Finite Group Databasebeta

← #28483502

Subgroups — #28483502

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837679C211yesmaximal, characteristic
2#21427043C211yesmaximal, characteristic
3#21425058C211yesmaximal, characteristic, complemented
4C2×#16837129C211yesmaximal, characteristic
5#21421980C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1981C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9C2×#14273685—21no—
10#16837679—22no—
11#16837679C414yescomplemented
12C2×#14273660—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273647—21no—
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837304—21no—
17#16837294—22no—
18#16837272C2211yescharacteristic
19#16837259—21no—
20C4×C4⋊Dic59—21no—
21#16837164—22no—
22#16837129—22no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×#103×C59—21no—
26C2×#103×C59—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32C4×D236—21no—
33C4×D236—21no—
34C4×D236C2×C412yes—
35C4×D236D412yes—
36#14273685—42no—
37#14273685—42no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59D411yescharacteristic
41C2×C4⋊Dic59D412yes—
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—22no—
44C2×(C2×D59)⋊C4—22no—
45C2×C4×Dic59—22no—
46#14273660—24no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)C2×C412yes—
49#14273647—24no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C59⋊#95—22no—
53C22×Q8×C59—21no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59C2×C412yes—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59D411yescharacteristic
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67#103×C59—42no—
68#103×C59—42no—
69#95×C59—21no—
70C23×D59—21no—
71C2×Q8×C59—22no—
72C2×Q8×C59—41no—
73C2×Q8×C59—41no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—41no—
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×Dic59—21no—
84C22×Dic59—22no—
85C2×D236—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236—21no—
89C2×D236—21no—
90C2×D236C22⋊C412yes—
91C2×C4×D59—22no—
92C2×C4×D59—22no—
93C2×C4×D59—41no—
94C2×C4×D59—41no—
95C2×C4×D59—41no—
96C2×C4×D59—41no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59D812yes—
108C42×C59—41no—
109C42×C59—41no—
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59#4111yescharacteristic
113C42×C59#4111yescharacteristic
114(C2×D59)⋊C4—22no—
115(C2×D59)⋊C4—22no—
116(C2×D59)⋊C4—44no—
117(C2×D59)⋊C4—82no—
118C4⋊Dic59—21no—
119C4⋊Dic59—41no—
120C4⋊Dic59—41no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123C4⋊Dic59—42no—
124C4⋊Dic59C22⋊C412yes—
125C4×Dic59—44no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C23×C59#12011yescharacteristic
133C22×D59—22no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C22×D59—41no—
140Q8×C59—42no—
141Q8×C59—42no—
142Q8×C59—81no—
143Q8×C59—81no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59#12111yescharacteristic
159C2×C4×C59#12111yescharacteristic
160C2×C4×C59#12311yescharacteristic
161C2×C4×C59C2×D811yescharacteristic
162C2×C4×C59#13511yescharacteristic
163C2×C4×C59#13511yescharacteristic
164C2×C4×C59#13511yescharacteristic
165C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C59#13711yescharacteristic
168C2×C4×C59#13711yescharacteristic
169C2×Dic59—22no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—42no—
173C2×Dic59—42no—
174C2×Dic59—42no—
175D236—22no—
176D236—22no—
177D236—41no—
178D236—41no—
179C4×D59—42no—
180C4×D59—42no—
181C4×D59—42no—
182C4×D59—42no—
183C8×C59—81no—
184C59⋊C8—42no—
185C59⋊C8—42no—
186#8984—591nomaximal, Sylow
187C22×C59#38611yescharacteristic
188C22×C59#44711yescharacteristic
189C22×C59#44911yescharacteristic
190C22×C59#45211yescharacteristic
191C22×C59#45211yescharacteristic
192C22×C59#45511yescharacteristic
193C22×C59#45611yescharacteristic
194C2×D59—42no—
195C2×D59—42no—
196C2×D59—42no—
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—81no—
200C2×D59—81no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—42no—
204C4×C59—82no—
205C4×C59—81no—
206C4×C59#41611yescharacteristic
207C4×C59#41911yescharacteristic
208C4×C59C4×D811yescharacteristic
209C4×C59#44211yescharacteristic
210C4×C59#45911yescharacteristic
211C4×C59#46311yescharacteristic
212C4×C59#48511yescharacteristic
213C4×C59#48511yescharacteristic
214Dic59—42no—
215Dic59—82no—
216C2×#474—591no—
217#2354—591no—
218#1990—591no—
219#1981D5911yescharacteristic, complemented
220C2×D4⋊C8—591no—
221#1296—591no—
222#1279—591no—
223C2×C59#164111yescharacteristic
224C2×C59#164611yescharacteristic
225C2×C59#164711yescharacteristic
226C2×C59#165211yescharacteristic
227C2×C59#187311yescharacteristic
228C2×C59#187911yescharacteristic
229C2×C59#188211yescharacteristic
230D59—82no—
231C2×C4⋊Q8C2×D5911yescharacteristic
232C2×C4×D4—591no—
233C2×#117—1181no—
234#474—594no—
235#474—1182no—
236C2×#107—1181no—
237C2×C4⋊C8—591no—
238C2×C4⋊C8C2×D5911yescharacteristic
239C2×#103—21no—
240C2×#103—21no—
241C2×#102—1181no—
242C2×C4×C8—591no—
243#390—1181no—
244#388—1182no—
245#384—591no—
246#380—1181no—
247C4×C4⋊C4—1181no—
248C4×C4⋊C4C2×D5911yescharacteristic
249#340—1182no—
250D4⋊C8—1182no—
251C59#898411yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×D4—591no—
254C23×C4—1181no—
255C22×C8—21no—
256C22×C8—1181no—
257C22×C8—1181no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8C4×D5912yes—
262C4×D4—592no—
263C4×D4—592no—
264C4×D4—1181no—
265C4×D4—1181no—
266#117—2362no—
267#117—2362no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—591no—
271C2×C4⋊C4—592no—
272C2×C4⋊C4—1181no—
273C2×C4⋊C4—1181no—
274C2×C4⋊C4C22⋊D5911yescharacteristic
275C2×C4⋊C4C22⋊D5911yescharacteristic
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1182no—
278C2×C22⋊C4—1182no—
279C2×C42—21no—
280C2×C42—1182no—
281C2×C42C22×D5911yescharacteristic
282#107—1184no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—592no—
286C4⋊C8—1181no—
287#103—42no—
288#103—42no—
289#102—1184no—
290C4×C8—1181no—
291C4×C8—1181no—
292#95—21no—
293#95—1182no—
294C24—1181no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—592no—
302C2×D4—1181no—
303C2×D4—1181no—
304C2×D4—1181no—
305C2×D4—1181no—
306C2×D4—1181no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—1181no—
311C22×C4—1182no—
312C22×C4—1182no—
313C22×C4—1182no—
314C22×C4—2361no—
315C22×C4—2361no—
316C22×C4—2361no—
317C22×C4—2361no—
318C22×C4D4×D5911yescharacteristic
319C22×C4D4×D5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—1182no—
325C2×C8—1182no—
326C2×C8—2361no—
327C2×C8—2361no—
328C2×C8—2361no—
329C2×C8—2361no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—592no—
338C4⋊C4—1181no—
339C4⋊C4—2362no—
340C4⋊C4—2361no—
341C4⋊C4—2361no—
342C4⋊C4—2361no—
343C4⋊C4—2361no—
344C4⋊C4#1174566912yes—
345C22⋊C4—1182no—
346C22⋊C4—1182no—
347C22⋊C4—4722no—
348C22⋊C4—2364no—
349C42—41no—
350C42—41no—
351C42—2364no—
352C42C2×C4×D5911yescharacteristic
353C42#1174568511yescharacteristic
354C42D4×D5911yescharacteristic
355C42#1174568711yescharacteristic
356C23—1182no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23#1427369511yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—81no—
367Q8—81no—
368D4—1182no—
369D4—1182no—
370D4—2361no—
371D4—2361no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—1182no—
384C2×C4—2362no—
385C2×C4—2362no—
386C2×C4—2362no—
387C2×C4—2361no—
388C2×C4—2361no—
389C2×C4—2362no—
390C2×C4—2362no—
391C2×C4—2362no—
392C2×C4—2362no—
393C2×C4C22⋊C4×D5911yescharacteristic
394C2×C4(C22⋊D59)⋊C411yescharacteristic
395C2×C4C4×C22⋊D5911yescharacteristic
396C2×C4#1427369811yescharacteristic
397C2×C4#1427370011yescharacteristic
398C2×C4#1427370911yescharacteristic
399C2×C4C2×#1174566911yescharacteristic
400C2×C4#1427374711yescharacteristic
401C2×C4#1427374811yescharacteristic
402C2×C4#1427375011yescharacteristic
403C2×C4#1427375211yescharacteristic
404C2×C4#1427375311yescharacteristic
405C2×C4#1427375611yescharacteristic
406C2×C4#1427375911yescharacteristic
407C8—81no—
408C8—2362no—
409C8—2362no—
410C22—4721no—
411C22—4721no—
412C22—4721no—
413C22—4721no—
414C22—2362no—
415C22—2362no—
416C22—2362no—
417C22#1683727811yescharacteristic
418C22#1683753511yescharacteristic
419C22#1683754511yescharacteristic
420C22#1683756111yescharacteristic
421C22#1683756311yescharacteristic
422C22#1683757811yescharacteristic
423C22#1683758311yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—82no—
428C4—81no—
429C4—4722no—
430C4—2362no—
431C4#1683739711yescharacteristic
432C4#1683740911yescharacteristic
433C4C4×#1174566911yescharacteristic
434C4#1683751211yescharacteristic
435C4#1683759411yescharacteristic
436C4#1683761411yescharacteristic
437C4#1683774111yescharacteristic
438C4#1683774211yescharacteristic
439C2—4722no—
440C2#2142328711yescharacteristic
441C2#2142332211yescharacteristic
442C2#2142332911yescharacteristic
443C2#2142336411yescharacteristic
444C2#2142440511yescharacteristic
445C2#2142444711yescharacteristic
446C2#2142446211yescharacteristic
subgroupG/H
centerC23#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1981C2
socleC23×C59#120
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenter#1981D59
hypocenterC59#8984
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. #21425058C2semidirect
5. #21421980C2semidirect
6. #21421937C2semidirect
7. C59×#1981C2semidirect
11. #16837679C4semidirect
219. #1981D59semidirect
251. C59#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#168376792
#214270432
#214250582
C2×#168371292
#214219802
#214219372
C59×#19812
#898459