Finite Group Databasebeta

← #28477027

Subgroups — #28477027

Order 15104.

518 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432435C211yesmaximal, characteristic
2C2×#16837157C211yesmaximal, characteristic
3#21421921C212yesmaximal, complemented
4#21421920C212yesmaximal, complemented
5C59×#1817C211yesmaximal, characteristic
6C23×C59⋊C8—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—
19C22×C42×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#336—21no—
22C59×#336C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×C59⋊C8—24no—
25C22×C59⋊C8—22no—
26C22×C59⋊C8—42no—
27C22×C59⋊C8—42no—
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—22no—
48C2×C42×C59—42no—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59C2311yescharacteristic
53C22⋊C8×C59—42no—
54C24×C59C22⋊C411yescharacteristic
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59SD1612yes—
67C22×C4×C59Q1612yes—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C2×Dic118—22no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×C8×C59—42no—
79C2×C8×C59—42no—
80C2×C8×C59—42no—
81C42×C59—41no—
82C42×C59—41no—
83C42×C59—41no—
84C42×C59—41no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C4⋊Dic59—22no—
88C4⋊Dic59—41no—
89C4⋊Dic59—41no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92Dic59⋊C4—42no—
93C2×C59⋊C8—24no—
94C2×C59⋊C8—48no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8—42no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59#9912yes—
102C23×C59#10312yes—
103C23×C59#10412yes—
104C23×C59C2×C22⋊C411yescharacteristic
105C2×C4×C59—24no—
106C2×C4×C59—24no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
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—44no—
122C2×C4×C59—44no—
123C2×C4×C59#10312yes—
124C2×C4×C59#13112yes—
125C2×C4×C59C2×SD1611yescharacteristic
126C2×C4×C59C2×Q1611yescharacteristic
127C2×C4×C59#13512yes—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140Dic118—82no—
141C8×C59—82no—
142C59⋊C8—44no—
143#7655—591nomaximal, Sylow
144C22×C59—22no—
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—22no—
148C22×C59—22no—
149C22×C59—22no—
150C22×C59#32814yes—
151C22×C59#40811yescharacteristic
152C22×C59C2×#9911yescharacteristic
153C22×C59#41311yescharacteristic
154C22×C59C2×#10311yescharacteristic
155C22×C59#41611yescharacteristic
156C22×C59C2×#10411yescharacteristic
157C22×C59#42111yescharacteristic
158C4×C59—22no—
159C4×C59—21no—
160C4×C59—22no—
161C4×C59—21no—
162C4×C59—22no—
163C4×C59—41no—
164C4×C59—41no—
165Dic59—81no—
166Dic59—81no—
167Dic59—81no—
168Dic59—81no—
169#2855—591no—
170#1817D5911yescharacteristic
171C2×#336—591no—
172#1272—592no—
173#1271—592no—
174C2×C59—24no—
175C2×C59#146811yescharacteristic
176C2×C59#148111yescharacteristic
177C2×C59#149411yescharacteristic
178C2×C59C2×#32811yescharacteristic
179C2×C59#151611yescharacteristic
180C2×C59#153211yescharacteristic
181C2×C59#154211yescharacteristic
182C23×C8—1181no—
183C2×C22⋊Q8—591no—
184C22×C42C2×D5911yescharacteristic
185C2×C22⋊C8—21no—
186#391—591no—
187#391—591no—
188#389—591no—
189#389—591no—
190#387—1181no—
191#387—1181no—
192#385—1181no—
193#381—1181no—
194#380—591no—
195#336—21no—
196#336—592no—
197#336—1181no—
198#336C2×D5912yes—
199C59#765511yescharacteristic, complemented, Sylow
200C22×Q8—591no—
201C23×C4C4×D5911yescharacteristic
202C23×C4D23611yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C22×C8—22no—
205C22×C8—1184no—
206C22×C8—1182no—
207C22×C8—2362no—
208C22×C8—2362no—
209C22⋊Q8—2362no—
210C2×C4⋊C4—591no—
211C2×C4⋊C4—591no—
212C2×C4⋊C4—591no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C42—42no—
219C2×C42C4×D5911yescharacteristic
220C2×C42D23611yescharacteristic
221C2×C42C22⋊D5911yescharacteristic
222C2×C42C22×D5911yescharacteristic
223C22⋊C8—42no—
224#95—1181no—
225#95—1181no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231C24(C2×D59)⋊C411yescharacteristic
232C2×Q8—1182no—
233C2×Q8—2361no—
234C2×Q8—2361no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4#1174566012yes—
248C22×C4Dic23612yes—
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×C4×D5911yescharacteristic
253C22×C4C2×D23611yescharacteristic
254C22×C4C2×C22⋊D5911yescharacteristic
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—1184no—
259C2×C8—2368no—
260C2×C8—2362no—
261C2×C8—2362no—
262C4⋊C4—1182no—
263C4⋊C4—2361no—
264C4⋊C4—2361no—
265C4⋊C4—2362no—
266C4⋊C4—2362no—
267C4⋊C4—2362no—
268C22⋊C4—2362no—
269C22⋊C4—2362no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1427364112yes—
279C23#1427365212yes—
280C23#1427365312yes—
281C23C2×(C2×D59)⋊C411yescharacteristic
282Q8—4722no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4#1427365212yes—
314C2×C4#1427373512yes—
315C2×C4C2×#1174566011yescharacteristic
316C2×C4C2×Dic23611yescharacteristic
317C2×C4#1427374712yes—
318C8—82no—
319C8—2364no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#1683714014yes—
327C22#1683737011yescharacteristic
328C22C2×#1427364111yescharacteristic
329C22#1683738411yescharacteristic
330C22C2×#1427365211yescharacteristic
331C22#1683739611yescharacteristic
332C22C2×#1427365311yescharacteristic
333C22#1683742011yescharacteristic
334C4—22no—
335C4—22no—
336C4—21no—
337C4—22no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—4721no—
342C4—4721no—
343C4—4721no—
344C4—4721no—
345C2—24no—
346C2#2142244511yescharacteristic
347C2#2142250011yescharacteristic
348C2#2142255411yescharacteristic
349C2C2×#1683714011yescharacteristic
350C2#2142265011yescharacteristic
351C2#2142271711yescharacteristic
352C2#2142275111yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1817D59
hypocenterC59#7655
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #21421921C2semidirect
4. #21421920C2semidirect
199. C59#7655semidirect

Maximal subgroups

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

subgroupindex
#214324352
C2×#168371572
#214219212
#214219202
C59×#18172
#765559