Finite Group Databasebeta

← #28485395

Subgroups — #28485395

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435240C211yesmaximal, characteristic
2#21425077C211yesmaximal, characteristic
3#21422768C211yesmaximal, characteristic
4#21422659C211yesmaximal, characteristic
5#21421954C211yesmaximal, characteristic
6#21421925C211yesmaximal, characteristic
7C59×#2327C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837798C412yescomplemented
10#16837754C412yescomplemented
11#16837732—22no—
12#16837684—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16#16837162—21no—
17#16837162—21no—
18#16837148—22no—
19#16837138—22no—
20C2×C4⋊Q8×C59C2211yescharacteristic
21C2×C4×Q8×C59—21no—
22C59×#391—21no—
23C59×#387—22no—
24C59×#387—22no—
25C59×#382C2211yescharacteristic
26C59×#376C2211yescharacteristic
27C2×#11745664—21no—
28#14273727C2×C411yescharacteristic
29#14273727C2×C411yescharacteristic
30#14273719—21no—
31#14273706—21no—
32#14273706—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35#14273686—21no—
36#14273658—24no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41#14273650—24no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C4×C59⋊C8—21no—
45C4×C59⋊C8C2×C412yes—
46C22×Q8×C59—21no—
47C4⋊Q8×C59—21no—
48C4⋊Q8×C59—21no—
49C4⋊Q8×C59C2×C412yes—
50C4⋊Q8×C59D412yes—
51C4×Q8×C59—42no—
52C4×Q8×C59—42no—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—22no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59D412yes—
57C2×C42×C59—22no—
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61#95×C59—24no—
62#95×C59—21no—
63#95×C59—21no—
64#95×C59—21no—
65#95×C59D412yes—
66C2×Q8×C59—22no—
67C2×Q8×C59—22no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C22×C4×C59—21no—
73C22×C4×C59—22no—
74C22×C4×C59—21no—
75C22×C4×C59—22no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C2×Dic118—21no—
80C2×Dic118—21no—
81C4⋊C4×C59—22no—
82C4⋊C4×C59—22no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—44no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59C22⋊C414yes—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—44no—
92C42×C59SD1612yes—
93C42×C59C22×C411yescharacteristic
94C42×C59C2×D411yescharacteristic
95C42×C59#4111yescharacteristic
96C42×C59#4111yescharacteristic
97C22⋊Dic59—41no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102Dic59⋊C4—41no—
103Dic59⋊C4—41no—
104C4×Dic59—41no—
105#11745664—41no—
106#11745664—41no—
107C2×C59⋊C8—22no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C23×C59#12011yescharacteristic
111Q8×C59—42no—
112Q8×C59—42no—
113Q8×C59—42no—
114Q8×C59—42no—
115C2×C4×C59—24no—
116C2×C4×C59—22no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59C4×D411yescharacteristic
131C2×C4×C59C4×D411yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12311yescharacteristic
135C2×C4×C59C2×SD1611yescharacteristic
136C2×C4×C59#13511yescharacteristic
137C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C59#13711yescharacteristic
139C2×Dic59—41no—
140C2×Dic59—41no—
141Dic118—42no—
142Dic118—81no—
143C59⋊C8—42no—
144C59⋊C8—81no—
145C59⋊C8—81no—
146#9296—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59#38611yescharacteristic
150C22×C59#45011yescharacteristic
151C22×C59#45211yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—42no—
157C4×C59—44no—
158C4×C59—82no—
159C4×C59—81no—
160C4×C59C4×SD1611yescharacteristic
161C4×C59#44011yescharacteristic
162C4×C59#46111yescharacteristic
163C4×C59#46311yescharacteristic
164Dic59—81no—
165Dic59—81no—
166#3229—591no—
167#2327D5911yescharacteristic
168#1995—591no—
169#1546—591no—
170#1518—591no—
171#1286—591no—
172#1274—591no—
173C2×C59—21no—
174C2×C59—21no—
175C2×C59#187711yescharacteristic
176C2×C59#190511yescharacteristic
177C2×C59#191211yescharacteristic
178#533—591no—
179C2×C4⋊Q8C2×D5911yescharacteristic
180C2×C4×Q8—21no—
181#498—592no—
182#487—592no—
183#483—1182no—
184#475—1182no—
185#431—591no—
186#431—591no—
187#423—591no—
188#391—21no—
189#387—22no—
190#387—22no—
191#382C2×D5911yescharacteristic
192#376C2×D5911yescharacteristic
193#339—1181no—
194#339—1181no—
195#332—1182no—
196#328—1182no—
197C59#929611yescharacteristic, complemented, Sylow
198C22×Q8—21no—
199C2×M16—1181no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—591no—
203C4⋊Q8—591no—
204C4⋊Q8C4×D5912yes—
205C4⋊Q8D23612yes—
206#125—1181no—
207C22⋊Q8—1181no—
208C22⋊Q8—1181no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—1181no—
212C4×Q8—1181no—
213#117—1181no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22⋊D5912yes—
218C2×C42—22no—
219C2×C42C22⋊D5911yescharacteristic
220C2×C42C22⋊D5911yescharacteristic
221C2×C42C22×D5911yescharacteristic
222#106—1184no—
223C4⋊C8—1181no—
224C4⋊C8—1181no—
225C4⋊C8—1181no—
226C4⋊C8—1181no—
227#103—1184no—
228C22⋊C8—1181no—
229C22⋊C8—1181no—
230C4×C8—592no—
231C4×C8—1181no—
232#95—24no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C22⋊D5912yes—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—1181no—
244C2×Q8—1181no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4C2×C22⋊D5911yescharacteristic
250C22×C4C2×C22⋊D5911yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252M16—2361no—
253M16—2361no—
254C2×C8—1182no—
255C2×C8—2361no—
256C2×C8—2361no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—44no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1181no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—1181no—
267C4⋊C4—2361no—
268C4⋊C4—2361no—
269C4⋊C4(C2×D59)⋊C414yes—
270C22⋊C4—2361no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—44no—
275C42—2361no—
276C42#1174567012yes—
277C42C2×C4×D5911yescharacteristic
278C42C2×D23611yescharacteristic
279C42#1174568511yescharacteristic
280C42#1174568511yescharacteristic
281C23#1427369611yescharacteristic
282Q8—42no—
283Q8—42no—
284Q8—42no—
285Q8—42no—
286Q8—4721no—
287Q8—2362no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4C2×(C2×D59)⋊C411yescharacteristic
305C2×C4C4×C22⋊D5911yescharacteristic
306C2×C4C4×C22⋊D5911yescharacteristic
307C2×C4#1427370111yescharacteristic
308C2×C4#1427370111yescharacteristic
309C2×C4#1427371011yescharacteristic
310C2×C4C2×#1174567011yescharacteristic
311C2×C4#1427374711yescharacteristic
312C2×C4#1427375311yescharacteristic
313C2×C4#1427375711yescharacteristic
314C8—4721no—
315C8—4721no—
316C8—2362no—
317C22—21no—
318C22—21no—
319C22#1683728111yescharacteristic
320C22#1683755311yescharacteristic
321C22#1683756411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—44no—
328C4—82no—
329C4—81no—
330C4—4721no—
331C4—4721no—
332C4C4×#1174567011yescharacteristic
333C4#1683750211yescharacteristic
334C4#1683760711yescharacteristic
335C4#1683761711yescharacteristic
336C2—21no—
337C2—21no—
338C2#2142443011yescharacteristic
339C2#2142459011yescharacteristic
340C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2327C2
socleC22×C59#386
prime radicalC23×C59#120
cosocleC2×C42×C59C23
hypercenter#2327D59
hypocenterC59#9296
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
9. #16837798C4semidirect
10. #16837754C4semidirect
197. C59#9296semidirect

Maximal subgroups

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

subgroupindex
#214352402
#214250772
#214227682
#214226592
#214219542
#214219252
C59×#23272
#929659