Finite Group Databasebeta

← #28485621

Subgroups — #28485621

Order 15104.

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

normal subgroups only

This group has 444 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#21426747C211yesmaximal, characteristic
2#21425041C211yesmaximal, characteristic
3#21422608C211yesmaximal, characteristic
4#21422543C211yesmaximal, characteristic
5#21421988C211yesmaximal, characteristic
6#21421953C211yesmaximal, characteristic
7C59×#3113C211yesmaximal, characteristic
8#16837459C2211yescharacteristic
9C4×#11745664C2211yescharacteristic
10#16837309—21no—
11#16837301—21no—
12#16837288—22no—
13#16837270—21no—
14#16837268—21no—
15C4×C4⋊Dic59C2211yescharacteristic
16#16837248C2211yescharacteristic
17#16837161—22no—
18#16837139—21no—
19#16837139—21no—
20Q8⋊(C59⋊C8)—22no—
21C59×#533C2211yescharacteristic
22C2×C4×Q8×C59C2211yescharacteristic
23C59×#483—21no—
24C59×#475C412yescomplemented
25C59×#462C412yescomplemented
26C59×#451—21no—
27C59×#441—22no—
28C59×#432C2211yescharacteristic
29C2×#11745664—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—22no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59D412yes—
34C4⋊(C59⋊C8)—21no—
35C22⋊(C59⋊C8)—21no—
36#14273638—21no—
37#14273638—21no—
38C4×C59⋊C8—21no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95D411yescharacteristic
45C59⋊#95D411yescharacteristic
46C22×Q8×C59D411yescharacteristic
47C2×Q16×C59—22no—
48C4⋊Q8×C59C2×C411yescharacteristic
49C4⋊Q8×C59C2×C411yescharacteristic
50#125×C59—21no—
51C22⋊Q8×C59—21no—
52C22⋊Q8×C59—21no—
53C4×Q8×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59C2×C412yes—
58C4×Q8×C59D412yes—
59#117×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59D411yescharacteristic
62C2×C42×C59—21no—
63C2×C42×C59C2311yescharacteristic
64#106×C59—22no—
65C4⋊C8×C59C2×C412yes—
66#103×C59—22no—
67#103×C59—22no—
68C22⋊C8×C59—21no—
69#97×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59—42no—
74C2×Q8×C59C22⋊C412yes—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×Dic59—21no—
81C22×Dic59—22no—
82C22×Dic59—21no—
83Q16×C59—44no—
84C2×C8×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59C22⋊C412yes—
95C22⋊C4×C59—41no—
96C42×C59—22no—
97C42×C59—22no—
98C42×C59—41no—
99C42×C59C22×C411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59#4111yescharacteristic
102C42×C59#4111yescharacteristic
103C4⋊Dic59—42no—
104C4⋊Dic59—82no—
105C4×Dic59—42no—
106C4×Dic59—42no—
107#11745664—22no—
108#11745664—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C23×C59#12011yescharacteristic
112Q8×C59—22no—
113Q8×C59—22no—
114Q8×C59—42no—
115Q8×C59—42no—
116Q8×C59—81no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59C2×C22⋊C411yescharacteristic
129C2×C4×C59C4×D411yescharacteristic
130C2×C4×C59C4×D411yescharacteristic
131C2×C4×C59#12111yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12311yescharacteristic
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—42no—
137C2×Dic59—41no—
138C2×Dic59—42no—
139C2×Dic59—41no—
140C2×Dic59—42no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C8×C59—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—81no—
161C4×C59#42112yes—
162Dic59—81no—
163Dic59—82no—
164Dic59—81no—
165#3113D5911yescharacteristic
166#2320—591no—
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#533C2×D5911yescharacteristic
178C2×C4×Q8C2×D5911yescharacteristic
179#483—21no—
180#475Dic5912yescomplemented
181#462Dic5912yescomplemented
182#451—21no—
183#441—22no—
184#432—591no—
185#432C2×D5911yescharacteristic
186C4×M16—591no—
187#391—1181no—
188#389—1181no—
189#387—1182no—
190#383—1181no—
191#382—1181no—
192C4×C4⋊C4—591no—
193#376—591no—
194#339—1182no—
195#328—1181no—
196#328—1181no—
197Q8⋊C8—1182no—
198C59#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D5911yescharacteristic
200C2×Q16—22no—
201C2×M16—1181no—
202C4⋊Q8C2×Dic5911yescharacteristic
203C4⋊Q8C2×Dic5911yescharacteristic
204#125—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8C2×Dic5912yes—
212C4×Q8C22⋊D5912yes—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1182no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4C22⋊D5911yescharacteristic
218C2×C42—21no—
219C2×C42—592no—
220C2×C42—1181no—
221C2×C42C22×D5911yescharacteristic
222#106—22no—
223C4⋊C8—1181no—
224C4⋊C8C2×Dic5912yes—
225#103—22no—
226#103—22no—
227C22⋊C8—21no—
228C22⋊C8—1181no—
229#97—21no—
230#97—1181no—
231#97—1181no—
232C4×C8—1181no—
233#95—591no—
234#95—591no—
235#95—1182no—
236#95—1182no—
237#95—1182no—
238#95—1181no—
239#95—1181no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic5912yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1182no—
250C22×C4D4×D5911yescharacteristic
251C22×C4D4×D5911yescharacteristic
252C22×C4C2×C22⋊D5911yescharacteristic
253Q16—44no—
254M16—1182no—
255M16—2361no—
256C2×C8—22no—
257C2×C8—2361no—
258C2×C8—2361no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—4722no—
269C4⋊C4—2362no—
270C4⋊C4C22⋊Dic5912yes—
271C22⋊C4—41no—
272C42—22no—
273C42—22no—
274C42—41no—
275C42—2362no—
276C42—2362no—
277C42#1174568711yescharacteristic
278C42#1174568711yescharacteristic
279C42C22×Dic5911yescharacteristic
280C42C2×C22⋊D5911yescharacteristic
281C23#1427369511yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—42no—
286Q8—81no—
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—2361no—
299C2×C4—2361no—
300C2×C4—2362no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2362no—
304C2×C4—2362no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4C2×C22⋊Dic5911yescharacteristic
308C2×C4D4×Dic5911yescharacteristic
309C2×C4D4×Dic5911yescharacteristic
310C2×C4#1427369911yescharacteristic
311C2×C4#1427369911yescharacteristic
312C2×C4#1427370811yescharacteristic
313C8—42no—
314C8—4721no—
315C8—2362no—
316C22—21no—
317C22—21no—
318C22#1683728211yescharacteristic
319C22#1683757811yescharacteristic
320C22#1683758311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—4721no—
331C4—4722no—
332C4—4721no—
333C4#1683741912yes—
334C2—21no—
335C2—21no—
336C2#2142446711yescharacteristic
337C2#2142454511yescharacteristic
338C2#2142459211yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3113C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#3113D59
hypocenterC59#9328
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
24. C59×#475C4semidirect
25. C59×#462C4semidirect
180. #475Dic59semidirect
181. #462Dic59semidirect
198. C59#9328semidirect

Maximal subgroups

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

subgroupindex
#214267472
#214250412
#214226082
#214225432
#214219882
#214219532
C59×#31132
#932859