Finite Group Databasebeta

← #28492084

Subgroups — #28492084

Order 15104.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429265C211yesmaximal, characteristic
2#21424897C211yesmaximal, characteristic, complemented
3#21424093C211yesmaximal, characteristic
4C2×#16837150C211yesmaximal, characteristic, complemented
5C2×#16837129C211yesmaximal, characteristic
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1947C211yesmaximal, characteristic, complemented
8C2×#14273727C2211yescharacteristic
9C2×C4×D236C2211yescharacteristic
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11C2×#14273650—22no—
12C2×#14273647—21no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837315—22no—
15#16837304—21no—
16#16837268—21no—
17#16837259—21no—
18#16837259—21no—
19#16837164—21no—
20#16837150—22no—
21#16837129—22no—
22C2×#107×C59—21no—
23C2×C4⋊C8×C59C2211yescharacteristic
24C59×#384C2211yescharacteristic
25C4×C4⋊C4×C59C2211yescharacteristic
26C59×#340—21no—
27C22×Dic118D412yes—
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273727—21no—
33#14273727—21no—
34#14273727—21no—
35#14273727—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C4×D236—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59D411yescharacteristic
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—22no—
45C2×(C2×D59)⋊C4—22no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48#14273650—44no—
49#14273647—42no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C59⋊#95—22no—
53C59⋊#95—22no—
54C22×C8×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C4⋊C4×C59D411yescharacteristic
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62#107×C59—42no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59D412yes—
65#95×C59—21no—
66C23×D59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—22no—
75C2×D236—21no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236—21no—
79C2×D236—21no—
80C2×D236Q1612yes—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59—41no—
87C2×Dic118—22no—
88C2×Dic118—22no—
89C2×Dic118—42no—
90C2×Dic118—42no—
91C2×C8×C59—22no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—42no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C42×C59—41no—
104C42×C59—41no—
105C42×C59C2×D411yescharacteristic
106C42×C59#4111yescharacteristic
107C42×C59#4111yescharacteristic
108C42×C59#4111yescharacteristic
109(C2×D59)⋊C4—41no—
110(C2×D59)⋊C4—41no—
111(C2×D59)⋊C4—82no—
112(C2×D59)⋊C4—82no—
113C4⋊Dic59—21no—
114C4⋊Dic59—21no—
115C4⋊Dic59—41no—
116C4⋊Dic59—41no—
117C4⋊Dic59—41no—
118C4⋊Dic59—41no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8—41no—
125C23×C59#12011yescharacteristic
126C22×D59—22no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131C22×D59—41no—
132C22×D59—41no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—81no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12311yescharacteristic
148C2×C4×C59#12311yescharacteristic
149C2×C4×C59#12411yescharacteristic
150C2×C4×C59C2×Q1611yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59#13511yescharacteristic
153C2×C4×C59#13511yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C59#13711yescharacteristic
157C2×C4×C59#13711yescharacteristic
158C2×Dic59—41no—
159C2×Dic59—42no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—42no—
163C2×Dic59—42no—
164D236—22no—
165D236—22no—
166D236—41no—
167D236—41no—
168C4×D59—81no—
169C4×D59—81no—
170C4×D59—81no—
171C4×D59—81no—
172Dic118—44no—
173Dic118—82no—
174C8×C59—42no—
175C59⋊C8—42no—
176C59⋊C8—81no—
177#10413—591nomaximal, Sylow
178C22×C59#39411yescharacteristic
179C22×C59#44911yescharacteristic
180C22×C59#44911yescharacteristic
181C22×C59#45111yescharacteristic
182C22×C59#45211yescharacteristic
183C22×C59#45311yescharacteristic
184C22×C59#45511yescharacteristic
185C2×D59—42no—
186C2×D59—42no—
187C2×D59—42no—
188C2×D59—81no—
189C2×D59—81no—
190C2×D59—81no—
191C2×D59—81no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—81no—
195C4×C59—81no—
196C4×C59—161no—
197C4×C59#46311yescharacteristic
198C4×C59#46411yescharacteristic
199C4×C59#46711yescharacteristic
200C4×C59#46811yescharacteristic
201C4×C59#48411yescharacteristic
202C4×C59#48511yescharacteristic
203C4×C59#48811yescharacteristic
204C4×C59#48911yescharacteristic
205Dic59—82no—
206Dic59—81no—
207#2563—591no—
208#1956—591no—
209#1947D5911yescharacteristic, complemented
210#1802—591no—
211C2×#333—591no—
212C2×D4⋊C8—591no—
213#1279—591no—
214C2×C59#162411yescharacteristic
215C2×C59#163211yescharacteristic
216C2×C59#164411yescharacteristic
217C2×C59#164611yescharacteristic
218C2×C59#203611yescharacteristic
219C2×C59#203711yescharacteristic
220C2×C59#204211yescharacteristic
221D59—82no—
222C2×C4⋊Q8—591no—
223C2×C4×D4—591no—
224C2×#107—21no—
225C2×C4⋊C8—591no—
226C2×C4⋊C8C2×D5911yescharacteristic
227C2×#103—1182no—
228C2×#102—1181no—
229C2×C4×C8—591no—
230#393—1182no—
231#390—1181no—
232#384C2×D5911yescharacteristic
233#382—1181no—
234#380—1181no—
235#380—1181no—
236C4×C4⋊C4C2×D5911yescharacteristic
237#340—21no—
238#340—1181no—
239#333—1182no—
240D4⋊C8—1182no—
241C59#1041311yescharacteristic, complemented, Sylow
242C22×Q8—592no—
243C22×D4—591no—
244C23×C4—1181no—
245C22×C8—21no—
246C22×C8—1181no—
247C22×C8—1181no—
248C4⋊Q8—1181no—
249C4⋊Q8—1181no—
250C4⋊Q8—1181no—
251C4⋊Q8—1181no—
252C4×D4—1181no—
253C4×D4—1181no—
254C4×D4—1181no—
255C4×D4—1181no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—591no—
260C2×C4⋊C4—1181no—
261C2×C4⋊C4—1181no—
262C2×C4⋊C4C22⋊D5911yescharacteristic
263C2×C4⋊C4C22⋊D5911yescharacteristic
264C2×C22⋊C4—1182no—
265C2×C22⋊C4—1182no—
266C2×C22⋊C4—1181no—
267C2×C42—21no—
268C2×C42C22×D5911yescharacteristic
269#107—42no—
270C4⋊C8—21no—
271C4⋊C8—1181no—
272C4⋊C8—1181no—
273C4⋊C8D23612yes—
274#103—2364no—
275#102—2362no—
276C4×C8—1181no—
277C4×C8—1181no—
278#95—21no—
279#95—1182no—
280#95—1182no—
281C24—1181no—
282C2×Q8—1182no—
283C2×Q8—1182no—
284C2×Q8—2362no—
285C2×Q8—2362no—
286C2×D4—592no—
287C2×D4—1181no—
288C2×D4—1181no—
289C2×D4—1181no—
290C2×D4—1181no—
291C2×D4—1181no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—1181no—
296C22×C4—1182no—
297C22×C4—2361no—
298C22×C4—2361no—
299C22×C4—2361no—
300C22×C4—2361no—
301C22×C4—2361no—
302C22×C4—2361no—
303C22×C4D4×D5911yescharacteristic
304C22×C4D4×D5911yescharacteristic
305C22×C4C2×C22⋊D5911yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—1182no—
310C2×C8—2361no—
311C2×C8—2361no—
312C2×C8—2361no—
313C2×C8—2361no—
314C2×C8—2361no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—1181no—
325C4⋊C4—1181no—
326C4⋊C4—2361no—
327C4⋊C4—2361no—
328C4⋊C4—2361no—
329C4⋊C4—2361no—
330C22⋊C4—4722no—
331C22⋊C4—4722no—
332C22⋊C4—2361no—
333C22⋊C4—2361no—
334C42—41no—
335C42—41no—
336C42C2×D23611yescharacteristic
337C42#1174568511yescharacteristic
338C42#1174568911yescharacteristic
339C42#1174568911yescharacteristic
340C23—1182no—
341C23—2361no—
342C23—2361no—
343C23—2361no—
344C23—2361no—
345C23—2361no—
346C23—2361no—
347C23#1427369511yescharacteristic
348Q8—4722no—
349Q8—2364no—
350D4—1182no—
351D4—1182no—
352D4—2361no—
353D4—2361no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—4721no—
366C2×C4—4721no—
367C2×C4—4721no—
368C2×C4—4721no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2362no—
372C2×C4—2361no—
373C2×C4—2362no—
374C2×C4—2362no—
375C2×C4#1427370111yescharacteristic
376C2×C4#1427370211yescharacteristic
377C2×C4#1427370211yescharacteristic
378C2×C4#1427371211yescharacteristic
379C2×C4#1427371211yescharacteristic
380C2×C4#1427371611yescharacteristic
381C2×C4Q16×D5911yescharacteristic
382C2×C4#1427374611yescharacteristic
383C2×C4#1427374711yescharacteristic
384C2×C4#1427374711yescharacteristic
385C2×C4#1427375111yescharacteristic
386C2×C4#1427375311yescharacteristic
387C2×C4#1427375811yescharacteristic
388C2×C4#1427375911yescharacteristic
389C8—42no—
390C8—4721no—
391C8—2362no—
392C22—4721no—
393C22—4721no—
394C22—4721no—
395C22—4721no—
396C22—2362no—
397C22—2362no—
398C22—2362no—
399C22#1683731911yescharacteristic
400C22#1683754411yescharacteristic
401C22#1683754511yescharacteristic
402C22#1683755611yescharacteristic
403C22#1683756211yescharacteristic
404C22#1683756811yescharacteristic
405C22#1683757811yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—161no—
411C4—4722no—
412C4—4721no—
413C4#1683761711yescharacteristic
414C4#1683762411yescharacteristic
415C4#1683764411yescharacteristic
416C4#1683764911yescharacteristic
417C4#1683773911yescharacteristic
418C4#1683774511yescharacteristic
419C4#1683775911yescharacteristic
420C4#1683776411yescharacteristic
421C2—4722no—
422C2#2142317811yescharacteristic
423C2#2142322911yescharacteristic
424C2#2142330811yescharacteristic
425C2#2142332311yescharacteristic
426C2#2142530111yescharacteristic
427C2#2142530811yescharacteristic
428C2#2142533711yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1947C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenter#1947D59
hypocenterC59#10413
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #21424897C2semidirect
4. C2×#16837150C2semidirect
6. #21421937C2semidirect
7. C59×#1947C2semidirect
209. #1947D59semidirect
241. C59#10413semidirect

Maximal subgroups

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

subgroupindex
#214292652
#214248972
#214240932
C2×#168371502
C2×#168371292
#214219372
C59×#19472
#1041359