Finite Group Databasebeta

← #28592005

Subgroups — #28592005

Order 15104.

1002 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837430C211yesmaximal, characteristic
2C2×#16837429C211yesmaximal, characteristic
3C2×#16837425C211yesmaximal, characteristic
4C2×#16837357C211yesmaximal, characteristic
5#21426578C211yesmaximal, characteristic
6#21424818C214yesmaximal, complemented
7#21424813C214yesmaximal, complemented
8C2×#16837156C211yesmaximal, characteristic
9C59×#1729C211yesmaximal, characteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C22×C4×Dic59C2211yescharacteristic
14#16837430—24no—
15#16837429—24no—
16#16837425—24no—
17C2×C4⋊(C59⋊C8)C2214yes—
18C2×Dic59⋊C8C2214yes—
19C2×Dic59⋊C8C2214yes—
20#16837357—24no—
21C2×#14273637C2212yes—
22C2×C8×Dic59C2212yes—
23#16837298C412yes—
24#16837298C412yes—
25#16837274C414yes—
26C4×Dic59⋊C4C2214yes—
27#16837156C2212yes—
28#16837156C2212yes—
29C22×M16×C59C2211yescharacteristic
30C23×C8×C59C2211yescharacteristic
31C2×#117×C59C2211yescharacteristic
32C59×#336C2212yes—
33C59×#336C2212yes—
34C23×Dic59C2×C412yes—
35C2×#11745664—22no—
36C2×#11745664—22no—
37C2×#11745664D414yes—
38C22×C59⋊C8—22no—
39C22×C59⋊C8—22no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8C2312yes—
42C22×C59⋊C8C2312yes—
43#14273686D414yes—
44#14273686Q814yes—
45C2×C4⋊Dic59C2×C412yes—
46C2×Dic59⋊C4C2×C414yes—
47C2×Dic59⋊C4C2×C414yes—
48C2×C22⋊Dic59C2×C412yes—
49C2×C4×Dic59—22no—
50C2×C4×Dic59—22no—
51C2×C4×Dic59C2×C411yescharacteristic
52C2×C4×Dic59C2×C411yescharacteristic
53C2×C4×Dic59C2312yes—
54C2×C4×Dic59C2312yes—
55C4⋊(C59⋊C8)—28no—
56Dic59⋊C8—28no—
57Dic59⋊C8—28no—
58#14273637—24no—
59C8×Dic59—24no—
60C59⋊#95C2×C414yes—
61C59⋊#95C2×C414yes—
62C23×C4×C59C2311yescharacteristic
63C2×M16×C59—22no—
64C2×M16×C59—22no—
65C2×M16×C59—22no—
66C22×C8×C59—22no—
67C22×C8×C59—22no—
68C22×C8×C59Q814yes—
69C22×C8×C59C2312yes—
70C22×C8×C59C2311yescharacteristic
71C22×C8×C59C2311yescharacteristic
72#117×C59—22no—
73#117×C59—22no—
74C2×C4⋊C4×C59C2×C412yes—
75C2×C22⋊C4×C59C2×C412yes—
76C2×C42×C59C2312yes—
77C24×C59C22×C411yescharacteristic
78C22×C4×C59C22×C414yes—
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2×Q812yes—
83C22×C4×C59C2×Q812yes—
84C22×C4×C59#4112yes—
85C22×C4×C59#4112yes—
86C22×C4×C59C2411yescharacteristic
87C22×Dic59—24no—
88C22×Dic59—24no—
89C22×Dic59—42no—
90C22×Dic59C22×C412yes—
91C22×Dic59C22×C414yes—
92C22×Dic59C22×C412yes—
93M16×C59—44no—
94C2×C8×C59—28no—
95C2×C8×C59—21no—
96C2×C8×C59—21no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—22no—
102C2×C8×C59C2×Q812yes—
103C2×C8×C59C2×Q812yes—
104C4⋊C4×C59—24no—
105C22⋊C4×C59—24no—
106C42×C59—22no—
107C42×C59—22no—
108C22⋊Dic59C4⋊C418yes—
109C4⋊Dic59C4⋊C418yes—
110Dic59⋊C4—28no—
111Dic59⋊C4—44no—
112C4×Dic59—22no—
113C4×Dic59—22no—
114C4×Dic59—22no—
115C4×Dic59—22no—
116C4×Dic59C2×D414yes—
117C4×Dic59C2×Q814yes—
118#11745664—28no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—44no—
125C2×C59⋊C8C2×D414yes—
126C23×C59—22no—
127C23×C59—22no—
128C23×C59C2×C4⋊C412yes—
129C23×C59C4×D412yes—
130C23×C59C4×Q812yes—
131C23×C59C23×C411yescharacteristic
132C2×C4×C59—24no—
133C2×C4×C59—24no—
134C2×C4×C59—24no—
135C2×C4×C59C2×C4⋊C412yes—
136C2×C4×C59C4×D412yes—
137C2×C4×C59C4×Q812yes—
138C2×C4×C59#12114yes—
139C2×C4×C59C22⋊Q814yes—
140C2×C4×C59#12514yes—
141C2×C4×C59C4⋊Q814yes—
142C2×C4×C59C22×D411yescharacteristic
143C2×C4×C59C22×D411yescharacteristic
144C2×C4×C59C22×Q811yescharacteristic
145C2×C4×C59C22×Q811yescharacteristic
146C2×C4×C59C2×#4111yescharacteristic
147C2×C4×C59C2×#4111yescharacteristic
148C2×Dic59—22no—
149C2×Dic59—24no—
150C2×Dic59—24no—
151C2×Dic59—22no—
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—22no—
155C2×Dic59—48no—
156C2×Dic59C2×C4⋊C418yes—
157C2×Dic59#13114yes—
158C8×C59—22no—
159C8×C59—22no—
160C8×C59—42no—
161C59⋊C8—24no—
162C59⋊C8—42no—
163#21148—591nomaximal, Sylow
164C22×C59—24no—
165C22×C59—24no—
166C22×C59—22no—
167C22×C59—22no—
168C22×C59#38914yes—
169C22×C59C22×C4⋊C411yescharacteristic
170C22×C59C2×C4×D411yescharacteristic
171C22×C59C2×C4×Q811yescharacteristic
172C22×C59C2×#13112yes—
173C22×C59#56812yes—
174C4×C59—44no—
175C4×C59#38914yes—
176C4×C59C2×#12111yescharacteristic
177C4×C59C2×C22⋊Q811yescharacteristic
178C4×C59C2×#12511yescharacteristic
179C4×C59C2×C4⋊Q811yescharacteristic
180Dic59—28no—
181Dic59—24no—
182Dic59—42no—
183C2×#424—591no—
184C2×#424—591no—
185C2×#423—591no—
186C2×#405—591no—
187#2296—591no—
188#1940—594no—
189#1939—594no—
190#1729D5911yescharacteristic
191C2×#336—591no—
192C2×C59—24no—
193C2×C59C2×#38911yescharacteristic
194C2×C59#290612yes—
195C2×C59#293312yes—
196C2×C59#296011yescharacteristic
197C2×C59#296111yescharacteristic
198C22×M16—591no—
199C22×M16C2×D5911yescharacteristic
200C23×C8—591no—
201C23×C8C2×D5911yescharacteristic
202C2×#117—591no—
203C2×#117C2×D5911yescharacteristic
204C22×C42—591no—
205#424—1184no—
206#424—1184no—
207#423—1184no—
208C2×C4⋊C8—594no—
209C2×C4⋊C8—594no—
210C2×C4⋊C8—594no—
211#405—1184no—
212C2×#97—592no—
213C2×C4×C8—592no—
214#389—592no—
215#389—592no—
216#385—594no—
217C4×C4⋊C4—594no—
218#336—592no—
219#336—592no—
220#336C2×D5912yes—
221#336C2×D5912yes—
222C59#2114811yescharacteristic, complemented, Sylow
223C23×C4—592no—
224C23×C4C22×D5911yescharacteristic
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—594no—
229C2×M16—1182no—
230C2×M16—1182no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—592no—
234C22×C8—592no—
235C22×C8—1182no—
236C22×C8—1182no—
237C22×C8—1182no—
238C22×C8Dic11814yes—
239C22×C8C22×D5912yes—
240C22×C8C22×D5911yescharacteristic
241C22×C8C22×D5911yescharacteristic
242#117—22no—
243#117—22no—
244#117—594no—
245#117—594no—
246C2×C4⋊C4—592no—
247C2×C4⋊C4—594no—
248C2×C4⋊C4—594no—
249C2×C4⋊C4C4×D5912yes—
250C2×C22⋊C4—592no—
251C2×C22⋊C4C4×D5912yes—
252C2×C42—592no—
253C2×C42—591no—
254C2×C42—591no—
255C2×C42—592no—
256C2×C42—1182no—
257C2×C42—1182no—
258C2×C42C22×D5912yes—
259C4⋊C8—1188no—
260C4⋊C8—1188no—
261C4⋊C8—1188no—
262#97—1184no—
263C4×C8—1184no—
264#95—594no—
265#95—594no—
266C24C2×C4×D5911yescharacteristic
267C22×C4—592no—
268C22×C4—594no—
269C22×C4—592no—
270C22×C4—1184no—
271C22×C4—1184no—
272C22×C4—2362no—
273C22×C4C2×Dic11812yes—
274C22×C4C2×C4×D5914yes—
275C22×C4C2×C4×D5911yescharacteristic
276C22×C4#1174568512yes—
277C22×C4D4×D5912yes—
278C22×C4D4×D5912yes—
279C22×C4#1174568712yes—
280C22×C4Q8×D5912yes—
281C22×C4C23×D5911yescharacteristic
282M16—44no—
283M16—1188no—
284C2×C8—28no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—594no—
293C2×C8—1182no—
294C2×C8—1182no—
295C2×C8—1182no—
296C2×C8—1182no—
297C2×C8—1182no—
298C2×C8—2364no—
299C2×C8C2×Dic11812yes—
300C2×C8C2×Dic11812yes—
301C4⋊C4—24no—
302C4⋊C4—598no—
303C4⋊C4—1188no—
304C4⋊C4—2364no—
305C22⋊C4—24no—
306C22⋊C4—598no—
307C42—22no—
308C42—22no—
309C42—594no—
310C42—594no—
311C42—1182no—
312C42—1182no—
313C42—1182no—
314C42—1182no—
315C23—22no—
316C23—22no—
317C23C4⋊C4×D5912yes—
318C23(C22⋊D59)⋊C412yes—
319C23C4×Dic11812yes—
320C23C22×C4×D5911yescharacteristic
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—594no—
325C2×C4—598no—
326C2×C4—1184no—
327C2×C4—1184no—
328C2×C4—1182no—
329C2×C4—1182no—
330C2×C4—1182no—
331C2×C4—1182no—
332C2×C4—1182no—
333C2×C4—2368no—
334C2×C4C4⋊C4×D5912yes—
335C2×C4(C22⋊D59)⋊C412yes—
336C2×C4C4×Dic11812yes—
337C2×C4#1427370014yes—
338C2×C4#1427370414yes—
339C2×C4#1427371714yes—
340C2×C4#1427372614yes—
341C2×C4C2×D4×D5911yescharacteristic
342C2×C4C2×D4×D5911yescharacteristic
343C2×C4C2×Q8×D5911yescharacteristic
344C2×C4C22×Dic11811yescharacteristic
345C2×C4C2×#1174568711yescharacteristic
346C2×C4C2×#1174568511yescharacteristic
347C8—22no—
348C8—22no—
349C8—42no—
350C8—1184no—
351C8—2362no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22#1683729914yes—
357C22C2×C4⋊C4×D5911yescharacteristic
358C22C2×(C22⋊D59)⋊C411yescharacteristic
359C22C2×C4×Dic11811yescharacteristic
360C22#131×D5912yes—
361C22#1683816712yes—
362C4—44no—
363C4—1188no—
364C4—1184no—
365C4—2362no—
366C4#1683729914yes—
367C4C2×#1427370011yescharacteristic
368C4C2×#1427370411yescharacteristic
369C4C2×#1427371711yescharacteristic
370C4C2×#1427372611yescharacteristic
371C2—24no—
372C2C2×#1683729911yescharacteristic
373C2#2143266312yes—
374C2#2143282612yes—
375C2#2143302811yescharacteristic
376C2#2143303411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D59
commutator, socleC23×C59C23×C4
FittingC59×#1729C2
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenter#1729D59
hypocenterC59#21148
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
6. #21424818C2semidirect
7. #21424813C2semidirect
222. C59#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#168374302
C2×#168374292
C2×#168374252
C2×#168373572
#214265782
#214248182
#214248132
C2×#168371562
C59×#17292
#2114859