Finite Group Databasebeta

← #28483496

Subgroups — #28483496

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#21427047C211yesmaximal, characteristic
3#21425020C211yesmaximal, characteristic, complemented
4C2×#16837131C211yesmaximal, characteristic
5#21421979C211yesmaximal, characteristic, complemented
6#21421936C211yesmaximal, characteristic, complemented
7C59×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9C2×#14273685—21no—
10#16837679—22no—
11#16837679C414yescomplemented
12C2×#14273660—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273651—21no—
16C2×#14273651—21no—
17C2×#14273647—21no—
18#16837303—21no—
19#16837292—22no—
20#16837259—21no—
21C4⋊C4×Dic59—21no—
22C4×C4⋊Dic59C2211yescharacteristic
23#16837131—22no—
24C2×C4⋊Q8×C59C2211yescharacteristic
25C2×C4×C8×C59C2211yescharacteristic
26C59×#384C2211yescharacteristic
27C59×#340—22no—
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⋊Dic59D411yescharacteristic
39C2×C4⋊Dic59D411yescharacteristic
40C2×C4⋊Dic59D411yescharacteristic
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—22no—
44C2×C4×Dic59—22no—
45C2×C4×Dic59—21no—
46#14273660—24no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)C2×C412yes—
51#14273651—42no—
52#14273651—42no—
53#14273647—24no—
54C59⋊#95—22no—
55C59⋊#95—21no—
56C22×Q8×C59—21no—
57C22×C8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59C2×C412yes—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59D412yes—
67C2×C42×C59C2311yescharacteristic
68C4×C8×C59—21no—
69C4×C8×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—22no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
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—22no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—42no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C42×C59C22×C411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C42×C59#4111yescharacteristic
113(C2×D59)⋊C4—22no—
114(C2×D59)⋊C4—22no—
115(C2×D59)⋊C4—44no—
116(C2×D59)⋊C4—82no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59—21no—
121C4⋊Dic59C22⋊C412yes—
122C4⋊Dic59D812yes—
123C4×Dic59—44no—
124C4×Dic59—41no—
125C4×Dic59—41no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—41no—
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—42no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59C2×C22⋊C411yescharacteristic
153C2×C4×C59C4×D411yescharacteristic
154C2×C4×C59C4×D411yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12311yescharacteristic
158C2×C4×C59C2×D811yescharacteristic
159C2×C4×C59#13511yescharacteristic
160C2×C4×C59#13511yescharacteristic
161C2×C4×C59#13511yescharacteristic
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C59#13711yescharacteristic
165C2×C4×C59#13711yescharacteristic
166C2×Dic59—22no—
167C2×Dic59—22no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175D236—22no—
176D236—22no—
177D236—41no—
178D236—41no—
179C4×D59—42no—
180C4×D59—42no—
181C4×D59—42no—
182C4×D59—42no—
183C8×C59—42no—
184C59⋊C8—42no—
185C59⋊C8—81no—
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—82no—
204C4×C59—82no—
205C4×C59#41611yescharacteristic
206C4×C59#41911yescharacteristic
207C4×C59C4×D811yescharacteristic
208C4×C59#44211yescharacteristic
209C4×C59#45911yescharacteristic
210C4×C59#46311yescharacteristic
211C4×C59#48511yescharacteristic
212C4×C59#48511yescharacteristic
213Dic59—42no—
214Dic59—42no—
215Dic59—81no—
216C2×#474—591no—
217#2354—591no—
218#1990D5911yescharacteristic, complemented
219#1981—591no—
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⋊C8—591no—
239C2×#103—1181no—
240C2×#103—1181no—
241C2×#102—1181no—
242C2×C4×C8C2×D5911yescharacteristic
243#390—1181no—
244#388—1182no—
245#384C2×D5911yescharacteristic
246#380—1181no—
247C4×C4⋊C4—591no—
248C4×C4⋊C4—1181no—
249#340—22no—
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—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—591no—
274C2×C4⋊C4—591no—
275C2×C4⋊C4C22⋊D5912yes—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1182no—
278C2×C22⋊C4—1182no—
279C2×C42—1182no—
280C2×C42—1181no—
281C2×C42C22×D5911yescharacteristic
282#107—1184no—
283C4⋊C8—592no—
284C4⋊C8—1181no—
285C4⋊C8—1181no—
286C4⋊C8—1181no—
287#103—2362no—
288#103—2362no—
289#102—1184no—
290C4×C8—21no—
291C4×C8—21no—
292#95—1182no—
293#95—1181no—
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—22no—
308C22×C4—41no—
309C22×C4—1181no—
310C22×C4—1181no—
311C22×C4—1182no—
312C22×C4—1182no—
313C22×C4—1181no—
314C22×C4—2361no—
315C22×C4—2361no—
316C22×C4—2361no—
317C22×C4—2361no—
318C22×C4D4×D5911yescharacteristic
319C22×C4D4×D5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—1182no—
325C2×C8—2361no—
326C2×C8—2361no—
327C2×C8—2361no—
328C2×C8—2361no—
329C2×C8—2361no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—592no—
340C4⋊C4—592no—
341C4⋊C4—1181no—
342C4⋊C4—1181no—
343C4⋊C4—1181no—
344C4⋊C4—1181no—
345C22⋊C4—1182no—
346C22⋊C4—1182no—
347C22⋊C4—4722no—
348C22⋊C4—2364no—
349C42—2364no—
350C42—2361no—
351C42—2361no—
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—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—1182no—
381C2×C4—1182no—
382C2×C4—2361no—
383C2×C4—2361no—
384C2×C4—2361no—
385C2×C4—2361no—
386C2×C4—2361no—
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×C4D8×D5911yescharacteristic
400C2×C4#1427374811yescharacteristic
401C2×C4#1427374811yescharacteristic
402C2×C4#1427375011yescharacteristic
403C2×C4#1427375111yescharacteristic
404C2×C4#1427375411yescharacteristic
405C2×C4#1427375611yescharacteristic
406C2×C4#1427375911yescharacteristic
407C8—42no—
408C8—4721no—
409C8—2362no—
410C22—4721no—
411C22—4721no—
412C22—4721no—
413C22—4721no—
414C22—2362no—
415C22—2362no—
416C22—2362no—
417C22#1683727811yescharacteristic
418C22#1683753411yescharacteristic
419C22#1683754611yescharacteristic
420C22#1683756111yescharacteristic
421C22#1683756311yescharacteristic
422C22#1683757911yescharacteristic
423C22#1683758411yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—82no—
428C4—4721no—
429C4—2362no—
430C4—2362no—
431C4#1683739711yescharacteristic
432C4#1683740911yescharacteristic
433C4#1683748211yescharacteristic
434C4#1683751111yescharacteristic
435C4#1683759611yescharacteristic
436C4#1683761611yescharacteristic
437C4#1683774111yescharacteristic
438C4#1683774211yescharacteristic
439C2—4722no—
440C2#2142328611yescharacteristic
441C2#2142332111yescharacteristic
442C2#2142332811yescharacteristic
443C2#2142336311yescharacteristic
444C2#2142440511yescharacteristic
445C2#2142444111yescharacteristic
446C2#2142446811yescharacteristic
subgroupG/H
centerC23#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1990C2
socleC23×C59#120
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenter#1990D59
hypocenterC59#8984
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #21425020C2semidirect
5. #21421979C2semidirect
6. #21421936C2semidirect
7. C59×#1990C2semidirect
11. #16837679C4semidirect
218. #1990D59semidirect
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
#214270472
#214250202
C2×#168371312
#214219792
#214219362
C59×#19902
#898459