Finite Group Databasebeta

← #28477144

Subgroups — #28477144

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432435C211yesmaximal, characteristic
2#21423790C211yesmaximal, characteristic
3#21421921C212yesmaximal, complemented
4#21421920C212yesmaximal, complemented
5C2×C59×#336C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273706C411yescharacteristic
8#16837308C411yescharacteristic
9#16837308C2211yescharacteristic
10#16837300C411yescharacteristic
11#16837300C2211yescharacteristic
12#16837286—21no—
13#16837286—21no—
14#16837275—21no—
15#16837265—21no—
16#16837262C411yescharacteristic
17#16837157—21no—
18#16837157C2212yes—
19C23×C8×C59—21no—
20C22×C42×C59C2211yescharacteristic
21C59×#336—21no—
22C59×#336C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C2×#11745664—24no—
25C2×#11745664—42no—
26C2×#11745664—42no—
27C22×C59⋊C8—22no—
28#14273706—42no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C22×C8×C59—24no—
48C22×C8×C59—22no—
49C22×C8×C59—42no—
50C22×C8×C59—42no—
51C2×C42×C59—42no—
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59C2311yescharacteristic
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—41no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59SD1612yes—
70C22×C4×C59Q1612yes—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C2×Dic118—22no—
79C2×Dic118—41no—
80C2×Dic118—41no—
81C2×C8×C59—24no—
82C2×C8×C59—48no—
83C2×C8×C59—42no—
84C2×C8×C59—42no—
85C42×C59—41no—
86C42×C59—41no—
87C42×C59—41no—
88C42×C59—41no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C4⋊Dic59—22no—
92C4⋊Dic59—41no—
93C4⋊Dic59—41no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96Dic59⋊C4—41no—
97Dic59⋊C4—41no—
98#11745664—48no—
99C2×C59⋊C8—24no—
100C2×C59⋊C8—42no—
101C2×C59⋊C8—42no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59#9912yes—
107C23×C59#10312yes—
108C23×C59#10412yes—
109C23×C59C2×C22⋊C411yescharacteristic
110C2×C4×C59—24no—
111C2×C4×C59—24no—
112C2×C4×C59—24no—
113C2×C4×C59—24no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—44no—
128C2×C4×C59#10312yes—
129C2×C4×C59#10412yes—
130C2×C4×C59#13112yes—
131C2×C4×C59C2×SD1611yescharacteristic
132C2×C4×C59C2×Q1611yescharacteristic
133C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C59#13711yescharacteristic
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147Dic118—82no—
148C8×C59—44no—
149C59⋊C8—44no—
150#7681—591nomaximal, Sylow
151C22×C59—22no—
152C22×C59—22no—
153C22×C59—22no—
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59#32814yes—
158C22×C59#40811yescharacteristic
159C22×C59C2×#9911yescharacteristic
160C22×C59#41311yescharacteristic
161C22×C59C2×#10311yescharacteristic
162C22×C59#41711yescharacteristic
163C22×C59C2×#10411yescharacteristic
164C22×C59C2×#10411yescharacteristic
165C4×C59—22no—
166C4×C59—21no—
167C4×C59—22no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—22no—
171C4×C59—21no—
172C4×C59—41no—
173Dic59—81no—
174Dic59—81no—
175Dic59—81no—
176Dic59—81no—
177#2855—591no—
178#1728—591no—
179C2×#336D5911yescharacteristic
180#1272—592no—
181#1271—592no—
182C2×C59—24no—
183C2×C59#146811yescharacteristic
184C2×C59#148311yescharacteristic
185C2×C59#149311yescharacteristic
186C2×C59C2×#32811yescharacteristic
187C2×C59#150411yescharacteristic
188C2×C59#152411yescharacteristic
189C2×C59#155011yescharacteristic
190C22×M16—1181no—
191C23×C8—21no—
192C2×C22⋊Q8—591no—
193C22×C42C2×D5911yescharacteristic
194#391—591no—
195#391—591no—
196#389—591no—
197#389—591no—
198#387—1181no—
199#387—1181no—
200#385—1181no—
201#381—1181no—
202#380—591no—
203#336—21no—
204#336—592no—
205#336—1181no—
206#336C2×D5912yes—
207C59#768111yescharacteristic, complemented, Sylow
208C22×Q8—591no—
209C23×C4C4×D5911yescharacteristic
210C23×C4D23611yescharacteristic
211C23×C4C22⋊D5911yescharacteristic
212C2×M16—1184no—
213C2×M16—2362no—
214C2×M16—2362no—
215C22×C8—24no—
216C22×C8—22no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—1182no—
220C22⋊Q8—2362no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—591no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4—1181no—
227C2×C22⋊C4—1181no—
228C2×C22⋊C4—1181no—
229C2×C42—42no—
230C2×C42C4×D5911yescharacteristic
231C2×C42D23611yescharacteristic
232C2×C42C22⋊D5911yescharacteristic
233C2×C42C22×D5911yescharacteristic
234#95—1181no—
235#95—1181no—
236#95—1181no—
237#95—1181no—
238#95—1181no—
239#95—1181no—
240#95—1181no—
241C24(C2×D59)⋊C411yescharacteristic
242C2×Q8—1182no—
243C2×Q8—2361no—
244C2×Q8—2361no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
258C22×C4#1174566012yes—
259C22×C4Dic23612yes—
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4(C2×D59)⋊C411yescharacteristic
262C22×C4(C2×D59)⋊C411yescharacteristic
263C22×C4C2×C4×D5911yescharacteristic
264C22×C4C2×D23611yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266M16—2368no—
267C2×C8—24no—
268C2×C8—48no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—1184no—
272C2×C8—2362no—
273C2×C8—2362no—
274C4⋊C4—1182no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2362no—
278C4⋊C4—2362no—
279C4⋊C4—2361no—
280C4⋊C4—2361no—
281C22⋊C4—2362no—
282C22⋊C4—2362no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1427364112yes—
292C23#1427365212yes—
293C23#1427365312yes—
294C23C2×(C2×D59)⋊C411yescharacteristic
295Q8—4722no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4#1427365212yes—
327C2×C4#1427365512yes—
328C2×C4#1427373312yes—
329C2×C4C2×#1174566011yescharacteristic
330C2×C4C2×Dic23611yescharacteristic
331C2×C4#1427375311yescharacteristic
332C2×C4#1427375811yescharacteristic
333C8—44no—
334C8—2364no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#1683714014yes—
342C22#1683736811yescharacteristic
343C22C2×#1427364111yescharacteristic
344C22#1683738411yescharacteristic
345C22C2×#1427365211yescharacteristic
346C22#1683740111yescharacteristic
347C22C2×#1427365311yescharacteristic
348C22C2×#1427365511yescharacteristic
349C4—21no—
350C4—22no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—41no—
357C4—4721no—
358C4—4721no—
359C4—4721no—
360C4—4721no—
361C2—24no—
362C2#2142244211yescharacteristic
363C2#2142250711yescharacteristic
364C2#2142255211yescharacteristic
365C2C2×#1683714011yescharacteristic
366C2#2142259811yescharacteristic
367C2#2142268211yescharacteristic
368C2#2142278411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#336C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenterC2×#336D59
hypocenterC59#7681
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
3. #21421921C2semidirect
4. #21421920C2semidirect
207. C59#7681semidirect

Maximal subgroups

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

subgroupindex
#214324352
#214237902
#214219212
#214219202
C2×C59×#3362
#768159