Finite Group Databasebeta

← #28484299

Subgroups — #28484299

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432406C211yesmaximal, characteristic
2#21423858C211yesmaximal, characteristic
3C2×#16837143C212yesmaximal, complemented
4#21421978C212yesmaximal, complemented
5C2×C4×M16×C59C211yesmaximal, characteristic
6C2×#14273719C2211yescharacteristic
7C2×C22⋊(C59⋊C8)—21no—
8C2×#14273638C2212yes—
9#16837300C411yescharacteristic
10#16837300C411yescharacteristic
11#16837286—21no—
12#16837286—21no—
13#16837275—21no—
14#16837275—21no—
15#16837268C2211yescharacteristic
16#16837262C411yescharacteristic
17#16837262C411yescharacteristic
18#16837157—21no—
19#16837143—24no—
20C22×M16×C59—21no—
21C22×C42×C59C2211yescharacteristic
22C4×M16×C59—24no—
23C2×#97×C59C2212yes—
24C2×C4×C8×C59—21no—
25C22×C59⋊C8—22no—
26#14273719—22no—
27#14273719—22no—
28C2×C4⋊Dic59C2×C411yescharacteristic
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×Dic59⋊C4—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C22⋊(C59⋊C8)—24no—
39#14273638—22no—
40#14273638—22no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C2×M16×C59—24no—
51C2×M16×C59—24no—
52C2×M16×C59—24no—
53C22×C8×C59—22no—
54C2×C42×C59—22no—
55C2×C42×C59—22no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59C2311yescharacteristic
60#97×C59—22no—
61#97×C59D414yes—
62C4×C8×C59—24no—
63C24×C59C22⋊C411yescharacteristic
64C22×C4×C59—22no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84M16×C59—216no—
85C2×C8×C59—24no—
86C2×C8×C59—24no—
87C2×C8×C59—24no—
88C42×C59—22no—
89C42×C59—21no—
90C42×C59—22no—
91C42×C59—21no—
92C42×C59C2×D412yes—
93C42×C59#4112yes—
94C22⋊Dic59—42no—
95C22⋊Dic59—42no—
96C4⋊Dic59—22no—
97C4⋊Dic59—22no—
98C4⋊Dic59—41no—
99C4⋊Dic59—41no—
100Dic59⋊C4—42no—
101Dic59⋊C4—42no—
102Dic59⋊C4—41no—
103Dic59⋊C4—42no—
104Dic59⋊C4—41no—
105C2×C59⋊C8—24no—
106C2×C59⋊C8—42no—
107C2×C59⋊C8—42no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59#10412yes—
115C23×C59C2×C22⋊C411yescharacteristic
116C2×C4×C59—24no—
117C2×C4×C59—24no—
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—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—44no—
131C2×C4×C59—44no—
132C2×C4×C59#10112yes—
133C2×C4×C59#10412yes—
134C2×C4×C59#10412yes—
135C2×C4×C59C4×D412yes—
136C2×C4×C59#12011yescharacteristic
137C2×C4×C59#12112yes—
138C2×C4×C59#12311yescharacteristic
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C8×C59—28no—
152C59⋊C8—44no—
153#9122—591nomaximal, Sylow
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59#38611yescharacteristic
162C22×C59C2×#10111yescharacteristic
163C22×C59#41311yescharacteristic
164C22×C59C2×#10411yescharacteristic
165C22×C59C2×#10411yescharacteristic
166C22×C59C2×#10411yescharacteristic
167C22×C59#42111yescharacteristic
168C4×C59—22no—
169C4×C59—22no—
170C4×C59—21no—
171C4×C59—22no—
172C4×C59—21no—
173C4×C59—22no—
174C4×C59—22no—
175Dic59—81no—
176Dic59—81no—
177Dic59—81no—
178Dic59—81no—
179C2×C4×M16D5911yescharacteristic
180#2851—591no—
181#1749—591no—
182C2×#330—592no—
183#1295—592no—
184C2×C59—42no—
185C2×C59#152411yescharacteristic
186C2×C59#152611yescharacteristic
187C2×C59#152811yescharacteristic
188C2×C59#152911yescharacteristic
189C2×C59#187111yescharacteristic
190C2×C59#189611yescharacteristic
191C2×C59#190111yescharacteristic
192C22×M16—21no—
193C2×#125—591no—
194C22×C42C2×D5911yescharacteristic
195C2×C22⋊C8—1181no—
196C4×M16—24no—
197C2×#97—592no—
198C2×#97C2×D5912yes—
199C2×C4×C8—21no—
200#389—591no—
201#389—591no—
202#387—1181no—
203#387—1181no—
204#385—1181no—
205#385—1181no—
206#382—591no—
207#380—591no—
208#380—591no—
209#336—1181no—
210#330—1184no—
211C59#912211yescharacteristic, complemented, Sylow
212C23×C4C4×D5911yescharacteristic
213C23×C4D23611yescharacteristic
214C23×C4C22⋊D5911yescharacteristic
215C2×M16—24no—
216C2×M16—24no—
217C2×M16—24no—
218C22×C8—22no—
219C22×C8—1182no—
220#125—1182no—
221#125—1182no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—591no—
224C2×C4⋊C4—591no—
225C2×C4⋊C4—591no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4—1181no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C4×D5911yescharacteristic
235C2×C42D23611yescharacteristic
236C2×C42C22⋊D5911yescharacteristic
237C2×C42C22×D5911yescharacteristic
238C22⋊C8—1184no—
239#97—22no—
240#97—1182no—
241#97—1182no—
242#97C22⋊D5914yes—
243C4×C8—24no—
244#95—1181no—
245#95—1181no—
246#95—1181no—
247#95—1181no—
248#95—1181no—
249#95—1181no—
250C24(C2×D59)⋊C411yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4—1181no—
265C22×C4(C2×D59)⋊C411yescharacteristic
266C22×C4(C2×D59)⋊C411yescharacteristic
267C22×C4(C2×D59)⋊C411yescharacteristic
268C22×C4C2×C4×D5911yescharacteristic
269C22×C4C2×D23611yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271M16—216no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—1184no—
276C2×C8—2362no—
277C2×C8—2362no—
278C4⋊C4—1182no—
279C4⋊C4—1182no—
280C4⋊C4—2361no—
281C4⋊C4—2361no—
282C4⋊C4—2362no—
283C4⋊C4—2362no—
284C4⋊C4—2361no—
285C4⋊C4—2362no—
286C4⋊C4—2361no—
287C22⋊C4—2362no—
288C22⋊C4—2362no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42#1174568512yes—
294C42C2×C22⋊D5912yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#1427365512yes—
302C23C2×(C2×D59)⋊C411yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
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—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—44no—
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—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4#1427364512yes—
332C2×C4#1427365312yes—
333C2×C4#1427365512yes—
334C2×C4C4×C22⋊D5912yes—
335C2×C4#1427369611yescharacteristic
336C2×C4#1427370112yes—
337C2×C4#1427371011yescharacteristic
338C8—28no—
339C8—2364no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#1683728111yescharacteristic
348C22C2×#1427364511yescharacteristic
349C22#1683738411yescharacteristic
350C22C2×#1427365311yescharacteristic
351C22C2×#1427365511yescharacteristic
352C22C2×#1427365511yescharacteristic
353C22#1683741811yescharacteristic
354C4—22no—
355C4—22no—
356C4—21no—
357C4—22no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—4721no—
362C4—4721no—
363C4—4721no—
364C4—4721no—
365C2—42no—
366C2#2142268511yescharacteristic
367C2#2142269011yescharacteristic
368C2#2142269711yescharacteristic
369C2#2142270111yescharacteristic
370C2#2142439811yescharacteristic
371C2#2142453911yescharacteristic
372C2#2142456411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C4×M16×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenterC2×C4×M16D59
hypocenterC59#9122
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16837143C2semidirect
4. #21421978C2semidirect
211. C59#9122semidirect

Maximal subgroups

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

subgroupindex
#214324062
#214238582
C2×#168371432
#214219782
C2×C4×M16×C592
#912259