Finite Group Databasebeta

← #28504659

Subgroups — #28504659

Order 15104.

600 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 103 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#21426750C211yesmaximal, characteristic
2C2×#16837188C212yesmaximal
3#21424088C211yesmaximal, characteristic, complemented
4#21422213C212yesmaximal, complemented
5C59×#1799C211yesmaximal, characteristic, complemented
6C2×#14273695C2211yescharacteristic
7C22×(C2×D59)⋊C4C411yescharacteristic, complemented
8#16837426—21no—
9C2×#14273644C2212yes—
10#16837302—21no—
11#16837293—21no—
12#16837292—21no—
13#16837282C2211yescharacteristic
14#16837281—21no—
15#16837274C411yescharacteristic, complemented
16#16837257C411yescharacteristic, complemented
17C4×C22⋊Dic59C411yescharacteristic, complemented
18#16837188—24no—
19#16837171—21no—
20C2×C4×D4×C59C2211yescharacteristic
21C2×#99×C59C2212yes—
22C59×#380—21no—
23C59×#342—21no—
24C24×D59C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C23×Dic59C2×C411yescharacteristic
27C2×#11745664—22no—
28#14273695—21no—
29#14273695—21no—
30#14273695D412yes—
31#14273695Q812yes—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59C2×C411yescharacteristic
35C2×C22⋊Dic59C2×C411yescharacteristic
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—41no—
42C2×(C2×D59)⋊C4—41no—
43C2×C4×Dic59—21no—
44C4⋊(C59⋊C8)—41no—
45C4⋊(C59⋊C8)—41no—
46#14273644—22no—
47#14273644—22no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59C2×C411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C4×D4×C59—41no—
55C4×D4×C59—41no—
56C2×C4⋊C4×C59—41no—
57C2×C4⋊C4×C59D411yescharacteristic
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59C2×C411yescharacteristic
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C42×C59D411yescharacteristic
62#99×C59—22no—
63#99×C59—22no—
64#95×C59—41no—
65C24×C59C22×C411yescharacteristic
66C24×C59C2×D411yescharacteristic
67C23×D59—21no—
68C23×D59—21no—
69C23×D59—41no—
70C23×D59—41no—
71C23×D59—81no—
72C23×D59—81no—
73C23×D59C4⋊C412yescomplemented
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59C2×D411yescharacteristic
77C2×D4×C59C2×Q811yescharacteristic
78C2×D4×C59#4111yescharacteristic
79C2×D4×C59#4111yescharacteristic
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—42no—
85C22×C4×C59—81no—
86C22×C4×C59—81no—
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C22⋊C411yescharacteristic
90C22×C4×C59C22⋊C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C2×C22⋊D59—22no—
93C2×C22⋊D59—22no—
94C2×C22⋊D59—81no—
95C2×C22⋊D59—81no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—21no—
99C22×Dic59—41no—
100C22×Dic59—81no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—82no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C42×C59—21no—
108C42×C59—21no—
109C22⋊Dic59—21no—
110C22⋊Dic59—41no—
111C22⋊Dic59—81no—
112C22⋊Dic59C4⋊C412yescomplemented
113(C2×D59)⋊C4—22no—
114(C2×D59)⋊C4—22no—
115(C2×D59)⋊C4—42no—
116(C2×D59)⋊C4—41no—
117(C2×D59)⋊C4—41no—
118(C2×D59)⋊C4—81no—
119(C2×D59)⋊C4—81no—
120Dic59⋊C4—81no—
121C4×Dic59—81no—
122#11745664—42no—
123#11745664—42no—
124C2×C59⋊C8—42no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59C2×C22⋊C411yescharacteristic
134C23×C59C2×C4⋊C411yescharacteristic
135C23×C59#11711yescharacteristic
136C23×C59#12111yescharacteristic
137C23×C59C22⋊Q811yescharacteristic
138C22×D59—22no—
139C22×D59—41no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143C22×D59—42no—
144C22×D59—41no—
145C22×D59—41no—
146C22×D59—42no—
147C22×D59—82no—
148C22×D59—82no—
149C22×D59—84no—
150C22×D59—82no—
151C22×D59—82no—
152D4×C59—41no—
153D4×C59—41no—
154D4×C59—41no—
155D4×C59—41no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—41no—
162C2×C4×C59—42no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—81no—
167C2×C4×C59—81no—
168C2×C4×C59—81no—
169C2×C4×C59—82no—
170C2×C4×C59—82no—
171C2×C4×C59—82no—
172C2×C4×C59#9911yescharacteristic
173C2×C4×C59#9911yescharacteristic
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59C22⋊Q811yescharacteristic
176C22⋊D59—82no—
177C22⋊D59—82no—
178C2×Dic59—22no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—41no—
182C2×Dic59—41no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—82no—
188C59⋊C8—82no—
189#12795—591nomaximal, Sylow
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—42no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—41no—
205C22×C59—81no—
206C22×C59#38011yescharacteristic
207C22×C59C2×#9911yescharacteristic
208C22×C59#41011yescharacteristic
209C2×D59—42no—
210C2×D59—42no—
211C2×D59—42no—
212C2×D59—81no—
213C2×D59—81no—
214C2×D59—84no—
215C2×D59—81no—
216C2×D59—81no—
217C2×D59—84no—
218C4×C59—21no—
219C4×C59—21no—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—41no—
223C4×C59—41no—
224C4×C59—41no—
225C4×C59—162no—
226Dic59—42no—
227Dic59—81no—
228Dic59—81no—
229#2321—591no—
230C2×AΣL(1,𝔽16)—592no—
231#1800—591no—
232#1799D5911yescharacteristic, complemented
233#1390—592no—
234C2×C59—21no—
235C2×C59—21no—
236C2×C59—41no—
237C2×C59—41no—
238C2×C59—41no—
239C2×C59—41no—
240C2×C59#179911yescharacteristic
241C2×C59#212111yescharacteristic
242C2×C59#212211yescharacteristic
243D59—82no—
244C2×#120—591no—
245C2×C4×D4C2×D5911yescharacteristic
246C22×C22⋊C4—591no—
247#423—1181no—
248C2×#100—592no—
249C2×#99C2×D5912yes—
250#390—1181no—
251#388—1181no—
252#388—1181no—
253#386—591no—
254#386—1181no—
255#385—591no—
256#380—21no—
257#379—591no—
258C4×C22⋊C4—591no—
259AΣL(1,𝔽16)—1184no—
260#343—1181no—
261#342—21no—
262C59#1279511yescharacteristic, complemented, Sylow
263C25—591no—
264C22×D4—1181no—
265C22×D4C22×D5911yescharacteristic
266C23×C4—591no—
267C23×C4C4×D5911yescharacteristic
268C23×C4C22⋊D5911yescharacteristic
269C2×M16—1182no—
270#120—592no—
271#120—592no—
272#120—1181no—
273#120—1181no—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—1181no—
278C2×C4⋊C4D23611yescharacteristic
279C2×C22⋊C4—22no—
280C2×C22⋊C4—591no—
281C2×C22⋊C4—591no—
282C2×C22⋊C4—1181no—
283C2×C22⋊C4—1181no—
284C2×C22⋊C4—1181no—
285C2×C22⋊C4—1181no—
286C2×C22⋊C4—1181no—
287C2×C22⋊C4—1181no—
288C2×C22⋊C4—2361no—
289C2×C22⋊C4—2361no—
290C2×C22⋊C4C4×D5911yescharacteristic
291C2×C22⋊C4C22⋊D5911yescharacteristic
292C2×C42—1181no—
293C2×C42D23611yescharacteristic
294C4⋊C8—2361no—
295C4⋊C8—2361no—
296#100—1182no—
297#100—1182no—
298#99—22no—
299#99—22no—
300#95—41no—
301#95—1181no—
302#95—1181no—
303#95—1181no—
304C24—592no—
305C24—1181no—
306C24—1181no—
307C24—4721no—
308C24—4721no—
309C24—2361no—
310C24—2361no—
311C24C2×C4×D5911yescharacteristic
312C24C2×C22⋊D5911yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—1182no—
316C2×D4—1182no—
317C2×D4—4721no—
318C2×D4—4721no—
319C2×D4D4×D5911yescharacteristic
320C2×D4#1174568711yescharacteristic
321C2×D4Q8×D5911yescharacteristic
322C2×D4#1174568911yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—1181no—
331C22×C4—1181no—
332C22×C4—1181no—
333C22×C4—4721no—
334C22×C4—2361no—
335C22×C4(C2×D59)⋊C411yescharacteristic
336C22×C4(C2×D59)⋊C411yescharacteristic
337C22×C4(C2×D59)⋊C411yescharacteristic
338C22×C4(C2×D59)⋊C411yescharacteristic
339C22×C4C2×D23611yescharacteristic
340M16—2362no—
341M16—2362no—
342C2×C8—2362no—
343C4⋊C4—41no—
344C4⋊C4—82no—
345C4⋊C4—4721no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—592no—
351C22⋊C4—1181no—
352C22⋊C4—1182no—
353C22⋊C4—1182no—
354C22⋊C4—4721no—
355C22⋊C4—4721no—
356C22⋊C4—4721no—
357C22⋊C4—2362no—
358C22⋊C4—2361no—
359C22⋊C4—2361no—
360C22⋊C4—2361no—
361C42—21no—
362C42—21no—
363C42—4721no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—1182no—
373C23—4722no—
374C23—4722no—
375C23—4724no—
376C23—4722no—
377C23—4722no—
378C23—2362no—
379C23—2361no—
380C23—2361no—
381C23—2361no—
382C23—2361no—
383C23—2362no—
384C23—2361no—
385C23—2361no—
386C23C2×(C2×D59)⋊C411yescharacteristic
387C23C4⋊C4×D5911yescharacteristic
388C23#1427368511yescharacteristic
389C23#1427369811yescharacteristic
390C23#1427370311yescharacteristic
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—4722no—
396D4—4722no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—1182no—
414C2×C4—4722no—
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#1427364111yescharacteristic
424C2×C4#1427364111yescharacteristic
425C2×C4#1427370211yescharacteristic
426C2×C4#1427370711yescharacteristic
427C8—4722no—
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—4721no—
445C22—4721no—
446C22—4724no—
447C22—4724no—
448C22—4721no—
449C22—4721no—
450C22—2362no—
451C22—2362no—
452C22—2362no—
453C22#1683725911yescharacteristic
454C22C2×#1427364111yescharacteristic
455C22#1683737511yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—162no—
464C4—4721no—
465C4—4721no—
466C4—2362no—
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—4722no—
474C2#2142408111yescharacteristic
475C2#2142574011yescharacteristic
476C2#2142574511yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radicalC2×#14273695C22
cosocleC22×D4×C59C23
hypercenter#1799D59
hypocenterC59#12795
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21424088C2semidirect
4. #21422213C2semidirect
5. C59×#1799C2semidirect
7. C22×(C2×D59)⋊C4C4semidirect
15. #16837274C4semidirect
16. #16837257C4semidirect
17. C4×C22⋊Dic59C4semidirect
73. C23×D59C4⋊C4semidirect
112. C22⋊Dic59C4⋊C4semidirect
232. #1799D59semidirect
262. C59#12795semidirect

Maximal subgroups

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

subgroupindex
#214267502
C2×#168371882
#214240882
#214222132
C59×#17992
#1279559