Finite Group Databasebeta

← #28744267

Subgroups — #28744267

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434666C211yesmaximal, characteristic
2#21426379C211yesmaximal, characteristic, complemented
3#21426365C211yesmaximal, characteristic, complemented
4#21426341C211yesmaximal, characteristic, complemented
5#21426335C211yesmaximal, characteristic, complemented
6#21426211C212yesmaximal
7#21426202C212yesmaximal, complemented
8C4×#14273673C212yesmaximal, complemented
9C4×#14273670C212yesmaximal, complemented
10C4×#14273667C211yesmaximal, characteristic, complemented
11C59×#3170C211yesmaximal, characteristic, complemented
12#16837936—21no—
13C2×#14273673C2212yes—
14C2×#14273670C2212yes—
15C2×#14273667C2212yes—
16#16837804C2211yescharacteristic
17#16837786C2211yescharacteristic
18#16837757—21no—
19#16837643C2211yescharacteristic
20#16837638C2211yescharacteristic
21#16837631—21no—
22C4×Dic236C2211yescharacteristic
23C4×#11745660—21no—
24C4×D472C2211yescharacteristic
25#16837229C2212yes—
26#16837227C2212yes—
27#16837206C2212yes—
28#16837206C2212yes—
29#16837205C2212yes—
30#16837202C2212yes—
31#16837178C2211yescharacteristic
32#16837178C2211yescharacteristic
33C4×C59⋊C16C2212yes—
34C59×#533—21no—
35C59×#500C2211yescharacteristic
36C59×#494C2211yescharacteristic
37C59×#487—21no—
38C59×#467—21no—
39C59×#467C2212yes—
40C4×Q16×C59—21no—
41C4×Q16×C59C2212yes—
42C2×C4×C8×C59C2211yescharacteristic
43C2×Dic236C2311yescharacteristic
44C2×#11745660—21no—
45C2×D472C2311yescharacteristic
46#14273721—41no—
47#14273719—21no—
48#14273715—21no—
49#14273710—41no—
50#14273706—21no—
51#14273701—21no—
52C4×Dic118—21no—
53C4×D236—21no—
54C4×C22⋊D59—41no—
55#14273684—41no—
56#14273673—24no—
57#14273670—24no—
58#14273667—24no—
59C2×C59⋊C16C2312yes—
60C2×C59⋊C16C2312yes—
61#14273661C2311yescharacteristic
62#14273661C2311yescharacteristic
63#14273659—21no—
64#14273652—21no—
65#14273652—21no—
66#14273649—21no—
67#14273649—21no—
68C2×Q16×C59—21no—
69C2×Q16×C59C2312yes—
70C22×C8×C59D411yescharacteristic
71C22×C8×C59C2311yescharacteristic
72C4⋊Q8×C59—21no—
73C4⋊Q8×C59—21no—
74#125×C59—41no—
75C22⋊Q8×C59—22no—
76C22⋊Q8×C59—41no—
77C4×Q8×C59—22no—
78C4×Q8×C59—41no—
79#117×C59—41no—
80C2×C42×C59D411yescharacteristic
81#107×C59—21no—
82#107×C59C2312yes—
83#103×C59—22no—
84#103×C59—22no—
85C4×C8×C59D411yescharacteristic
86C4×C8×C59D411yescharacteristic
87C4×C8×C59C2311yescharacteristic
88C4×C8×C59C2311yescharacteristic
89C2×Q8×C59—22no—
90C2×Q8×C59—41no—
91C22×C4×C59D811yescharacteristic
92C22×C4×C59D811yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C2×C22⋊D59—41no—
95C2×D236—21no—
96C2×C4×D59—41no—
97C2×Dic118—21no—
98Q16×C59—22no—
99Q16×C59—22no—
100Q16×C59—42no—
101C2×C8×C59—22no—
102C2×C8×C59—22no—
103C2×C8×C59C2×D411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59C2411yescharacteristic
107C4⋊C4×C59—22no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—42no—
111C4⋊C4×C59—42no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—42no—
114C42×C59—42no—
115C42×C59D811yescharacteristic
116C42×C59D811yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C22⋊Dic59—41no—
120(C2×D59)⋊C4—41no—
121(C2×D59)⋊C4—41no—
122(C2×D59)⋊C4—41no—
123(C2×D59)⋊C4—41no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126Dic59⋊C4—41no—
127Dic59⋊C4—41no—
128Dic59⋊C4—41no—
129Dic59⋊C4—41no—
130C4×Dic59—41no—
131Dic236—22no—
132D472—22no—
133#11745660—42no—
134C59⋊C16—22no—
135C59⋊C16#4114yes—
136C23×C59C2×D811yescharacteristic
137C22×D59—41no—
138Q8×C59—42no—
139Q8×C59—42no—
140Q8×C59—82no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59C2×D811yescharacteristic
148C2×C4×C59C2×D811yescharacteristic
149C2×C4×C59C2×D811yescharacteristic
150C2×C4×C59C2×D811yescharacteristic
151C2×C4×C59C2×D811yescharacteristic
152C2×C4×C59C22×D411yescharacteristic
153C22⋊D59—82no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157D236—41no—
158D236—41no—
159C4×D59—82no—
160Dic118—41no—
161Dic118—41no—
162C8×C59—21no—
163C8×C59—21no—
164C8×C59#13512yes—
165C8×C59C2×#4112yes—
166#33738—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59C22×D811yescharacteristic
171C2×D59—81no—
172C2×D59—81no—
173C2×D59—81no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—82no—
179C4×C59—82no—
180C4×C59#51012yes—
181C4×C59#53211yescharacteristic
182C4×C59C2×#13511yescharacteristic
183Dic59—81no—
184Dic59—81no—
185Dic59—81no—
186#3170D5911yescharacteristic, complemented
187#3168—591no—
188#2255—591no—
189#2251—591no—
190#2245—591no—
191#2243—591no—
192#2220—592no—
193#2219—592no—
194C4×Q32—592no—
195C4×SD32—592no—
196C4×M32—591no—
197C2×C59—41no—
198C2×C59#316911yescharacteristic
199C2×C59C2×#51011yescharacteristic
200C2×C59#341411yescharacteristic
201D59—161no—
202#533—21no—
203#529—1181no—
204C2×Q32—592no—
205C2×SD32—592no—
206C2×M32—592no—
207#500C2×D5911yescharacteristic
208#499—591no—
209#495—591no—
210#494C2×D5911yescharacteristic
211#488—1181no—
212#487—21no—
213#467—21no—
214#467—591no—
215#467C2×D5912yes—
216#466—591no—
217#465—1181no—
218C4×Q16—21no—
219C4×Q16—591no—
220C4×Q16C2×D5912yes—
221C4×SD16—1181no—
222C4×D8—591no—
223C2×C4×C8C2×D5911yescharacteristic
224#367—592no—
225#366—592no—
226#358—592no—
227#358—592no—
228#358—592no—
229#357—592no—
230#346—591no—
231#346—591no—
232C4×C16—592no—
233C59#3373811yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×Q16—591no—
236C2×Q16C22×D5912yes—
237C2×SD16—1181no—
238C2×D8—591no—
239C22×C8C22⋊D5911yescharacteristic
240C22×C8C22×D5911yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243#126—2361no—
244#125—41no—
245#125—1181no—
246#124—1181no—
247#123—2361no—
248C22⋊Q8—22no—
249C22⋊Q8—41no—
250C22⋊Q8—1181no—
251#121—1181no—
252C4×Q8—22no—
253C4×Q8—41no—
254C4×Q8—1181no—
255C4×D4—1181no—
256C4×D4—2361no—
257#117—41no—
258#117—2361no—
259C2×C42C22⋊D5911yescharacteristic
260Q32—1184no—
261SD32—1184no—
262M32—1184no—
263C2×C16—592no—
264C2×C16—592no—
265#107—21no—
266#107—591no—
267#107—591no—
268#107C22×D5912yes—
269#106—1181no—
270#103—22no—
271#103—22no—
272#103—1181no—
273#103—1181no—
274#102—1181no—
275#102—1181no—
276C4×C8C22⋊D5911yescharacteristic
277C4×C8C22⋊D5911yescharacteristic
278C4×C8C22×D5911yescharacteristic
279C4×C8C22×D5911yescharacteristic
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—1181no—
283C2×D4—1181no—
284C2×D4—2361no—
285C22×C4—2361no—
286C22×C4#1174566911yescharacteristic
287C22×C4#1174566911yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—42no—
292Q16—1182no—
293SD16—2362no—
294D8—1182no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D5911yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C2×C8C2×C22⋊D5911yescharacteristic
300C2×C8C23×D5911yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—1181no—
308C4⋊C4—1181no—
309C4⋊C4—2361no—
310C4⋊C4—2361no—
311C4⋊C4—2361no—
312C4⋊C4—2361no—
313C22⋊C4—42no—
314C22⋊C4—2361no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C22⋊C4—2361no—
319C42—42no—
320C42—2361no—
321C42#1174566911yescharacteristic
322C42#1174566911yescharacteristic
323C42C2×C22⋊D5911yescharacteristic
324C42C2×C22⋊D5911yescharacteristic
325C16—594no—
326C16—1182no—
327C23—2361no—
328C23C2×#1174566911yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—82no—
332Q8—2361no—
333Q8—2361no—
334D4—4722no—
335D4—2361no—
336D4—2361no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—4722no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4C2×#1174566911yescharacteristic
348C2×C4C2×#1174566911yescharacteristic
349C2×C4C2×#1174566911yescharacteristic
350C2×C4C2×#1174566911yescharacteristic
351C2×C4C2×#1174566911yescharacteristic
352C2×C4C22×C22⋊D5911yescharacteristic
353C8—21no—
354C8—21no—
355C8#1427374712yes—
356C8#41×D5912yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22C22×#1174566911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—4721no—
371C4—4721no—
372C4—4721no—
373C4#1683784312yes—
374C4#1683794811yescharacteristic
375C4C2×#1427374711yescharacteristic
376C2—41no—
377C2—9441no—
378C2#2143467111yescharacteristic
379C2C2×#1683784311yescharacteristic
380C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3170C2
socleC22×C59C22×D8
prime radical#16837638C22
hypercenter#3170D59
hypocenterC59#33738
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #21426379C2semidirect
3. #21426365C2semidirect
4. #21426341C2semidirect
5. #21426335C2semidirect
7. #21426202C2semidirect
8. C4×#14273673C2semidirect
9. C4×#14273670C2semidirect
10. C4×#14273667C2semidirect
11. C59×#3170C2semidirect
186. #3170D59semidirect
233. C59#33738semidirect

Maximal subgroups

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

subgroupindex
#214346662
#214263792
#214263652
#214263412
#214263352
#214262112
#214262022
C4×#142736732
C4×#142736702
C4×#142736672
C59×#31702
#3373859