Finite Group Databasebeta

← #28589192

Subgroups — #28589192

Order 15104.

854 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837463C212yesmaximal
2#21424248C212yesmaximal, complemented
3#21424216C212yesmaximal, complemented
4#21424080C212yesmaximal
5#21424076C212yesmaximal
6#21423953C212yesmaximal, complemented
7#21423949C211yesmaximal, characteristic, complemented
8C4×C4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
9C4×#117×C59C211yesmaximal, characteristic
10#16837463—28no—
11C2×C4⋊(C59⋊C8)C2212yes—
12C2×C4⋊(C59⋊C8)C2212yes—
13C2×C4⋊(C59⋊C8)C2214yes—
14C2×C4⋊(C59⋊C8)C2214yes—
15C2×C22⋊(C59⋊C8)C2214yes—
16C2×#14273638C2212yes—
17C2×C4×C59⋊C8C2212yes—
18#16837157C2212yes—
19#16837157C2212yes—
20#16837157C2214yes—
21C2×#117×C59C2211yescharacteristic
22C2×#117×C59C2211yescharacteristic
23C22×C42×C59C2211yescharacteristic
24C4×C4⋊C4×C59C412yes—
25C4×C4⋊C4×C59C412yes—
26C4×C22⋊C4×C59C412yes—
27C4×C22⋊C4×C59C412yes—
28C59×#376C2211yescharacteristic
29C59×#376C2211yescharacteristic
30C43×C59C2211yescharacteristic
31C43×C59C2211yescharacteristic
32C22×C59⋊C8C2314yes—
33C22×C59⋊C8C2314yes—
34C4⋊(C59⋊C8)—28no—
35C4⋊(C59⋊C8)—24no—
36C4⋊(C59⋊C8)—24no—
37C4⋊(C59⋊C8)—28no—
38C22⋊(C59⋊C8)—28no—
39#14273638—24no—
40C4×C59⋊C8—24no—
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59C2×C411yescharacteristic
43C23×C4×C59C2311yescharacteristic
44#117×C59—24no—
45#117×C59D414yes—
46#117×C59Q814yes—
47C2×C4⋊C4×C59C2×C412yes—
48C2×C4⋊C4×C59C2×C412yes—
49C2×C22⋊C4×C59C2×C412yes—
50C2×C22⋊C4×C59C2×C412yes—
51C2×C42×C59—24no—
52C2×C42×C59C2×C412yes—
53C2×C42×C59C2×C412yes—
54C2×C42×C59C2×C411yescharacteristic
55C2×C42×C59C2×C411yescharacteristic
56C2×C42×C59C2×C412yes—
57C2×C42×C59C2×C412yes—
58C2×C42×C59C2311yescharacteristic
59C2×C42×C59C2311yescharacteristic
60C2×C42×C59C2311yescharacteristic
61C2×C42×C59C2311yescharacteristic
62C2×C42×C59C2311yescharacteristic
63C2×C42×C59C2311yescharacteristic
64#95×C59C2×C412yes—
65#95×C59C2×C412yes—
66#95×C59C2×C412yes—
67#95×C59C2×C412yes—
68C24×C59C22×C411yescharacteristic
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—22no—
73C22×C4×C59—24no—
74C22×C4×C59—42no—
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C22×C412yes—
78C22×C4×C59C22×C412yes—
79C22×C4×C59C22×C412yes—
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C22×C412yes—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D412yes—
85C22×C4×C59C2×Q812yes—
86C22×C4×C59C2411yescharacteristic
87C4⋊C4×C59—24no—
88C4⋊C4×C59C4⋊C418yes—
89C22⋊C4×C59—24no—
90C22⋊C4×C59C4⋊C418yes—
91C42×C59—22no—
92C42×C59—22no—
93C42×C59—24no—
94C42×C59—24no—
95C42×C59—22no—
96C42×C59—22no—
97C42×C59—42no—
98C42×C59—42no—
99C42×C59C2×D412yes—
100C42×C59C2×D412yes—
101C42×C59C2×Q812yes—
102C42×C59C2×Q812yes—
103C42×C59#4114yes—
104C42×C59#4114yes—
105C2×C59⋊C8—24no—
106C2×C59⋊C8—24no—
107C2×C59⋊C8—24no—
108C2×C59⋊C8—24no—
109C2×C59⋊C8—24no—
110C2×C59⋊C8—24no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59C2×C4⋊C412yes—
116C23×C59C23×C411yescharacteristic
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—22no—
125C2×C4×C59—22no—
126C2×C4×C59—24no—
127C2×C4×C59—24no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—44no—
136C2×C4×C59—44no—
137C2×C4×C59C2×C4⋊C414yes—
138C2×C4×C59C2×C4⋊C414yes—
139C2×C4×C59C2×C4⋊C412yes—
140C2×C4×C59#11712yes—
141C2×C4×C59#11714yes—
142C2×C4×C59#11712yes—
143C2×C4×C59#13114yes—
144C2×C4×C59C22×D411yescharacteristic
145C2×C4×C59C22×Q811yescharacteristic
146C2×C4×C59C2×#4112yes—
147C2×C4×C59C2×#4112yes—
148C59⋊C8—44no—
149C59⋊C8—44no—
150#20812—591nomaximal, Sylow
151C22×C59—24no—
152C22×C59—22no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—42no—
156C22×C59C22×C4⋊C411yescharacteristic
157C22×C59C2×#11711yescharacteristic
158C22×C59C2×#11711yescharacteristic
159C22×C59C2×#13112yes—
160C22×C59#56812yes—
161C4×C59—24no—
162C4×C59—24no—
163C4×C59—22no—
164C4×C59—24no—
165C4×C59—22no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—42no—
169C4×C59#43314yescomplemented
170C4×C59#53212yes—
171C4×C59#53312yes—
172C2×#433—592no—
173C4×#117D5911yescharacteristic
174#1842—592no—
175#1832—592no—
176#1798—592no—
177#1797—592no—
178#1768—592no—
179#1767—591no—
180C4×C4⋊C8—591no—
181C2×C59—42no—
182C2×C59#228511yescharacteristic
183C2×C59#290612yes—
184C2×C59C2×#43311yescharacteristic
185C2×C59#292111yescharacteristic
186C2×C59#292512yes—
187C2×#117C2×D5911yescharacteristic
188C2×#117C2×D5911yescharacteristic
189C22×C42C2×D5911yescharacteristic
190#433—1188no—
191C2×C4⋊C8—592no—
192C2×C4⋊C8—592no—
193C2×C4⋊C8—594no—
194C2×C4⋊C8—594no—
195C2×C22⋊C8—594no—
196C2×#97—592no—
197C2×C4×C8—592no—
198C4×C4⋊C4Dic5912yes—
199C4×C4⋊C4Dic5912yes—
200C4×C22⋊C4Dic5912yes—
201C4×C22⋊C4Dic5912yes—
202#376C2×D5911yescharacteristic
203#376C2×D5911yescharacteristic
204C43C2×D5911yescharacteristic
205C43C2×D5911yescharacteristic
206#336—594no—
207#336—592no—
208#336—592no—
209C59#2081211yescharacteristic, complemented, Sylow
210C23×C4C2×Dic5911yescharacteristic
211C23×C4C2×Dic5911yescharacteristic
212C23×C4C22×D5911yescharacteristic
213C22×C8—594no—
214C22×C8—594no—
215#117—24no—
216#117Dic11814yes—
217#117D23614yes—
218C2×C4⋊C4C2×Dic5912yes—
219C2×C4⋊C4C2×Dic5912yes—
220C2×C22⋊C4C2×Dic5912yes—
221C2×C22⋊C4C2×Dic5912yes—
222C2×C42—24no—
223C2×C42C2×Dic5912yes—
224C2×C42C2×Dic5912yes—
225C2×C42C2×Dic5911yescharacteristic
226C2×C42C2×Dic5911yescharacteristic
227C2×C42C2×Dic5912yes—
228C2×C42C2×Dic5912yes—
229C2×C42C22×D5911yescharacteristic
230C2×C42C22×D5911yescharacteristic
231C2×C42C22×D5911yescharacteristic
232C2×C42C22×D5911yescharacteristic
233C2×C42C22×D5911yescharacteristic
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—1188no—
236C4⋊C8—1184no—
237C4⋊C8—1184no—
238C4⋊C8—1188no—
239C22⋊C8—1188no—
240#97—1184no—
241C4×C8—1184no—
242#95C2×Dic5912yes—
243#95C2×Dic5912yes—
244#95C2×Dic5912yes—
245#95C2×Dic5912yes—
246C24C22×Dic5911yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4C2×Dic11812yes—
254C22×C4C2×D23612yes—
255C22×C4C22×Dic5911yescharacteristic
256C22×C4C22×Dic5911yescharacteristic
257C22×C4C22×Dic5912yes—
258C22×C4C22×Dic5912yes—
259C22×C4C22×Dic5912yes—
260C22×C4C22×Dic5911yescharacteristic
261C22×C4C22×Dic5912yes—
262C22×C4C22×Dic5911yescharacteristic
263C22×C4C22×Dic5911yescharacteristic
264C22×C4C23×D5911yescharacteristic
265C2×C8—1184no—
266C2×C8—1184no—
267C2×C8—1184no—
268C2×C8—1184no—
269C2×C8—1184no—
270C2×C8—1184no—
271C4⋊C4—24no—
272C4⋊C4C4⋊Dic5918yes—
273C22⋊C4—24no—
274C22⋊C4C4⋊Dic5918yes—
275C42—22no—
276C42—22no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42C2×Dic11812yes—
284C42C2×Dic11812yes—
285C42C2×D23612yes—
286C42C2×D23612yes—
287C42#1174568514yes—
288C42#1174568514yes—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—21no—
293C23C2×C4⋊Dic5912yes—
294C23C23×Dic5911yescharacteristic
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4C2×C4⋊Dic5914yes—
316C2×C4C2×C4⋊Dic5914yes—
317C2×C4C2×C4⋊Dic5912yes—
318C2×C4#1427368612yes—
319C2×C4#1427368614yes—
320C2×C4#1427368612yes—
321C2×C4#1427373414yes—
322C2×C4C22×D23611yescharacteristic
323C2×C4C22×Dic11811yescharacteristic
324C2×C4C2×#1174568512yes—
325C2×C4C2×#1174568512yes—
326C8—2364no—
327C8—2364no—
328C22—22no—
329C22—24no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22C22×C4⋊Dic5911yescharacteristic
334C22C2×#1427368611yescharacteristic
335C22C2×#1427368611yescharacteristic
336C22C2×#1427373412yes—
337C22#1683816912yes—
338C4—22no—
339C4—24no—
340C4—24no—
341C4—22no—
342C4—24no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4#1683746314yescomplemented
347C4#1683795212yes—
348C4#1683795912yes—
349C2—42no—
350C2#2142650311yescharacteristic
351C2#2143266412yes—
352C2C2×#1683746311yescharacteristic
353C2#2143273411yescharacteristic
354C2#2143276212yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutator, socleC23×C59C23×C4
FittingC4×#117×C59C2
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenterC4×#117D59
hypocenterC59#20812
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #21424248C2semidirect
3. #21424216C2semidirect
6. #21423953C2semidirect
7. #21423949C2semidirect
8. C4×C4⋊(C59⋊C8)C2semidirect
169. C4×C59#433semidirect
209. C59#20812semidirect
346. C4#16837463semidirect

Maximal subgroups

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

subgroupindex
C2×#168374632
#214242482
#214242162
#214240802
#214240762
#214239532
#214239492
C4×C4⋊(C59⋊C8)2
C4×#117×C592
#2081259