Finite Group Databasebeta

← #28744273

Subgroups — #28744273

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#21434666C211yesmaximal, characteristic
3#21426382C211yesmaximal, characteristic, complemented
4#21426367C211yesmaximal, characteristic, complemented
5#21426342C211yesmaximal, characteristic, complemented
6#21426333C211yesmaximal, characteristic, complemented
7#21426209C212yesmaximal
8#21426204C212yesmaximal, complemented
9C4×Dic472C212yesmaximal, complemented
10C4×#14273672C212yesmaximal, complemented
11C4×M32×C59C211yesmaximal, characteristic, complemented
12#16837959—21no—
13#16837936—21no—
14C2×Dic472C2212yes—
15C2×#14273672C2212yes—
16#16837807C2211yescharacteristic
17#16837804C2211yescharacteristic
18#16837786C2211yescharacteristic
19#16837782C2211yescharacteristic
20#16837757—21no—
21#16837753—21no—
22#16837643—21no—
23#16837643C2211yescharacteristic
24#16837643C2212yes—
25#16837638C2211yescharacteristic
26#16837631—21no—
27C4×Dic236—21no—
28C4×Dic236C2212yes—
29C4×Dic236C2211yescharacteristic
30C4×#11745660—21no—
31C4×D472C2211yescharacteristic
32#16837230C2212yes—
33#16837228C2212yes—
34#16837207C2212yes—
35#16837207C2212yes—
36#16837207C2212yes—
37#16837204C2212yes—
38C2×M32×C59C2212yes—
39C2×C4×C8×C59C2211yescharacteristic
40C59×#346C2211yescharacteristic
41C59×#346C2211yescharacteristic
42C4×C16×C59C2212yes—
43C2×Dic236—21no—
44C2×Dic236C2312yes—
45C2×Dic236C2311yescharacteristic
46C2×#11745660—21no—
47C2×D472C2311yescharacteristic
48#14273727—21no—
49#14273727—21no—
50#14273721—41no—
51#14273719—21no—
52#14273719—41no—
53#14273715—21no—
54#14273710—41no—
55#14273706—22no—
56#14273706—21no—
57#14273706—41no—
58#14273701—21no—
59C4×Dic118—21no—
60C4×Dic118—22no—
61C4×Dic118—41no—
62C4×D236—21no—
63C4×C22⋊D59—41no—
64#14273686—41no—
65#14273684—41no—
66Dic472—24no—
67#14273672—24no—
68#14273661—21no—
69#14273661C2311yescharacteristic
70#14273661C2312yes—
71#14273661C2311yescharacteristic
72#14273659—21no—
73#14273652—22no—
74#14273652—21no—
75#14273652—22no—
76#14273652—21no—
77#14273649—21no—
78#14273649—21no—
79C22×C8×C59D411yescharacteristic
80C22×C8×C59C2311yescharacteristic
81C2×C42×C59D411yescharacteristic
82M32×C59—24no—
83C2×C16×C59C2312yes—
84C2×C16×C59C2312yes—
85C4×C8×C59D411yescharacteristic
86C4×C8×C59D411yescharacteristic
87C4×C8×C59C2311yescharacteristic
88C4×C8×C59C2311yescharacteristic
89C22×C4×C59D811yescharacteristic
90C22×C4×C59D811yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C2×C22⋊D59—41no—
93C2×D236—21no—
94C2×C4×D59—41no—
95C2×Dic118—21no—
96C2×Dic118—22no—
97C2×Dic118—41no—
98C2×C8×C59—22no—
99C2×C8×C59—22no—
100C2×C8×C59C2×D411yescharacteristic
101C2×C8×C59C2×D411yescharacteristic
102C2×C8×C59C2×D411yescharacteristic
103C2×C8×C59C2411yescharacteristic
104C42×C59D811yescharacteristic
105C42×C59D811yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C22⋊Dic59—42no—
109C22⋊Dic59—41no—
110(C2×D59)⋊C4—41no—
111(C2×D59)⋊C4—41no—
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59—22no—
117C4⋊Dic59—41no—
118C4⋊Dic59—41no—
119C4⋊Dic59—41no—
120Dic59⋊C4—41no—
121Dic59⋊C4—41no—
122Dic59⋊C4—41no—
123Dic59⋊C4—42no—
124Dic59⋊C4—41no—
125Dic59⋊C4—42no—
126C4×Dic59—41no—
127C4×Dic59—42no—
128Dic236—22no—
129Dic236—22no—
130Dic236—22no—
131Dic236—42no—
132D472—22no—
133#11745660—42no—
134C16×C59—22no—
135C16×C59#4114yes—
136C23×C59C2×D811yescharacteristic
137C22×D59—41no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—22no—
141C2×C4×C59—41no—
142C2×C4×C59C2×D811yescharacteristic
143C2×C4×C59C2×D811yescharacteristic
144C2×C4×C59C2×D811yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59C2×D811yescharacteristic
147C2×C4×C59C22×D411yescharacteristic
148C22⋊D59—82no—
149C2×Dic59—41no—
150C2×Dic59—42no—
151C2×Dic59—42no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154D236—41no—
155D236—41no—
156C4×D59—82no—
157Dic118—42no—
158Dic118—41no—
159Dic118—41no—
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×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#51012yes—
179C4×C59#53211yescharacteristic
180C4×C59C2×#13511yescharacteristic
181Dic59—81no—
182Dic59—82no—
183Dic59—81no—
184Dic59—81no—
185Dic59—82no—
186#3170—591no—
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×M32D5911yescharacteristic, complemented
197C2×C59—41no—
198C2×C59#316911yescharacteristic
199C2×C59C2×#51011yescharacteristic
200C2×C59#341411yescharacteristic
201D59—161no—
202#533—1181no—
203#529—1181no—
204C2×Q32—592no—
205C2×SD32—592no—
206C2×M32C2×D5912yes—
207#500—591no—
208#499—591no—
209#495—591no—
210#494—591no—
211#488—1181no—
212#487—1181no—
213#467—591no—
214#467—592no—
215#467—1181no—
216#466—591no—
217#465—1181no—
218C4×Q16—591no—
219C4×Q16—592no—
220C4×Q16—1181no—
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#346C2×D5911yescharacteristic
231#346C2×D5911yescharacteristic
232C4×C16C2×D5912yes—
233C59#3373811yescharacteristic, complemented, Sylow
234C2×Q16—592no—
235C2×Q16—591no—
236C2×Q16—1181no—
237C2×SD16—1181no—
238C2×D8—591no—
239C22×C8D23611yescharacteristic
240C22×C8C22×D5911yescharacteristic
241C4⋊Q8—1181no—
242C4⋊Q8—1181no—
243#126—2361no—
244#125—1181no—
245#125—2361no—
246#124—1181no—
247#123—2361no—
248C22⋊Q8—1181no—
249C22⋊Q8—1182no—
250C22⋊Q8—2361no—
251#121—1181no—
252C4×Q8—1181no—
253C4×Q8—1182no—
254C4×Q8—2361no—
255C4×D4—1181no—
256C4×D4—2361no—
257#117—2361no—
258#117—2361no—
259C2×C42D23611yescharacteristic
260Q32—1184no—
261SD32—1184no—
262M32—24no—
263C2×C16C22×D5912yes—
264C2×C16C22×D5912yes—
265#107—592no—
266#107—591no—
267#107—591no—
268#107—1181no—
269#106—1181no—
270#103—1181no—
271#103—1181no—
272#103—1182no—
273#103—1182no—
274#102—1181no—
275#102—1181no—
276C4×C8D23611yescharacteristic
277C4×C8D23611yescharacteristic
278C4×C8C22×D5911yescharacteristic
279C4×C8C22×D5911yescharacteristic
280C2×Q8—1181no—
281C2×Q8—1182no—
282C2×Q8—2361no—
283C2×D4—1181no—
284C2×D4—2361no—
285C22×C4—2361no—
286C22×C4D47211yescharacteristic
287C22×C4D47211yescharacteristic
288C22×C4C2×D23611yescharacteristic
289Q16—1182no—
290Q16—1182no—
291Q16—1182no—
292Q16—2362no—
293SD16—2362no—
294D8—1182no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×D23611yescharacteristic
298C2×C8C2×D23611yescharacteristic
299C2×C8C2×D23611yescharacteristic
300C2×C8C23×D5911yescharacteristic
301C4⋊C4—1181no—
302C4⋊C4—1181no—
303C4⋊C4—1182no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C4⋊C4—2361no—
307C4⋊C4—2361no—
308C4⋊C4—2361no—
309C4⋊C4—2362no—
310C4⋊C4—2361no—
311C4⋊C4—2362no—
312C4⋊C4—2361no—
313C22⋊C4—2362no—
314C22⋊C4—2361no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C22⋊C4—2361no—
319C42—2361no—
320C42—2362no—
321C42D47211yescharacteristic
322C42D47211yescharacteristic
323C42C2×D23611yescharacteristic
324C42C2×D23611yescharacteristic
325C16—22no—
326C16#1174568514yes—
327C23—2361no—
328C23C2×D47211yescharacteristic
329Q8—4722no—
330Q8—2361no—
331Q8—2362no—
332Q8—2362no—
333Q8—2361no—
334D4—4722no—
335D4—2361no—
336D4—2361no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—4722no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2362no—
345C2×C4—2362no—
346C2×C4—2361no—
347C2×C4C2×D47211yescharacteristic
348C2×C4C2×D47211yescharacteristic
349C2×C4C2×D47211yescharacteristic
350C2×C4C2×D47211yescharacteristic
351C2×C4C2×D47211yescharacteristic
352C2×C4C22×D23611yescharacteristic
353C8—21no—
354C8—21no—
355C8#1427374912yes—
356C8C2×#1174568512yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22C22×D47211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—4721no—
369C4—4721no—
370C4—4722no—
371C4—4721no—
372C4—4722no—
373C4#1683784512yes—
374C4#1683795211yescharacteristic
375C4C2×#1427374911yescharacteristic
376C2—41no—
377C2—9441no—
378C2#2143467511yescharacteristic
379C2C2×#1683784511yescharacteristic
380C2#2143723211yescharacteristic
subgroupG/H
centerC2×C4C2×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC4×M32×C59C2
socleC22×C59C22×D8
prime radical#16837638C22
hypercenterC4×M32D59
hypocenterC59#33738
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×M32

Complements

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

NH ≅ G/Ntype
1. #21434682C2semidirect
3. #21426382C2semidirect
4. #21426367C2semidirect
5. #21426342C2semidirect
6. #21426333C2semidirect
8. #21426204C2semidirect
9. C4×Dic472C2semidirect
10. C4×#14273672C2semidirect
11. C4×M32×C59C2semidirect
196. C4×M32D59semidirect
233. C59#33738semidirect

Maximal subgroups

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

subgroupindex
#214346822
#214346662
#214263822
#214263672
#214263422
#214263332
#214262092
#214262042
C4×Dic4722
C4×#142736722
C4×M32×C592
#3373859