Finite Group Databasebeta

← #28485445

Subgroups — #28485445

Order 15104.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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#21433895C211yesmaximal, characteristic
2#21433119C211yesmaximal, characteristic
3#21424107C211yesmaximal, characteristic
4#21422661C211yesmaximal, characteristic, complemented
5#21422498C211yesmaximal, characteristic, complemented
6#21421956C211yesmaximal, characteristic, complemented
7C59×#1301C211yesmaximal, characteristic, complemented
8#16837959C2211yescharacteristic
9C2×C4×D236C2211yescharacteristic
10#16837703—22no—
11#16837693C412yescomplemented
12#16837683C412yescomplemented
13C8×D236—22no—
14#16837467—22no—
15#16837455C2211yescharacteristic
16#16837444—21no—
17#16837408—21no—
18C2×#14273649—21no—
19C22⋊C8×D59—22no—
20#16837268C2211yescharacteristic
21#16837165—21no—
22#16837140—22no—
23#16837132—22no—
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C22×D236D411yescharacteristic
28C22×C4×D59—21no—
29C2×C8×D59—24no—
30#14273727C2×C411yescharacteristic
31#14273727C2×C411yescharacteristic
32#14273719—21no—
33#14273706—21no—
34#14273706—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37C4×D236—21no—
38C4×D236—21no—
39C4×D236C2×C412yes—
40C4×D236D412yes—
41#14273686—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59D411yescharacteristic
44C2×(C2×D59)⋊C4—21no—
45#14273661—22no—
46#14273659—22no—
47C4⋊(C59⋊C8)—21no—
48#14273649—22no—
49#14273649—22no—
50#14273649—42no—
51C22⋊(C59⋊C8)—21no—
52(C2×D59)⋊C8—24no—
53C4×C59⋊C8D412yes—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C2×M16×C59—21no—
57C22×C8×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59C2×C412yes—
64C22⋊C8×C59D412yes—
65C4×C8×C59—21no—
66C23×D59—21no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—41no—
72C2×D236—21no—
73C2×D236—21no—
74C2×D236—21no—
75C2×D236C22⋊C412yes—
76C2×D236SD1612yes—
77C2×D236Q1612yes—
78C2×C4×D59—22no—
79C2×C4×D59—22no—
80C2×C4×D59—82no—
81C2×Dic118—21no—
82C2×Dic118—21no—
83M16×C59—41no—
84M16×C59—41no—
85C2×C8×C59—22no—
86C2×C8×C59—22no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C42×C59C22×C411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59C2×D411yescharacteristic
93C42×C59#4111yescharacteristic
94C22⋊Dic59—41no—
95(C2×D59)⋊C4—22no—
96(C2×D59)⋊C4—22no—
97C4⋊Dic59—21no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—41no—
103C4⋊Dic59—41no—
104C4⋊Dic59C22⋊C412yes—
105Dic59⋊C4—41no—
106Dic59⋊C4—41no—
107C4×Dic59—41no—
108C2×C59⋊C8—22no—
109C8×D59—84no—
110C23×C59#12111yescharacteristic
111C22×D59—22no—
112C22×D59—22no—
113C22×D59—22no—
114C22×D59—81no—
115C22×D59—81no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59C2×C22⋊C411yescharacteristic
119C2×C4×C59C4×D411yescharacteristic
120C2×C4×C59C4×D411yescharacteristic
121C2×C4×C59#12011yescharacteristic
122C2×C4×C59#12111yescharacteristic
123C2×C4×C59#12311yescharacteristic
124C2×C4×C59C2×SD1611yescharacteristic
125C2×C4×C59C2×Q1611yescharacteristic
126C2×C4×C59#13511yescharacteristic
127C2×C4×C59#13511yescharacteristic
128C2×Dic59—22no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C2×Dic59—81no—
135D236—22no—
136D236—22no—
137D236—22no—
138D236#10314yes—
139C4×D59—44no—
140C4×D59—82no—
141Dic118—42no—
142Dic118—81no—
143C8×C59—42no—
144C8×C59—42no—
145C8×C59—81no—
146C59⋊C8—42no—
147#9303—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59#38611yescharacteristic
151C22×C59#46511yescharacteristic
152C22×C59#46711yescharacteristic
153C2×D59—24no—
154C2×D59—42no—
155C2×D59—42no—
156C2×D59—82no—
157C2×D59—82no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59C2×#10311yescharacteristic
163C4×C59#41611yescharacteristic
164C4×C59#46511yescharacteristic
165C4×C59#46711yescharacteristic
166Dic59—42no—
167Dic59—81no—
168Dic59—81no—
169Dic59—161no—
170#3077—591no—
171#2972—591no—
172#1804—591no—
173#1518—591no—
174#1481—591no—
175#1301D5911yescharacteristic, complemented
176#1286—591no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59#188111yescharacteristic
180C2×C59#189211yescharacteristic
181C2×C59#190411yescharacteristic
182D59—44no—
183#533—591no—
184C2×C4×D4—591no—
185#478—1182no—
186#476—592no—
187#474—592no—
188C8×D4—1182no—
189C8×D4—1182no—
190#431—591no—
191#431C2×D5911yescharacteristic
192#428—1181no—
193#423C2×D5911yescharacteristic
194C2×C4⋊C8C2×D5911yescharacteristic
195#418—1181no—
196C2×#102—1181no—
197C2×C22⋊C8—1182no—
198#382—591no—
199#340—1181no—
200#328—1182no—
201Q8⋊C8—1182no—
202C59#930311yescharacteristic, complemented, Sylow
203C22×D4—591no—
204C23×C4—1181no—
205C2×M16—21no—
206C22×C8—21no—
207C22×C8—1184no—
208C4⋊Q8—591no—
209C4⋊Q8—591no—
210#125—1181no—
211C22⋊Q8—1181no—
212C22⋊Q8—1181no—
213C4×Q8—1181no—
214C4×Q8—1181no—
215C4×D4—592no—
216C4×D4—592no—
217C4×D4—1181no—
218C4×D4—1181no—
219#117—1181no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C42C22×D5911yescharacteristic
224#107—1182no—
225#106—1182no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—1181no—
231C4⋊C8C4×D5912yes—
232#102—1182no—
233#102—1182no—
234#102—2362no—
235C22⋊C8—1184no—
236C22⋊C8—1181no—
237C22⋊C8D23612yes—
238C4×C8—21no—
239C4×C8—592no—
240#95—1181no—
241#95—1181no—
242C24—1181no—
243C2×Q8—1181no—
244C2×Q8—1181no—
245C2×D4—592no—
246C2×D4—592no—
247C2×D4—592no—
248C2×D4—1181no—
249C2×D4—1181no—
250C2×D4—1181no—
251C22×C4—1181no—
252C22×C4—1182no—
253C22×C4—1182no—
254C22×C4—4722no—
255C22×C4—2361no—
256C22×C4C2×D23611yescharacteristic
257C22×C4D4×D5911yescharacteristic
258C22×C4#1174568911yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—1182no—
267C2×C8—4724no—
268C4⋊C4—592no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—1181no—
273C4⋊C4—1181no—
274C4⋊C4—2361no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2361no—
278C22⋊C4—1182no—
279C22⋊C4—1182no—
280C22⋊C4—2361no—
281C42—2361no—
282C42C2×C4×D5911yescharacteristic
283C42#1174568511yescharacteristic
284C42D4×D5911yescharacteristic
285C42D4×D5911yescharacteristic
286C23—1182no—
287C23—1182no—
288C23—1182no—
289C23—4721no—
290C23—4721no—
291C23#1427370211yescharacteristic
292Q8—4721no—
293Q8—2362no—
294D4—594no—
295D4—1182no—
296D4—1182no—
297D4—1182no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—1182no—
301C2×C4—4722no—
302C2×C4—4721no—
303C2×C4—4721no—
304C2×C4—4721no—
305C2×C4—4721no—
306C2×C4—2364no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4C22⋊C4×D5911yescharacteristic
310C2×C4#1427368911yescharacteristic
311C2×C4C4×D23611yescharacteristic
312C2×C4#1427369711yescharacteristic
313C2×C4#1427370011yescharacteristic
314C2×C4#1427371211yescharacteristic
315C2×C4SD16×D5911yescharacteristic
316C2×C4Q16×D5911yescharacteristic
317C2×C4#1427374611yescharacteristic
318C2×C4#1427374811yescharacteristic
319C8—42no—
320C8—42no—
321C8—81no—
322C8—2362no—
323C22—21no—
324C22—21no—
325C22—1184no—
326C22—4722no—
327C22—4722no—
328C22—2362no—
329C22—2362no—
330C22#1683728011yescharacteristic
331C22#1683763211yescharacteristic
332C22#1683764411yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—9441no—
338C4—4721no—
339C4—4721no—
340C4—2362no—
341C4#103×D5911yescharacteristic
342C4#1683739511yescharacteristic
343C4#1683763011yescharacteristic
344C4#1683764211yescharacteristic
345C2—21no—
346C2—21no—
347C2—2364no—
348C2#2142445911yescharacteristic
349C2#2142451811yescharacteristic
350C2#2142458211yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1301C2
socleC22×C59#386
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenter#1301D59
hypocenterC59#9303
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
4. #21422661C2semidirect
5. #21422498C2semidirect
6. #21421956C2semidirect
7. C59×#1301C2semidirect
11. #16837693C4semidirect
12. #16837683C4semidirect
175. #1301D59semidirect
202. C59#9303semidirect

Maximal subgroups

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

subgroupindex
#214338952
#214331192
#214241072
#214226612
#214224982
#214219562
C59×#13012
#930359