Finite Group Databasebeta

← #28477257

Subgroups — #28477257

Order 15104.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#21432463C211yesmaximal, characteristic
2#21423845C211yesmaximal, characteristic, complemented
3#21421918C212yesmaximal, complemented
4#21421916C212yesmaximal
5C59×#1831C211yesmaximal, characteristic, complemented
6C2×#14273721—21no—
7C2×#14273638—21no—
8#16837304C411yescharacteristic
9#16837304C411yescharacteristic
10#16837304C2211yescharacteristic
11#16837300C411yescharacteristic
12#16837300C411yescharacteristic
13#16837300C2211yescharacteristic
14#16837293—21no—
15#16837293—21no—
16#16837293—21no—
17#16837157—21no—
18#16837157C2212yes—
19C2×C4⋊C8×C59—21no—
20C43×C59C2211yescharacteristic
21C59×#336—21no—
22C59×#336C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C59⋊C8—22no—
25#14273721—44no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38#14273638—24no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×C8×C59—22no—
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C2×C42×C59C2311yescharacteristic
50C4⋊C8×C59—42no—
51C23×D59—21no—
52C22×C4×C59C22⋊C411yescharacteristic
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—21no—
60C22×Dic59—21no—
61C22×Dic59—21no—
62C2×D236—22no—
63C2×D236—41no—
64C2×D236—41no—
65C2×C8×C59—42no—
66C2×C8×C59—42no—
67C2×C8×C59—42no—
68C42×C59—22no—
69C42×C59—22no—
70C42×C59—21no—
71C42×C59—22no—
72C42×C59—21no—
73C42×C59—41no—
74C42×C59—41no—
75C42×C59—41no—
76(C2×D59)⋊C4—42no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79(C2×D59)⋊C4—42no—
80(C2×D59)⋊C4—41no—
81(C2×D59)⋊C4—42no—
82(C2×D59)⋊C4—41no—
83C4⋊Dic59—22no—
84C4⋊Dic59—41no—
85C4⋊Dic59—41no—
86Dic59⋊C4—42no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89C2×C59⋊C8—24no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—42no—
92C23×C59C2×C22⋊C411yescharacteristic
93C22×D59—41no—
94C22×D59—41no—
95C22×D59—41no—
96C22×D59—41no—
97C22×D59—41no—
98C22×D59—41no—
99C22×D59—41no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
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#10412yes—
118C2×C4×C59#13112yes—
119C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C59AGL(1,ℤ/8)12yes—
121C2×C4×C59#13711yescharacteristic
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131D236—82no—
132C8×C59—82no—
133C59⋊C8—44no—
134#7706—591nomaximal, Sylow
135C22×C59#40811yescharacteristic
136C22×C59#41011yescharacteristic
137C22×C59#41311yescharacteristic
138C22×C59#41711yescharacteristic
139C22×C59#41811yescharacteristic
140C22×C59C2×#10411yescharacteristic
141C22×C59#42111yescharacteristic
142C2×D59—81no—
143C2×D59—81no—
144C2×D59—81no—
145C2×D59—81no—
146C2×D59—81no—
147C2×D59—81no—
148C2×D59—81no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—22no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157Dic59—81no—
158Dic59—81no—
159Dic59—81no—
160#2860—591no—
161#1831D5911yescharacteristic, complemented
162#1745—591no—
163#1271—592no—
164#1270—592no—
165C2×C59#147311yescharacteristic
166C2×C59#148311yescharacteristic
167C2×C59#149611yescharacteristic
168C2×C59#150511yescharacteristic
169C2×C59#151211yescharacteristic
170C2×C59#152911yescharacteristic
171C2×C59#155711yescharacteristic
172D59—161no—
173C2×#126—1181no—
174C2×C4⋊C8—21no—
175C2×#97—1181no—
176#390—591no—
177#390—591no—
178#390—591no—
179#389—591no—
180#389—591no—
181#389—591no—
182#388—1181no—
183#388—1181no—
184#388—1181no—
185C43C2×D5911yescharacteristic
186#336—21no—
187#336—592no—
188#336—1181no—
189#336C2×D5912yes—
190C59#770611yescharacteristic, complemented, Sylow
191C22×D4—591no—
192C22×C8—22no—
193C22×C8—1182no—
194#126—2364no—
195C2×C4⋊C4—591no—
196C2×C4⋊C4—591no—
197C2×C4⋊C4—591no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1181no—
201C2×C22⋊C4—1181no—
202C2×C22⋊C4—1181no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4—1181no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C42C4×D5911yescharacteristic
208C2×C42C4×D5911yescharacteristic
209C2×C42D23611yescharacteristic
210C2×C42D23611yescharacteristic
211C2×C42C22⋊D5911yescharacteristic
212C2×C42C22⋊D5911yescharacteristic
213C2×C42C22×D5911yescharacteristic
214C4⋊C8—42no—
215#97—1184no—
216#95—1181no—
217#95—1181no—
218#95—1181no—
219C24—1181no—
220C2×D4—1182no—
221C2×D4—2361no—
222C2×D4—2361no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4—1181no—
226C22×C4(C2×D59)⋊C411yescharacteristic
227C22×C4(C2×D59)⋊C411yescharacteristic
228C22×C4(C2×D59)⋊C411yescharacteristic
229C22×C4(C2×D59)⋊C411yescharacteristic
230C22×C4C2×C4×D5911yescharacteristic
231C22×C4C2×D23611yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—1184no—
237C2×C8—2362no—
238C2×C8—2362no—
239C4⋊C4—1182no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C4⋊C4—2362no—
243C4⋊C4—2362no—
244C4⋊C4—2362no—
245C22⋊C4—2362no—
246C22⋊C4—2362no—
247C22⋊C4—2362no—
248C22⋊C4—2362no—
249C22⋊C4—2361no—
250C22⋊C4—2362no—
251C22⋊C4—2361no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—22no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C23—2361no—
261C23—2361no—
262C23—2361no—
263C23—2361no—
264C23—2361no—
265C23—2361no—
266C23—2361no—
267C23C2×(C2×D59)⋊C411yescharacteristic
268D4—4722no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4#1427365312yes—
296C2×C4#1427373312yes—
297C2×C4#1427375311yescharacteristic
298C2×C4#1427375512yes—
299C2×C4#1427375811yescharacteristic
300C8—82no—
301C8—2364no—
302C22—4721no—
303C22—4721no—
304C22—4721no—
305C22—4721no—
306C22—4721no—
307C22—4721no—
308C22—4721no—
309C22#1683736811yescharacteristic
310C22#1683737511yescharacteristic
311C22#1683738411yescharacteristic
312C22#1683740111yescharacteristic
313C22#1683740811yescharacteristic
314C22C2×#1427365311yescharacteristic
315C22#1683742011yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4721no—
325C4—4721no—
326C4—4721no—
327C2—9441no—
328C2#2142246311yescharacteristic
329C2#2142250711yescharacteristic
330C2#2142256611yescharacteristic
331C2#2142260511yescharacteristic
332C2#2142263211yescharacteristic
333C2#2142270411yescharacteristic
334C2#2142281611yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1831C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1831D59
hypocenterC59#7706
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
2. #21423845C2semidirect
3. #21421918C2semidirect
5. C59×#1831C2semidirect
161. #1831D59semidirect
190. C59#7706semidirect

Maximal subgroups

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

subgroupindex
#214324632
#214238452
#214219182
#214219162
C59×#18312
#770659