—1181no—
348C2×C4—2362no—
349C2×C4—2362no—
350C2×C4—2362no—
351C2×C4—2362no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4#1427373311yescharacteristic
359C2×C4#1427373511yescharacteristic
360C2×C4#1427374611yescharacteristic
361C2×C4#1427374811yescharacteristic
362C2×C4#1427375411yescharacteristic
363C2×C4#1427375911yescharacteristic
364C8—42no—
365C8—42no—
366C8—2362no—
367C8—2362no—
368C22—594no—
369C22—1182no—
370C22—1182no—
371C22—1182no—
372C22—1182no—
373C22—1182no—
374C22—1182no—
375C22#1683736711yescharacteristic
376C22#99×D5911yescharacteristic
377C22#101×D5911yescharacteristic
378C22#1683739511yescharacteristic
379C22#1683740211yescharacteristic
380C22#104×D5911yescharacteristic
381C22#104×D5911yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—1182no—
391C4—4721no—
392C4—4721no—
393C4—2361no—
394C4—2361no—
395C2—1184no—
396C2#2142245811yescharacteristic
397C2#2142250811yescharacteristic
398C2#2142255311yescharacteristic
399C2#2142258411yescharacteristic
400C2#2142259911yescharacteristic
401C2#330×D5911yescharacteristic
402C2#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#7691
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
200. #1271D59semidirect
228. C59#7691semidirect

Maximal subgroups

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

subgroupindex
#214324052
#214238572
#336×D592
#214219202
#214219192
#214219182
C59×#12712
#769159