Finite Group Databasebeta

← #28479596

Subgroups — #28479596

Order 15104.

724 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273701C211yesmaximal, characteristic
2C2×#16837137C212yesmaximal
3#21422323C211yesmaximal, characteristic, complemented
4#21421976C212yesmaximal, complemented
5C4×C22⋊C8×C59C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7C2×C4×D236C411yescharacteristic, complemented
8C2×C4×C22⋊D59—21no—
9#16837304C411yescharacteristic, complemented
10#16837300C411yescharacteristic, complemented
11#16837293—21no—
12#16837281—21no—
13#16837262C2211yescharacteristic
14C4×C4⋊Dic59C411yescharacteristic, complemented
15C4×(C2×D59)⋊C4—21no—
16C2×C59⋊#95—21no—
17C2×C59⋊#95—21no—
18C2×C59⋊#95C2212yes—
19#16837137C418yescomplemented
20C22×C42×C59C2211yescharacteristic
21C2×C22⋊C8×C59C2212yes—
22C2×C4×C8×C59—21no—
23C59×#336—21no—
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C23×Dic59—22no—
27C22×C4×D59—21no—
28#14273701—22no—
29#14273701C2×C414yes—
30C4×D236—42no—
31C4×C22⋊D59—42no—
32C4×C22⋊D59—42no—
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×C22⋊Dic59—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×C4×Dic59—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—22no—
47C59⋊#95—42no—
48C59⋊#95—42no—
49C59⋊#95C2×C414yescomplemented
50C23×C4×C59D411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C22×C8×C59—22no—
54C2×C42×C59—42no—
55C2×C42×C59C2×C411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C22⋊C8×C59—22no—
60C22⋊C8×C59C2×C414yescomplemented
61C4×C8×C59—42no—
62C24×C59C2×D411yescharacteristic
63C23×D59—21no—
64C22×C4×C59—22no—
65C22×C4×C59—22no—
66C22×C4×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59D812yes—
76C22×C4×C59SD1612yes—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C22×C412yes—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4112yes—
81C2×C22⋊D59—24no—
82C2×C22⋊D59—41no—
83C2×C22⋊D59—41no—
84C2×C22⋊D59—42no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—24no—
89C22×Dic59—42no—
90C22×Dic59—42no—
91C22×Dic59—42no—
92C22×Dic59—42no—
93C22×Dic59—42no—
94C22×Dic59—42no—
95C2×D236—21no—
96C2×D236—41no—
97C2×D236—41no—
98C2×D236C4212yes—
99C2×C4×D59—42no—
100C2×C4×D59—42no—
101C2×C4×D59—42no—
102C2×C8×C59—24no—
103C2×C8×C59—42no—
104C2×C8×C59—42no—
105C42×C59—22no—
106C42×C59—22no—
107C42×C59—41no—
108C42×C59—41no—
109C22⋊Dic59—41no—
110C22⋊Dic59—41no—
111(C2×D59)⋊C4—24no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—42no—
115C4⋊Dic59—21no—
116C4⋊Dic59—41no—
117C4⋊Dic59—41no—
118C4⋊Dic59C4212yes—
119Dic59⋊C4—42no—
120C4×Dic59—42no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59C2×C22⋊C411yescharacteristic
126C23×C59C4×D412yes—
127C23×C59C2×D811yescharacteristic
128C23×C59C2×SD1611yescharacteristic
129C23×C59#13512yes—
130C22×D59—22no—
131C22×D59—41no—
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—42no—
151C2×C4×C59—42no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59—44no—
155C2×C4×C59#9912yes—
156C2×C4×C59#10212yes—
157C2×C4×C59#10212yes—
158C2×C4×C59C2×C4211yescharacteristic
159C2×C4×C59#11711yescharacteristic
160C2×C4×C59C4×D412yes—
161C22⋊D59—44no—
162C22⋊D59—44no—
163C2×Dic59—22no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—44no—
167C2×Dic59—44no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—44no—
175C2×Dic59—82no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179D236—42no—
180D236—42no—
181C4×D59—84no—
182C8×C59—44no—
183#8220—591nomaximal, Sylow
184C22×C59—22no—
185C22×C59—22no—
186C22×C59—22no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—22no—
192C22×C59C4×C22⋊C411yescharacteristic
193C22×C59C2×#9911yescharacteristic
194C22×C59#41011yescharacteristic
195C22×C59C2×#10211yescharacteristic
196C22×C59#41611yescharacteristic
197C22×C59#42111yescharacteristic
198C22×C59#42111yescharacteristic
199C22×C59C4×D812yes—
200C22×C59C4×SD1612yes—
201C2×D59—42no—
202C2×D59—42no—
203C2×D59—42no—
204C2×D59—81no—
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C4×C59—22no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—22no—
212C4×C59—22no—
213C4×C59—22no—
214C4×C59—41no—
215Dic59—42no—
216Dic59—84no—
217Dic59—81no—
218Dic59—81no—
219C4×#121—591no—
220C4×C22⋊C8D5911yescharacteristic, complemented
221C2×#327—592no—
222#1434—591no—
223#1294—592no—
224C2×C59—22no—
225C2×C59—22no—
226C2×C59#151611yescharacteristic
227C2×C59#151711yescharacteristic
228C2×C59#151911yescharacteristic
229C2×C59#152111yescharacteristic
230C2×C59C4×#9911yescharacteristic
231C2×C59#175911yescharacteristic
232C2×C59C4×#10211yescharacteristic
233D59—82no—
234C2×#121—591no—
235C2×C4×D4—591no—
236C2×C4×D4—1181no—
237C22×C42C2×D5911yescharacteristic
238C2×C22⋊C8C2×D5912yes—
239C2×C4×C8—21no—
240#390—591no—
241#389—591no—
242#388—1181no—
243#386—1181no—
244#380—591no—
245C4×C4⋊C4—591no—
246C4×C22⋊C4—1181no—
247C2×#95—592no—
248C2×#95—1181no—
249C2×#95—1181no—
250#336—21no—
251#327—598no—
252C59#822011yescharacteristic, complemented, Sylow
253C22×D4—591no—
254C22×D4—1181no—
255C23×C4—1182no—
256C23×C4—1181no—
257C23×C4D23611yescharacteristic
258C23×C4D23611yescharacteristic
259C23×C4C22×D5911yescharacteristic
260C22×C8—22no—
261#121—594no—
262#121—1182no—
263C4×D4—2362no—
264C4×D4—2362no—
265C4×D4—2362no—
266C2×C4⋊C4—591no—
267C2×C4⋊C4—591no—
268C2×C4⋊C4—591no—
269C2×C4⋊C4—1181no—
270C2×C22⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C42—42no—
276C2×C42—1181no—
277C2×C42C4×D5911yescharacteristic
278C2×C42C4×D5911yescharacteristic
279C2×C42C22⋊D5911yescharacteristic
280C2×C42C22⋊D5911yescharacteristic
281C22⋊C8—22no—
282C22⋊C8C4×D5914yescomplemented
283C4×C8—42no—
284#95—594no—
285#95—1181no—
286#95—1181no—
287#95—1182no—
288#95—1181no—
289#95—2362no—
290#95—2362no—
291C24—1181no—
292C24C2×D23611yescharacteristic
293C2×D4—592no—
294C2×D4—1184no—
295C2×D4—1181no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4—2361no—
300C2×D4—2362no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1181no—
309C22×C4—1181no—
310C22×C4—1181no—
311C22×C4—1184no—
312C22×C4—2362no—
313C22×C4—2362no—
314C22×C4—2362no—
315C22×C4—2362no—
316C22×C4—2362no—
317C22×C4—2362no—
318C22×C4—2362no—
319C22×C4—2362no—
320C22×C4—2362no—
321C22×C4#1174566012yes—
322C22×C4D47212yes—
323C22×C4(C2×D59)⋊C411yescharacteristic
324C22×C4(C2×D59)⋊C411yescharacteristic
325C22×C4(C2×D59)⋊C411yescharacteristic
326C22×C4(C2×D59)⋊C411yescharacteristic
327C22×C4C2×C4×D5911yescharacteristic
328C22×C4C2×C4×D5912yes—
329C22×C4#1174568512yes—
330C22×C4C2×C22⋊D5911yescharacteristic
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—592no—
335C4⋊C4—1181no—
336C4⋊C4—2361no—
337C4⋊C4—2361no—
338C4⋊C4—2362no—
339C22⋊C4—1184no—
340C22⋊C4—2362no—
341C22⋊C4—2361no—
342C22⋊C4—2361no—
343C22⋊C4—2362no—
344C22⋊C4—2362no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42—41no—
349C42—2362no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—1182no—
355C23—2361no—
356C23—2361no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23C2×(C2×D59)⋊C411yescharacteristic
362C23C4×D23612yes—
363C23C2×D47211yescharacteristic
364C23C2×#1174566011yescharacteristic
365C23#1427374912yes—
366D4—2362no—
367D4—2362no—
368D4—2364no—
369D4—2364no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—1182no—
389C2×C4—4724no—
390C2×C4—4722no—
391C2×C4—4722no—
392C2×C4—4722no—
393C2×C4—4722no—
394C2×C4—2364no—
395C2×C4—2364no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2364no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4#1427364112yes—
406C2×C4#1427364912yes—
407C2×C4#1427364912yes—
408C2×C4C42×D5911yescharacteristic
409C2×C4#1427368411yescharacteristic
410C2×C4C4×C22⋊D5912yes—
411C8—44no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—4721no—
421C22—4721no—
422C22—4721no—
423C22—4721no—
424C22—2362no—
425C22—2362no—
426C22—2362no—
427C22C4×(C2×D59)⋊C411yescharacteristic
428C22C2×#1427364111yescharacteristic
429C22#1683737511yescharacteristic
430C22C2×#1427364911yescharacteristic
431C22#1683739911yescharacteristic
432C22#1683741811yescharacteristic
433C22#1683741811yescharacteristic
434C22C4×D47212yes—
435C22C4×#1174566012yes—
436C4—22no—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—4724no—
444C4—4721no—
445C4—4721no—
446C4—2362no—
447C2—22no—
448C2—22no—
449C2—4722no—
450C2#2142265311yescharacteristic
451C2#2142265711yescharacteristic
452C2#2142266511yescharacteristic
453C2#2142267411yescharacteristic
454C2C4×#1427364111yescharacteristic
455C2#2142390911yescharacteristic
456C2C4×#1427364911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC2×C4×C59C2×C42
FrattiniC23×C4C22×D59
FittingC4×C22⋊C8×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenterC4×C22⋊C8D59
hypocenterC59#8220
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #21422323C2semidirect
4. #21421976C2semidirect
5. C4×C22⋊C8×C59C2semidirect
7. C2×C4×D236C4semidirect
9. #16837304C4semidirect
10. #16837300C4semidirect
14. C4×C4⋊Dic59C4semidirect
19. #16837137C4semidirect
49. C59⋊#95C2×C4semidirect
60. C22⋊C8×C59C2×C4semidirect
220. C4×C22⋊C8D59semidirect
252. C59#8220semidirect
282. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C4×#142737012
C2×#168371372
#214223232
#214219762
C4×C22⋊C8×C592
#822059