Finite Group Databasebeta

← #28499498

Subgroups — #28499498

Order 15104.

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

normal subgroups only

This group has 420 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#21427355C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422122C212yesmaximal, complemented
4#21422101C212yesmaximal, complemented
5C59×#1800C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic
7#16837366—21no—
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837330—21no—
12#16837327C411yescharacteristic
13#16837316—21no—
14C4×Dic59⋊C4—21no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#423—21no—
21C2×#100×C59—21no—
22C59×#343C2212yes—
23C2×#11745664—41no—
24#14273686—41no—
25#14273686—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×D4×C59D411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C4⋊C8×C59—41no—
56#100×C59—42no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C2×D4×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78M16×C59—42no—
79M16×C59—42no—
80C2×C8×C59—42no—
81C4⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—41no—
91C4⋊Dic59—41no—
92Dic59⋊C4—42no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95C4×Dic59—42no—
96C4×Dic59—41no—
97C4×Dic59—41no—
98#11745664—82no—
99C2×C59⋊C8—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59C2×C22⋊C411yescharacteristic
111D4×C59—42no—
112D4×C59—42no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—81no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C59#13711yescharacteristic
130C2×Dic59—22no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C8×C59—82no—
143C59⋊C8—82no—
144#11830—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59C2×#9911yescharacteristic
158C22×C59#41711yescharacteristic
159C22×C59#42111yescharacteristic
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59#35212yes—
167Dic59—42no—
168Dic59—81no—
169Dic59—81no—
170Dic59—81no—
171#2395—591no—
172#1800D5911yescharacteristic
173#1465—591no—
174#1353—592no—
175#1346—592no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—81no—
179C2×C59—81no—
180C2×C59#151311yescharacteristic
181C2×C59C2×#35211yescharacteristic
182C2×C59#213211yescharacteristic
183C2×C4×D4C2×D5911yescharacteristic
184C2×#117—591no—
185#423—21no—
186C2×#100—21no—
187#407—1181no—
188#400—591no—
189#399—591no—
190#398—591no—
191#397—591no—
192#397—1181no—
193#394—1181no—
194C4×C4⋊C4—1181no—
195C4×C22⋊C4—591no—
196C4×C22⋊C4—1181no—
197#343C2×D5912yes—
198#324—592no—
199#323—1181no—
200C59#1183011yescharacteristic, complemented, Sylow
201C22×D4C22⋊D5911yescharacteristic
202C23×C4C4×D5911yescharacteristic
203C23×C4D23611yescharacteristic
204C2×M16—22no—
205C2×M16—2361no—
206C4×D4—41no—
207C4×D4—41no—
208#117—2361no—
209#117—2361no—
210C2×C4⋊C4—591no—
211C2×C4⋊C4—591no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4C22×D5911yescharacteristic
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4C4×D5911yescharacteristic
220C2×C22⋊C4D23611yescharacteristic
221C2×C42—1181no—
222C2×C42—1181no—
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226#100—42no—
227C22⋊C8—1182no—
228C22⋊C8—2361no—
229#95—591no—
230#95—591no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235C24(C2×D59)⋊C411yescharacteristic
236C24(C2×D59)⋊C411yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4C2×C4×D5911yescharacteristic
254C22×C4C2×D23611yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256M16—42no—
257M16—42no—
258M16—4722no—
259C2×C8—42no—
260C2×C8—2362no—
261C4⋊C4—41no—
262C4⋊C4—2361no—
263C4⋊C4—2361no—
264C4⋊C4—2362no—
265C4⋊C4—2361no—
266C4⋊C4—2361no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C22⋊C4—2362no—
273C42—21no—
274C42—21no—
275C42—2362no—
276C42—2361no—
277C42—2361no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×(C2×D59)⋊C411yescharacteristic
289D4—42no—
290D4—42no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—1182no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4#1427364111yescharacteristic
317C2×C4#1427364111yescharacteristic
318C2×C4#1427375311yescharacteristic
319C2×C4#1427375811yescharacteristic
320C8—82no—
321C8—4722no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1427364111yescharacteristic
335C22#1683740111yescharacteristic
336C22#1683742011yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4721no—
344C4—4721no—
345C4—4721no—
346C4—2362no—
347C4#1683719312yes—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—81no—
352C2#2142263711yescharacteristic
353C2C2×#1683719311yescharacteristic
354C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1800C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C4⋊C4×C59C23
hypercenter#1800D59
hypocenterC59#11830
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #21422122C2semidirect
4. #21422101C2semidirect
200. C59#11830semidirect

Maximal subgroups

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

subgroupindex
#214273552
#214224342
#214221222
#214221012
C59×#18002
#1183059