Finite Group Databasebeta

← #28477000

Subgroups — #28477000

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273706C211yesmaximal, characteristic
2#21424076C211yesmaximal, characteristic
3#21421921C212yesmaximal, complemented
4#21421920C212yesmaximal, complemented
5C59×#1817C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic
7C2×C4×Dic118C411yescharacteristic
8C2×C4⋊(C59⋊C8)—21no—
9C2×C22⋊(C59⋊C8)—21no—
10#16837308C2211yescharacteristic
11#16837300C2211yescharacteristic
12#16837286—21no—
13#16837275—21no—
14#16837262C411yescharacteristic
15C4×Dic59⋊C4—21no—
16C4×C4⋊Dic59C411yescharacteristic
17C4×C22⋊Dic59—21no—
18#16837157C2212yes—
19C22×C42×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#336—21no—
22C59×#336C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×C59⋊C8—22no—
25#14273706—22no—
26#14273706—22no—
27C4×Dic118—42no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59—21no—
38C4⋊(C59⋊C8)—42no—
39C22⋊(C59⋊C8)—42no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C22×C8×C59—22no—
49C2×C42×C59—42no—
50C2×C42×C59C2×C411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59C2311yescharacteristic
54C22⋊C8×C59—42no—
55C24×C59C22⋊C411yescharacteristic
56C22×C4×C59—22no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—41no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59SD1612yes—
69C22×C4×C59Q1612yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×Dic118—21no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118M1612yes—
81C2×C8×C59—42no—
82C2×C8×C59—42no—
83C2×C8×C59—42no—
84C42×C59—22no—
85C42×C59—22no—
86C42×C59—41no—
87C42×C59—41no—
88C22⋊Dic59—24no—
89C22⋊Dic59—42no—
90C4⋊Dic59—21no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59M1612yes—
94Dic59⋊C4—24no—
95Dic59⋊C4—42no—
96Dic59⋊C4—42no—
97C4×Dic59—42no—
98C4×Dic59—42no—
99C2×C59⋊C8—42no—
100C2×C59⋊C8—42no—
101C2×C59⋊C8—42no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59#10312yes—
109C23×C59C2×C22⋊C411yescharacteristic
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—21no—
122C2×C4×C59—21no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—44no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59#9912yes—
129C2×C4×C59#10312yes—
130C2×C4×C59C2×M1611yescharacteristic
131C2×C4×C59C2×M1611yescharacteristic
132C2×C4×C59C2×SD1611yescharacteristic
133C2×C4×C59C2×Q1611yescharacteristic
134C2×C4×C59#13512yes—
135C2×Dic59—22no—
136C2×Dic59—22no—
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—42no—
148Dic118—42no—
149C8×C59—82no—
150C59⋊C8—82no—
151#7648—591nomaximal, Sylow
152C22×C59—22no—
153C22×C59—22no—
154C22×C59—22no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59#40711yescharacteristic
160C22×C59C2×#9911yescharacteristic
161C22×C59#41311yescharacteristic
162C22×C59C2×#10311yescharacteristic
163C22×C59#41611yescharacteristic
164C22×C59#42111yescharacteristic
165C22×C59#42111yescharacteristic
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—22no—
169C4×C59—22no—
170C4×C59—21no—
171C4×C59—22no—
172C4×C59—21no—
173C4×C59—41no—
174Dic59—42no—
175Dic59—42no—
176Dic59—81no—
177Dic59—81no—
178C4×C22⋊Q8—591no—
179#1817D5911yescharacteristic
180#1797—591no—
181#1272—592no—
182#1271—592no—
183C2×C59—42no—
184C2×C59#146511yescharacteristic
185C2×C59#148711yescharacteristic
186C2×C59#148811yescharacteristic
187C2×C59#151611yescharacteristic
188C2×C59#152011yescharacteristic
189C2×C59#153511yescharacteristic
190C2×C59#154211yescharacteristic
191C2×C22⋊Q8—591no—
192C2×C4×Q8—591no—
193C22×C42C2×D5911yescharacteristic
194C2×C4⋊C8—1181no—
195C2×C22⋊C8—21no—
196C2×C22⋊C8—1181no—
197#391—591no—
198#389—591no—
199#387—1181no—
200#385—1181no—
201#380—591no—
202C4×C4⋊C4—591no—
203C4×C4⋊C4—1181no—
204C4×C22⋊C4—1181no—
205#336—21no—
206#336—592no—
207#336C2×D5912yes—
208C59#764811yescharacteristic, complemented, Sylow
209C22×Q8—591no—
210C23×C4C4×D5911yescharacteristic
211C23×C4D23611yescharacteristic
212C23×C4C22⋊D5911yescharacteristic
213C22×C8—22no—
214C22×C8—1182no—
215C22⋊Q8—1182no—
216C22⋊Q8—1182no—
217C4×Q8—2362no—
218C2×C4⋊C4—591no—
219C2×C4⋊C4—591no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4—1181no—
226C2×C42—42no—
227C2×C42—1181no—
228C2×C42—1181no—
229C2×C42C4×D5911yescharacteristic
230C2×C42D23611yescharacteristic
231C2×C42C22⋊D5911yescharacteristic
232C2×C42C22×D5911yescharacteristic
233C4⋊C8—2362no—
234C22⋊C8—42no—
235C22⋊C8—2362no—
236#95—1181no—
237#95—1181no—
238#95—1181no—
239#95—1181no—
240#95—1181no—
241C24(C2×D59)⋊C411yescharacteristic
242C2×Q8—592no—
243C2×Q8—1181no—
244C2×Q8—2361no—
245C2×Q8—2361no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
258C22×C4—1181no—
259C22×C4#1174566012yes—
260C22×C4Dic23612yes—
261C22×C4(C2×D59)⋊C411yescharacteristic
262C22×C4(C2×D59)⋊C411yescharacteristic
263C22×C4(C2×D59)⋊C411yescharacteristic
264C22×C4C2×C4×D5911yescharacteristic
265C22×C4C2×D23611yescharacteristic
266C22×C4C2×C22⋊D5911yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—2362no—
271C2×C8—2362no—
272C2×C8—2362no—
273C4⋊C4—592no—
274C4⋊C4—1184no—
275C4⋊C4—1181no—
276C4⋊C4—2361no—
277C4⋊C4—2361no—
278C4⋊C4—2362no—
279C4⋊C4—2362no—
280C22⋊C4—1184no—
281C22⋊C4—2362no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—2362no—
287C42—2362no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1427365212yes—
295C23C2×(C2×D59)⋊C411yescharacteristic
296Q8—2362no—
297Q8—2362no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—1182no—
317C2×C4—1182no—
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—2361no—
327C2×C4—2361no—
328C2×C4#1427364112yes—
329C2×C4#1427365212yes—
330C2×C4M16×D5911yescharacteristic
331C2×C4M16×D5911yescharacteristic
332C2×C4C2×#1174566011yescharacteristic
333C2×C4C2×Dic23611yescharacteristic
334C2×C4#1427374712yes—
335C8—82no—
336C8—4722no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1683736511yescharacteristic
345C22C2×#1427364111yescharacteristic
346C22#1683738411yescharacteristic
347C22C2×#1427365211yescharacteristic
348C22#1683739611yescharacteristic
349C22#1683741811yescharacteristic
350C22#1683742011yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—4721no—
360C4—4721no—
361C4—2362no—
362C4—2362no—
363C2—42no—
364C2#2142243211yescharacteristic
365C2#2142252611yescharacteristic
366C2#2142252811yescharacteristic
367C2#2142265011yescharacteristic
368C2#2142266911yescharacteristic
369C2#2142272811yescharacteristic
370C2#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#7648
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
208. C59#7648semidirect

Maximal subgroups

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

subgroupindex
C4×#142737062
#214240762
#214219212
#214219202
C59×#18172
#764859