Finite Group Databasebeta

← #28475779

Subgroups — #28475779

Order 15104.

376 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 87 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#21424891C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3C59×#1799C211yesmaximal, characteristic
4C2×C4⋊(C59⋊C8)C411yescharacteristic, complemented
5C2×#14273642—21no—
6C2×C22⋊(C59⋊C8)—21no—
7C2×C22⋊(C59⋊C8)—21no—
8C2×C4×C59⋊C8C411yescharacteristic, complemented
9#16837262—21no—
10#16837169C412yescomplemented
11C2×C4×D4×C59C2211yescharacteristic
12C2×#99×C59—21no—
13C59×#380—21no—
14C59×#342C412yescomplemented
15C22×C59⋊C8—21no—
16C22×C59⋊C8—21no—
17C2×C4⋊Dic59—41no—
18C2×C22⋊Dic59—22no—
19C4⋊(C59⋊C8)—41no—
20#14273642—41no—
21#14273642—41no—
22C22⋊(C59⋊C8)—42no—
23C22⋊(C59⋊C8)—42no—
24C4×C59⋊C8—41no—
25C59⋊#95—41no—
26C22×D4×C59D411yescharacteristic
27C23×C4×C59D411yescharacteristic
28C23×C4×C59D411yescharacteristic
29C4×D4×C59—41no—
30C4×D4×C59—41no—
31C2×C4⋊C4×C59—41no—
32C2×C4⋊C4×C59Q811yescharacteristic
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59C2×C411yescharacteristic
35C2×C22⋊C4×C59C2×C411yescharacteristic
36C2×C42×C59C2×C411yescharacteristic
37#99×C59—41no—
38#99×C59—41no—
39#95×C59—41no—
40C24×C59C22⋊C411yescharacteristic
41C24×C59C22⋊C411yescharacteristic
42C2×D4×C59—41no—
43C2×D4×C59—41no—
44C2×D4×C59D811yescharacteristic
45C2×D4×C59SD1611yescharacteristic
46C2×D4×C59SD1611yescharacteristic
47C2×D4×C59Q1611yescharacteristic
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59—41no—
55C22×C4×C59—41no—
56C22×C4×C59—42no—
57C22×C4×C59C4211yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C4⋊C411yescharacteristic
60C22×C4×C59C4⋊C411yescharacteristic
61C22×C4×C59C4⋊C411yescharacteristic
62C22×Dic59—42no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—81no—
65C4⋊C4×C59—81no—
66C22⋊C4×C59—41no—
67C22⋊C4×C59—41no—
68C22⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C42×C59—21no—
71C42×C59—21no—
72C22⋊Dic59—42no—
73C22⋊Dic59—42no—
74C4⋊Dic59—81no—
75C4⋊Dic59—81no—
76C2×C59⋊C8—22no—
77C2×C59⋊C8—41no—
78C2×C59⋊C8—41no—
79C2×C59⋊C8—41no—
80C2×C59⋊C8—41no—
81C2×C59⋊C8—41no—
82C23×C59—21no—
83C23×C59—21no—
84C23×C59—21no—
85C23×C59—21no—
86C23×C59—41no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59—41no—
90C23×C59#9511yescharacteristic
91C23×C59#10211yescharacteristic
92C23×C59#10211yescharacteristic
93C23×C59#10311yescharacteristic
94C23×C59#10311yescharacteristic
95D4×C59—41no—
96D4×C59—41no—
97D4×C59—41no—
98D4×C59—41no—
99C2×C4×C59—21no—
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—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—81no—
114C2×C4×C59—81no—
115C2×C4×C59—82no—
116C2×C4×C59—82no—
117C2×C4×C59—82no—
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#10611yescharacteristic
121C2×C4×C59#10711yescharacteristic
122C2×Dic59—82no—
123C2×Dic59—82no—
124C2×Dic59—82no—
125C59⋊C8—42no—
126C59⋊C8—81no—
127#7237—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—81no—
143C22×C59—81no—
144C22×C59H(3,ℤ/4)11yescharacteristic
145C22×C59#34011yescharacteristic
146C22×C59#34211yescharacteristic
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—162no—
154C4×C59#35311yescharacteristic
155C4×C59#35411yescharacteristic
156Dic59—162no—
157#1954—591no—
158#1799—591no—
159#1799D5911yescharacteristic
160C2×C59—21no—
161C2×C59—21no—
162C2×C59—41no—
163C2×C59—41no—
164C2×C59—41no—
165C2×C59—41no—
166C2×C59#128911yescharacteristic
167C2×C59#139111yescharacteristic
168C2×C59#139311yescharacteristic
169C2×C4×D4C2×D5911yescharacteristic
170C2×C4⋊C8—591no—
171C2×#99—21no—
172C2×#99—1181no—
173C2×C22⋊C8—1181no—
174C2×C22⋊C8—1181no—
175C2×C4×C8—591no—
176#380—21no—
177#380—1181no—
178#342—592no—
179#342Dic5912yescomplemented
180C59#723711yescharacteristic, complemented, Sylow
181C22×D4C22⋊D5911yescharacteristic
182C23×C4D23611yescharacteristic
183C23×C4C22⋊D5911yescharacteristic
184C22×C8—1181no—
185C22×C8—1181no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—2361no—
190C2×C4⋊C4Dic11811yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1182no—
193C2×C22⋊C4C4×D5911yescharacteristic
194C2×C22⋊C4C2×Dic5911yescharacteristic
195C2×C42C4×D5911yescharacteristic
196C4⋊C8—2361no—
197#99—41no—
198#99—41no—
199#99—2361no—
200#99—2361no—
201C22⋊C8—2362no—
202C22⋊C8—2362no—
203C4×C8—2361no—
204#95—41no—
205#95—2361no—
206C24(C2×D59)⋊C411yescharacteristic
207C24C22⋊Dic5911yescharacteristic
208C2×D4—41no—
209C2×D4—41no—
210C2×D4#1174566911yescharacteristic
211C2×D4#1174567011yescharacteristic
212C2×D4#1174567111yescharacteristic
213C2×D4#1174567211yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—2362no—
224C22×C4C4×Dic5911yescharacteristic
225C22×C4Dic59⋊C411yescharacteristic
226C22×C4Dic59⋊C411yescharacteristic
227C22×C4C4⋊Dic5911yescharacteristic
228C22×C4(C2×D59)⋊C411yescharacteristic
229C2×C8—1182no—
230C2×C8—2361no—
231C2×C8—2361no—
232C2×C8—2361no—
233C2×C8—2361no—
234C2×C8—2361no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—4721no—
239C4⋊C4—4721no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C59⋊#9511yescharacteristic
257C23#1427364711yescharacteristic
258C23#1427364811yescharacteristic
259C23#1427365011yescharacteristic
260C23#1427365111yescharacteristic
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—4722no—
285C2×C4—4722no—
286C2×C4—4722no—
287C2×C4#1427364111yescharacteristic
288C2×C4#1427364111yescharacteristic
289C2×C4#1427365811yescharacteristic
290C2×C4#1427366011yescharacteristic
291C8—4721no—
292C8—2362no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1683715811yescharacteristic
310C22#1683716411yescharacteristic
311C22#1683716811yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—9442no—
320C4#1683719411yescharacteristic
321C4#1683719611yescharacteristic
322C2—21no—
323C2—21no—
324C2—41no—
325C2—41no—
326C2—41no—
327C2—41no—
328C2#2142196311yescharacteristic
329C2#2142221611yescharacteristic
330C2#2142222011yescharacteristic
subgroupG/H
centerC22#16837168
commutatorC22×C4×C59C42
FrattiniC2×C4×D4C2×D59
FittingC59×#1799C2
socleC22×C59#342
prime radicalC22×D4×C59D4
cosocleC2×C4×D4×C59C22
hypercenter#1799D59
hypocenterC59#7237
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(C59⋊C8)C4semidirect
8. C2×C4×C59⋊C8C4semidirect
10. #16837169C4semidirect
14. C59×#342C4semidirect
179. #342Dic59semidirect
180. C59#7237semidirect

Maximal subgroups

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

subgroupindex
#214248912
#214240842
C59×#17992
#723759