Finite Group Databasebeta

← #28506935

Subgroups — #28506935

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434520C211yesmaximal, characteristic
2C2×#16837213C211yesmaximal, characteristic, complemented
3C2×#16837212C211yesmaximal, characteristic, complemented
4C2×#16837211C211yesmaximal, characteristic, complemented
5M32×Dic59C211yesmaximal, characteristic, complemented
6#21424251C211yesmaximal, characteristic, complemented
7C2×C59×#360C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9C2×#14273747—21no—
10#16837948—21no—
11C2×#14273667C2211yescharacteristic
12#16837787C411yescharacteristic, complemented
13#16837785C411yescharacteristic, complemented
14#16837783—21no—
15#16837780C411yescharacteristic, complemented
16#16837777C411yescharacteristic, complemented
17#16837773—21no—
18C2×#14273659C2211yescharacteristic
19C2×#14273658—21no—
20C2×C8×Dic59C2211yescharacteristic
21#16837271—21no—
22#16837213—22no—
23#16837212—22no—
24#16837211—22no—
25#16837177—21no—
26C16×Dic59—21no—
27C2×#135×C59C2211yescharacteristic
28C2×M32×C59C2211yescharacteristic
29C2×#106×C59C2211yescharacteristic
30C59×#360—22no—
31C2×#11745685—21no—
32#14273749—21no—
33#14273749—21no—
34#14273749—21no—
35#14273749—21no—
36#14273747—42no—
37#14273747—42no—
38C2×Dic236C2×C411yescharacteristic
39C2×#11745672—21no—
40C2×#11745660—21no—
41C2×#11745670—21no—
42C2×#11745671—21no—
43C2×D472C2×C411yescharacteristic
44C2×#11745669—21no—
45C22×C59⋊C8D411yescharacteristic
46#14273725—21no—
47#14273723—21no—
48#14273716—21no—
49#14273714—21no—
50#14273699—42no—
51C4×C22⋊D59—42no—
52C2×C4⋊Dic59—21no—
53C2×Dic59⋊C4—42no—
54C2×C4×Dic59D411yescharacteristic
55#14273667—21no—
56#14273667D412yescomplemented
57C2×C59⋊C16—21no—
58#14273659—21no—
59#14273659—21no—
60#14273658—42no—
61C8×Dic59C2×C411yescharacteristic
62C8×Dic59C2×C411yescharacteristic
63C8×Dic59D411yescharacteristic
64C8×Dic59D411yescharacteristic
65C2×#41×C59—21no—
66#135×C59—21no—
67#135×C59—21no—
68#135×C59—21no—
69#135×C59—21no—
70C2×Q16×C59C2×C411yescharacteristic
71C2×SD16×C59—21no—
72C2×D8×C59C2×C411yescharacteristic
73C22×C8×C59C2311yescharacteristic
74C2×C4⋊C4×C59—21no—
75M32×C59—21no—
76M32×C59D412yescomplemented
77C2×C16×C59—21no—
78#106×C59—21no—
79#106×C59—21no—
80#41×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83#41×C59—41no—
84C2×Q8×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59C2×D411yescharacteristic
90C2×C22⋊D59—41no—
91C2×C22⋊D59—42no—
92C22×Dic59—41no—
93C22×Dic59D812yes—
94#11745685—41no—
95#11745685—41no—
96#11745685—41no—
97#11745685—41no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×Dic118—21no—
101Q16×C59—21no—
102Q16×C59—21no—
103SD16×C59—41no—
104SD16×C59—41no—
105D8×C59—21no—
106D8×C59—21no—
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59#4111yescharacteristic
112C2×C8×C59#4111yescharacteristic
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C22⋊Dic59—42no—
116#11745672—42no—
117#11745671—42no—
118#11745670—42no—
119#11745669—42no—
120(C2×D59)⋊C4—42no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123Dic59⋊C4—42no—
124Dic59⋊C4—84no—
125C4×Dic59C22⋊C411yescharacteristic
126C4×Dic59C22⋊C411yescharacteristic
127C4×Dic59SD1612yes—
128C2×C59⋊C8C22⋊C411yescharacteristic
129C2×C59⋊C8C22⋊C411yescharacteristic
130C2×C59⋊C8D812yes—
131C2×C59⋊C8SD1612yes—
132Dic236—21no—
133Dic236—21no—
134D472—21no—
135D472—21no—
136#11745660—41no—
137#11745660—41no—
138C16×C59—22no—
139C59⋊C16—22no—
140C23×C59—41no—
141C23×C59C2×D811yescharacteristic
142C22×D59—41no—
143Q8×C59—41no—
144Q8×C59—41no—
145D4×C59—41no—
146D4×C59—41no—
147D4×C59—81no—
148D4×C59—81no—
149C2×C4×C59—41no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59—82no—
153C2×C4×C59—81no—
154C2×C4×C59C2×C22⋊C411yescharacteristic
155C2×C4×C59#12411yescharacteristic
156C2×C4×C59#12711yescharacteristic
157C2×C4×C59C2×D811yescharacteristic
158C2×C4×C59C2×SD1611yescharacteristic
159C2×C4×C59C2×SD1611yescharacteristic
160C22⋊D59—81no—
161C22⋊D59—81no—
162C22⋊D59—84no—
163C2×Dic59—24no—
164C2×Dic59—41no—
165C2×Dic59—82no—
166C2×Dic59—81no—
167C2×Dic59#10212yes—
168C2×Dic59#10212yes—
169D236—41no—
170D236—41no—
171C4×D59—81no—
172C4×D59—81no—
173Dic118—41no—
174Dic118—41no—
175C8×C59C4×D411yescharacteristic
176C8×C59C4×D411yescharacteristic
177C8×C59#12111yescharacteristic
178C8×C59#12111yescharacteristic
179C59⋊C8#10212yes—
180C59⋊C8#10212yes—
181#13210—591nomaximal, Sylow
182C22×C59—22no—
183C22×C59—82no—
184C22×C59—81no—
185C22×C59C2×#10211yescharacteristic
186C22×C59#48611yescharacteristic
187C22×C59#49311yescharacteristic
188C2×D59—82no—
189C2×D59—81no—
190C4×C59—81no—
191C4×C59—161no—
192C4×C59#39011yescharacteristic
193C4×C59C2×#10211yescharacteristic
194C4×C59#48611yescharacteristic
195C4×C59#49211yescharacteristic
196Dic59—22no—
197Dic59—81no—
198Dic59—161no—
199Dic59#36014yescomplemented
200#3147—591no—
201C2×#360—591no—
202C2×#360—591no—
203C2×#360—591no—
204C2×#360D5911yescharacteristic, complemented
205C4×M32—591no—
206#1844—591no—
207C2×C59—21no—
208C2×C59—21no—
209C2×C59—161no—
210C2×C59#195711yescharacteristic
211C2×C59C2×#36012yes—
212D59—161no—
213C2×#135—591no—
214C2×#135—1181no—
215C2×#135C2×D5911yescharacteristic
216#532—1181no—
217C2×M32—591no—
218C2×M32C2×D5911yescharacteristic
219#495—591no—
220#495—591no—
221#495—1181no—
222#494—591no—
223#493—591no—
224#492—1181no—
225C2×#106—591no—
226C2×#106—1181no—
227C2×#106C2×D5911yescharacteristic
228C2×C4×C8—591no—
229#384—1181no—
230#360—22no—
231#360—1182no—
232#360—1182no—
233#360—1182no—
234#346—1181no—
235C4×C16—1181no—
236C59#1321011yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C2×#41—1181no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135—21no—
243#135—1181no—
244#135—1181no—
245#135—1181no—
246#135—1181no—
247#135—2362no—
248#135—2362no—
249C2×Q16—591no—
250C2×Q16—1181no—
251C2×Q16C4×D5911yescharacteristic
252C2×SD16—21no—
253C2×SD16—1181no—
254C2×SD16—1181no—
255C2×SD16—1181no—
256C2×D8—591no—
257C2×D8—1181no—
258C2×D8C4×D5911yescharacteristic
259C22×C8—591no—
260C22×C8C22×D5911yescharacteristic
261C4⋊Q8—1181no—
262#127—1181no—
263#124—1181no—
264#124—1181no—
265#121—2362no—
266C4×D4—2362no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1181no—
269C2×C4⋊C4—2362no—
270C2×C42—591no—
271M32—21no—
272M32—592no—
273M32—1181no—
274M32C22⋊D5912yescomplemented
275C2×C16—21no—
276C2×C16—1181no—
277#106—21no—
278#106—21no—
279#106—1181no—
280#106—1181no—
281#106—2362no—
282C4×C8—591no—
283C4×C8—591no—
284C4×C8—591no—
285C4×C8—591no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—2361no—
291#41—2361no—
292#41—2361no—
293#41—2361no—
294C2×Q8—21no—
295C2×Q8—1181no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—1181no—
299C2×D4—2362no—
300C2×D4—2361no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—592no—
304C22×C4—2361no—
305C22×C4—2361no—
306C22×C4D4×D5911yescharacteristic
307Q16—21no—
308Q16—21no—
309Q16—1181no—
310Q16—1181no—
311Q16—2362no—
312SD16—41no—
313SD16—41no—
314SD16—2362no—
315SD16—2362no—
316SD16—2361no—
317SD16—2361no—
318D8—21no—
319D8—21no—
320D8—1181no—
321D8—1181no—
322D8—2362no—
323C2×C8—591no—
324C2×C8—591no—
325C2×C8—592no—
326C2×C8—592no—
327C2×C8C2×C4×D5911yescharacteristic
328C2×C8#1174568511yescharacteristic
329C2×C8D4×D5911yescharacteristic
330C2×C8D4×D5911yescharacteristic
331C2×C8#1174568711yescharacteristic
332C2×C8C2×C22⋊D5911yescharacteristic
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—4724no—
336C4⋊C4—2362no—
337C4⋊C4—2361no—
338C4⋊C4—2361no—
339C22⋊C4—2362no—
340C22⋊C4—2362no—
341C42—591no—
342C42—591no—
343C42—592no—
344C16—22no—
345C16—1182no—
346C23—41no—
347C23—2361no—
348C23D8×D5911yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—2361no—
352Q8—2361no—
353D4—41no—
354D4—41no—
355D4—81no—
356D4—81no—
357D4—4724no—
358D4—4721no—
359D4—4721no—
360D4—2361no—
361D4—2361no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—592no—
368C2×C4—592no—
369C2×C4—1184no—
370C2×C4—4722no—
371C2×C4—4721no—
372C2×C4—4721no—
373C2×C4—4721no—
374C2×C4—2361no—
375C2×C4C22⋊C4×D5911yescharacteristic
376C2×C4#1427371311yescharacteristic
377C2×C4#1427372311yescharacteristic
378C2×C4D8×D5911yescharacteristic
379C2×C4SD16×D5911yescharacteristic
380C2×C4SD16×D5911yescharacteristic
381C8—592no—
382C8—592no—
383C8(C22⋊D59)⋊C411yescharacteristic
384C8C4×C22⋊D5911yescharacteristic
385C8#1427369911yescharacteristic
386C8#1427370011yescharacteristic
387C22—22no—
388C22—81no—
389C22—82no—
390C22—4721no—
391C22—4722no—
392C22#102×D5911yescharacteristic
393C22#1683774711yescharacteristic
394C22#1683777711yescharacteristic
395C4—81no—
396C4—161no—
397C4—594no—
398C4—1182no—
399C4—9441no—
400C4—4721no—
401C4#1683730211yescharacteristic
402C4#102×D5911yescharacteristic
403C4#1683774711yescharacteristic
404C4#1683777311yescharacteristic
405C2—21no—
406C2—21no—
407C2—161no—
408C2—9441no—
409C2#2142490011yescharacteristic
410C2#360×D5912yes—
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#360C2
socleC22×C59C2×#102
prime radical#21434520C2
cosocleC22×C8×C59C23
hypercenterC2×#360D59
hypocenterC59#13210
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
2. C2×#16837213C2semidirect
3. C2×#16837212C2semidirect
4. C2×#16837211C2semidirect
5. M32×Dic59C2semidirect
6. #21424251C2semidirect
7. C2×C59×#360C2semidirect
12. #16837787C4semidirect
13. #16837785C4semidirect
15. #16837780C4semidirect
16. #16837777C4semidirect
56. #14273667D4semidirect
76. M32×C59D4semidirect
199. Dic59#360semidirect
204. C2×#360D59semidirect
236. C59#13210semidirect
274. M32C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214345202
C2×#168372132
C2×#168372122
C2×#168372112
M32×Dic592
#214242512
C2×C59×#3602
#1321059