Finite Group Databasebeta

← #28499193

Subgroups — #28499193

Order 15104.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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#21422430C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422101C212yesmaximal, complemented
5C59×#1742C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7#16837459—21no—
8#16837332C2211yescharacteristic
9#16837327C411yescharacteristic
10#16837322C2211yescharacteristic
11#16837314C411yescharacteristic
12#16837290—21no—
13#16837286—21no—
14#16837283—21no—
15#16837275—21no—
16#16837270C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C22×M16×C59—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C4×Dic59—21no—
33C4⋊(C59⋊C8)—41no—
34C22⋊(C59⋊C8)—22no—
35#14273638—41no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C2×M16×C59—24no—
46C2×M16×C59—22no—
47C2×M16×C59—41no—
48C2×M16×C59—42no—
49C22×C8×C59—41no—
50#117×C59—41no—
51#117×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C22×C4×C59—22no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×Dic118—41no—
75C2×Dic118—42no—
76M16×C59—24no—
77M16×C59—44no—
78M16×C59—42no—
79C2×C8×C59—44no—
80C2×C8×C59—42no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C4⋊Dic59—41no—
92Dic59⋊C4—42no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97C4×Dic59—42no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59#10412yes—
104C23×C59C2×C22⋊C411yescharacteristic
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C59#13711yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—41no—
127C2×Dic59—41no—
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—
137Dic118—42no—
138Dic118—42no—
139C8×C59—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—41no—
158C4×C59—41no—
159C4×C59—41no—
160Dic59—42no—
161Dic59—81no—
162Dic59—81no—
163Dic59—81no—
164#2599—591no—
165#1742D5911yescharacteristic
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—21no—
176C2×C4×Q8—591no—
177C2×#117C2×D5911yescharacteristic
178#432—1181no—
179#398—591no—
180#397—591no—
181#395—591no—
182#393—591no—
183#388—1181no—
184#387—1181no—
185#387—1181no—
186#385—1181no—
187#383—591no—
188#343—21no—
189#343C2×D5912yes—
190#324—592no—
191#323—1181no—
192C59#1176311yescharacteristic, complemented, Sylow
193C22×Q8—591no—
194C23×C4C22⋊D5911yescharacteristic
195C2×M16—24no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—42no—
199C22×C8—41no—
200C4×Q8—2361no—
201C4×Q8—2361no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C4⋊C4C22×D5911yescharacteristic
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4D23611yescharacteristic
214C2×C42—1181no—
215C2×C42C4×D5911yescharacteristic
216C2×C42D23611yescharacteristic
217C4⋊C8—2361no—
218C22⋊C8—1182no—
219#97—2361no—
220#95—591no—
221#95—591no—
222#95—1181no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226#95—1181no—
227#95—1181no—
228C24(C2×D59)⋊C411yescharacteristic
229C2×Q8—2361no—
230C2×Q8—2362no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1181no—
237C22×C4—1181no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4C2×C4×D5911yescharacteristic
244C22×C4C2×D23611yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246M16—24no—
247M16—44no—
248M16—42no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2361no—
255C4⋊C4—2362no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C4⋊C4—2361no—
259C4⋊C4—2361no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—2362no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1427365512yes—
274C23C2×(C2×D59)⋊C411yescharacteristic
275Q8—2362no—
276Q8—2362no—
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—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—1182no—
294C2×C4—2361no—
295C2×C4—2361no—
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#1427364111yescharacteristic
306C2×C4#1427364111yescharacteristic
307C2×C4#1427375311yescharacteristic
308C2×C4#1427375811yescharacteristic
309C8—44no—
310C8—4722no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1427364111yescharacteristic
319C22#1683740111yescharacteristic
320C22C2×#1427365511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4721no—
330C4—4721no—
331C4—4721no—
332C4—2362no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2142259811yescharacteristic
337C2#2142573811yescharacteristic
338C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1742C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1742D59
hypocenterC59#11763
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422101C2semidirect
192. C59#11763semidirect

Maximal subgroups

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

subgroupindex
#214296732
#214224302
#214221252
#214221012
C59×#17422
#1176359