Finite Group Databasebeta

← #28506931

Subgroups — #28506931

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434520C211yesmaximal, characteristic
2C2×#16837210C211yesmaximal, characteristic, complemented
3C2×#16837209C211yesmaximal, characteristic, complemented
4C2×#16837208C211yesmaximal, characteristic, complemented
5C2×#16837177C211yesmaximal, characteristic, complemented
6#21424266C211yesmaximal, characteristic, complemented
7C2×C59×#359C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9C2×#14273747—21no—
10#16837948—21no—
11C22×C59⋊C16C2211yescharacteristic
12#16837787C411yescharacteristic
13#16837785C411yescharacteristic
14#16837783—21no—
15#16837780C411yescharacteristic
16#16837777C411yescharacteristic
17#16837773—21no—
18C2×#14273663C2211yescharacteristic
19C2×#14273662—21no—
20#16837425—21no—
21C2×C8×Dic59C2211yescharacteristic
22#16837210—22no—
23#16837209—22no—
24#16837208—22no—
25#16837177—21no—
26#16837177—21no—
27C2×#135×C59C2211yescharacteristic
28C22×C16×C59C2211yescharacteristic
29C2×#108×C59C2211yescharacteristic
30C59×#359—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—
45C2×#11745664—21no—
46C22×C59⋊C8D411yescharacteristic
47#14273725—21no—
48#14273723—21no—
49#14273716—21no—
50#14273714—21no—
51#14273699—42no—
52C4×C22⋊D59—42no—
53C2×C4×Dic59D411yescharacteristic
54C2×C59⋊C16—21no—
55C2×C59⋊C16—21no—
56C2×C59⋊C16D412yescomplemented
57#14273663—21no—
58#14273663—21no—
59#14273662—42no—
60Dic59⋊C8—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
73C2×M16×C59—21no—
74C22×C8×C59C2311yescharacteristic
75C2×C16×C59—21no—
76C2×C16×C59—21no—
77C2×C16×C59D412yescomplemented
78#108×C59—21no—
79#108×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×C59C2×D411yescharacteristic
89C2×C22⋊D59—41no—
90C2×C22⋊D59—42no—
91C22×Dic59SD1612yes—
92#11745685—41no—
93#11745685—41no—
94#11745685—41no—
95#11745685—41no—
96C2×D236—21no—
97C2×C4×D59—41no—
98C2×Dic118—21no—
99Q16×C59—21no—
100Q16×C59—21no—
101SD16×C59—41no—
102SD16×C59—41no—
103D8×C59—21no—
104D8×C59—21no—
105M16×C59—41no—
106M16×C59—41no—
107C2×C8×C59—41no—
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59#4111yescharacteristic
113C2×C8×C59#4111yescharacteristic
114C22⋊Dic59—42no—
115#11745672—42no—
116#11745671—42no—
117#11745670—42no—
118#11745669—42no—
119(C2×D59)⋊C4—42no—
120Dic59⋊C4—42no—
121C4×Dic59C22⋊C411yescharacteristic
122C4×Dic59C22⋊C411yescharacteristic
123C4×Dic59D812yes—
124#11745664—41no—
125#11745664—41no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8C22⋊C411yescharacteristic
128C2×C59⋊C8C22⋊C411yescharacteristic
129C2×C59⋊C8D812yes—
130C2×C59⋊C8SD1612yes—
131Dic236—21no—
132Dic236—21no—
133D472—21no—
134D472—21no—
135#11745660—41no—
136#11745660—41no—
137C16×C59—22no—
138C59⋊C16—22no—
139C23×C59—41no—
140C23×C59C2×SD1611yescharacteristic
141C22×D59—41no—
142Q8×C59—41no—
143Q8×C59—41no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—81no—
147D4×C59—81no—
148C2×C4×C59—41no—
149C2×C4×C59—81no—
150C2×C4×C59—81no—
151C2×C4×C59C2×C22⋊C411yescharacteristic
152C2×C4×C59#12411yescharacteristic
153C2×C4×C59#12711yescharacteristic
154C2×C4×C59C2×D811yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59C2×SD1611yescharacteristic
157C22⋊D59—84no—
158C22⋊D59—81no—
159C22⋊D59—81no—
160C2×Dic59—42no—
161C2×Dic59—41no—
162C2×Dic59#10212yes—
163C2×Dic59#10212yes—
164D236—41no—
165D236—41no—
166C4×D59—81no—
167C4×D59—81no—
168Dic118—41no—
169Dic118—41no—
170C8×C59—81no—
171C8×C59C4×D411yescharacteristic
172C8×C59C4×D411yescharacteristic
173C8×C59#12111yescharacteristic
174C8×C59#12111yescharacteristic
175C59⋊C8—81no—
176C59⋊C8#10212yes—
177C59⋊C8#10212yes—
178#13209—591nomaximal, Sylow
179C22×C59—22no—
180C22×C59—82no—
181C22×C59—81no—
182C22×C59C2×#10211yescharacteristic
183C22×C59#48611yescharacteristic
184C22×C59#49211yescharacteristic
185C2×D59—82no—
186C2×D59—81no—
187C4×C59—81no—
188C4×C59#39011yescharacteristic
189C4×C59C2×#10211yescharacteristic
190C4×C59#48611yescharacteristic
191C4×C59#49311yescharacteristic
192Dic59—22no—
193Dic59—22no—
194Dic59—81no—
195#3147—591no—
196C2×#359—591no—
197C2×#359—591no—
198C2×#359—591no—
199C2×#359D5911yescharacteristic, complemented
200C2×#346—591no—
201#1848—591no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59—161no—
205C2×C59#195711yescharacteristic
206C2×C59#214312yes—
207D59—161no—
208C2×#135—591no—
209C2×#135—1181no—
210C2×#135C2×D5911yescharacteristic
211#532—1181no—
212C22×C16—591no—
213C22×C16C2×D5911yescharacteristic
214#495—591no—
215#495—591no—
216#495—1181no—
217#494—591no—
218#493—591no—
219#492—1181no—
220C2×#108—591no—
221C2×#108—1181no—
222C2×#108C2×D5911yescharacteristic
223#423—1181no—
224C2×C4×C8—591no—
225#359—22no—
226#359—1182no—
227#359—1182no—
228#359—1182no—
229#346—1181no—
230#346—1181no—
231C59#1320911yescharacteristic, complemented, Sylow
232C2×#41—21no—
233C2×#41—1181no—
234#135—21no—
235#135—21no—
236#135—21no—
237#135—21no—
238#135—1181no—
239#135—1181no—
240#135—1181no—
241#135—1181no—
242#135—2362no—
243#135—2362no—
244C2×Q16—591no—
245C2×Q16—1181no—
246C2×Q16C4×D5911yescharacteristic
247C2×SD16—21no—
248C2×SD16—1181no—
249C2×SD16—1181no—
250C2×SD16—1181no—
251C2×D8—591no—
252C2×D8—1181no—
253C2×D8C4×D5911yescharacteristic
254C2×M16—21no—
255C2×M16—1181no—
256C22×C8—591no—
257C22×C8C22×D5911yescharacteristic
258C4⋊Q8—1181no—
259#127—1181no—
260#124—1181no—
261#124—1181no—
262#121—2362no—
263C4×D4—2362no—
264C2×C42—591no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—592no—
268C2×C16—1181no—
269C2×C16—1181no—
270C2×C16C22⋊D5912yescomplemented
271#108—21no—
272#108—21no—
273#108—1181no—
274#108—1181no—
275#108—2362no—
276C4⋊C8—2362no—
277C4×C8—591no—
278C4×C8—591no—
279C4×C8—591no—
280C4×C8—591no—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—2361no—
286#41—2361no—
287#41—2361no—
288#41—2361no—
289C2×Q8—21no—
290C2×Q8—1181no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1181no—
294C2×D4—2362no—
295C2×D4—2361no—
296C22×C4—41no—
297C22×C4—592no—
298C22×C4—2361no—
299C22×C4D4×D5911yescharacteristic
300Q16—21no—
301Q16—21no—
302Q16—1181no—
303Q16—1181no—
304Q16—2362no—
305SD16—41no—
306SD16—41no—
307SD16—2362no—
308SD16—2362no—
309SD16—2361no—
310SD16—2361no—
311D8—21no—
312D8—21no—
313D8—1181no—
314D8—1181no—
315D8—2362no—
316M16—41no—
317M16—41no—
318M16—2361no—
319M16—2361no—
320C2×C8—41no—
321C2×C8—591no—
322C2×C8—591no—
323C2×C8—592no—
324C2×C8—592no—
325C2×C8—2361no—
326C2×C8C2×C4×D5911yescharacteristic
327C2×C8#1174568511yescharacteristic
328C2×C8D4×D5911yescharacteristic
329C2×C8D4×D5911yescharacteristic
330C2×C8#1174568711yescharacteristic
331C2×C8C2×C22⋊D5911yescharacteristic
332C4⋊C4—2362no—
333C22⋊C4—2362no—
334C22⋊C4—2362no—
335C42—591no—
336C42—591no—
337C42—592no—
338C16—22no—
339C16—1182no—
340C23—41no—
341C23—2361no—
342C23SD16×D5911yescharacteristic
343Q8—41no—
344Q8—41no—
345Q8—2361no—
346Q8—2361no—
347D4—41no—
348D4—41no—
349D4—81no—
350D4—81no—
351D4—4724no—
352D4—4721no—
353D4—4721no—
354D4—2361no—
355D4—2361no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—592no—
360C2×C4—592no—
361C2×C4—4721no—
362C2×C4—4721no—
363C2×C4—2362no—
364C2×C4—2361no—
365C2×C4C22⋊C4×D5911yescharacteristic
366C2×C4#1427371311yescharacteristic
367C2×C4#1427372311yescharacteristic
368C2×C4D8×D5911yescharacteristic
369C2×C4D8×D5911yescharacteristic
370C2×C4SD16×D5911yescharacteristic
371C8—81no—
372C8—592no—
373C8—592no—
374C8—4721no—
375C8(C22⋊D59)⋊C411yescharacteristic
376C8C4×C22⋊D5911yescharacteristic
377C8#1427369911yescharacteristic
378C8#1427370011yescharacteristic
379C22—22no—
380C22—81no—
381C22—82no—
382C22—4721no—
383C22—4722no—
384C22#102×D5911yescharacteristic
385C22#1683774711yescharacteristic
386C22#1683777311yescharacteristic
387C4—81no—
388C4—1182no—
389C4—1182no—
390C4—4721no—
391C4#1683730211yescharacteristic
392C4#102×D5911yescharacteristic
393C4#1683774711yescharacteristic
394C4#1683777711yescharacteristic
395C2—21no—
396C2—21no—
397C2—161no—
398C2—9441no—
399C2#2142490011yescharacteristic
400C2#2142582212yes—
subgroupG/H
centerC2×C4D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#359C2
socleC22×C59C2×#102
prime radical#21434520C2
cosocleC22×C8×C59C23
hypercenterC2×#359D59
hypocenterC59#13209
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C2×#16837210C2semidirect
3. C2×#16837209C2semidirect
4. C2×#16837208C2semidirect
5. C2×#16837177C2semidirect
6. #21424266C2semidirect
7. C2×C59×#359C2semidirect
56. C2×C59⋊C16D4semidirect
77. C2×C16×C59D4semidirect
199. C2×#359D59semidirect
231. C59#13209semidirect
270. C2×C16C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214345202
C2×#168372102
C2×#168372092
C2×#168372082
C2×#168371772
#214242662
C2×C59×#3592
#1320959