Finite Group Databasebeta

← #28499195

Subgroups — #28499195

Order 15104.

428 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 91 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#21423835C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422126C212yesmaximal, complemented
4#21422102C212yesmaximal, complemented
5C59×#2599C211yesmaximal, characteristic
6C22×#11745664—21no—
7#16837459—21no—
8#16837171—21no—
9#16837171C2212yes—
10#16837128C2212yes—
11#16837126—21no—
12C2×C4×Q8×C59C411yescharacteristic
13C2×#117×C59C2211yescharacteristic
14C59×#398C2211yescharacteristic
15C59×#397C411yescharacteristic
16C59×#395C2211yescharacteristic
17C59×#393C411yescharacteristic
18C59×#388—21no—
19C59×#387—21no—
20C59×#387—21no—
21C59×#385—21no—
22C59×#383C411yescharacteristic
23C2×#11745664—24no—
24C2×#11745664—22no—
25C2×#11745664—41no—
26C2×#11745664—42no—
27C22×C59⋊C8—41no—
28C4⋊(C59⋊C8)—41no—
29C22⋊(C59⋊C8)—22no—
30#14273638—41no—
31C22×Q8×C59C2×C411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C4×Q8×C59—41no—
34C4×Q8×C59—41no—
35#117×C59—41no—
36#117×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C42×C59—21no—
48C2×C42×C59C2×C411yescharacteristic
49C2×C42×C59D411yescharacteristic
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C2×Q8×C59—42no—
60C2×Q8×C59—41no—
61C22×C4×C59—22no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59—42no—
93#11745664—24no—
94#11745664—44no—
95#11745664—42no—
96C2×C59⋊C8—44no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59#10412yes—
104C23×C59C2×C22⋊C411yescharacteristic
105Q8×C59—42no—
106Q8×C59—42no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
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—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59#9911yescharacteristic
136C2×C4×C59#9911yescharacteristic
137C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C59#13711yescharacteristic
139C59⋊C8—44no—
140C59⋊C8—82no—
141#11763—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59C2×#9911yescharacteristic
150C22×C59#41711yescharacteristic
151C22×C59C2×#10411yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—41no—
157C4×C59—42no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—81no—
162C4×C59—81no—
163C4×C59—81no—
164#2599D5911yescharacteristic
165#1742—591no—
166#1464—591no—
167#1354—592no—
168#1346—592no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—42no—
172C2×C59#150411yescharacteristic
173C2×C59#212011yescharacteristic
174C2×C59#213511yescharacteristic
175C22×M16—1181no—
176C2×C4×Q8Dic5911yescharacteristic
177C2×#117C2×D5911yescharacteristic
178#432—1181no—
179#398C2×D5911yescharacteristic
180#397Dic5911yescharacteristic
181#395C2×D5911yescharacteristic
182#393Dic5911yescharacteristic
183#388—21no—
184#387—21no—
185#387—21no—
186#385—21no—
187#383Dic5911yescharacteristic
188#343—592no—
189#343—1181no—
190#324—592no—
191#323—1181no—
192C59#1176311yescharacteristic, complemented, Sylow
193C22×Q8C2×Dic5911yescharacteristic
194C23×C4C22⋊D5911yescharacteristic
195C2×M16—1184no—
196C2×M16—1182no—
197C2×M16—2361no—
198C2×M16—2362no—
199C22×C8—2361no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×Dic5911yescharacteristic
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C4⋊C4C22×D5911yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4C2×Dic5911yescharacteristic
213C2×C22⋊C4C22⋊D5911yescharacteristic
214C2×C42—21no—
215C2×C42C2×Dic5911yescharacteristic
216C2×C42C22⋊D5911yescharacteristic
217C4⋊C8—2361no—
218C22⋊C8—1182no—
219#97—2361no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C2×Dic5911yescharacteristic
227#95C2×Dic5911yescharacteristic
228C24C22⋊Dic5911yescharacteristic
229C2×Q8—42no—
230C2×Q8—41no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4C22⋊Dic5911yescharacteristic
241C22×C4C22⋊Dic5911yescharacteristic
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22×Dic5911yescharacteristic
244C22×C4C2×C22⋊D5911yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246M16—1184no—
247M16—2364no—
248M16—2362no—
249C2×C8—2362no—
250C2×C8—2364no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1427365412yes—
274C23C2×C22⋊Dic5911yescharacteristic
275Q8—42no—
276Q8—42no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4#1427364211yescharacteristic
306C2×C4#1427364211yescharacteristic
307C2×C4#1427375211yescharacteristic
308C2×C4#1427375711yescharacteristic
309C8—4722no—
310C8—2364no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1427364211yescharacteristic
319C22#1683740311yescharacteristic
320C22C2×#1427365411yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2142260011yescharacteristic
337C2#2142573711yescharacteristic
338C2#2142579111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2599C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2599D59
hypocenterC59#11763
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
3. #21422126C2semidirect
4. #21422102C2semidirect
192. C59#11763semidirect

Maximal subgroups

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

subgroupindex
#214238352
#214224302
#214221262
#214221022
C59×#25992
#1176359