Finite Group Databasebeta

← #28484591

Subgroups — #28484591

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#21432762C211yesmaximal, characteristic, complemented
2#21422764C211yesmaximal, characteristic, complemented
3C2×#16837146C211yesmaximal, characteristic, complemented
4#21422483C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421970C211yesmaximal, characteristic, complemented
7C59×#3038C211yesmaximal, characteristic
8#16837463—22no—
9#16837459—21no—
10#16837455—21no—
11#16837455C2211yescharacteristic
12#16837426C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837369—21no—
16#16837358—21no—
17#16837146—21no—
18#16837146—21no—
19#16837146—22no—
20Q8⋊(C59⋊C8)—21no—
21D4⋊(C59⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C59—21no—
23C2×#127×C59C2211yescharacteristic
24C4×#41×C59C2211yescharacteristic
25C59×#460—21no—
26C59×#460C412yes—
27C59×#459—21no—
28C59×#459C412yes—
29C59×#423C2211yescharacteristic
30C59×#418—21no—
31C2×#11745664—21no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34C4⋊(C59⋊C8)—21no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40C4⋊(C59⋊C8)D411yescharacteristic
41C4⋊(C59⋊C8)D412yes—
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—22no—
44C22⋊(C59⋊C8)—21no—
45#14273638—22no—
46C4×C59⋊C8—22no—
47C4×C59⋊C8—21no—
48C2×#41×C59D411yescharacteristic
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C22×D4×C59—21no—
52AGL(1,ℤ/8)×C59—42no—
53AGL(1,ℤ/8)×C59—42no—
54C2×SD16×C59—22no—
55C2×D8×C59—22no—
56C2×M16×C59—21no—
57#127×C59—21no—
58#127×C59—21no—
59#127×C59—21no—
60#127×C59C2×C412yes—
61C4×Q8×C59C2×C411yescharacteristic
62C4×Q8×C59D411yescharacteristic
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59C2×C411yescharacteristic
66C4×D4×C59D411yescharacteristic
67#117×C59—21no—
68#117×C59D411yescharacteristic
69C2×C42×C59—21no—
70C2×C42×C59C2311yescharacteristic
71C4⋊C8×C59—21no—
72C4⋊C8×C59C2×C412yes—
73#102×C59—22no—
74#102×C59—22no—
75C24×C59—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×C59—22no—
83C2×D4×C59—22no—
84C2×D4×C59—22no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59—81no—
93C2×D4×C59—81no—
94C2×D4×C59—82no—
95C2×D4×C59C22⋊C411yescharacteristic
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—21no—
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59#4111yescharacteristic
101C22×C4×C59#4111yescharacteristic
102SD16×C59—44no—
103D8×C59—44no—
104M16×C59—41no—
105M16×C59—41no—
106C2×C8×C59—22no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59C22⋊C411yescharacteristic
110C4⋊C4×C59C22⋊C411yescharacteristic
111C22⋊C4×C59—21no—
112C22⋊C4×C59—21no—
113C22⋊C4×C59—21no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—21no—
117C42×C59C22×C411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59C2×D411yescharacteristic
121#11745664—41no—
122#11745664—41no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—81no—
131C23×C59—21no—
132C23×C59—42no—
133C23×C59—82no—
134C23×C59—82no—
135C23×C59—81no—
136C23×C59—81no—
137C23×C59#12311yescharacteristic
138Q8×C59—21no—
139Q8×C59—21no—
140D4×C59—21no—
141D4×C59—21no—
142D4×C59—41no—
143D4×C59—41no—
144D4×C59—42no—
145D4×C59—42no—
146D4×C59—82no—
147D4×C59—82no—
148D4×C59—82no—
149D4×C59—81no—
150D4×C59—81no—
151D4×C59—81no—
152D4×C59—81no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—81no—
164C2×C4×C59C2×C22⋊C411yescharacteristic
165C2×C4×C59C4×D411yescharacteristic
166C2×C4×C59C4×D411yescharacteristic
167C2×C4×C59#12011yescharacteristic
168C2×C4×C59#12111yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#13511yescharacteristic
171C2×C4×C59#13511yescharacteristic
172C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
174C8×C59—42no—
175C59⋊C8—41no—
176C59⋊C8—41no—
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—82no—
187C22×C59—82no—
188C22×C59—82no—
189C22×C59—162no—
190C22×C59—162no—
191C22×C59#38611yescharacteristic
192C22×C59#48211yescharacteristic
193C22×C59#48211yescharacteristic
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#3038D5911yescharacteristic
207#2925—591no—
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—162no—
217C2×C59#187511yescharacteristic
218C2×C59#190311yescharacteristic
219C2×C59#190711yescharacteristic
220C2×AGL(1,ℤ/8)—21no—
221C2×#127C2×D5911yescharacteristic
222C4×#41C2×D5911yescharacteristic
223#460—21no—
224#460Dic5912yes—
225#459—21no—
226#459Dic5912yes—
227#433—1182no—
228#432—1181no—
229#431—591no—
230#431—1181no—
231#423—591no—
232#423C2×D5911yescharacteristic
233C2×C4⋊C8—591no—
234C2×C4⋊C8—591no—
235#418—21no—
236#408—1181no—
237#405—1181no—
238#331—1181no—
239#331—1181no—
240#331—1182no—
241Q8⋊C8—1181no—
242D4⋊C8—1181no—
243C59#917411yescharacteristic, complemented, Sylow
244C2×#41C22⋊D5911yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16—21no—
253C2×M16—1181no—
254C22×C8—1181no—
255C22×C8—1181no—
256#127—21no—
257#127—21no—
258#127—21no—
259#127C2×Dic5912yes—
260C4×Q8C2×Dic5911yescharacteristic
261C4×Q8C22⋊D5911yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C2×Dic5911yescharacteristic
265C4×D4C22⋊D5911yescharacteristic
266#117—21no—
267#117C22⋊D5911yescharacteristic
268C2×C42—21no—
269C2×C42C22×D5911yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—591no—
272C4⋊C8—591no—
273C4⋊C8—592no—
274C4⋊C8—1181no—
275C4⋊C8—1181no—
276C4⋊C8—1181no—
277C4⋊C8—1181no—
278C4⋊C8—1181no—
279C4⋊C8C2×Dic5912yes—
280#102—22no—
281#102—22no—
282C22⋊C8—1182no—
283C22⋊C8—1182no—
284C22⋊C8—1181no—
285#97—1182no—
286C4×C8—1182no—
287C4×C8—1181no—
288C24—41no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8C22⋊Dic5911yescharacteristic
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—81no—
306C2×D4—81no—
307C2×D4—82no—
308C2×D4C22⋊Dic5911yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#1174568711yescharacteristic
313C22×C4#1174568711yescharacteristic
314C22×C4C2×C22⋊D5911yescharacteristic
315SD16—44no—
316D8—44no—
317M16—41no—
318M16—41no—
319M16—2361no—
320M16—2361no—
321C2×C8—22no—
322C2×C8—1181no—
323C2×C8—1182no—
324C2×C8—1181no—
325C2×C8—4721no—
326C2×C8—2361no—
327C2×C8—2361no—
328C2×C8—2361no—
329C2×C8—2361no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4C22⋊Dic5911yescharacteristic
333C4⋊C4C22⋊Dic5911yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42D4×D5911yescharacteristic
341C42D4×D5911yescharacteristic
342C42C22×Dic5911yescharacteristic
343C42C2×C22⋊D5911yescharacteristic
344C23—21no—
345C23—42no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23#1427370811yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
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×C22⋊Dic5911yescharacteristic
378C2×C4D4×Dic5911yescharacteristic
379C2×C4D4×Dic5911yescharacteristic
380C2×C4#1427369511yescharacteristic
381C2×C4#1427369911yescharacteristic
382C2×C4#1427369911yescharacteristic
383C2×C4#1427374711yescharacteristic
384C2×C4#1427374711yescharacteristic
385C2×C4#1427375311yescharacteristic
386C2×C4#1427375311yescharacteristic
387C8—42no—
388C8—4721no—
389C8—4721no—
390C8—2361no—
391C8—2361no—
392C8—2362no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—162no—
402C22—162no—
403C22#1683728211yescharacteristic
404C22#1683772311yescharacteristic
405C22#1683772311yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1683739811yescharacteristic
413C4#1683740711yescharacteristic
414C4#1683754511yescharacteristic
415C4#1683754511yescharacteristic
416C4#1683766311yescharacteristic
417C4#1683767011yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—162no—
422C2#2142441911yescharacteristic
423C2#2142457811yescharacteristic
424C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3038C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3038D59
hypocenterC59#9174
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21432762C2semidirect
2. #21422764C2semidirect
3. C2×#16837146C2semidirect
4. #21422483C2semidirect
5. #21421991C2semidirect
6. #21421970C2semidirect
243. C59#9174semidirect

Maximal subgroups

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

subgroupindex
#214327622
#214227642
C2×#168371462
#214224832
#214219912
#214219702
C59×#30382
#917459