Finite Group Databasebeta

← #28744274

Subgroups — #28744274

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#21434682C211yesmaximal, characteristic, complemented
2#21426383C211yesmaximal, characteristic, complemented
3#21426368C211yesmaximal, characteristic, complemented
4#21426343C211yesmaximal, characteristic, complemented
5#21426332C211yesmaximal, characteristic, complemented
6#21426207C212yesmaximal
7#21426202C212yesmaximal, complemented
8C4×#14273673C212yesmaximal, complemented
9C4×#14273671C212yesmaximal, complemented
10C4×#14273667C211yesmaximal, characteristic, complemented
11C59×#3168C211yesmaximal, characteristic
12#16837959—21no—
13C2×#14273673C2212yes—
14C2×#14273671C2212yes—
15C2×#14273667C2212yes—
16#16837807C2211yescharacteristic
17#16837782C2211yescharacteristic
18#16837753—21no—
19#16837643—21no—
20#16837643C2212yes—
21C4×Dic236—21no—
22C4×Dic236C2212yes—
23#16837229C2212yes—
24#16837227C2212yes—
25#16837206C2212yes—
26#16837205C2212yes—
27#16837205C2212yes—
28#16837203C2212yes—
29#16837178C2211yescharacteristic
30#16837178C2211yescharacteristic
31C4×C59⋊C16C2212yes—
32C59×#529—21no—
33C59×#499C2211yescharacteristic
34C59×#495C2211yescharacteristic
35C59×#488—21no—
36C59×#467C2211yescharacteristic
37C59×#466C2211yescharacteristic
38C59×#465—21no—
39C4×Q16×C59C2211yescharacteristic
40C4×SD16×C59—21no—
41C4×D8×C59C2211yescharacteristic
42C2×C4×C8×C59C2211yescharacteristic
43C2×Dic236—21no—
44C2×Dic236C2312yes—
45#14273727—21no—
46#14273727—21no—
47#14273719—41no—
48#14273706—22no—
49#14273706—41no—
50C4×Dic118—22no—
51C4×Dic118—41no—
52#14273686—41no—
53#14273673—24no—
54#14273671—24no—
55#14273667—24no—
56C2×C59⋊C16C2312yes—
57C2×C59⋊C16C2312yes—
58#14273661—21no—
59#14273661C2312yes—
60#14273652—22no—
61#14273652—22no—
62C2×Q16×C59C2311yescharacteristic
63C2×SD16×C59—21no—
64C2×D8×C59C2311yescharacteristic
65C22×C8×C59D411yescharacteristic
66C22×C8×C59C2311yescharacteristic
67#126×C59—41no—
68#125×C59—21no—
69#124×C59—21no—
70#123×C59—41no—
71C22⋊Q8×C59—21no—
72#121×C59—21no—
73C4×Q8×C59—21no—
74C4×D4×C59—21no—
75C4×D4×C59—41no—
76#117×C59—41no—
77C2×C42×C59D411yescharacteristic
78#107×C59C2311yescharacteristic
79#107×C59C2311yescharacteristic
80#106×C59—21no—
81#103×C59—21no—
82#103×C59—21no—
83#102×C59—21no—
84#102×C59—21no—
85C4×C8×C59D411yescharacteristic
86C4×C8×C59D411yescharacteristic
87C4×C8×C59C2311yescharacteristic
88C4×C8×C59C2311yescharacteristic
89C2×Q8×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59D811yescharacteristic
94C22×C4×C59D811yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C2×Dic118—22no—
97C2×Dic118—41no—
98Q16×C59—22no—
99SD16×C59—42no—
100D8×C59—22no—
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—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C42×C59—41no—
119C42×C59D811yescharacteristic
120C42×C59D811yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C22⋊Dic59—42no—
124C4⋊Dic59—22no—
125C4⋊Dic59—41no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128Dic59⋊C4—42no—
129Dic59⋊C4—42no—
130C4×Dic59—42no—
131Dic236—22no—
132Dic236—22no—
133Dic236—42no—
134C59⋊C16—22no—
135C59⋊C16#4114yes—
136C23×C59—41no—
137C23×C59C2×D811yescharacteristic
138Q8×C59—41no—
139Q8×C59—41no—
140D4×C59—41no—
141D4×C59—41no—
142D4×C59—82no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—82no—
151C2×C4×C59C2×D811yescharacteristic
152C2×C4×C59C2×D811yescharacteristic
153C2×C4×C59C2×D811yescharacteristic
154C2×C4×C59C2×D811yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59C22×D411yescharacteristic
157C2×Dic59—42no—
158C2×Dic59—42no—
159Dic118—42no—
160Dic118—42no—
161Dic118—82no—
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×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59C22×D811yescharacteristic
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—81no—
179C4×C59—81no—
180C4×C59—81no—
181C4×C59#51012yes—
182C4×C59#53211yescharacteristic
183C4×C59C2×#13511yescharacteristic
184Dic59—82no—
185Dic59—82no—
186#3170—591no—
187#3168D5911yescharacteristic
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—161no—
199C2×C59#316911yescharacteristic
200C2×C59C2×#51011yescharacteristic
201C2×C59#341411yescharacteristic
202#533—1181no—
203#529—21no—
204C2×Q32—592no—
205C2×SD32—592no—
206C2×M32—592no—
207#500—591no—
208#499C2×D5911yescharacteristic
209#495C2×D5911yescharacteristic
210#494—591no—
211#488—21no—
212#487—1181no—
213#467—592no—
214#467—1181no—
215#467C2×D5911yescharacteristic
216#466C2×D5911yescharacteristic
217#465—21no—
218C4×Q16—592no—
219C4×Q16—1181no—
220C4×Q16C2×D5911yescharacteristic
221C4×SD16—21no—
222C4×D8C2×D5911yescharacteristic
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—592no—
235C2×Q16—1181no—
236C2×Q16C22×D5911yescharacteristic
237C2×SD16—21no—
238C2×D8C22×D5911yescharacteristic
239C22×C8C22⋊D5911yescharacteristic
240C22×C8C22×D5911yescharacteristic
241C4⋊Q8—1181no—
242C4⋊Q8—1181no—
243#126—41no—
244#125—21no—
245#125—2361no—
246#124—21no—
247#123—41no—
248C22⋊Q8—21no—
249C22⋊Q8—1182no—
250C22⋊Q8—2361no—
251#121—21no—
252C4×Q8—21no—
253C4×Q8—1182no—
254C4×Q8—2361no—
255C4×D4—21no—
256C4×D4—41no—
257#117—41no—
258#117—2361no—
259C2×C42C22⋊D5911yescharacteristic
260Q32—1184no—
261SD32—1184no—
262M32—1184no—
263C2×C16—592no—
264C2×C16—592no—
265#107—592no—
266#107—1181no—
267#107C22×D5911yescharacteristic
268#107C22×D5911yescharacteristic
269#106—21no—
270#103—21no—
271#103—21no—
272#103—1182no—
273#103—1182no—
274#102—21no—
275#102—21no—
276C4×C8C22⋊D5911yescharacteristic
277C4×C8C22⋊D5911yescharacteristic
278C4×C8C22×D5911yescharacteristic
279C4×C8C22×D5911yescharacteristic
280C2×Q8—21no—
281C2×Q8—1182no—
282C2×Q8—2361no—
283C2×D4—21no—
284C2×D4—41no—
285C22×C4—41no—
286C22×C4#1174566911yescharacteristic
287C22×C4#1174566911yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289Q16—22no—
290Q16—1182no—
291Q16—1182no—
292Q16—2362no—
293SD16—42no—
294D8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D5911yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C2×C8C2×C22⋊D5911yescharacteristic
300C2×C8C23×D5911yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1182no—
308C4⋊C4—2361no—
309C4⋊C4—2361no—
310C4⋊C4—2362no—
311C4⋊C4—2362no—
312C4⋊C4—2361no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—2362no—
319C42—41no—
320C42—2362no—
321C42#1174566911yescharacteristic
322C42#1174566911yescharacteristic
323C42C2×C22⋊D5911yescharacteristic
324C42C2×C22⋊D5911yescharacteristic
325C16—594no—
326C16—1182no—
327C23—41no—
328C23C2×#1174566911yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—4722no—
332Q8—2362no—
333Q8—2362no—
334D4—41no—
335D4—41no—
336D4—82no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—2362no—
346C2×C4—2362no—
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—81no—
361C22—81no—
362C22—81no—
363C22C22×#1174566911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—4722no—
372C4—4722no—
373C4#1683784212yes—
374C4#1683794811yescharacteristic
375C4C2×#1427374711yescharacteristic
376C2—41no—
377C2—161no—
378C2#2143467111yescharacteristic
379C2C2×#1683784211yescharacteristic
380C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3168C2
socleC22×C59C22×D8
prime radicalC59×#466C22
hypercenter#3168D59
hypocenterC59#33738
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #21434682C2semidirect
2. #21426383C2semidirect
3. #21426368C2semidirect
4. #21426343C2semidirect
5. #21426332C2semidirect
7. #21426202C2semidirect
8. C4×#14273673C2semidirect
9. C4×#14273671C2semidirect
10. C4×#14273667C2semidirect
233. C59#33738semidirect

Maximal subgroups

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

subgroupindex
#214346822
#214263832
#214263682
#214263432
#214263322
#214262072
#214262022
C4×#142736732
C4×#142736712
C4×#142736672
C59×#31682
#3373859