Finite Group Databasebeta

← #28599133

Subgroups — #28599133

Order 15104.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837455C211yesmaximal, characteristic
2C22×C4⋊(C59⋊C8)C211yesmaximal, characteristic
3#21424815C214yesmaximal, complemented
4#21424276C214yesmaximal
5#21424248C212yesmaximal, complemented
6#21424213C212yesmaximal, complemented
7C4×#117×C59C211yesmaximal, characteristic
8C23×C59⋊C8C2212yes—
9#16837455—24no—
10C2×C4⋊(C59⋊C8)—24no—
11C2×C4⋊(C59⋊C8)C2212yes—
12C2×C4⋊(C59⋊C8)C2212yes—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2218yes—
16C2×C22⋊(C59⋊C8)C2212yes—
17C2×C4×C59⋊C8C2212yes—
18#16837157C2218yes—
19C2×#117×C59C411yescharacteristic
20C2×#117×C59C411yescharacteristic
21C22×C42×C59C2211yescharacteristic
22C4×C4⋊C4×C59C412yes—
23C4×C4⋊C4×C59C2212yes—
24C4×C22⋊C4×C59C412yes—
25C4×C22⋊C4×C59C2212yes—
26C59×#376C412yes—
27C43×C59C2212yes—
28C22×C59⋊C8—24no—
29C22×C59⋊C8—44no—
30C22×C59⋊C8C2314yes—
31C22×C59⋊C8C2312yes—
32C22×C59⋊C8C2312yes—
33C4⋊(C59⋊C8)—24no—
34C4⋊(C59⋊C8)—24no—
35C4⋊(C59⋊C8)—22no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—48no—
38C22⋊(C59⋊C8)—24no—
39C4×C59⋊C8—24no—
40C23×C4×C59C2×C411yescharacteristic
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59C2311yescharacteristic
43#117×C59—24no—
44#117×C59C818yescomplemented
45C2×C4⋊C4×C59C2×C412yes—
46C2×C4⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59C2×C412yes—
48C2×C22⋊C4×C59C2×C412yes—
49C2×C42×C59C2×C412yes—
50C2×C42×C59C2×C414yes—
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59C2×C412yes—
54C2×C42×C59D414yes—
55C2×C42×C59Q814yes—
56C2×C42×C59C2314yes—
57C2×C42×C59C2311yescharacteristic
58C2×C42×C59C2311yescharacteristic
59#95×C59C2×C414yes—
60#95×C59C2×C414yes—
61C24×C59C22×C411yescharacteristic
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—42no—
65C22×C4×C59C4⋊C414yes—
66C22×C4×C59C4⋊C414yes—
67C22×C4×C59C2×C814yes—
68C22×C4×C59M1614yes—
69C22×C4×C59C22×C414yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C22×C414yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D412yes—
77C22×C4×C59C2×Q812yes—
78C22×C4×C59C2411yescharacteristic
79C4⋊C4×C59—24no—
80C4⋊C4×C59C2×C818yescomplemented
81C22⋊C4×C59—24no—
82C22⋊C4×C59C2×C818yes—
83C42×C59—24no—
84C42×C59—24no—
85C42×C59—24no—
86C42×C59—44no—
87C42×C59C4⋊C414yes—
88C42×C59C4⋊C414yes—
89C42×C59C2×C818yescomplemented
90C42×C59C2×D412yes—
91C42×C59C2×D412yes—
92C42×C59C2×Q812yes—
93C42×C59C2×Q812yes—
94C2×C59⋊C8—22no—
95C2×C59⋊C8—22no—
96C2×C59⋊C8—24no—
97C2×C59⋊C8—24no—
98C2×C59⋊C8—28no—
99C2×C59⋊C8—24no—
100C2×C59⋊C8—48no—
101C23×C59—22no—
102C23×C59—22no—
103C23×C59C2×C4⋊C412yes—
104C23×C59C22×C812yes—
105C23×C59C2×M1612yes—
106C23×C59C23×C411yescharacteristic
107C2×C4×C59—24no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—24no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—24no—
115C2×C4×C59—24no—
116C2×C4×C59—24no—
117C2×C4×C59—24no—
118C2×C4×C59—44no—
119C2×C4×C59C4⋊C8116yescomplemented
120C2×C4×C59C2×C4⋊C412yes—
121C2×C4×C59C2×C4⋊C412yes—
122C2×C4×C59C2×C4⋊C412yes—
123C2×C4×C59C2×C4⋊C412yes—
124C2×C4×C59C2×C4⋊C412yes—
125C2×C4×C59C22×C818yes—
126C2×C4×C59C22×C812yes—
127C2×C4×C59C22×C812yes—
128C2×C4×C59C2×M1612yes—
129C2×C4×C59C2×M1612yes—
130C2×C4×C59C22×D411yescharacteristic
131C2×C4×C59C22×Q811yescharacteristic
132C2×C4×C59H(5,𝔽2)12yes—
133C2×C4×C592−1+412yes—
134C59⋊C8—42no—
135C59⋊C8—44no—
136C59⋊C8—42no—
137#21972—591nomaximal, Sylow
138C22×C59—22no—
139C22×C59—22no—
140C22×C59—24no—
141C22×C59—24no—
142C22×C59C2×C4⋊C814yes—
143C22×C59C22×C4⋊C411yescharacteristic
144C22×C59#52012yes—
145C22×C59C23×C811yescharacteristic
146C22×C59C22×M1611yescharacteristic
147C22×C59#56812yes—
148C4×C59—22no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—28no—
152C4×C59—44no—
153C4×C59C2×C4⋊C814yes—
154C4×C59C2×C4⋊C814yes—
155C4×C59#53511yescharacteristic
156C4×C59#53611yescharacteristic
157C4×C59#54412yes—
158C2×#431—591no—
159C22×C4⋊C8—591no—
160C4×#117D5911yescharacteristic
161#1939—594no—
162#1850—594no—
163#1842—592no—
164#1831—592no—
165C2×C59—24no—
166C2×C59#231711yescharacteristic
167C2×C59C22×C4⋊C811yescharacteristic
168C2×C59#290511yescharacteristic
169C2×C59#298812yes—
170C2×C59#298912yes—
171C23×C8—592no—
172C2×#117Dic5911yescharacteristic
173C2×#117Dic5911yescharacteristic
174C22×C42C2×D5911yescharacteristic
175#431—1184no—
176C2×C4⋊C8—591no—
177C2×C4⋊C8—592no—
178C2×C4⋊C8—591no—
179C2×C4⋊C8—592no—
180C2×C4⋊C8—598no—
181C2×C4⋊C8—1184no—
182C2×C22⋊C8—592no—
183C2×C4×C8—592no—
184C4×C4⋊C4Dic5912yes—
185C4×C4⋊C4C2×D5912yes—
186C4×C22⋊C4Dic5912yes—
187C4×C22⋊C4C2×D5912yes—
188#376Dic5912yes—
189C43C2×D5912yes—
190#336—598no—
191C59#2197211yescharacteristic, complemented, Sylow
192C23×C4C2×Dic5911yescharacteristic
193C23×C4C2×Dic5911yescharacteristic
194C23×C4C22×D5911yescharacteristic
195C22×C8—594no—
196C22×C8—592no—
197C22×C8—592no—
198C22×C8—1184no—
199C22×C8—2364no—
200#117—24no—
201#117C59⋊C818yescomplemented
202C2×C4⋊C4C2×Dic5912yes—
203C2×C4⋊C4C2×Dic5912yes—
204C2×C22⋊C4C2×Dic5912yes—
205C2×C22⋊C4C2×Dic5912yes—
206C2×C42Dic11814yes—
207C2×C42D23614yes—
208C2×C42C2×Dic5912yes—
209C2×C42C2×Dic5914yes—
210C2×C42C2×Dic5912yes—
211C2×C42C2×Dic5911yescharacteristic
212C2×C42C2×Dic5911yescharacteristic
213C2×C42C22×D5914yes—
214C2×C42C22×D5911yescharacteristic
215C2×C42C22×D5911yescharacteristic
216C4⋊C8—1184no—
217C4⋊C8—1184no—
218C4⋊C8—1182no—
219C4⋊C8—1182no—
220C4⋊C8—2368no—
221C22⋊C8—1184no—
222C4×C8—1184no—
223#95C2×Dic5914yes—
224#95C2×Dic5914yes—
225C24C22×Dic5911yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4C2×C59⋊C814yes—
230C22×C4#1174566414yes—
231C22×C4C4⋊Dic5914yes—
232C22×C4C4⋊Dic5914yes—
233C22×C4C2×Dic11812yes—
234C22×C4C2×D23612yes—
235C22×C4C22×Dic5914yes—
236C22×C4C22×Dic5911yescharacteristic
237C22×C4C22×Dic5911yescharacteristic
238C22×C4C22×Dic5911yescharacteristic
239C22×C4C22×Dic5914yes—
240C22×C4C22×Dic5911yescharacteristic
241C22×C4C22×Dic5911yescharacteristic
242C22×C4C23×D5911yescharacteristic
243C2×C8—1182no—
244C2×C8—1182no—
245C2×C8—1184no—
246C2×C8—1184no—
247C2×C8—1188no—
248C2×C8—1184no—
249C2×C8—2368no—
250C4⋊C4—24no—
251C4⋊C4C2×C59⋊C818yescomplemented
252C22⋊C4—24no—
253C22⋊C4C2×C59⋊C818yes—
254C42—24no—
255C42—24no—
256C42—24no—
257C42—44no—
258C42C2×C59⋊C818yescomplemented
259C42C4⋊Dic5914yes—
260C42C4⋊Dic5914yes—
261C42C2×Dic11812yes—
262C42C2×Dic11812yes—
263C42C2×D23612yes—
264C42C2×D23612yes—
265C23—22no—
266C23—22no—
267C23C2×C4⋊Dic5912yes—
268C23C22×C59⋊C812yes—
269C23C2×#1174566412yes—
270C23C23×Dic5911yescharacteristic
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—44no—
283C2×C4C4⋊(C59⋊C8)116yescomplemented
284C2×C4C2×C4⋊Dic5912yes—
285C2×C4C2×C4⋊Dic5912yes—
286C2×C4C2×C4⋊Dic5912yes—
287C2×C4C2×C4⋊Dic5912yes—
288C2×C4C2×C4⋊Dic5912yes—
289C2×C4C22×C59⋊C818yes—
290C2×C4C22×C59⋊C812yes—
291C2×C4C22×C59⋊C812yes—
292C2×C4C2×#1174566412yes—
293C2×C4C2×#1174566412yes—
294C2×C4C22×D23611yescharacteristic
295C2×C4C22×Dic11811yescharacteristic
296C2×C4#1427377312yes—
297C2×C4#1427377412yes—
298C8—2362no—
299C8—2364no—
300C8—2362no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22C2×C4⋊(C59⋊C8)14yes—
306C22C22×C4⋊Dic5911yescharacteristic
307C22#1683788512yes—
308C22C23×C59⋊C811yescharacteristic
309C22C22×#1174566411yescharacteristic
310C22#1683816912yes—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—28no—
315C4—44no—
316C4C2×C4⋊(C59⋊C8)14yes—
317C4C2×C4⋊(C59⋊C8)14yes—
318C4#1683797111yescharacteristic
319C4#1683797811yescharacteristic
320C4#1683803012yes—
321C2—24no—
322C2#2142671911yescharacteristic
323C2C22×C4⋊(C59⋊C8)11yescharacteristic
324C2#2143266011yescharacteristic
325C2#2143320712yes—
326C2#2143321312yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C23×C8
FittingC4×#117×C59C2
socleC23×C59C23×C4
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenterC4×#117D59
hypocenterC59#21972
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #21424815C2semidirect
5. #21424248C2semidirect
6. #21424213C2semidirect
44. #117×C59C8semidirect
80. C4⋊C4×C59C2×C8semidirect
89. C42×C59C2×C8semidirect
119. C2×C4×C59C4⋊C8semidirect
191. C59#21972semidirect
201. #117C59⋊C8semidirect
251. C4⋊C4C2×C59⋊C8semidirect
258. C42C2×C59⋊C8semidirect
283. C2×C4C4⋊(C59⋊C8)semidirect

Maximal subgroups

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

subgroupindex
C2×#168374552
C22×C4⋊(C59⋊C8)2
#214248152
#214242762
#214242482
#214242132
C4×#117×C592
#2197259