Finite Group Databasebeta

← #28618500

Subgroups — #28618500

Order 15104.

1314 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837445C212yesmaximal, complemented
2C2×#16837406C212yesmaximal
3C2×#16837304C211yesmaximal, characteristic
4#21425630C212yesmaximal, complemented
5#21425625C212yesmaximal, complemented
6#21425620C214yesmaximal, complemented
7#21423790C211yesmaximal, characteristic, complemented
8C59×#2285C211yesmaximal, characteristic, complemented
9C23×D236C2211yescharacteristic
10C22×#11745664C2212yes—
11C22×C4⋊Dic59C2211yescharacteristic
12C22×(C2×D59)⋊C4—22no—
13#16837445—28no—
14C2×#14273660C2214yescomplemented
15C2×#14273658C2214yescomplemented
16#16837406—28no—
17C2×#14273647C2214yes—
18C2×#14273647C2214yes—
19#16837304—22no—
20#16837304C2212yes—
21#16837304C2212yes—
22#16837157C2212yescomplemented
23#16837157C2212yescomplemented
24C2×#117×C59C2212yes—
25C22×C42×C59C2211yescharacteristic
26C59×#384C2211yescharacteristic, complemented
27C59×#384C2211yescharacteristic, complemented
28C59×#382C2212yescomplemented
29C59×#380—22no—
30C4×C4⋊C4×C59—22no—
31C24×D59—21no—
32C22×D236—24no—
33C22×D236—24no—
34C22×D236—21no—
35C22×D236—22no—
36C22×D236Q814yes—
37C22×D236C2312yes—
38C23×Dic59—21no—
39C2×#11745664—24no—
40C2×#11745664—24no—
41C2×#11745664D418yes—
42C22×C59⋊C8C2314yes—
43C2×C4⋊Dic59—24no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59C2312yes—
46C2×(C2×D59)⋊C4—24no—
47C2×(C2×D59)⋊C4—24no—
48C2×(C2×D59)⋊C4—48no—
49#14273660—28no—
50#14273658—28no—
51#14273647—28no—
52#14273647—28no—
53C23×C4×C59D412yes—
54C23×C4×C59C2311yescharacteristic
55#117×C59—24no—
56#117×C59—24no—
57C2×C4⋊C4×C59—22no—
58C2×C4⋊C4×C59—22no—
59C2×C4⋊C4×C59C2312yes—
60C2×C4⋊C4×C59C2312yes—
61C2×C22⋊C4×C59—22no—
62C2×C42×C59—24no—
63C2×C42×C59—22no—
64C2×C42×C59D412yes—
65C2×C42×C59C2311yescharacteristic
66C2×C42×C59C2311yescharacteristic
67#95×C59—24no—
68C24×C59C2×D411yescharacteristic
69C23×D59—24no—
70C23×D59—22no—
71C23×D59—44no—
72C23×D59—44no—
73C23×D59—42no—
74C23×D59—42no—
75C22×C4×C59—24no—
76C22×C4×C59—24no—
77C22×C4×C59—22no—
78C22×C4×C59—22no—
79C22×C4×C59—42no—
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D414yes—
84C22×C4×C59C2×Q812yes—
85C22×C4×C59#4114yes—
86C22×C4×C59#4112yes—
87C22×C4×C59C2411yescharacteristic
88C22×Dic59—22no—
89C22×Dic59—44no—
90C22×Dic59—42no—
91C2×D236—216no—
92C2×D236—24no—
93C2×D236—24no—
94C2×D236—24no—
95C2×D236—22no—
96C2×D236—48no—
97C2×D236—42no—
98C2×D236—42no—
99C2×D236C2×Q814yes—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—44no—
105C4⋊C4×C59—44no—
106C22⋊C4×C59—44no—
107C42×C59—24no—
108C42×C59—22no—
109C42×C59—22no—
110C42×C59—44no—
111(C2×D59)⋊C4—48no—
112(C2×D59)⋊C4—48no—
113C4⋊Dic59—24no—
114C4⋊Dic59—42no—
115#11745664—216no—
116C2×C59⋊C8—24no—
117C2×C59⋊C8—24no—
118C2×C59⋊C8C2×D418yes—
119C23×C59—22no—
120C23×C59—22no—
121C23×C59C22⋊Q812yes—
122C23×C59#12312yes—
123C23×C59#12712yes—
124C23×C59C22×D411yescharacteristic
125C22×D59—24no—
126C22×D59—416no—
127C22×D59—42no—
128C22×D59—42no—
129C22×D59—44no—
130C22×D59—44no—
131C22×D59—48no—
132C22×D59—88no—
133C22×D59—84no—
134C22×D59—84no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—48no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—44no—
144C2×C4×C59—44no—
145C2×C4×C59C22⋊Q812yes—
146C2×C4×C59C22⋊Q818yes—
147C2×C4×C59#12312yes—
148C2×C4×C59#12318yes—
149C2×C4×C59#12712yes—
150C2×C4×C59AGL(1,ℤ/8)14yes—
151C2×C4×C59C22×D412yes—
152C2×C4×C59C22×Q811yescharacteristic
153C2×C4×C59C2×#4112yes—
154C2×C4×C59C2×#4111yescharacteristic
155C2×Dic59—42no—
156C2×Dic59—44no—
157C2×Dic59—84no—
158D236—28no—
159D236—44no—
160D236—48no—
161C59⋊C8—28no—
162#23731—591nomaximal, Sylow
163C22×C59—24no—
164C22×C59—24no—
165C22×C59—22no—
166C22×C59—22no—
167C22×C59#39914yes—
168C22×C59C2×C22⋊Q811yescharacteristic
169C22×C59C2×#12311yescharacteristic
170C22×C59C2×#12711yescharacteristic
171C22×C59C2×AGL(1,ℤ/8)12yes—
172C22×C59#57512yes—
173C2×D59—48no—
174C2×D59—44no—
175C2×D59—816no—
176C2×D59—84no—
177C2×D59—82no—
178C2×D59—82no—
179C4×C59—24no—
180C4×C59—42no—
181C4×C59—82no—
182C4×C59—82no—
183C4×C59#39914yes—
184C4×C59C2×C22⋊Q812yes—
185C4×C59C2×#12312yes—
186Dic59—82no—
187C2×#428—592no—
188C2×#418—592no—
189C2×#390—591no—
190#2285D5911yescharacteristic, complemented
191#2094—592no—
192#2093—592no—
193#2092—594no—
194#1728—591no—
195C2×C59—24no—
196C2×C59C2×#39911yescharacteristic
197C2×C59#307712yes—
198C2×C59#309412yes—
199C2×C59#315111yescharacteristic
200C2×C59#315311yescharacteristic
201D59—84no—
202C23×D4—591no—
203C22×M16—592no—
204C2×#117C2×D5912yes—
205C22×C4⋊C4—591no—
206C22×C22⋊C4—1182no—
207C22×C42C2×D5911yescharacteristic
208#428—1188no—
209C2×#107—594no—
210C2×#106—594no—
211#418—1188no—
212C2×#102—594no—
213C2×#102—594no—
214#390—592no—
215#390—592no—
216#390—1182no—
217#384C2×D5911yescharacteristic, complemented
218#384C2×D5911yescharacteristic, complemented
219#382C2×D5912yescomplemented
220#380—22no—
221C4×C4⋊C4—22no—
222#336—592no—
223#336—592no—
224C59#2373111yescharacteristic, complemented, Sylow
225C25—1181no—
226C22×D4—594no—
227C22×D4—592no—
228C22×D4—1184no—
229C22×D4—1184no—
230C22×D4—1182no—
231C22×D4—1181no—
232C23×C4—1181no—
233C23×C4C22⋊D5912yes—
234C23×C4C22×D5911yescharacteristic
235C2×M16—598no—
236C2×M16—1184no—
237C2×M16—1184no—
238C22×C8—594no—
239#117—24no—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—592no—
244C2×C4⋊C4—1184no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4C22×D5912yes—
247C2×C4⋊C4C22×D5912yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—1184no—
250C2×C22⋊C4—1184no—
251C2×C22⋊C4—2368no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42C22⋊D5912yes—
255C2×C42C22×D5911yescharacteristic
256C2×C42C22×D5911yescharacteristic
257#107—1188no—
258#106—1188no—
259#102—1188no—
260#102—1188no—
261#95—24no—
262C24—1184no—
263C24—1182no—
264C24—2364no—
265C24—2364no—
266C24—2362no—
267C24—2362no—
268C24C2×C22⋊D5911yescharacteristic
269C2×D4—594no—
270C2×D4—1184no—
271C2×D4—11816no—
272C2×D4—1184no—
273C2×D4—1184no—
274C2×D4—1182no—
275C2×D4—2362no—
276C2×D4—2368no—
277C2×D4—2362no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—1182no—
284C22×C4—2364no—
285C22×C4—2362no—
286C22×C4#1174568514yes—
287C22×C4D4×D5914yes—
288C22×C4Q8×D5912yes—
289C22×C4#1174568912yes—
290C22×C4C2×C22⋊D5912yes—
291C22×C4C2×C22⋊D5912yes—
292C22×C4C2×C22⋊D5911yescharacteristic
293C22×C4C23×D5911yescharacteristic
294M16—11816no—
295C2×C8—598no—
296C2×C8—1184no—
297C2×C8—1184no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—1184no—
305C4⋊C4—2362no—
306C22⋊C4—44no—
307C22⋊C4—2368no—
308C22⋊C4—2368no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—44no—
313C23—22no—
314C23—22no—
315C23—1184no—
316C23—4728no—
317C23—4724no—
318C23—4724no—
319C23—23616no—
320C23—2362no—
321C23—2364no—
322C23—2362no—
323C23—2364no—
324C23—2368no—
325C23#1427370312yes—
326C23#1427371012yes—
327C23#1427372312yes—
328C23C22×C22⋊D5911yescharacteristic
329D4—1188no—
330D4—2364no—
331D4—2368no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—4724no—
343C2×C4—2364no—
344C2×C4—2362no—
345C2×C4#1427370312yes—
346C2×C4#1427370518yes—
347C2×C4#1427371012yes—
348C2×C4#1427371218yes—
349C2×C4#1427372312yes—
350C2×C4#1427375214yes—
351C2×C4C2×D4×D5912yes—
352C2×C4C2×Q8×D5911yescharacteristic
353C2×C4C2×#1174568512yes—
354C2×C4C2×#1174568911yescharacteristic
355C8—1188no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22—47216no—
361C22—4724no—
362C22—4722no—
363C22—4722no—
364C22—2368no—
365C22—2364no—
366C22#1683733714yes—
367C22C2×#1427370311yescharacteristic
368C22C2×#1427371011yescharacteristic
369C22C2×#1427372311yescharacteristic
370C22C2×#1427375212yes—
371C22#1683820012yes—
372C4—24no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—4722no—
377C4#1683733714yes—
378C4C2×#1427370512yes—
379C4C2×#1427371212yes—
380C2—24no—
381C2—4724no—
382C2C2×#1683733711yescharacteristic
383C2#2143388912yes—
384C2#2143403612yes—
385C2#2143454911yescharacteristic
386C2#2143456411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2285C2
socleC23×C59C22×D4
prime radicalC23×D236C22
hypercenter#2285D59
hypocenterC59#23731
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C2×#16837445C2semidirect
4. #21425630C2semidirect
5. #21425625C2semidirect
6. #21425620C2semidirect
7. #21423790C2semidirect
8. C59×#2285C2semidirect
14. C2×#14273660C22semidirect
15. C2×#14273658C22semidirect
22. #16837157C22semidirect
23. #16837157C22semidirect
26. C59×#384C22semidirect
27. C59×#384C22semidirect
28. C59×#382C22semidirect
190. #2285D59semidirect
217. #384C2×D59semidirect
218. #384C2×D59semidirect
219. #382C2×D59semidirect
224. C59#23731semidirect

Maximal subgroups

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

subgroupindex
C2×#168374452
C2×#168374062
C2×#168373042
#214256302
#214256252
#214256202
#214237902
C59×#22852
#2373159