Finite Group Databasebeta

← #28485619

Subgroups — #28485619

Order 15104.

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

normal subgroups only

This group has 444 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#21434226C211yesmaximal, characteristic
2#21425039C211yesmaximal, characteristic
3#21422607C211yesmaximal, characteristic
4#21422543C211yesmaximal, characteristic
5#21421989C211yesmaximal, characteristic
6#21421952C211yesmaximal, characteristic
7C59×#2320C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732—21no—
10#16837687C412yescomplemented
11#16837612C412yescomplemented
12#16837558—21no—
13#16837507—22no—
14#16837459C2211yescharacteristic
15#16837459C2211yescharacteristic
16C4×#11745664C2211yescharacteristic
17#16837162—22no—
18#16837138—21no—
19#16837138—21no—
20Q8⋊(C59⋊C8)—22no—
21C2×C4×Q8×C59C2211yescharacteristic
22C59×#391—21no—
23C59×#389—21no—
24C59×#387—22no—
25C59×#383—21no—
26C59×#382—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C59×#376C2211yescharacteristic
29C2×#11745672—22no—
30C2×#11745664—21no—
31#14273727C2×C411yescharacteristic
32#14273727C2×C411yescharacteristic
33#14273719—21no—
34#14273706—21no—
35#14273706—21no—
36C4×Dic118—21no—
37C4×Dic118—21no—
38#14273686—21no—
39#14273658—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)C2×C412yes—
42#14273651—22no—
43#14273650—22no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46#14273638—21no—
47#14273638—21no—
48#14273638—21no—
49C4×C59⋊C8—21no—
50C22×Q8×C59D411yescharacteristic
51C4×Q8×C59—21no—
52C4×Q8×C59—21no—
53C4×Q8×C59C2×C412yes—
54C4×Q8×C59D412yes—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—22no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C42×C59—21no—
60C2×C42×C59—21no—
61C2×C42×C59D412yes—
62C2×C42×C59C2311yescharacteristic
63#95×C59—22no—
64#95×C59—22no—
65#95×C59—22no—
66#95×C59—21no—
67#95×C59—21no—
68#95×C59D411yescharacteristic
69#95×C59D411yescharacteristic
70C2×Q8×C59—21no—
71C2×Q8×C59—42no—
72C2×Q8×C59C22⋊C412yes—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—22no—
77C22×C4×C59—21no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C2×Dic118—21no—
82C2×Dic118—21no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—21no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—82no—
88C4⋊C4×C59C22⋊C412yes—
89C42×C59—22no—
90C42×C59—22no—
91C42×C59—42no—
92C42×C59—42no—
93C42×C59C22×C411yescharacteristic
94C42×C59C2×D411yescharacteristic
95C42×C59#4111yescharacteristic
96C42×C59#4111yescharacteristic
97C22⋊Dic59—41no—
98#11745672—44no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103Dic59⋊C4—41no—
104Dic59⋊C4—41no—
105C4×Dic59—41no—
106#11745664—22no—
107#11745664—41no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C23×C59#12011yescharacteristic
112Q8×C59—22no—
113Q8×C59—22no—
114Q8×C59—42no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—41no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—42no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59C2×C22⋊C411yescharacteristic
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59C4×D411yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59#12111yescharacteristic
138C2×C4×C59#12311yescharacteristic
139C2×Dic59—41no—
140C2×Dic59—41no—
141Dic118—42no—
142Dic118—81no—
143C59⋊C8—42no—
144C59⋊C8—42no—
145C59⋊C8—81no—
146#9328—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59#38611yescharacteristic
150C22×C59#45511yescharacteristic
151C22×C59#45611yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—42no—
158C4×C59—42no—
159C4×C59—81no—
160C4×C59—82no—
161C4×C59—81no—
162C4×C59#42112yes—
163Dic59—81no—
164Dic59—81no—
165#3113—591no—
166#2320D5911yescharacteristic
167#1986—591no—
168#1506—591no—
169#1491—591no—
170#1300—591no—
171#1285—591no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#188211yescharacteristic
175C2×C59#189711yescharacteristic
176C2×C59#190511yescharacteristic
177#533—591no—
178C2×C4×Q8C2×D5911yescharacteristic
179#483—1181no—
180#475—592no—
181#462—592no—
182#451—1181no—
183#441—1182no—
184#432—591no—
185#432—591no—
186C4×M16—591no—
187#391—21no—
188#389—21no—
189#387—22no—
190#383—21no—
191#382—21no—
192C4×C4⋊C4C2×D5911yescharacteristic
193#376C2×D5911yescharacteristic
194#339—1182no—
195#328—1181no—
196#328—1181no—
197Q8⋊C8—1182no—
198C59#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D5911yescharacteristic
200C2×Q16—1182no—
201C2×M16—1181no—
202C4⋊Q8—591no—
203C4⋊Q8—591no—
204#125—1181no—
205C22⋊Q8—1181no—
206C22⋊Q8—1181no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—1181no—
210C4×Q8—1181no—
211C4×Q8C4×D5912yes—
212C4×Q8D23612yes—
213#117—1181no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊D5911yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22⋊D5912yes—
221C2×C42C22×D5911yescharacteristic
222#106—1182no—
223C4⋊C8—592no—
224C4⋊C8—1181no—
225#103—1182no—
226#103—1182no—
227C22⋊C8—1181no—
228C22⋊C8—1181no—
229#97—1181no—
230#97—1181no—
231#97—1181no—
232C4×C8—1181no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95C22⋊D5911yescharacteristic
239#95C22⋊D5911yescharacteristic
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—1181no—
243C2×Q8—1181no—
244C2×Q8(C2×D59)⋊C412yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×C22⋊D5911yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252C22×C4C2×C22⋊D5911yescharacteristic
253Q16—2364no—
254M16—1182no—
255M16—2361no—
256C2×C8—1182no—
257C2×C8—2361no—
258C2×C8—2361no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—1181no—
267C4⋊C4—1181no—
268C4⋊C4—2361no—
269C4⋊C4—2361no—
270C4⋊C4(C2×D59)⋊C412yes—
271C22⋊C4—2361no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—42no—
276C42—2361no—
277C42C2×C4×D5911yescharacteristic
278C42C2×D23611yescharacteristic
279C42#1174568511yescharacteristic
280C42#1174568511yescharacteristic
281C23#1427369611yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—4721no—
286Q8—2362no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4C2×(C2×D59)⋊C411yescharacteristic
308C2×C4C4×C22⋊D5911yescharacteristic
309C2×C4C4×C22⋊D5911yescharacteristic
310C2×C4#1427370111yescharacteristic
311C2×C4#1427370111yescharacteristic
312C2×C4#1427371011yescharacteristic
313C8—4721no—
314C8—2362no—
315C8—2362no—
316C22—21no—
317C22—21no—
318C22#1683728111yescharacteristic
319C22#1683758011yescharacteristic
320C22#1683758511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—4721no—
332C4—4721no—
333C4#1683742012yes—
334C2—21no—
335C2—21no—
336C2#2142446511yescharacteristic
337C2#2142454411yescharacteristic
338C2#2142459011yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2320C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#2320D59
hypocenterC59#9328
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2320

Complements

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

NH ≅ G/Ntype
10. #16837687C4semidirect
11. #16837612C4semidirect
198. C59#9328semidirect

Maximal subgroups

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

subgroupindex
#214342262
#214250392
#214226072
#214225432
#214219892
#214219522
C59×#23202
#932859