Finite Group Databasebeta

← #28484589

Subgroups — #28484589

Order 15104.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433576C211yesmaximal, characteristic
2#21422765C211yesmaximal, characteristic, complemented
3C2×#16837145C211yesmaximal, characteristic, complemented
4#21422483C211yesmaximal, characteristic, complemented
5#21421993C211yesmaximal, characteristic, complemented
6#21421970C211yesmaximal, characteristic, complemented
7C59×#2925C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C2×#14273724C2211yescharacteristic
10#16837602—21no—
11#16837602C412yes—
12#16837597—21no—
13#16837597C412yes—
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17#16837406—21no—
18#16837145—21no—
19#16837145—21no—
20#16837145—22no—
21Q8⋊(C59⋊C8)—21no—
22D4⋊(C59⋊C8)—21no—
23C4×#41×C59C2211yescharacteristic
24C59×#433—22no—
25C59×#432—21no—
26C59×#431—21no—
27C59×#423C2211yescharacteristic
28C2×C4⋊C8×C59C2211yescharacteristic
29C59×#408—21no—
30C59×#405—21no—
31C22×D236—21no—
32C22×D236—21no—
33C22×D236—21no—
34#14273752—42no—
35#14273752—42no—
36C2×#11745671—22no—
37C2×#11745669—22no—
38C2×#11745664—21no—
39C22×C59⋊C8—21no—
40#14273724—21no—
41#14273724—21no—
42#14273724—21no—
43#14273724C2×C412yes—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)C2×C412yes—
49#14273647—22no—
50#14273647—22no—
51C22⋊(C59⋊C8)—21no—
52C4×C59⋊C8—21no—
53C2×#41×C59D411yescharacteristic
54C2×M16×C59—21no—
55C22×C8×C59—21no—
56C4×Q8×C59C2×C411yescharacteristic
57C4×Q8×C59D411yescharacteristic
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59C2×C411yescharacteristic
61C4×D4×C59D411yescharacteristic
62#117×C59—21no—
63#117×C59D411yescharacteristic
64C2×C42×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59D411yescharacteristic
69C4⋊C8×C59D411yescharacteristic
70C4⋊C8×C59D412yes—
71C22⋊C8×C59—22no—
72C22⋊C8×C59—22no—
73#97×C59—22no—
74C4×C8×C59—22no—
75C23×D59—41no—
76#41×C59—21no—
77#41×C59—21no—
78#41×C59—21no—
79#41×C59—21no—
80C2×Q8×C59C22⋊C411yescharacteristic
81C2×D4×C59—21no—
82C2×D4×C59C22⋊C411yescharacteristic
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59#4111yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C2×D236—22no—
90C2×D236—22no—
91C2×D236—22no—
92C2×D236—41no—
93C2×D236—41no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—41no—
98C2×D236—41no—
99C2×D236—82no—
100C2×D236—81no—
101C2×D236—81no—
102M16×C59—41no—
103M16×C59—41no—
104C2×C8×C59—21no—
105C2×C8×C59—22no—
106C2×C8×C59—21no—
107C2×C8×C59—41no—
108C2×C8×C59—41no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59C22⋊C411yescharacteristic
112C4⋊C4×C59C22⋊C411yescharacteristic
113C22⋊C4×C59—21no—
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C42×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59C22×C411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123#11745671—44no—
124#11745669—44no—
125#11745664—41no—
126#11745664—41no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—81no—
131C23×C59—21no—
132C23×C59#12311yescharacteristic
133C22×D59—42no—
134C22×D59—82no—
135C22×D59—82no—
136C22×D59—81no—
137C22×D59—81no—
138Q8×C59—21no—
139Q8×C59—21no—
140D4×C59—21no—
141D4×C59—21no—
142D4×C59—41no—
143D4×C59—41no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—81no—
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59#12011yescharacteristic
159C2×C4×C59#12111yescharacteristic
160C2×C4×C59#12111yescharacteristic
161C2×C4×C59#13511yescharacteristic
162C2×C4×C59#13511yescharacteristic
163C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
165D236—42no—
166D236—42no—
167D236—82no—
168D236—82no—
169D236—82no—
170D236—81no—
171D236—81no—
172D236—81no—
173D236—81no—
174C8×C59—41no—
175C8×C59—41no—
176C8×C59—42no—
177C59⋊C8—42no—
178C59⋊C8—81no—
179C59⋊C8—81no—
180#9174—591nomaximal, Sylow
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59#38611yescharacteristic
187C22×C59#48211yescharacteristic
188C22×C59#48211yescharacteristic
189C2×D59—82no—
190C2×D59—82no—
191C2×D59—82no—
192C2×D59—162no—
193C2×D59—162no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59#41611yescharacteristic
201C4×C59#41811yescharacteristic
202C4×C59#44911yescharacteristic
203C4×C59#44911yescharacteristic
204C4×C59#47111yescharacteristic
205C4×C59#47211yescharacteristic
206#3038—591no—
207#2925D5911yescharacteristic, complemented
208#1545—591no—
209C2×#331—591no—
210#1477—591no—
211#1301—591no—
212#1292—591no—
213C2×C59—21no—
214C2×C59—21no—
215C2×C59—81no—
216C2×C59#187511yescharacteristic
217C2×C59#190311yescharacteristic
218C2×C59#190711yescharacteristic
219D59—162no—
220C2×AGL(1,ℤ/8)—1181no—
221C2×#127—591no—
222C4×#41C2×D5911yescharacteristic
223#460—592no—
224#460—1181no—
225#459—592no—
226#459—1181no—
227#433—22no—
228#432—21no—
229#431—21no—
230#431—591no—
231#423—591no—
232#423C2×D5911yescharacteristic
233C2×C4⋊C8—591no—
234C2×C4⋊C8C2×D5911yescharacteristic
235#418—1181no—
236#408—21no—
237#405—21no—
238#331—1181no—
239#331—1181no—
240#331—1182no—
241Q8⋊C8—1181no—
242D4⋊C8—1181no—
243C59#917411yescharacteristic, complemented, Sylow
244C2×#41C22⋊D5911yescharacteristic
245C22×D4—1181no—
246C22×D4—1181no—
247C22×D4—1181no—
248AGL(1,ℤ/8)—2362no—
249AGL(1,ℤ/8)—2362no—
250C2×SD16—1182no—
251C2×D8—1182no—
252C2×M16—21no—
253C2×M16—1181no—
254C22×C8—21no—
255C22×C8—1181no—
256#127—592no—
257#127—1181no—
258#127—1181no—
259#127—1181no—
260C4×Q8C4×D5911yescharacteristic
261C4×Q8D23611yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C4×D5911yescharacteristic
265C4×D4D23611yescharacteristic
266#117—21no—
267#117C22⋊D5911yescharacteristic
268C2×C42—21no—
269C2×C42C22×D5911yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—592no—
273C4⋊C8—1181no—
274C4⋊C8—1181no—
275C4⋊C8—1181no—
276C4⋊C8—1181no—
277C4⋊C8C22⋊D5911yescharacteristic
278C4⋊C8C22⋊D5911yescharacteristic
279C4⋊C8C22⋊D5912yes—
280#102—1182no—
281#102—1182no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—1181no—
285#97—22no—
286C4×C8—22no—
287C4×C8—1181no—
288C24—2361no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8(C2×D59)⋊C411yescharacteristic
294C2×D4—21no—
295C2×D4—1182no—
296C2×D4—1182no—
297C2×D4—1182no—
298C2×D4—4721no—
299C2×D4—4721no—
300C2×D4—4722no—
301C2×D4—2361no—
302C2×D4—2361no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2361no—
306C2×D4—2361no—
307C2×D4—2361no—
308C2×D4(C2×D59)⋊C411yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#1174568511yescharacteristic
313C22×C4#1174568511yescharacteristic
314C22×C4C2×C22⋊D5911yescharacteristic
315SD16—2364no—
316D8—2364no—
317M16—41no—
318M16—41no—
319M16—2361no—
320M16—2361no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—1182no—
327C2×C8—4721no—
328C2×C8—2361no—
329C2×C8—2361no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4(C2×D59)⋊C411yescharacteristic
333C4⋊C4(C2×D59)⋊C411yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C2×C4×D5911yescharacteristic
341C42C2×D23611yescharacteristic
342C42C2×C22⋊D5911yescharacteristic
343C42C2×C22⋊D5911yescharacteristic
344C23—21no—
345C23—4722no—
346C23—4722no—
347C23—4721no—
348C23—4721no—
349C23—2362no—
350C23#1427371011yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—4722no—
358D4—4722no—
359D4—4722no—
360D4—4721no—
361D4—4721no—
362D4—4721no—
363D4—4721no—
364D4—2362no—
365D4—2362no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×(C2×D59)⋊C411yescharacteristic
378C2×C4C4×C22⋊D5911yescharacteristic
379C2×C4C4×C22⋊D5911yescharacteristic
380C2×C4#1427369611yescharacteristic
381C2×C4#1427370111yescharacteristic
382C2×C4#1427370111yescharacteristic
383C2×C4#1427374711yescharacteristic
384C2×C4#1427374711yescharacteristic
385C2×C4#1427375211yescharacteristic
386C2×C4#1427375211yescharacteristic
387C8—41no—
388C8—41no—
389C8—42no—
390C8—4721no—
391C8—4721no—
392C8—2362no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—9442no—
399C22—9442no—
400C22—4722no—
401C22—4722no—
402C22—4722no—
403C22#1683728111yescharacteristic
404C22#1683772611yescharacteristic
405C22#1683772611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1683739611yescharacteristic
413C4#1683740611yescharacteristic
414C4#1683754711yescharacteristic
415C4#1683754711yescharacteristic
416C4#1683766511yescharacteristic
417C4#1683767211yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—9442no—
422C2#2142441811yescharacteristic
423C2#2142457611yescharacteristic
424C2#2142460411yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2925C2
socleC22×C59#386
prime radical#21433576C2
cosocleC2×C42×C59C23
hypercenter#2925D59
hypocenterC59#9174
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #21422765C2semidirect
3. C2×#16837145C2semidirect
4. #21422483C2semidirect
5. #21421993C2semidirect
6. #21421970C2semidirect
7. C59×#2925C2semidirect
207. #2925D59semidirect
243. C59#9174semidirect

Maximal subgroups

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

subgroupindex
#214335762
#214227652
C2×#168371452
#214224832
#214219932
#214219702
C59×#29252
#917459