Finite Group Databasebeta

← #28483311

Subgroups — #28483311

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433207C211yesmaximal, characteristic, complemented
2C4×#14273752C211yesmaximal, characteristic
3#21424890C211yesmaximal, characteristic, complemented
4#21422488C211yesmaximal, characteristic, complemented
5#21421945C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C59×#1476C211yesmaximal, characteristic, complemented
8C2×#14273752C411yescharacteristic
9C2×C4×D236C2211yescharacteristic
10Q8×C59⋊C8—21no—
11#16837512—21no—
12#16837500—21no—
13C4×#11745671—21no—
14C4×#11745669—21no—
15D4×C59⋊C8—22no—
16D4×C59⋊C8—21no—
17#16837455—21no—
18#16837455C2211yescharacteristic
19#16837445C411yescharacteristic
20C2×C4⋊(C59⋊C8)—21no—
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22#16837406C411yescharacteristic
23#16837403C411yescharacteristic
24C2×(C2×D59)⋊C8—22no—
25C4×#11745664C2211yescharacteristic
26#16837131—21no—
27#16837131—21no—
28C4×#41×C59C2211yescharacteristic
29C59×#431C2211yescharacteristic
30C2×C4×C8×C59C2211yescharacteristic
31C59×Q8⋊C8—21no—
32C59×D4⋊C8—21no—
33C22×D236C2×C411yescharacteristic
34C22×C4×D59—21no—
35#14273752—22no—
36#14273752C814yescomplemented
37C2×#11745671—21no—
38C2×#11745669—21no—
39C2×#11745664C2×C411yescharacteristic
40C2×#11745664C2×C411yescharacteristic
41C22×C59⋊C8—22no—
42C22×C59⋊C8D412yes—
43C4×D236—21no—
44C4×D236—21no—
45C4×D236—21no—
46C4×D236—21no—
47C2×C4⋊Dic59C2×C411yescharacteristic
48C2×(C2×D59)⋊C4—21no—
49#14273660—21no—
50#14273658—21no—
51C4⋊(C59⋊C8)—22no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)D411yescharacteristic
55C4⋊(C59⋊C8)D411yescharacteristic
56#14273651—21no—
57#14273648—21no—
58#14273647—21no—
59#14273647—21no—
60C22⋊(C59⋊C8)—22no—
61C22⋊(C59⋊C8)—21no—
62(C2×D59)⋊C8—44no—
63#14273638—21no—
64C4×C59⋊C8—22no—
65C4×C59⋊C8—21no—
66C4×C59⋊C8—21no—
67C2×#41×C59C2×C411yescharacteristic
68C22×C8×C59—21no—
69C4×Q8×C59D411yescharacteristic
70C4×Q8×C59D411yescharacteristic
71C4×D4×C59—21no—
72C4×D4×C59—21no—
73C4×D4×C59D411yescharacteristic
74C4×D4×C59D411yescharacteristic
75#117×C59—21no—
76#117×C59C2×C411yescharacteristic
77C2×C42×C59—21no—
78C2×C42×C59C2311yescharacteristic
79C4⋊C8×C59—21no—
80C22⋊C8×C59—21no—
81C4×C8×C59—21no—
82C4×C8×C59—21no—
83C4×C8×C59—21no—
84C23×D59—21no—
85#41×C59—21no—
86#41×C59—21no—
87#41×C59C2×C812yes—
88#41×C59M1612yes—
89C2×Q8×C59C22⋊C411yescharacteristic
90C2×D4×C59—21no—
91C2×D4×C59C22⋊C411yescharacteristic
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59C22×C411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59#4111yescharacteristic
98C22×Dic59—21no—
99C2×D236—21no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×D236—21no—
104C2×D236C2×C812yescomplemented
105C2×C4×D59—42no—
106C2×C4×D59—42no—
107C2×C4×D59—41no—
108C2×C4×D59—41no—
109C2×C8×C59—22no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59C22⋊C411yescharacteristic
116C4⋊C4×C59C22⋊C411yescharacteristic
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59C2×D411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59#4111yescharacteristic
127#11745671—24no—
128#11745669—24no—
129(C2×D59)⋊C4—41no—
130(C2×D59)⋊C4—41no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133#11745664—21no—
134#11745664—21no—
135#11745664—21no—
136#11745664C2×C812yescomplemented
137C2×C59⋊C8—21no—
138C2×C59⋊C8—21no—
139C2×C59⋊C8—21no—
140C2×C59⋊C8—22no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—42no—
143C2×C59⋊C8—82no—
144C23×C59—21no—
145C23×C59C4×D411yescharacteristic
146C22×D59—22no—
147C22×D59—42no—
148C22×D59—42no—
149C22×D59—41no—
150C22×D59—41no—
151Q8×C59—21no—
152Q8×C59C22⋊C812yescomplemented
153D4×C59—22no—
154D4×C59—22no—
155D4×C59—21no—
156D4×C59C22⋊C812yescomplemented
157C2×C4×C59—21no—
158C2×C4×C59—22no—
159C2×C4×C59—22no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—81no—
168C2×C4×C59C2×C22⋊C411yescharacteristic
169C2×C4×C59C4×D411yescharacteristic
170C2×C4×C59#12011yescharacteristic
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12311yescharacteristic
174C2×C4×C59C22×C811yescharacteristic
175C2×C4×C59C2×M1611yescharacteristic
176C2×C4×C59#13111yescharacteristic
177C2×C4×C59#13111yescharacteristic
178C2×Dic59—42no—
179C2×Dic59—41no—
180D236—22no—
181D236—22no—
182D236—41no—
183D236—41no—
184C4×D59—82no—
185C4×D59—81no—
186C4×D59—81no—
187C8×C59—42no—
188C8×C59—81no—
189C59⋊C8—22no—
190C59⋊C8—41no—
191C59⋊C8—42no—
192C59⋊C8—41no—
193C59⋊C8—41no—
194#8954—591nomaximal, Sylow
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—22no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59#38611yescharacteristic
201C22×C59C8×D411yescharacteristic
202C22×C59#43511yescharacteristic
203C2×D59—44no—
204C2×D59—42no—
205C2×D59—82no—
206C2×D59—81no—
207C2×D59—81no—
208C4×C59—22no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—41no—
213C4×C59—41no—
214C4×C59C2×C22⋊C811yescharacteristic
215C4×C59#40811yescharacteristic
216C4×C59C8×D411yescharacteristic
217C4×C59#43511yescharacteristic
218C4×C59#45311yescharacteristic
219C4×C59#45411yescharacteristic
220Dic59—81no—
221#2988—591no—
222C4×AGL(1,ℤ/8)—591no—
223#1954—591no—
224#1478—591no—
225#1476D5911yescharacteristic, complemented
226#1283—591no—
227#1278—591no—
228C2×C59—21no—
229C2×C59—21no—
230C2×C59—42no—
231C2×C59#186911yescharacteristic
232C2×C59#188111yescharacteristic
233C2×C59#188311yescharacteristic
234D59—82no—
235C2×AGL(1,ℤ/8)—591no—
236C4×#41C2×D5911yescharacteristic
237C2×C4×D4—591no—
238C8×Q8—1181no—
239#442—1181no—
240#440—1181no—
241C4×SD16—1181no—
242C4×D8—1181no—
243C8×D4—1182no—
244C8×D4—1181no—
245#431—591no—
246#431—1181no—
247#431C2×D5911yescharacteristic
248#428—591no—
249C2×C4⋊C8—591no—
250C2×C4⋊C8—1181no—
251#418—591no—
252#417—591no—
253C2×C22⋊C8—1182no—
254C4×M16—591no—
255C2×C4×C8C2×D5911yescharacteristic
256Q8⋊C8—21no—
257D4⋊C8—21no—
258D4⋊C8—1181no—
259D4⋊C8—1181no—
260C59#895411yescharacteristic, complemented, Sylow
261C2×#41C4×D5911yescharacteristic
262C22×D4—591no—
263C23×C4—1181no—
264AGL(1,ℤ/8)—594no—
265AGL(1,ℤ/8)—1182no—
266C2×SD16—1181no—
267C2×D8—1181no—
268C2×M16—591no—
269C2×M16—591no—
270C22×C8—21no—
271C22×C8—592no—
272C22×C8—1182no—
273C4×Q8D23611yescharacteristic
274C4×Q8C22⋊D5911yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—1181no—
278C4×D4—1181no—
279C4×D4—1181no—
280C4×D4—1181no—
281C4×D4D23611yescharacteristic
282C4×D4C22⋊D5911yescharacteristic
283#117—21no—
284#117C4×D5911yescharacteristic
285C2×C4⋊C4—591no—
286C2×C22⋊C4—1181no—
287C2×C42—21no—
288C2×C42C22×D5911yescharacteristic
289#107—1181no—
290#106—1181no—
291C4⋊C8—21no—
292C4⋊C8—591no—
293C4⋊C8—591no—
294C4⋊C8—1182no—
295C4⋊C8—1181no—
296C4⋊C8—1181no—
297#103—1181no—
298#102—1181no—
299#102—1181no—
300#102—1181no—
301C22⋊C8—21no—
302C22⋊C8—1182no—
303C22⋊C8—1181no—
304C22⋊C8—2364no—
305#97—1181no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—1182no—
310C4×C8—1181no—
311C4×C8—1181no—
312C24—1181no—
313#41—21no—
314#41—21no—
315#41C8×D5912yes—
316#41#1174565912yes—
317C2×Q8(C2×D59)⋊C411yescharacteristic
318C2×D4—21no—
319C2×D4—592no—
320C2×D4—1181no—
321C2×D4—1181no—
322C2×D4—1181no—
323C2×D4—1181no—
324C2×D4—1181no—
325C2×D4(C2×D59)⋊C411yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—1181no—
330C22×C4—2362no—
331C22×C4—2362no—
332C22×C4—2361no—
333C22×C4—2361no—
334C22×C4C2×C4×D5911yescharacteristic
335C22×C4D4×D5911yescharacteristic
336C22×C4#1174568711yescharacteristic
337SD16—1184no—
338D8—1184no—
339M16—592no—
340M16—1181no—
341M16—1181no—
342M16—1181no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—1181no—
348C2×C8—1181no—
349C2×C8—1181no—
350C2×C8—1181no—
351C2×C8—1182no—
352C2×C8—4722no—
353C2×C8—2362no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—1181no—
357C4⋊C4—1181no—
358C4⋊C4(C2×D59)⋊C411yescharacteristic
359C4⋊C4(C2×D59)⋊C411yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—2361no—
364C22⋊C4—2361no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42C2×D23611yescharacteristic
369C42D4×D5911yescharacteristic
370C42#1174568711yescharacteristic
371C42C2×C22⋊D5911yescharacteristic
372C23—21no—
373C23—1182no—
374C23—2362no—
375C23—2362no—
376C23—2361no—
377C23—2361no—
378C23(C22⋊D59)⋊C411yescharacteristic
379Q8—21no—
380Q8(C2×D59)⋊C812yescomplemented
381D4—22no—
382D4—22no—
383D4—21no—
384D4—1182no—
385D4—1182no—
386D4—2361no—
387D4—2361no—
388D4(C2×D59)⋊C812yescomplemented
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—4722no—
401C2×C4—4721no—
402C2×C4—4721no—
403C2×C4—2362no—
404C2×C4—2361no—
405C2×C4C2×(C2×D59)⋊C411yescharacteristic
406C2×C4(C22⋊D59)⋊C411yescharacteristic
407C2×C4#1427369711yescharacteristic
408C2×C4#1427369911yescharacteristic
409C2×C4#1427369911yescharacteristic
410C2×C4#1427371111yescharacteristic
411C2×C4C2×C8×D5911yescharacteristic
412C2×C4C2×#1174565911yescharacteristic
413C2×C4#1427373511yescharacteristic
414C2×C4#1427373511yescharacteristic
415C8—42no—
416C8—81no—
417C8—1182no—
418C8—2361no—
419C8—2362no—
420C8—2361no—
421C8—2361no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—4722no—
428C22—4721no—
429C22—4721no—
430C22—2364no—
431C22—2362no—
432C22#1683727911yescharacteristic
433C22(C22⋊D59)⋊C811yescharacteristic
434C22#1683747011yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—4721no—
442C4C2×(C2×D59)⋊C811yescharacteristic
443C4#1683737011yescharacteristic
444C4(C22⋊D59)⋊C811yescharacteristic
445C4#1683747011yescharacteristic
446C4#1683757111yescharacteristic
447C4#1683757611yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—4722no—
452C2#2142438611yescharacteristic
453C2#2142445611yescharacteristic
454C2#2142447011yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1476C2
socleC22×C59#386
prime radicalC2×#14273752C4
cosocleC2×C42×C59C23
hypercenter#1476D59
hypocenterC59#8954
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
1. #21433207C2semidirect
3. #21424890C2semidirect
4. #21422488C2semidirect
5. #21421945C2semidirect
6. #21421933C2semidirect
7. C59×#1476C2semidirect
36. #14273752C8semidirect
104. C2×D236C2×C8semidirect
136. #11745664C2×C8semidirect
152. Q8×C59C22⋊C8semidirect
156. D4×C59C22⋊C8semidirect
225. #1476D59semidirect
260. C59#8954semidirect
380. Q8(C2×D59)⋊C8semidirect
388. D4(C2×D59)⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214332072
C4×#142737522
#214248902
#214224882
#214219452
#214219332
C59×#14762
#895459