Finite Group Databasebeta

← #28745870

Subgroups — #28745870

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436499C211yesmaximal, characteristic
2#21426341C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426296C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426138C211yesmaximal, characteristic, complemented
9#21426013C211yesmaximal, characteristic, complemented
10#21426013C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425816C211yesmaximal, characteristic, complemented
14C2×#16837206C211yesmaximal, characteristic, complemented
15D4×Q16×C59C211yesmaximal, characteristic, complemented
16#16838098—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837727—21no—
24#16837721—21no—
25#16837703—21no—
26#16837638C2211yescharacteristic
27#16837638C2211yescharacteristic
28C4×D472C2211yescharacteristic
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837206C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837206C2211yescharacteristic
43#16837206C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C22×Q16×C59C2211yescharacteristic
49C22×Q16×C59C2211yescharacteristic
50D4×Q8×C59—21no—
51C59×#494C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#467C2211yescharacteristic
54C59×#462—21no—
55C59×#451—21no—
56C59×#451—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×D472C2311yescharacteristic
60#14273722—41no—
61#14273719—21no—
62#14273709—41no—
63#14273701—21no—
64#14273701—21no—
65#14273699—41no—
66C4×D236—21no—
67(C22⋊D59)⋊C4—41no—
68#14273686—21no—
69C2×C4⋊Dic59—21no—
70#14273670—22no—
71#14273670—41no—
72#14273670—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273659—21no—
84#14273659—21no—
85#14273649—21no—
86#14273649—21no—
87#14273649—21no—
88C22×Q8×C59—21no—
89C22×Q8×C59—21no—
90C2×Q16×C59—22no—
91C2×Q16×C59—22no—
92C2×Q16×C59—21no—
93C2×Q16×C59—21no—
94C2×Q16×C59—21no—
95C2×Q16×C59—41no—
96C2×Q16×C59—41no—
97C2×Q16×C59C2311yescharacteristic
98C2×Q16×C59C2311yescharacteristic
99C2×Q16×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C4⋊Q8×C59—21no—
103C4⋊Q8×C59—41no—
104C22⋊Q8×C59—21no—
105C22⋊Q8×C59—21no—
106C22⋊Q8×C59—41no—
107C22⋊Q8×C59—41no—
108C4×Q8×C59—21no—
109C4×D4×C59—41no—
110C4×D4×C59D411yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#103×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×Q8×C59—21no—
120C2×Q8×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—42no—
123C2×Q8×C59—42no—
124C2×Q8×C59—41no—
125C2×Q8×C59—41no—
126C2×Q8×C59—81no—
127C2×Q8×C59—81no—
128C2×D4×C59D811yescharacteristic
129C22×C4×C59—41no—
130C22×C4×C59—41no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C2×C22⋊D59—41no—
134C2×C22⋊D59—41no—
135C22×Dic59—41no—
136C2×D236—21no—
137C2×C4×D59—41no—
138Q16×C59—21no—
139Q16×C59—21no—
140Q16×C59—21no—
141Q16×C59—21no—
142Q16×C59—21no—
143Q16×C59—42no—
144Q16×C59—42no—
145Q16×C59—81no—
146Q16×C59#4112yes—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2411yescharacteristic
155C4⋊C4×C59—21no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59—41no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59—41no—
161C4⋊C4×C59D811yescharacteristic
162C22⋊C4×C59—41no—
163C22⋊C4×C59—41no—
164C22⋊C4×C59D811yescharacteristic
165C22⋊C4×C59D811yescharacteristic
166C42×C59—41no—
167C42×C59C2×D411yescharacteristic
168C22⋊Dic59—41no—
169(C2×D59)⋊C4—41no—
170(C2×D59)⋊C4—41no—
171(C2×D59)⋊C4—41no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—41no—
176Dic59⋊C4—41no—
177Dic59⋊C4—41no—
178C4×Dic59—41no—
179D472—21no—
180D472—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188Q8×C59—22no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—41no—
193Q8×C59—41no—
194Q8×C59—82no—
195Q8×C59—82no—
196D4×C59—22no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—82no—
206C2×C4×C59—82no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C22×D411yescharacteristic
212C22⋊D59—82no—
213C22⋊D59—161no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—161no—
218D236—41no—
219D236—41no—
220C4×D59—161no—
221C8×C59—21no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59C2×#4111yescharacteristic
225C8×C59H(5,𝔽2)11yescharacteristic
226#33887—591nomaximal, Sylow
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59#51012yes—
233C22×C59C22×D811yescharacteristic
234C2×D59—81no—
235C2×D59—81no—
236C2×D59—81no—
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—21no—
240C4×C59—21no—
241C4×C59—42no—
242C4×C59—81no—
243C4×C59—81no—
244C4×C59—81no—
245C4×C59#54611yescharacteristic
246C4×C59#57711yescharacteristic
247Dic59—81no—
248Dic59—81no—
249Dic59—81no—
250#3339—591no—
251D4×Q16D5911yescharacteristic, complemented
252#2245—591no—
253#2237—591no—
254#2236—591no—
255#2229—591no—
256#2220—591no—
257#2217—591no—
258#2208—591no—
259#2185—591no—
260#2185—591no—
261#2175—591no—
262#2168—591no—
263#2141—591no—
264C2×#358—591no—
265C2×C59—22no—
266C2×C59—41no—
267C2×C59#329411yescharacteristic
268C2×C59C2×#51011yescharacteristic
269C2×C59#341611yescharacteristic
270D59—161no—
271C22×Q16C2×D5911yescharacteristic
272C22×Q16C2×D5911yescharacteristic
273#553—1181no—
274D4×Q8—21no—
275C2×SD32—591no—
276C2×SD32—591no—
277C2×SD32—591no—
278C2×M32—591no—
279C22×C16—591no—
280#499—591no—
281#494C2×D5911yescharacteristic
282#482—1181no—
283#481—1181no—
284#478—1181no—
285#467C2×D5911yescharacteristic
286#467C2×D5911yescharacteristic
287#466—591no—
288#466—591no—
289#462—21no—
290#451—21no—
291#451—21no—
292C4×Q16C2×D5911yescharacteristic
293C4×D8—591no—
294C8×D4C2×D5911yescharacteristic
295#427—591no—
296C2×#107—591no—
297#367—591no—
298#366—591no—
299#366—591no—
300C4⋊C16—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#358—591no—
310#357—591no—
311C22⋊C16—591no—
312C22⋊C16—591no—
313#346—591no—
314C59#3388711yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8—21no—
317C2×Q16—22no—
318C2×Q16—22no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×Q16—21no—
322C2×Q16—41no—
323C2×Q16—41no—
324C2×Q16C22×D5911yescharacteristic
325C2×Q16C22×D5911yescharacteristic
326C2×Q16C22×D5911yescharacteristic
327C2×D8—591no—
328C22×C8C22×D5911yescharacteristic
329C22×C8C22×D5911yescharacteristic
330C4⋊Q8—21no—
331C4⋊Q8—41no—
332#126—2361no—
333#125—1181no—
334#123—2361no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—41no—
338C22⋊Q8—41no—
339#121—1181no—
340#121—1181no—
341#121—2361no—
342C4×Q8—21no—
343C4×D4—41no—
344C4×D4—1181no—
345C4×D4—2361no—
346C4×D4C22⋊D5911yescharacteristic
347#117—1181no—
348C2×C4⋊C4—1181no—
349SD32—1182no—
350SD32—2361no—
351SD32—2361no—
352M32—2361no—
353C2×C16—591no—
354C2×C16—591no—
355C2×C16—591no—
356C2×C16—591no—
357C2×C16—2361no—
358#107—591no—
359#107—591no—
360#107—591no—
361#107—1181no—
362#107C22×D5911yescharacteristic
363#106—1181no—
364#106—1181no—
365C4⋊C8C22⋊D5911yescharacteristic
366#103—21no—
367#103—21no—
368#103—21no—
369#102—1181no—
370#102—1181no—
371#102—1181no—
372C22⋊C8C22⋊D5911yescharacteristic
373C22⋊C8C22⋊D5911yescharacteristic
374C4×C8C22×D5911yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×D4—1181no—
385C2×D4—2361no—
386C2×D4—2361no—
387C2×D4#1174566911yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2361no—
391C22×C4—2361no—
392C22×C4C2×C22⋊D5911yescharacteristic
393C22×C4C2×C22⋊D5911yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—21no—
397Q16—21no—
398Q16—21no—
399Q16—42no—
400Q16—42no—
401Q16—81no—
402Q16#1174568712yes—
403D8—1181no—
404D8—1181no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D5911yescharacteristic
410C2×C8C2×C22⋊D5911yescharacteristic
411C2×C8C2×C22⋊D5911yescharacteristic
412C2×C8C23×D5911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—1181no—
420C4⋊C4—1181no—
421C4⋊C4—1181no—
422C4⋊C4—2361no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4#1174566911yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—2361no—
429C22⋊C4—2361no—
430C22⋊C4—2361no—
431C22⋊C4—2361no—
432C22⋊C4#1174566911yescharacteristic
433C22⋊C4#1174566911yescharacteristic
434C42—41no—
435C42—2361no—
436C42C2×C22⋊D5911yescharacteristic
437C16—1181no—
438C16—1181no—
439C16—1181no—
440C16—1181no—
441C23—2361no—
442C23C2×#1174566911yescharacteristic
443C23C2×#1174566911yescharacteristic
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—82no—
451Q8—82no—
452D4—22no—
453D4—9441no—
454D4—4722no—
455D4—2361no—
456D4—2361no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—9441no—
468C2×C4—9441no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4C2×#1174566911yescharacteristic
473C2×C4C2×#1174566911yescharacteristic
474C2×C4C2×#1174566911yescharacteristic
475C2×C4C2×#1174566911yescharacteristic
476C2×C4C22×C22⋊D5911yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8C2×#1174568711yescharacteristic
481C8#1427377211yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—4721no—
488C22—4721no—
489C22—4721no—
490C22#1683784312yes—
491C22C22×#1174566911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—4721no—
501C4—4721no—
502C4—4721no—
503C4#1683804611yescharacteristic
504C4#1683821311yescharacteristic
505C2—22no—
506C2—41no—
507C2—9441no—
508C2#2143590911yescharacteristic
509C2C2×#1683784311yescharacteristic
510C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×Q16×C59C2
socleC22×C59C22×D8
prime radical#21436499C2
hypercenterD4×Q16D59
hypocenterC59#33887
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #21426341C2semidirect
3. #21426301C2semidirect
4. #21426296C2semidirect
5. #21426260C2semidirect
6. #21426211C2semidirect
7. #21426194C2semidirect
8. #21426138C2semidirect
9. #21426013C2semidirect
10. #21426013C2semidirect
11. #21425958C2semidirect
12. #21425922C2semidirect
13. #21425816C2semidirect
14. C2×#16837206C2semidirect
15. D4×Q16×C59C2semidirect
251. D4×Q16D59semidirect
314. C59#33887semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214263412
#214263012
#214262962
#214262602
#214262112
#214261942
#214261382
#214260132
#214259582
#214259222
#214258162
C2×#168372062
D4×Q16×C592
#3388759