Finite Group Databasebeta

← #28504665

Subgroups — #28504665

Order 15104.

512 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426768C211yesmaximal, characteristic
2C2×#16837192C211yesmaximal, characteristic
3#21424086C211yesmaximal, characteristic
4#21424082C211yesmaximal, characteristic
5#21422215C211yesmaximal, characteristic
6#21422213C211yesmaximal, characteristic
7C2×C59×#352C211yesmaximal, characteristic
8C2×D4×Dic59C2211yescharacteristic
9C22×C4⋊Dic59C411yescharacteristic, complemented
10#16837425—21no—
11C2×#14273644C2211yescharacteristic
12C2×#14273642C2211yescharacteristic
13#16837301—21no—
14#16837295—21no—
15#16837288—21no—
16#16837282C2211yescharacteristic
17#16837277—21no—
18#16837277C411yescharacteristic, complemented
19#16837265C411yescharacteristic, complemented
20#16837260—21no—
21C22⋊C4×Dic59C411yescharacteristic, complemented
22#16837192—22no—
23#16837170—21no—
24#16837168—21no—
25C2×#123×C59C2211yescharacteristic
26C2×#100×C59C2211yescharacteristic
27C2×#99×C59C2211yescharacteristic
28C59×#352—22no—
29C23×Dic59C2×C411yescharacteristic
30C23×Dic59C2×C411yescharacteristic
31C23×Dic59D411yescharacteristic
32C2×#11745664—21no—
33D4×Dic59—41no—
34D4×Dic59—41no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59—41no—
39C2×C4⋊Dic59—41no—
40C2×C4⋊Dic59D411yescharacteristic
41C2×Dic59⋊C4—41no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59C2×C411yescharacteristic
45C2×C22⋊Dic59C2×C411yescharacteristic
46C2×C22⋊Dic59D411yescharacteristic
47C2×C4×Dic59—21no—
48C2×C4×Dic59D411yescharacteristic
49Dic59⋊C8—41no—
50Dic59⋊C8—41no—
51#14273644—21no—
52#14273644—21no—
53#14273642—21no—
54#14273642—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—21no—
59C59⋊#95—21no—
60C59⋊#95—41no—
61C22×D4×C59C2311yescharacteristic
62C23×C4×C59C2×C411yescharacteristic
63C2×M16×C59—21no—
64#123×C59—21no—
65#123×C59—21no—
66#123×C59D412yes—
67#123×C59Q812yes—
68C2×C4⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59C2×C411yescharacteristic
72#100×C59—21no—
73#100×C59—21no—
74#99×C59—21no—
75#99×C59—21no—
76C24×C59C22×C411yescharacteristic
77C24×C59C2×D411yescharacteristic
78C2×D4×C59—41no—
79C2×D4×C59—41no—
80C2×D4×C59C2×D411yescharacteristic
81C2×D4×C59C2×Q811yescharacteristic
82C2×D4×C59#4111yescharacteristic
83C2×D4×C59#4111yescharacteristic
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C4⋊C412yescomplemented
91C22×C4×C59C2×D411yescharacteristic
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99C22×Dic59—41no—
100C22×Dic59—81no—
101C22×Dic59—81no—
102C22×Dic59—81no—
103C22×Dic59C22⋊C411yescharacteristic
104C22×Dic59C22⋊C411yescharacteristic
105C22×Dic59C22⋊C411yescharacteristic
106C22×Dic59C22⋊C411yescharacteristic
107M16×C59—41no—
108M16×C59—41no—
109C2×C8×C59—41no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—22no—
112C22⋊C4×C59—21no—
113C22⋊C4×C59—22no—
114C22⋊C4×C59—22no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59C4⋊C412yescomplemented
118C22⋊Dic59—41no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121C22⋊Dic59—41no—
122C22⋊Dic59—41no—
123C22⋊Dic59—81no—
124C4⋊Dic59—41no—
125C4⋊Dic59—42no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128C4⋊Dic59—81no—
129Dic59⋊C4—82no—
130C4×Dic59—21no—
131C4×Dic59—21no—
132C4×Dic59—81no—
133#11745664—41no—
134#11745664—41no—
135C2×C59⋊C8—41no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59C2×C22⋊C411yescharacteristic
145C23×C59C2×C4⋊C411yescharacteristic
146C23×C59#11711yescharacteristic
147C23×C59#12111yescharacteristic
148C23×C59C22⋊Q811yescharacteristic
149D4×C59—41no—
150D4×C59—41no—
151D4×C59—41no—
152D4×C59—41no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—22no—
156C2×C4×C59—22no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—42no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—42no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59#12111yescharacteristic
167C2×C4×C59C22⋊Q811yescharacteristic
168C2×Dic59—21no—
169C2×Dic59—21no—
170C2×Dic59—21no—
171C2×Dic59—41no—
172C2×Dic59—42no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—41no—
182C2×Dic59—81no—
183C2×Dic59—81no—
184C2×Dic59—81no—
185C2×Dic59—82no—
186C2×Dic59—81no—
187C2×Dic59—81no—
188C2×Dic59—81no—
189C2×Dic59#9911yescharacteristic
190C2×Dic59#9911yescharacteristic
191C8×C59—81no—
192C59⋊C8—81no—
193#12796—591nomaximal, Sylow
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—21no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—42no—
205C22×C59—41no—
206C22×C59—41no—
207C22×C59—41no—
208C22×C59—41no—
209C22×C59—81no—
210C22×C59#38011yescharacteristic
211C22×C59C2×#9911yescharacteristic
212C22×C59#41011yescharacteristic
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—42no—
216C4×C59—42no—
217C4×C59—161no—
218Dic59—21no—
219Dic59—21no—
220Dic59—41no—
221Dic59—41no—
222Dic59—41no—
223Dic59—81no—
224Dic59—81no—
225Dic59—161no—
226#2322—591no—
227C2×#352—591no—
228C2×#352D5911yescharacteristic
229#1800—591no—
230#1799—591no—
231#1390—591no—
232#1390—591no—
233C2×C59—21no—
234C2×C59—21no—
235C2×C59—41no—
236C2×C59—41no—
237C2×C59—41no—
238C2×C59—41no—
239C2×C59#179911yescharacteristic
240C2×C59#212111yescharacteristic
241C2×C59#212311yescharacteristic
242C2×#123C2×D5911yescharacteristic
243C2×C4×D4—591no—
244C22×C4⋊C4—591no—
245#423—1181no—
246C2×#100—591no—
247C2×#100C2×D5911yescharacteristic
248C2×#99—591no—
249C2×#99C2×D5911yescharacteristic
250#389—1181no—
251#388—1181no—
252#387—1181no—
253#386—591no—
254#385—591no—
255#385—1181no—
256#381—591no—
257#380—1181no—
258C4×C22⋊C4—591no—
259#352—22no—
260#352—1182no—
261#343—1181no—
262#342—1181no—
263C59#1279611yescharacteristic, complemented, Sylow
264C22×D4C22×D5911yescharacteristic
265C23×C4—591no—
266C23×C4—591no—
267C23×C4—591no—
268C23×C4C4×D5911yescharacteristic
269C2×M16—21no—
270C2×M16—1181no—
271#123—21no—
272#123—21no—
273#123Dic11812yes—
274#123C22⋊D5912yes—
275C4×D4—2361no—
276C4×D4—2361no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—591no—
279C2×C4⋊C4—1181no—
280C2×C4⋊C4—1181no—
281C2×C4⋊C4—1181no—
282C2×C4⋊C4—2361no—
283C2×C4⋊C4—2361no—
284C2×C4⋊C4—2361no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—591no—
288C2×C22⋊C4—591no—
289C2×C22⋊C4—591no—
290C2×C22⋊C4—1181no—
291C2×C22⋊C4—1181no—
292C2×C22⋊C4C4×D5911yescharacteristic
293C2×C42—591no—
294C2×C42—1181no—
295C4⋊C8—2361no—
296C4⋊C8—2361no—
297#100—21no—
298#100—21no—
299#100—1181no—
300#100—1181no—
301#99—21no—
302#99—21no—
303#99—1181no—
304#99—1181no—
305#95—1181no—
306#95—1181no—
307#95—1181no—
308#95—1181no—
309#95—1181no—
310#95—2361no—
311C24C2×C4×D5911yescharacteristic
312C24D4×D5911yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4C2×Dic11811yescharacteristic
316C2×D4#1174568711yescharacteristic
317C2×D4#1174568711yescharacteristic
318C2×D4C2×C22⋊D5911yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—591no—
326C22×C4—591no—
327C22×C4—591no—
328C22×C4—591no—
329C22×C4—1181no—
330C22×C4—1181no—
331C22×C4—1181no—
332C22×C4—1181no—
333C22×C4—1181no—
334C22×C4—1181no—
335C22×C4—4721no—
336C22×C4—4721no—
337C22×C4—4721no—
338C22×C4—2361no—
339C22×C4—2361no—
340C22×C4Dic59⋊C412yescomplemented
341C22×C4D4×D5911yescharacteristic
342M16—41no—
343M16—41no—
344M16—2361no—
345M16—2361no—
346C2×C8—41no—
347C2×C8—2361no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—4721no—
351C4⋊C4—4722no—
352C4⋊C4—2361no—
353C4⋊C4—2361no—
354C4⋊C4—2361no—
355C4⋊C4—2362no—
356C22⋊C4—21no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—4721no—
362C22⋊C4—2361no—
363C22⋊C4—2361no—
364C22⋊C4—2361no—
365C22⋊C4—2361no—
366C22⋊C4—2361no—
367C22⋊C4Dic59⋊C412yescomplemented
368C42—1181no—
369C42—1181no—
370C42—4721no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C22⋊C4×D5911yescharacteristic
380C23C2×Dic59⋊C411yescharacteristic
381C23#1427368311yescharacteristic
382C23#1427369911yescharacteristic
383C23#1427370411yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—591no—
402C2×C4—591no—
403C2×C4—1181no—
404C2×C4—1181no—
405C2×C4—1181no—
406C2×C4—4721no—
407C2×C4—4721no—
408C2×C4—4721no—
409C2×C4—4721no—
410C2×C4—4721no—
411C2×C4—4721no—
412C2×C4—4722no—
413C2×C4—2361no—
414C2×C4—2362no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4—2361no—
418C2×C4—2361no—
419C2×C4—2361no—
420C2×C4—2361no—
421C2×C4—2361no—
422C2×C4—2361no—
423C2×C4—2361no—
424C2×C4#1427369911yescharacteristic
425C2×C4#1427370411yescharacteristic
426C8—81no—
427C8—4721no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22#1683726011yescharacteristic
445C22#99×D5911yescharacteristic
446C22#1683737411yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—161no—
452C4—1181no—
453C4—1181no—
454C4—9441no—
455C4—4721no—
456C4—4721no—
457C4—2361no—
458C4—2361no—
459C4—2361no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—41no—
464C2—41no—
465C2—41no—
466C2#2142408211yescharacteristic
467C2#2142574111yescharacteristic
468C2#2142574811yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#352C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC2×#352D59
hypocenterC59#12796
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#352

Complements

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

NH ≅ G/Ntype
9. C22×C4⋊Dic59C4semidirect
18. #16837277C4semidirect
19. #16837265C4semidirect
21. C22⋊C4×Dic59C4semidirect
90. C22×C4×C59C4⋊C4semidirect
117. C22⋊C4×C59C4⋊C4semidirect
263. C59#12796semidirect
340. C22×C4Dic59⋊C4semidirect
367. C22⋊C4Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
#214267682
C2×#168371922
#214240862
#214240822
#214222152
#214222132
C2×C59×#3522
#1279659