Finite Group Databasebeta

← #28481433

Subgroups — #28481433

Order 15104.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 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
5C59×#1797C211yesmaximal, 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#16837137—24no—
20C22×C42×C59C2211yescharacteristic
21C2×C4⋊C8×C59—21no—
22C2×C22⋊C8×C59C2212yes—
23C59×#336—21no—
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C23×Dic59—22no—
27C22×C4×D59—21no—
28#14273701—21no—
29#14273701—21no—
30#14273701D412yes—
31#14273701Q812yes—
32C4×D236—42no—
33C4×C22⋊D59—44no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×C4⋊Dic59C2×C411yescharacteristic
37C2×Dic59⋊C4—21no—
38C2×C22⋊Dic59—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×C4×Dic59—21no—
44C59⋊#95—24no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—22no—
49C59⋊#95—22no—
50C59⋊#95—42no—
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59C2311yescharacteristic
54C22×C8×C59—22no—
55C2×C42×C59—42no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59C2×C411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C4⋊C8×C59—42no—
61C22⋊C8×C59—22no—
62C22⋊C8×C59—22no—
63C24×C59C2×D411yescharacteristic
64C23×D59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59C22⋊C411yescharacteristic
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×C4×C59C2×Q811yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×C4×C59#4111yescharacteristic
83C2×C22⋊D59—22no—
84C2×C22⋊D59—22no—
85C2×C22⋊D59—42no—
86C2×C22⋊D59—42no—
87C22×Dic59—21no—
88C22×Dic59—24no—
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—42no—
92C22×Dic59—42no—
93C22×Dic59—42no—
94C22×Dic59—42no—
95C22×Dic59—42no—
96C22×Dic59—42no—
97C2×D236—21no—
98C2×D236—41no—
99C2×D236—41no—
100C2×D236C4⋊C412yes—
101C2×C4×D59—42no—
102C2×C4×D59—42no—
103C2×C4×D59—42no—
104C2×C8×C59—42no—
105C2×C8×C59—42no—
106C2×C8×C59—42no—
107C42×C59—22no—
108C42×C59—22no—
109C42×C59—41no—
110C42×C59—41no—
111C22⋊Dic59—42no—
112(C2×D59)⋊C4—22no—
113(C2×D59)⋊C4—22no—
114(C2×D59)⋊C4—42no—
115(C2×D59)⋊C4—42no—
116(C2×D59)⋊C4—42no—
117C4⋊Dic59—21no—
118C4⋊Dic59—41no—
119C4⋊Dic59—41no—
120C4⋊Dic59C4⋊C412yes—
121Dic59⋊C4—42no—
122C4×Dic59—42no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59#12111yescharacteristic
129C23×C59C22⋊Q811yescharacteristic
130C23×C59#13512yes—
131C23×C59AGL(1,ℤ/8)12yes—
132C22×D59—22no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
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—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—44no—
158C2×C4×C59—44no—
159C2×C4×C59#9912yes—
160C2×C4×C59#10412yes—
161C2×C4×C59C2×C4⋊C411yescharacteristic
162C2×C4×C59#11711yescharacteristic
163C2×C4×C59#12111yescharacteristic
164C2×C4×C59C22⋊Q811yescharacteristic
165C22⋊D59—44no—
166C22⋊D59—44no—
167C2×Dic59—22no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—44no—
171C2×Dic59—44no—
172C2×Dic59—44no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—82no—
180C2×Dic59—82no—
181C2×Dic59—82no—
182C2×Dic59—82no—
183D236—42no—
184D236—42no—
185C4×D59—84no—
186C8×C59—82no—
187#8575—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—22no—
193C22×C59—22no—
194C22×C59—22no—
195C22×C59—22no—
196C22×C59#38011yescharacteristic
197C22×C59C2×#9911yescharacteristic
198C22×C59#41011yescharacteristic
199C22×C59#41611yescharacteristic
200C22×C59#41811yescharacteristic
201C22×C59C2×#10411yescharacteristic
202C22×C59#42111yescharacteristic
203C22×C59#46312yes—
204C22×C59#47812yes—
205C2×D59—42no—
206C2×D59—42no—
207C2×D59—42no—
208C2×D59—81no—
209C2×D59—81no—
210C2×D59—81no—
211C2×D59—81no—
212C4×C59—21no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—22no—
217C4×C59—22no—
218C4×C59—41no—
219C4×C59—41no—
220Dic59—42no—
221Dic59—84no—
222Dic59—81no—
223Dic59—81no—
224C4×#121—591no—
225#1797D5911yescharacteristic, complemented
226C2×#327—592no—
227#1434—591no—
228#1294—592no—
229C2×C59—22no—
230C2×C59—22no—
231C2×C59#150111yescharacteristic
232C2×C59#150511yescharacteristic
233C2×C59#151011yescharacteristic
234C2×C59#152111yescharacteristic
235C2×C59#179911yescharacteristic
236C2×C59#180411yescharacteristic
237C2×C59#180611yescharacteristic
238D59—82no—
239C2×#121—591no—
240C2×C4×D4—591no—
241C2×C4×D4—1181no—
242C22×C42C2×D5911yescharacteristic
243C2×C4⋊C8—21no—
244C2×C22⋊C8C2×D5912yes—
245#390—591no—
246#389—591no—
247#388—1181no—
248#386—1181no—
249#380—591no—
250C4×C4⋊C4—591no—
251C4×C22⋊C4—1181no—
252C2×#95—592no—
253C2×#95—1181no—
254C2×#95—1181no—
255#336—21no—
256#327—1184no—
257C59#857511yescharacteristic, complemented, Sylow
258C22×D4—591no—
259C22×D4—1181no—
260C23×C4—1182no—
261C23×C4—1181no—
262C23×C4D23611yescharacteristic
263C23×C4D23611yescharacteristic
264C23×C4C22×D5911yescharacteristic
265C22×C8—22no—
266#121—592no—
267#121—592no—
268#121—1181no—
269#121—1181no—
270C4×D4—2362no—
271C4×D4—2364no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—591no—
274C2×C4⋊C4—591no—
275C2×C4⋊C4—1181no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4—1181no—
280C2×C22⋊C4—1181no—
281C2×C42—42no—
282C2×C42—1181no—
283C2×C42C4×D5911yescharacteristic
284C2×C42C4×D5911yescharacteristic
285C2×C42C22⋊D5911yescharacteristic
286C2×C42C22⋊D5911yescharacteristic
287C4⋊C8—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290#95—1184no—
291#95—1181no—
292#95—1181no—
293#95—1181no—
294#95—1182no—
295#95—1182no—
296#95—2362no—
297C24—1181no—
298C24C2×D23611yescharacteristic
299C2×D4—592no—
300C2×D4—1182no—
301C2×D4—1182no—
302C2×D4—1181no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2362no—
306C2×D4—2362no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—1181no—
316C22×C4—1184no—
317C22×C4—1181no—
318C22×C4—1181no—
319C22×C4—2362no—
320C22×C4—2362no—
321C22×C4—2362no—
322C22×C4—2362no—
323C22×C4—2362no—
324C22×C4—2362no—
325C22×C4—2362no—
326C22×C4—2362no—
327C22×C4—2362no—
328C22×C4(C2×D59)⋊C411yescharacteristic
329C22×C4(C2×D59)⋊C411yescharacteristic
330C22×C4(C2×D59)⋊C411yescharacteristic
331C22×C4(C2×D59)⋊C411yescharacteristic
332C22×C4C2×C4×D5911yescharacteristic
333C22×C4D4×D5911yescharacteristic
334C22×C4#1174568711yescharacteristic
335C22×C4Q8×D5911yescharacteristic
336C22×C4#1174568911yescharacteristic
337C22×C4C2×C22⋊D5911yescharacteristic
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—592no—
342C4⋊C4—1181no—
343C4⋊C4—2361no—
344C4⋊C4—2361no—
345C4⋊C4—2362no—
346C22⋊C4—1182no—
347C22⋊C4—1182no—
348C22⋊C4—2362no—
349C22⋊C4—2362no—
350C22⋊C4—2362no—
351C22⋊C4—2362no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—2362no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—1182no—
362C23—2361no—
363C23—2361no—
364C23—2361no—
365C23—2361no—
366C23—2361no—
367C23—2361no—
368C23C2×(C2×D59)⋊C411yescharacteristic
369C23#1427370211yescharacteristic
370C23#1427370711yescharacteristic
371C23#1427374912yes—
372C23#1427375512yes—
373D4—2362no—
374D4—2362no—
375D4—2364no—
376D4—2364no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—1182no—
398C2×C4—4724no—
399C2×C4—4722no—
400C2×C4—4722no—
401C2×C4—4722no—
402C2×C4—4722no—
403C2×C4—2364no—
404C2×C4—2364no—
405C2×C4—2364no—
406C2×C4—2361no—
407C2×C4—2361no—
408C2×C4—2361no—
409C2×C4—2361no—
410C2×C4—2361no—
411C2×C4—2361no—
412C2×C4—2361no—
413C2×C4—2361no—
414C2×C4#1427364112yes—
415C2×C4#1427365312yes—
416C2×C4C4⋊C4×D5911yescharacteristic
417C2×C4#1427368511yescharacteristic
418C2×C4#1427369811yescharacteristic
419C2×C4#1427370311yescharacteristic
420C8—82no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—4721no—
430C22—4721no—
431C22—4721no—
432C22—4721no—
433C22—2362no—
434C22—2362no—
435C22—2362no—
436C22#1683725911yescharacteristic
437C22C2×#1427364111yescharacteristic
438C22#1683737511yescharacteristic
439C22#1683739911yescharacteristic
440C22#1683740811yescharacteristic
441C22C2×#1427365311yescharacteristic
442C22#1683741811yescharacteristic
443C22#1683761812yes—
444C22#1683770312yes—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—4724no—
454C4—4721no—
455C4—4721no—
456C4—2362no—
457C2—22no—
458C2—22no—
459C2—4722no—
460C2#2142258811yescharacteristic
461C2#2142260511yescharacteristic
462C2#2142262611yescharacteristic
463C2#2142267411yescharacteristic
464C2#2142408111yescharacteristic
465C2#2142410711yescharacteristic
466C2#2142411311yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1797C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1797D59
hypocenterC59#8575
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #21422323C2semidirect
4. #21421976C2semidirect
5. C59×#1797C2semidirect
7. C2×C4×D236C4semidirect
9. #16837304C4semidirect
10. #16837300C4semidirect
14. C4×C4⋊Dic59C4semidirect
225. #1797D59semidirect
257. C59#8575semidirect

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
C59×#17972
#857559