Finite Group Databasebeta

← #28476830

Subgroups — #28476830

Order 15104.

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

normal subgroups only

This group has 484 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#21426560C211yesmaximal, characteristic
2#21423845C211yesmaximal, characteristic, complemented
3#21421920C212yesmaximal, complemented
4#21421916C212yesmaximal
5C59×#1839C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic
7C2×#14273684—21no—
8C2×#14273638—21no—
9#16837304C411yescharacteristic
10#16837304C2211yescharacteristic
11#16837300C411yescharacteristic
12#16837300C2211yescharacteristic
13#16837293—21no—
14#16837293—21no—
15C4×C4⋊Dic59C411yescharacteristic
16C4×(C2×D59)⋊C4—21no—
17#16837157—21no—
18#16837157C2212yes—
19C43×C59C2211yescharacteristic
20C59×#336—21no—
21C59×#336—21no—
22C59×#336C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C4×D59—21no—
25C22×C59⋊C8—22no—
26C4×D236—22no—
27C4×D236—22no—
28#14273684—44no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×C4×Dic59—21no—
40#14273638—24no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C22×C8×C59—22no—
45C2×C42×C59C2×C411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59C2311yescharacteristic
52C23×D59—21no—
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×Dic59—21no—
61C22×Dic59—21no—
62C22×Dic59—21no—
63C2×D236—21no—
64C2×D236—41no—
65C2×D236—41no—
66C2×D236M1612yes—
67C2×C4×D59—24no—
68C2×C4×D59—42no—
69C2×C4×D59—42no—
70C2×C8×C59—42no—
71C2×C8×C59—42no—
72C2×C8×C59—42no—
73C42×C59—22no—
74C42×C59—21no—
75C42×C59—21no—
76C42×C59—22no—
77C42×C59—22no—
78C42×C59—41no—
79C42×C59—41no—
80C42×C59—41no—
81(C2×D59)⋊C4—24no—
82(C2×D59)⋊C4—42no—
83(C2×D59)⋊C4—42no—
84(C2×D59)⋊C4—42no—
85(C2×D59)⋊C4—42no—
86C4⋊Dic59—21no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89C4⋊Dic59M1612yes—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92C4×Dic59—42no—
93C2×C59⋊C8—24no—
94C2×C59⋊C8—42no—
95C2×C59⋊C8—42no—
96C23×C59C2×C22⋊C411yescharacteristic
97C22×D59—22no—
98C22×D59—41no—
99C22×D59—41no—
100C22×D59—41no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—41no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
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—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59#10412yes—
122C2×C4×C59C2×M1611yescharacteristic
123C2×C4×C59C2×M1611yescharacteristic
124C2×C4×C59#13512yes—
125C2×C4×C59AGL(1,ℤ/8)12yes—
126C2×Dic59—22no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135D236—42no—
136D236—42no—
137C4×D59—48no—
138C8×C59—82no—
139C59⋊C8—44no—
140#7608—591nomaximal, Sylow
141C22×C59#40711yescharacteristic
142C22×C59#41011yescharacteristic
143C22×C59#41311yescharacteristic
144C22×C59#41611yescharacteristic
145C22×C59#41811yescharacteristic
146C22×C59C2×#10411yescharacteristic
147C22×C59#42111yescharacteristic
148C2×D59—42no—
149C2×D59—42no—
150C2×D59—42no—
151C2×D59—81no—
152C2×D59—81no—
153C2×D59—81no—
154C2×D59—81no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—22no—
158C4×C59—22no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163Dic59—42no—
164Dic59—81no—
165Dic59—81no—
166#2293—591no—
167#1839D5911yescharacteristic, complemented
168#1745—591no—
169#1271—592no—
170#1270—592no—
171C2×C59#146411yescharacteristic
172C2×C59#148811yescharacteristic
173C2×C59#149011yescharacteristic
174C2×C59#150011yescharacteristic
175C2×C59#151411yescharacteristic
176C2×C59#152911yescharacteristic
177C2×C59#155311yescharacteristic
178D59—82no—
179C2×C4×D4—591no—
180C2×#117—1181no—
181C2×#97—1181no—
182#390—591no—
183#390—591no—
184#389—591no—
185#389—591no—
186#388—1181no—
187#388—1181no—
188C4×C4⋊C4—591no—
189C4×C22⋊C4—1181no—
190C43C2×D5911yescharacteristic
191#336—21no—
192#336—21no—
193#336—592no—
194#336—1181no—
195#336C2×D5912yes—
196C59#760811yescharacteristic, complemented, Sylow
197C22×D4—591no—
198C23×C4—1181no—
199C22×C8—22no—
200C22×C8—1182no—
201C4×D4—1182no—
202C4×D4—1182no—
203#117—2364no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—591no—
206C2×C4⋊C4—591no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C22⋊C4—1181no—
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C42—1181no—
215C2×C42C4×D5911yescharacteristic
216C2×C42C4×D5911yescharacteristic
217C2×C42D23611yescharacteristic
218C2×C42D23611yescharacteristic
219C2×C42C22⋊D5911yescharacteristic
220C2×C42C22⋊D5911yescharacteristic
221C2×C42C22×D5911yescharacteristic
222#97—1184no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226C24—1181no—
227C2×D4—592no—
228C2×D4—1181no—
229C2×D4—2361no—
230C2×D4—2361no—
231C22×C4—1184no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—2362no—
236C22×C4—2362no—
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4C2×D23611yescharacteristic
243C22×C4C2×C22⋊D5911yescharacteristic
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—1184no—
248C2×C8—2362no—
249C2×C8—2362no—
250C4⋊C4—592no—
251C4⋊C4—1181no—
252C4⋊C4—2361no—
253C4⋊C4—2361no—
254C4⋊C4—2362no—
255C4⋊C4—2362no—
256C22⋊C4—1184no—
257C22⋊C4—2362no—
258C22⋊C4—2362no—
259C22⋊C4—2362no—
260C22⋊C4—2362no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—2362no—
270C23—1182no—
271C23—2361no—
272C23—2361no—
273C23—2361no—
274C23—2361no—
275C23—2361no—
276C23—2361no—
277C23C2×(C2×D59)⋊C411yescharacteristic
278D4—2362no—
279D4—2362no—
280C2×C4—21no—
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—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—1182no—
298C2×C4—2368no—
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#1427365312yes—
308C2×C4M16×D5911yescharacteristic
309C2×C4M16×D5911yescharacteristic
310C2×C4#1427374912yes—
311C2×C4#1427375212yes—
312C8—82no—
313C8—2364no—
314C22—4721no—
315C22—4721no—
316C22—4721no—
317C22—4721no—
318C22—2362no—
319C22—2362no—
320C22—2362no—
321C22#1683736511yescharacteristic
322C22#1683737511yescharacteristic
323C22#1683738411yescharacteristic
324C22#1683739911yescharacteristic
325C22#1683740611yescharacteristic
326C22C2×#1427365311yescharacteristic
327C22#1683742011yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—4721no—
337C4—4721no—
338C4—2362no—
339C2—4722no—
340C2#2142242811yescharacteristic
341C2#2142253111yescharacteristic
342C2#2142253811yescharacteristic
343C2#2142258311yescharacteristic
344C2#2142264211yescharacteristic
345C2#2142270411yescharacteristic
346C2#2142279911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1839C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1839D59
hypocenterC59#7608
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1839

Complements

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

NH ≅ G/Ntype
2. #21423845C2semidirect
3. #21421920C2semidirect
5. C59×#1839C2semidirect
167. #1839D59semidirect
196. C59#7608semidirect

Maximal subgroups

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

subgroupindex
#214265602
#214238452
#214219202
#214219162
C59×#18392
#760859