Finite Group Databasebeta

← #28477351

Subgroups — #28477351

Order 15104.

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

normal subgroups only

This group has 400 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#21424239C211yesmaximal, characteristic, complemented
2#21424239C211yesmaximal, characteristic, complemented
3#21421919C212yesmaximal, complemented
4#21421917C212yesmaximal
5C59×#2860C211yesmaximal, characteristic
6#16837157—21no—
7#16837157—21no—
8#16837157—21no—
9#16837157—21no—
10#16837157C2212yes—
11#16837157C2212yes—
12C2×#126×C59—21no—
13C59×#390C411yescharacteristic
14C59×#390C411yescharacteristic
15C59×#390C2211yescharacteristic
16C59×#389C411yescharacteristic
17C59×#389C411yescharacteristic
18C59×#389C2211yescharacteristic
19C59×#388—21no—
20C59×#388—21no—
21C59×#388—21no—
22C43×C59C2211yescharacteristic
23C22×C59⋊C8—22no—
24C22×C59⋊C8—22no—
25C22×D4×C59C2×C411yescharacteristic
26#126×C59—44no—
27C2×C4⋊C4×C59—21no—
28C2×C4⋊C4×C59—21no—
29C2×C4⋊C4×C59—21no—
30C2×C4⋊C4×C59C2×C411yescharacteristic
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59C2×C411yescharacteristic
33C2×C22⋊C4×C59—21no—
34C2×C22⋊C4×C59—21no—
35C2×C22⋊C4×C59—21no—
36C2×C22⋊C4×C59—21no—
37C2×C22⋊C4×C59—21no—
38C2×C22⋊C4×C59—21no—
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59C2×C411yescharacteristic
41C2×C42×C59D411yescharacteristic
42C2×C42×C59D411yescharacteristic
43C2×C42×C59D411yescharacteristic
44C2×C42×C59D411yescharacteristic
45C2×C42×C59C2311yescharacteristic
46#95×C59—21no—
47#95×C59—21no—
48#95×C59—21no—
49C24×C59—21no—
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C4⋊C4×C59—22no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—42no—
71C22⋊C4×C59—42no—
72C22⋊C4×C59—42no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—22no—
76C42×C59—22no—
77C42×C59—22no—
78C42×C59—41no—
79C42×C59—41no—
80C42×C59—41no—
81C42×C59—41no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—42no—
84C2×C59⋊C8—42no—
85C2×C59⋊C8—42no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—42no—
88C23×C59—41no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59C2×C22⋊C411yescharacteristic
96D4×C59—82no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
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—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—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59#13112yes—
125C2×C4×C59#13512yes—
126C2×C4×C59AGL(1,ℤ/8)12yes—
127C59⋊C8—82no—
128C59⋊C8—82no—
129#7726—591nomaximal, Sylow
130C22×C59—81no—
131C22×C59—81no—
132C22×C59—81no—
133C22×C59—81no—
134C22×C59—81no—
135C22×C59—81no—
136C22×C59—81no—
137C22×C59#40811yescharacteristic
138C22×C59#41011yescharacteristic
139C22×C59#41311yescharacteristic
140C22×C59#41611yescharacteristic
141C22×C59#41811yescharacteristic
142C22×C59#42111yescharacteristic
143C22×C59#42111yescharacteristic
144C4×C59—22no—
145C4×C59—22no—
146C4×C59—22no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—81no—
152C4×C59—81no—
153C4×C59—81no—
154#2860D5911yescharacteristic
155#1839—591no—
156#1839—591no—
157#1271—592no—
158#1270—592no—
159C2×C59—161no—
160C2×C59#147311yescharacteristic
161C2×C59#148211yescharacteristic
162C2×C59#149611yescharacteristic
163C2×C59#151411yescharacteristic
164C2×C59#152111yescharacteristic
165C2×C59#153511yescharacteristic
166C2×C59#155311yescharacteristic
167C2×#126—21no—
168#390Dic5911yescharacteristic
169#390Dic5911yescharacteristic
170#390C2×D5911yescharacteristic
171#389Dic5911yescharacteristic
172#389Dic5911yescharacteristic
173#389C2×D5911yescharacteristic
174#388—21no—
175#388—21no—
176#388—21no—
177C43C2×D5911yescharacteristic
178#336—592no—
179#336—592no—
180#336—1181no—
181#336—1181no—
182#336—1181no—
183#336—1181no—
184C59#772611yescharacteristic, complemented, Sylow
185C22×D4C2×Dic5911yescharacteristic
186C22×C8—1182no—
187C22×C8—1182no—
188#126—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic5911yescharacteristic
193C2×C4⋊C4C2×Dic5911yescharacteristic
194C2×C4⋊C4C2×Dic5911yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42C2×Dic5911yescharacteristic
202C2×C42C2×Dic5911yescharacteristic
203C2×C42C22⋊D5911yescharacteristic
204C2×C42C22⋊D5911yescharacteristic
205C2×C42C22⋊D5911yescharacteristic
206C2×C42C22⋊D5911yescharacteristic
207C2×C42C22×D5911yescharacteristic
208#95—21no—
209#95—21no—
210#95—21no—
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C22⋊Dic5911yescharacteristic
219C22×C4C22⋊Dic5911yescharacteristic
220C22×C4C22⋊Dic5911yescharacteristic
221C22×C4C22⋊Dic5911yescharacteristic
222C22×C4C22×Dic5911yescharacteristic
223C22×C4C2×C22⋊D5911yescharacteristic
224C22×C4C2×C22⋊D5911yescharacteristic
225C2×C8—2362no—
226C2×C8—2362no—
227C2×C8—2362no—
228C2×C8—2362no—
229C2×C8—2362no—
230C2×C8—2362no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊Dic5911yescharacteristic
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
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—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#1427373412yes—
287C2×C4#1427374712yes—
288C2×C4#1427375312yes—
289C8—4722no—
290C8—4722no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1683736911yescharacteristic
299C22#1683737611yescharacteristic
300C22#1683738611yescharacteristic
301C22#1683739811yescharacteristic
302C22#1683740711yescharacteristic
303C22#1683741911yescharacteristic
304C22#1683741911yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—161no—
316C2#2142246511yescharacteristic
317C2#2142250411yescharacteristic
318C2#2142256511yescharacteristic
319C2#2142264311yescharacteristic
320C2#2142267311yescharacteristic
321C2#2142272711yescharacteristic
322C2#2142280111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2860C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#2860D59
hypocenterC59#7726
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #21424239C2semidirect
2. #21424239C2semidirect
3. #21421919C2semidirect
184. C59#7726semidirect

Maximal subgroups

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

subgroupindex
#214242392
#214219192
#214219172
C59×#28602
#772659