Finite Group Databasebeta

← #28486568

Subgroups — #28486568

Order 15104.

544 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429673C211yesmaximal, characteristic
2#21422544C212yesmaximal, complemented
3#21422430C211yesmaximal, characteristic
4#21421961C212yesmaximal, complemented
5C59×#2873C211yesmaximal, characteristic
6C2×C4×Dic118C2211yescharacteristic
7#16837459C2212yes—
8#16837332C411yescharacteristic
9#16837327C411yescharacteristic
10#16837322C411yescharacteristic
11#16837314C411yescharacteristic
12#16837290—21no—
13#16837286—21no—
14#16837283—21no—
15#16837275—21no—
16#16837270C2211yescharacteristic
17#16837134—24no—
18#16837128—21no—
19#16837126—21no—
20C22×M16×C59—21no—
21C2×#117×C59C2211yescharacteristic
22C59×#405—24no—
23C4×M16×C59—21no—
24C4×M16×C59C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C4×Dic118—22no—
27C4×Dic118—22no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37#14273638—22no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C2×M16×C59—24no—
49C2×M16×C59—24no—
50C2×M16×C59—22no—
51C22×C8×C59—22no—
52#117×C59—22no—
53#117×C59—22no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59C2×C411yescharacteristic
57C2×C22⋊C4×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60#97×C59—22no—
61#97×C59—22no—
62C4×C8×C59—22no—
63C4×C8×C59D414yes—
64C24×C59C22⋊C411yescharacteristic
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C2×Dic118—22no—
80C2×Dic118—42no—
81M16×C59—28no—
82M16×C59—24no—
83M16×C59—24no—
84C2×C8×C59—28no—
85C2×C8×C59—24no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C22⋊C4×C59—22no—
89C22⋊C4×C59—22no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59C2×D412yes—
93C42×C59#4112yes—
94C22⋊Dic59—42no—
95C22⋊Dic59—42no—
96C4⋊Dic59—22no—
97Dic59⋊C4—42no—
98Dic59⋊C4—42no—
99Dic59⋊C4—41no—
100Dic59⋊C4—41no—
101C4×Dic59—42no—
102C2×C59⋊C8—42no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#10412yes—
108C23×C59C2×C22⋊C411yescharacteristic
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—24no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59#10412yes—
123C2×C4×C59C4×D412yes—
124C2×C4×C59#12011yescharacteristic
125C2×C4×C59#12112yes—
126C2×C4×C59#12311yescharacteristic
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139Dic118—24no—
140Dic118—42no—
141C8×C59—28no—
142C59⋊C8—82no—
143#9491—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59#38611yescharacteristic
152C22×C59C2×#10411yescharacteristic
153C22×C59C2×#10411yescharacteristic
154C4×C59—22no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—21no—
158C4×C59—22no—
159C4×C59—41no—
160C4×C59—41no—
161Dic59—42no—
162Dic59—81no—
163Dic59—81no—
164Dic59—81no—
165#2873D5911yescharacteristic
166#2599—591no—
167#1491—592no—
168#1464—591no—
169#1288—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—42no—
173C2×C59#189111yescharacteristic
174C2×C59#189611yescharacteristic
175C2×C59#190911yescharacteristic
176C22×M16—21no—
177C2×C4×Q8—591no—
178C2×#117C2×D5911yescharacteristic
179#432—592no—
180#405—24no—
181C4×M16—21no—
182C4×M16C2×D5912yes—
183#398—591no—
184#397—591no—
185#395—591no—
186#393—591no—
187#388—1181no—
188#387—1181no—
189#387—1181no—
190#385—1181no—
191#383—591no—
192Q8⋊C8—1184no—
193#324—1181no—
194#323—1181no—
195C59#949111yescharacteristic, complemented, Sylow
196C22×Q8—591no—
197C23×C4C22⋊D5911yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—22no—
201C2×M16—22no—
202C22×C8—22no—
203C4×Q8—1182no—
204C4×Q8—1182no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4C4×D5911yescharacteristic
212C2×C4⋊C4D23611yescharacteristic
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4C4×D5911yescharacteristic
216C2×C22⋊C4D23611yescharacteristic
217C2×C42—1181no—
218C2×C42C22⋊D5911yescharacteristic
219C2×C42C22×D5911yescharacteristic
220C4⋊C8—1182no—
221C22⋊C8—1182no—
222#97—22no—
223#97—22no—
224#97—1182no—
225C4×C8—22no—
226C4×C8C22⋊D5914yes—
227#95—591no—
228#95—591no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235C24(C2×D59)⋊C411yescharacteristic
236C2×Q8—1182no—
237C2×Q8—2362no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4(C2×D59)⋊C411yescharacteristic
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4C2×C4×D5911yescharacteristic
250C22×C4C2×D23611yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252M16—28no—
253M16—24no—
254M16—24no—
255C2×C8—28no—
256C2×C8—24no—
257C2×C8—2362no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—1182no—
261C4⋊C4—2362no—
262C4⋊C4—2362no—
263C4⋊C4—2361no—
264C4⋊C4—2361no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—2362no—
268C22⋊C4—2362no—
269C42—21no—
270C42—21no—
271C42—2362no—
272C42#1174568512yes—
273C42C2×C22⋊D5912yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1427365512yes—
279C23C2×(C2×D59)⋊C411yescharacteristic
280Q8—1184no—
281Q8—2362no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—1182no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4#1427365512yes—
308C2×C4C4×C22⋊D5912yes—
309C2×C4#1427369611yescharacteristic
310C2×C4#1427370112yes—
311C2×C4#1427371011yescharacteristic
312C8—28no—
313C8—4722no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683728111yescharacteristic
322C22C2×#1427365511yescharacteristic
323C22C2×#1427365511yescharacteristic
324C4—22no—
325C4—21no—
326C4—22no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—4721no—
332C4—4721no—
333C4—4721no—
334C4—2362no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2142451411yescharacteristic
339C2#2142453911yescharacteristic
340C2#2142461811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2873C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2873D59
hypocenterC59#9491
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. #21422544C2semidirect
4. #21421961C2semidirect
195. C59#9491semidirect

Maximal subgroups

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

subgroupindex
#214296732
#214225442
#214224302
#214219612
C59×#28732
#949159