Finite Group Databasebeta

← #28484744

Subgroups — #28484744

Order 15104.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 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#21429911C211yesmaximal, characteristic
2#21426698C211yesmaximal, characteristic
3#21422668C211yesmaximal, characteristic
4C2×#16837127C211yesmaximal, characteristic
5#21421964C211yesmaximal, characteristic
6#21421963C211yesmaximal, characteristic
7C59×#1520C211yesmaximal, characteristic
8C22×Dic59⋊C4C2211yescharacteristic
9C2×C22⋊(C59⋊C8)C2211yescharacteristic
10#16837340C412yescomplemented
11#16837335—22no—
12#16837321C412yescomplemented
13#16837289—22no—
14#16837283—22no—
15#16837274—21no—
16#16837263—21no—
17#16837263—21no—
18#16837262C2211yescharacteristic
19#16837262C2211yescharacteristic
20C22⋊C4×Dic59—21no—
21#16837139—22no—
22#16837127—22no—
23C2×C22⋊Q8×C59C2211yescharacteristic
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C59×#328—22no—
27C23×Dic59D411yescharacteristic
28C23×Dic59D411yescharacteristic
29C22×C59⋊C8—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—22no—
37C2×Dic59⋊C4C2×C412yes—
38C2×Dic59⋊C4D412yes—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C4×Dic59—22no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C59⋊#95—22no—
45C59⋊#95—24no—
46C59⋊#95—22no—
47C59⋊#95—22no—
48C59⋊#95—22no—
49C59⋊#95C2×C412yes—
50C59⋊#95D412yes—
51C22×Q8×C59C2×C411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C22×C8×C59—21no—
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56C22⋊Q8×C59—21no—
57C22⋊Q8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C22⋊C8×C59—21no—
64C22⋊C8×C59—21no—
65#95×C59D412yes—
66C24×C59C2×D411yescharacteristic
67C2×Q8×C59—21no—
68C2×Q8×C59—21no—
69C2×Q8×C59—81no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—22no—
73C22×C4×C59—81no—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—22no—
89C22×Dic59—22no—
90C22×Dic59—81no—
91C22×Dic59C22⋊C412yes—
92C22×Dic59C22⋊C412yes—
93C2×C8×C59—41no—
94C2×C8×C59—81no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—81no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊Dic59—41no—
105C22⋊Dic59—41no—
106C22⋊Dic59—41no—
107C22⋊Dic59—41no—
108C4⋊Dic59—81no—
109C4⋊Dic59—81no—
110Dic59⋊C4—22no—
111Dic59⋊C4—22no—
112Dic59⋊C4—22no—
113Dic59⋊C4—22no—
114Dic59⋊C4—82no—
115C4×Dic59—82no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—81no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#12011yescharacteristic
122C23×C59#12311yescharacteristic
123C23×C59#13511yescharacteristic
124C23×C59#13511yescharacteristic
125C23×C59#13711yescharacteristic
126C23×C59#13711yescharacteristic
127Q8×C59—81no—
128Q8×C59—81no—
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—42no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59C4×D411yescharacteristic
143C2×C4×C59C4×D411yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×Dic59—22no—
147C2×Dic59—22no—
148C2×Dic59—22no—
149C2×Dic59—22no—
150C2×Dic59—42no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—81no—
163C8×C59—81no—
164C59⋊C8—81no—
165#9202—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59#38611yescharacteristic
177C22×C59#41611yescharacteristic
178C22×C59#41911yescharacteristic
179C22×C59#45211yescharacteristic
180C22×C59#45211yescharacteristic
181C22×C59#48511yescharacteristic
182C22×C59#48511yescharacteristic
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—82no—
188C4×C59—81no—
189C4×C59—81no—
190Dic59—42no—
191Dic59—42no—
192Dic59—82no—
193Dic59—82no—
194#2627—591no—
195#2313—591no—
196#1520—591no—
197#1520D5911yescharacteristic
198C2×#324—591no—
199#1289—591no—
200#1289—591no—
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59—21no—
205C2×C59—21no—
206C2×C59—21no—
207C2×C59#187311yescharacteristic
208C2×C59#191211yescharacteristic
209C2×C59#191311yescharacteristic
210C2×C22⋊Q8C2×D5911yescharacteristic
211C22×C4⋊C4—591no—
212C2×C22⋊C8—591no—
213C2×C22⋊C8C2×D5911yescharacteristic
214#400—592no—
215#398—1182no—
216#395—592no—
217#387—1182no—
218#387—1182no—
219#385—1181no—
220#381—1181no—
221#381—1181no—
222#380—591no—
223#380—591no—
224#380C2×D5911yescharacteristic
225C4×C22⋊C4—1181no—
226#328—22no—
227#328—1182no—
228#324—1182no—
229C59#920211yescharacteristic, complemented, Sylow
230C22×Q8C4×D5911yescharacteristic
231C23×C4—591no—
232C23×C4—591no—
233C23×C4C22×D5911yescharacteristic
234C22×C8—21no—
235C22×C8—1181no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—592no—
243C2×C4⋊C4—592no—
244C2×C4⋊C4—1181no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—1182no—
249C2×C4⋊C4—1181no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4C4×D5911yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1181no—
256C2×C42—1182no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—1181no—
260C22⋊C8—1181no—
261#95—592no—
262#95—592no—
263#95—1182no—
264#95—1182no—
265#95—1182no—
266#95—1182no—
267#95—1184no—
268#95C22⋊D5912yes—
269C24D4×D5911yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—81no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—81no—
277C22×C4—592no—
278C22×C4—592no—
279C22×C4—1181no—
280C22×C4—1181no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4—1181no—
285C22×C4—1181no—
286C22×C4—1181no—
287C22×C4—1182no—
288C22×C4—1182no—
289C22×C4—4721no—
290C22×C4C2×C4×D5911yescharacteristic
291C22×C4#1174568511yescharacteristic
292C22×C4D4×D5911yescharacteristic
293C22×C4D4×D5911yescharacteristic
294C22×C4#1174568711yescharacteristic
295C22×C4C2×C22⋊D5911yescharacteristic
296C2×C8—41no—
297C2×C8—81no—
298C2×C8—4721no—
299C2×C8—2361no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—1182no—
306C4⋊C4—1182no—
307C4⋊C4—1182no—
308C4⋊C4—1182no—
309C4⋊C4—4722no—
310C4⋊C4—4721no—
311C4⋊C4—4721no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C22⋊C4—2361no—
319C22⋊C4—2361no—
320C42—4722no—
321C23—41no—
322C23—41no—
323C23C22⋊C4×D5911yescharacteristic
324C23#1427369511yescharacteristic
325C23#1427370911yescharacteristic
326C23#1427374811yescharacteristic
327C23#1427375011yescharacteristic
328C23#1427375611yescharacteristic
329C23#1427375911yescharacteristic
330Q8—81no—
331Q8—81no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—1182no—
346C2×C4—1182no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—4721no—
350C2×C4—2362no—
351C2×C4—2362no—
352C2×C4—2362no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2362no—
362C2×C4(C22⋊D59)⋊C411yescharacteristic
363C2×C4C4×C22⋊D5911yescharacteristic
364C2×C4#1427369811yescharacteristic
365C2×C4#1427370011yescharacteristic
366C8—81no—
367C8—4721no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1683727811yescharacteristic
379C22#1683739711yescharacteristic
380C22#1683740911yescharacteristic
381C22#1683756111yescharacteristic
382C22#1683756311yescharacteristic
383C22#1683774111yescharacteristic
384C22#1683774211yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—4722no—
393C4—4722no—
394C4—2362no—
395C4—2362no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#2142440511yescharacteristic
403C2#2142463011yescharacteristic
404C2#2142463511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1520C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#1520D59
hypocenterC59#9202
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
10. #16837340C4semidirect
12. #16837321C4semidirect
229. C59#9202semidirect

Maximal subgroups

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

subgroupindex
#214299112
#214266982
#214226682
C2×#168371272
#214219642
#214219632
C59×#15202
#920259