Finite Group Databasebeta

← #28485416

Subgroups — #28485416

Order 15104.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435014C211yesmaximal, characteristic
2#21424928C211yesmaximal, characteristic, complemented
3#21422764C211yesmaximal, characteristic, complemented
4#21422655C211yesmaximal, characteristic, complemented
5#21421954C211yesmaximal, characteristic, complemented
6#21421925C211yesmaximal, characteristic, complemented
7C59×#2328C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837778C412yescomplemented
10#16837748C412yescomplemented
11#16837594—22no—
12#16837536—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×#14273655—21no—
17C2×#14273655—21no—
18#16837146—22no—
19#16837135—22no—
20C2×#127×C59C2211yescharacteristic
21C2×C4×D4×C59—21no—
22C59×#390—21no—
23C59×#386—22no—
24C59×#386—22no—
25C59×#382C2211yescharacteristic
26C59×#376C2211yescharacteristic
27C2×#11745685—21no—
28C2×#11745669—24no—
29C2×#11745664—21no—
30#14273727C2×C411yescharacteristic, complemented
31#14273724C2×C411yescharacteristic, complemented
32#14273715—21no—
33#14273701—21no—
34#14273701—21no—
35C4×D236—21no—
36C4×D236—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41#14273655—42no—
42#14273655—42no—
43#14273648—24no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8C2×C412yescomplemented
48C22×D4×C59—21no—
49C22×D4×C59—21no—
50C22×D4×C59D412yes—
51C23×C4×C59—22no—
52#127×C59—21no—
53#127×C59—21no—
54#127×C59C2×C412yes—
55#127×C59D412yes—
56C4×D4×C59—42no—
57C4×D4×C59—42no—
58C2×C4⋊C4×C59—21no—
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59—22no—
61C2×C22⋊C4×C59—22no—
62C2×C42×C59D411yescharacteristic
63C2×C42×C59D411yescharacteristic
64C2×C42×C59C2311yescharacteristic
65#95×C59—21no—
66#95×C59—21no—
67#95×C59—21no—
68#95×C59D412yes—
69C24×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—22no—
72C2×D4×C59—22no—
73C2×D4×C59—22no—
74C2×D4×C59—22no—
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59—41no—
78C2×D4×C59—41no—
79C2×D4×C59—82no—
80C2×D4×C59—82no—
81C2×D4×C59C22⋊C414yes—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—44no—
86C22×C4×C59—82no—
87C22×C4×C59—82no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C2×C22⋊D59—41no—
92#11745685—41no—
93#11745685—41no—
94#11745685—41no—
95#11745685—41no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×Dic118—21no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C22⋊C4×C59—44no—
104C22⋊C4×C59—82no—
105C22⋊C4×C59—82no—
106C42×C59—21no—
107C42×C59—21no—
108C42×C59—21no—
109C42×C59D812yescomplemented
110C42×C59C22×C411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59#4111yescharacteristic
113C42×C59#4111yescharacteristic
114#11745669—84no—
115(C2×D59)⋊C4—41no—
116(C2×D59)⋊C4—41no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119#11745664—41no—
120#11745664—41no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C23×C59—24no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—42no—
128C23×C59—42no—
129C23×C59—42no—
130C23×C59—42no—
131C23×C59#12011yescharacteristic
132C22×D59—41no—
133D4×C59—44no—
134D4×C59—42no—
135D4×C59—42no—
136D4×C59—42no—
137D4×C59—42no—
138D4×C59—82no—
139D4×C59—82no—
140D4×C59—82no—
141C2×C4×C59—22no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—41no—
152C2×C4×C59—88no—
153C2×C4×C59C2×C22⋊C411yescharacteristic
154C2×C4×C59C4×D411yescharacteristic
155C2×C4×C59C4×D411yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59#12311yescharacteristic
159C2×C4×C59C2×D811yescharacteristic
160C2×C4×C59#13511yescharacteristic
161C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C22⋊D59—81no—
164C22⋊D59—81no—
165C2×Dic59—41no—
166D236—42no—
167D236—41no—
168D236—41no—
169D236—81no—
170C4×D59—81no—
171C4×D59—81no—
172Dic118—41no—
173Dic118—41no—
174C59⋊C8—42no—
175C59⋊C8—81no—
176C59⋊C8—81no—
177#9299—591nomaximal, Sylow
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—44no—
181C22×C59—44no—
182C22×C59—44no—
183C22×C59—82no—
184C22×C59—82no—
185C22×C59—82no—
186C22×C59—82no—
187C22×C59#38611yescharacteristic
188C22×C59#44711yescharacteristic
189C22×C59#44911yescharacteristic
190C2×D59—81no—
191C2×D59—81no—
192C2×D59—81no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—42no—
198C4×C59—81no—
199C4×C59—82no—
200C4×C59C4×D811yescharacteristic
201C4×C59#44211yescharacteristic
202C4×C59#45911yescharacteristic
203C4×C59#46311yescharacteristic
204Dic59—81no—
205#3205—591no—
206#2328D5911yescharacteristic, complemented
207#1963—591no—
208#1545—591no—
209#1517—591no—
210#1286—591no—
211#1274—591no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—84no—
215C2×C59#187911yescharacteristic
216C2×C59#190411yescharacteristic
217C2×C59#191111yescharacteristic
218D59—161no—
219#532—591no—
220C2×#127C2×D5911yescharacteristic
221C2×C4×D4—21no—
222#493—592no—
223#486—592no—
224#459—1182no—
225#447—1182no—
226#431—591no—
227#431—591no—
228#423—591no—
229C2×#104—1181no—
230C2×#104—1181no—
231#390—21no—
232#386—22no—
233#386—22no—
234#382C2×D5911yescharacteristic
235#376C2×D5911yescharacteristic
236#331—1182no—
237#327—1182no—
238C59#929911yescharacteristic, complemented, Sylow
239C2×#41—1181no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C22⋊D5912yes—
243C23×C4—22no—
244C2×D8—1184no—
245C2×M16—1181no—
246C4⋊Q8—591no—
247#127—21no—
248#127—21no—
249#127—591no—
250#127C4×D5912yes—
251#127D23612yes—
252#124—1181no—
253#121—1181no—
254#121—1181no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—1181no—
258C4×D4—1181no—
259C2×C4⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C42C22⋊D5911yescharacteristic
264C2×C42C22⋊D5911yescharacteristic
265C2×C42C22×D5911yescharacteristic
266C4⋊C8—1181no—
267C4⋊C8—1181no—
268C4⋊C8—1181no—
269C4⋊C8—1181no—
270#104—2362no—
271#104—2362no—
272#102—1184no—
273C22⋊C8—1181no—
274C22⋊C8—1181no—
275C4×C8—592no—
276C4×C8—1181no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95C22⋊D5912yes—
281C24—22no—
282#41—2361no—
283#41—2361no—
284#41—2361no—
285#41—2361no—
286C2×Q8—1181no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—1181no—
299C2×D4—1181no—
300C2×D4—1181no—
301C2×D4—2361no—
302C2×D4(C2×D59)⋊C414yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—44no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—2361no—
310C22×C4C2×C22⋊D5911yescharacteristic
311C22×C4C2×C22⋊D5911yescharacteristic
312C22×C4C2×C22⋊D5911yescharacteristic
313D8—4724no—
314M16—2361no—
315M16—2361no—
316C2×C8—1182no—
317C2×C8—2361no—
318C2×C8—2361no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—1181no—
322C4⋊C4—1181no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42#1174566912yescomplemented
332C42C2×C4×D5911yescharacteristic
333C42C2×D23611yescharacteristic
334C42#1174568511yescharacteristic
335C42#1174568511yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—2361no—
344C23#1427369611yescharacteristic
345Q8—2361no—
346Q8—2361no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—4721no—
356D4—4721no—
357D4—4721no—
358D4—2362no—
359D4—2361no—
360D4—2361no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—88no—
373C2×C4—4721no—
374C2×C4—4721no—
375C2×C4—2361no—
376C2×C4C2×(C2×D59)⋊C411yescharacteristic
377C2×C4C4×C22⋊D5911yescharacteristic
378C2×C4C4×C22⋊D5911yescharacteristic
379C2×C4#1427370111yescharacteristic
380C2×C4#1427370111yescharacteristic
381C2×C4#1427371011yescharacteristic
382C2×C4C2×#1174566911yescharacteristic
383C2×C4#1427374711yescharacteristic
384C2×C4#1427375211yescharacteristic
385C2×C4#1427375311yescharacteristic
386C8—4721no—
387C8—4721no—
388C8—2362no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—4721no—
399C22—4721no—
400C22—4721no—
401C22#1683728111yescharacteristic
402C22#1683753611yescharacteristic
403C22#1683754711yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—82no—
410C4—81no—
411C4—4721no—
412C4C4×#1174566911yescharacteristic
413C4#1683751211yescharacteristic
414C4#1683759711yescharacteristic
415C4#1683761711yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—9441no—
420C2#2142444411yescharacteristic
421C2#2142458311yescharacteristic
422C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2328C2
socleC22×C59#386
prime radical#21435014C2
cosocleC2×C42×C59C23
hypercenter#2328D59
hypocenterC59#9299
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #21424928C2semidirect
3. #21422764C2semidirect
4. #21422655C2semidirect
5. #21421954C2semidirect
6. #21421925C2semidirect
7. C59×#2328C2semidirect
9. #16837778C4semidirect
10. #16837748C4semidirect
30. #14273727C2×C4semidirect
31. #14273724C2×C4semidirect
47. C4×C59⋊C8C2×C4semidirect
109. C42×C59D8semidirect
206. #2328D59semidirect
238. C59#9299semidirect
331. C42#11745669semidirect

Maximal subgroups

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

subgroupindex
#214350142
#214249282
#214227642
#214226552
#214219542
#214219252
C59×#23282
#929959