Finite Group Databasebeta

← #28484742

Subgroups — #28484742

Order 15104.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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#21429901C211yesmaximal, characteristic
2#21422667C212yesmaximal, complemented
3C2×#16837126C211yesmaximal, characteristic
4#21421963C212yesmaximal, complemented
5C59×#2312C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837338C412yescomplemented
9#16837325—22no—
10#16837314C412yescomplemented
11#16837275—21no—
12#16837262C2211yescharacteristic
13#16837258—21no—
14#16837138—24no—
15#16837126—22no—
16C22×C22⋊C4×C59C2211yescharacteristic
17C59×#388—24no—
18C59×#380C2212yes—
19C59×#379—21no—
20C4×C22⋊C4×C59—21no—
21C22×Dic118C2×C411yescharacteristic
22C22×C59⋊C8—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C22⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—22no—
35C59⋊#95—22no—
36C59⋊#95C2×C412yes—
37C25×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59C2311yescharacteristic
40C2×C4⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59—22no—
49#95×C59—22no—
50#95×C59D414yes—
51C24×C59—21no—
52C24×C59—21no—
53C24×C59—22no—
54C24×C59—22no—
55C24×C59—81no—
56C24×C59—81no—
57C24×C59C22⋊C412yes—
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—24no—
62C22×C4×C59—81no—
63C22×C4×C59—81no—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D412yes—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59#4112yes—
69C22×Dic59—21no—
70C22×Dic59—22no—
71C22×Dic59—21no—
72C2×Dic118—22no—
73C2×Dic118—81no—
74C4⋊C4×C59—82no—
75C22⋊C4×C59—24no—
76C22⋊C4×C59—24no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—82no—
80C22⋊C4×C59—82no—
81C42×C59—82no—
82C22⋊Dic59—41no—
83C22⋊Dic59—41no—
84C22⋊Dic59—42no—
85C22⋊Dic59—82no—
86C22⋊Dic59—82no—
87C4⋊Dic59—22no—
88C4⋊Dic59—81no—
89Dic59⋊C4—42no—
90Dic59⋊C4—82no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—82no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—22no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59—42no—
104C23×C59—81no—
105C23×C59—81no—
106C23×C59—84no—
107C23×C59—84no—
108C23×C59C2×C22⋊C411yescharacteristic
109C23×C59#12011yescharacteristic
110C23×C59#12311yescharacteristic
111C23×C59#13512yes—
112C23×C59#13712yes—
113C2×C4×C59—22no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—44no—
121C2×C4×C59—44no—
122C2×C4×C59—44no—
123C2×C4×C59—81no—
124C2×C4×C59C4×D412yes—
125C2×C4×C59#12112yes—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—42no—
133Dic118—82no—
134C59⋊C8—82no—
135#9201—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59—44no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—82no—
149C22×C59—84no—
150C22×C59—84no—
151C22×C59#38611yescharacteristic
152C22×C59#41611yescharacteristic
153C22×C59#41911yescharacteristic
154C22×C59#45212yes—
155C22×C59#48512yes—
156C4×C59—22no—
157C4×C59—22no—
158C4×C59—42no—
159C4×C59—84no—
160Dic59—82no—
161Dic59—81no—
162Dic59—81no—
163#2626—591no—
164#2312D5911yescharacteristic
165#1520—592no—
166C2×#323—591no—
167#1289—592no—
168C2×C59—21no—
169C2×C59—22no—
170C2×C59—21no—
171C2×C59—22no—
172C2×C59—82no—
173C2×C59#187311yescharacteristic
174C2×C59#191111yescharacteristic
175C2×C59#191311yescharacteristic
176C2×C22⋊Q8—591no—
177C22×C22⋊C4C2×D5911yescharacteristic
178C2×C22⋊C8—592no—
179#399—592no—
180#396—1182no—
181#393—592no—
182#388—24no—
183#385—1181no—
184#380—591no—
185#380C2×D5912yes—
186#379—21no—
187#379—1181no—
188C4×C22⋊C4—21no—
189#328—1184no—
190#323—1182no—
191C59#920111yescharacteristic, complemented, Sylow
192C25C22⋊D5911yescharacteristic
193C22×Q8—591no—
194C23×C4C22⋊D5911yescharacteristic
195C23×C4C22×D5911yescharacteristic
196C22×C8—1182no—
197C22⋊Q8—1182no—
198C22⋊Q8—1182no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—591no—
201C2×C4⋊C4—1182no—
202C2×C4⋊C4—1181no—
203C2×C4⋊C4—1181no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1182no—
210C2×C22⋊C4—1182no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4C4×D5912yes—
214C2×C22⋊C4D23612yes—
215C2×C42—22no—
216C22⋊C8—1182no—
217C22⋊C8—1182no—
218#95—22no—
219#95—592no—
220#95—1182no—
221#95C22⋊D5914yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24(C2×D59)⋊C412yes—
229C24C2×C22⋊D5911yescharacteristic
230C2×Q8—1182no—
231C2×Q8—4721no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1182no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4(C2×D59)⋊C412yes—
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4C2×D23611yescharacteristic
243C22×C4#1174568512yes—
244C22×C4C2×C22⋊D5912yes—
245C2×C8—4722no—
246C2×C8—2362no—
247C4⋊C4—82no—
248C4⋊C4—1182no—
249C4⋊C4—4722no—
250C4⋊C4—4721no—
251C4⋊C4—2362no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—4722no—
259C22⋊C4—4722no—
260C22⋊C4—2361no—
261C22⋊C4—2361no—
262C22⋊C4—2362no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×(C2×D59)⋊C411yescharacteristic
280C23#1427369611yescharacteristic
281C23#1427371011yescharacteristic
282C23#1427374712yes—
283C23#1427375712yes—
284Q8—4722no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4C4×C22⋊D5912yes—
304C2×C4#1427370112yes—
305C8—4722no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1683728111yescharacteristic
322C22#1683739611yescharacteristic
323C22#1683741011yescharacteristic
324C22#1683756412yes—
325C22#1683774312yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—4722no—
331C4—4721no—
332C4—4721no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2142440811yescharacteristic
339C2#2142462811yescharacteristic
340C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2312C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2312D59
hypocenterC59#9201
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21422667C2semidirect
4. #21421963C2semidirect
8. #16837338C4semidirect
10. #16837314C4semidirect
191. C59#9201semidirect

Maximal subgroups

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

subgroupindex
#214299012
#214226672
C2×#168371262
#214219632
C59×#23122
#920159