Finite Group Databasebeta

← #28598098

Subgroups — #28598098

Order 15104.

1306 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic118C211yesmaximal, characteristic
2C2×#16837455C211yesmaximal, characteristic
3#21425032C214yesmaximal, complemented
4#21425031C214yesmaximal, complemented
5#21424162C214yesmaximal
6C2×C59×#431C211yesmaximal, characteristic
7C23×Dic118C411yescharacteristic
8C2×C4×Dic118—24no—
9C2×C4×Dic118—24no—
10C2×C4×Dic118C412yes—
11C2×C4×Dic118C412yes—
12C2×C4×Dic118C2212yes—
13C2×C4×Dic118C2212yes—
14C22×C4⋊Dic59C411yescharacteristic
15C22×Dic59⋊C4C412yes—
16C22×C4×Dic59C2212yes—
17#16837455—24no—
18C2×C4⋊(C59⋊C8)C2212yes—
19C2×C22⋊(C59⋊C8)C2212yes—
20C2×C4×C59⋊C8C2212yes—
21#16837156C22116yes—
22C22×C42×C59C2211yescharacteristic
23C59×#431—24no—
24C2×C4⋊C8×C59C2212yes—
25C2×C22⋊C8×C59C2212yes—
26C2×C4×C8×C59C2212yes—
27C22×Dic118—24no—
28C22×Dic118—24no—
29C22×Dic118C818yescomplemented
30C22×Dic118C2×C412yes—
31C22×Dic118C2×C412yes—
32C23×Dic59C2×C412yes—
33C23×Dic59C2×C412yes—
34C22×C59⋊C8C2314yes—
35C4×Dic118—28no—
36C4×Dic118—28no—
37C4×Dic118—24no—
38C4×Dic118—24no—
39C4×Dic118—24no—
40C4×Dic118—24no—
41C2×C4⋊Dic59—24no—
42C2×C4⋊Dic59C2×C412yes—
43C2×C4⋊Dic59C2×C412yes—
44C2×Dic59⋊C4—28no—
45C2×Dic59⋊C4C2×C414yes—
46C2×Dic59⋊C4C2×C414yes—
47C2×C4×Dic59—24no—
48C2×C4×Dic59—24no—
49C2×C4×Dic59C2×C414yes—
50C2×C4×Dic59C2314yes—
51C4⋊(C59⋊C8)—24no—
52C22⋊(C59⋊C8)—24no—
53C4×C59⋊C8—24no—
54C23×C4×C59C2×C411yescharacteristic
55C23×C4×C59C2×C411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C22×C8×C59C2314yes—
58C2×C42×C59C2×C411yescharacteristic
59C2×C42×C59C2×C411yescharacteristic
60C2×C42×C59D414yes—
61C2×C42×C59D414yes—
62C2×C42×C59C2311yescharacteristic
63C2×C42×C59C2311yescharacteristic
64C4⋊C8×C59—24no—
65C22⋊C8×C59—24no—
66C4×C8×C59—24no—
67C24×C59C22×C411yescharacteristic
68C22×C4×C59—24no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C414yes—
71C22×C4×C59C22⋊C414yes—
72C22×C4×C59C2×C814yes—
73C22×C4×C59M1614yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C2×D412yes—
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2411yescharacteristic
82C22×Dic59—24no—
83C22×Dic59—24no—
84C22×Dic59—24no—
85C22×Dic59—44no—
86C22×Dic59C2×C818yes—
87C22×Dic59C22×C414yes—
88C22×Dic59C22×C414yes—
89C2×Dic118—232no—
90C2×Dic118—24no—
91C2×Dic118—24no—
92C2×Dic118—24no—
93C2×Dic118—24no—
94C2×Dic118C2×C818yes—
95C2×Dic118C2×C818yes—
96C2×C8×C59—24no—
97C2×C8×C59—28no—
98C42×C59C22⋊C414yes—
99C42×C59C22⋊C414yes—
100C42×C59C2×D412yes—
101C42×C59C2×D412yes—
102C42×C59C2×D412yes—
103C42×C59C2×D412yes—
104C4⋊Dic59—24no—
105C4⋊Dic59—24no—
106Dic59⋊C4—28no—
107Dic59⋊C4—28no—
108C4×Dic59—28no—
109C4×Dic59—24no—
110C4×Dic59—24no—
111C2×C59⋊C8—24no—
112C2×C59⋊C8—28no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59C2×C22⋊C412yes—
116C23×C59C22×C812yes—
117C23×C59C2×M1612yes—
118C23×C59C23×C411yescharacteristic
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—24no—
123C2×C4×C59—24no—
124C2×C4×C59—44no—
125C2×C4×C59C22⋊C8116yes—
126C2×C4×C59C2×C22⋊C412yes—
127C2×C4×C59C2×C22⋊C412yes—
128C2×C4×C59C2×C22⋊C412yes—
129C2×C4×C59C2×C22⋊C412yes—
130C2×C4×C59C2×C22⋊C412yes—
131C2×C4×C59C22×C812yes—
132C2×C4×C59C22×C812yes—
133C2×C4×C59C2×M1612yes—
134C2×C4×C59C2×M1612yes—
135C2×C4×C59C22×D411yescharacteristic
136C2×C4×C59C22×D411yescharacteristic
137C2×C4×C592−1+414yes—
138C2×Dic59—24no—
139C2×Dic59—24no—
140C2×Dic59—216no—
141C2×Dic59—24no—
142C2×Dic59—28no—
143C2×Dic59—48no—
144C2×Dic59C22×C818yes—
145Dic118—216no—
146Dic118—216no—
147C8×C59—44no—
148C59⋊C8—44no—
149#21818—591nomaximal, Sylow
150C22×C59—22no—
151C22×C59—22no—
152C22×C59—24no—
153C22×C59—24no—
154C22×C59C2×C22⋊C814yes—
155C22×C59C22×C22⋊C411yescharacteristic
156C22×C59#51912yes—
157C22×C59C23×C811yescharacteristic
158C22×C59C22×M1611yescharacteristic
159C22×C59#56812yes—
160C4×C59—22no—
161C4×C59—22no—
162C4×C59—22no—
163C4×C59C2×C22⋊C814yes—
164C4×C59C2×C22⋊C814yes—
165C4×C59#53612yes—
166C4×C59#53612yes—
167Dic59—28no—
168Dic59—44no—
169C22×C4×Q8—591no—
170C2×#431—591no—
171C2×#431D5911yescharacteristic
172#1984—594no—
173#1984—594no—
174#1817—594no—
175C2×C59—24no—
176C2×C59#231011yescharacteristic
177C2×C59C22×C22⋊C811yescharacteristic
178C2×C59#287911yescharacteristic
179C2×C59#298212yes—
180C2×C59#298312yes—
181C23×Q8—591no—
182C2×C4×Q8—592no—
183C2×C4×Q8—592no—
184C2×C4×Q8—592no—
185C2×C4×Q8—592no—
186C2×C4×Q8—1184no—
187C2×C4×Q8—1184no—
188C22×C4⋊C4—592no—
189C22×C4⋊C4—591no—
190C22×C42—592no—
191C22×C42C2×D5911yescharacteristic
192#431—24no—
193#431—1184no—
194C2×C4⋊C8—592no—
195C2×C4⋊C8C2×D5912yes—
196C2×C22⋊C8—592no—
197C2×C22⋊C8C2×D5912yes—
198C2×C4×C8—592no—
199C2×C4×C8C2×D5912yes—
200#336—5916no—
201C59#2181811yescharacteristic, complemented, Sylow
202C22×Q8—598no—
203C22×Q8—592no—
204C22×Q8—592no—
205C22×Q8—1184no—
206C22×Q8—1184no—
207C23×C4—592no—
208C23×C4—592no—
209C23×C4C4×D5911yescharacteristic
210C23×C4C4×D5911yescharacteristic
211C23×C4C22×D5911yescharacteristic
212C22×C8—594no—
213C22×C8C22×D5914yes—
214C4×Q8—1188no—
215C4×Q8—1188no—
216C4×Q8—1184no—
217C4×Q8—1184no—
218C4×Q8—1184no—
219C4×Q8—1184no—
220C2×C4⋊C4—594no—
221C2×C4⋊C4—594no—
222C2×C4⋊C4—592no—
223C2×C4⋊C4—592no—
224C2×C4⋊C4—1188no—
225C2×C4⋊C4—1184no—
226C2×C42—594no—
227C2×C42—594no—
228C2×C42—1184no—
229C2×C42—1184no—
230C2×C42C4×D5911yescharacteristic
231C2×C42C4×D5911yescharacteristic
232C2×C42D23614yes—
233C2×C42C22⋊D5914yes—
234C2×C42C22×D5911yescharacteristic
235C2×C42C22×D5911yescharacteristic
236C4⋊C8—24no—
237C4⋊C8—1184no—
238C22⋊C8—24no—
239C22⋊C8—1184no—
240C4×C8—24no—
241C4×C8—1184no—
242C24C2×C4×D5911yescharacteristic
243C2×Q8—598no—
244C2×Q8—598no—
245C2×Q8—11832no—
246C2×Q8—1184no—
247C2×Q8—1184no—
248C2×Q8—1184no—
249C2×Q8—1184no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—598no—
253C22×C4—594no—
254C22×C4—594no—
255C22×C4—1184no—
256C22×C4—1184no—
257C22×C4—1184no—
258C22×C4—2364no—
259C22×C4C8×D5914yes—
260C22×C4#1174565914yes—
261C22×C4(C2×D59)⋊C414yes—
262C22×C4(C2×D59)⋊C414yes—
263C22×C4C2×C4×D5911yescharacteristic
264C22×C4C2×C4×D5911yescharacteristic
265C22×C4C2×C4×D5911yescharacteristic
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4C2×C4×D5911yescharacteristic
268C22×C4C2×D23612yes—
269C22×C4C2×C22⋊D5912yes—
270C22×C4C23×D5911yescharacteristic
271C2×C8—24no—
272C2×C8—28no—
273C2×C8—1184no—
274C2×C8—1188no—
275C4⋊C4—1188no—
276C4⋊C4—1188no—
277C4⋊C4—1184no—
278C4⋊C4—1184no—
279C42—1188no—
280C42—1184no—
281C42—1184no—
282C42(C2×D59)⋊C414yes—
283C42(C2×D59)⋊C414yes—
284C42C2×D23612yes—
285C42C2×D23612yes—
286C42C2×C22⋊D5912yes—
287C42C2×C22⋊D5912yes—
288C23—22no—
289C23—22no—
290C23C2×(C2×D59)⋊C412yes—
291C23C2×C8×D5912yes—
292C23C2×#1174565912yes—
293C23C22×C4×D5911yescharacteristic
294Q8—11816no—
295Q8—11816no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—44no—
302C2×C4—598no—
303C2×C4—1184no—
304C2×C4—1184no—
305C2×C4—11816no—
306C2×C4—1188no—
307C2×C4—1184no—
308C2×C4—2368no—
309C2×C4(C2×D59)⋊C8116yes—
310C2×C4C2×(C2×D59)⋊C412yes—
311C2×C4C2×(C2×D59)⋊C412yes—
312C2×C4C2×(C2×D59)⋊C412yes—
313C2×C4C2×(C2×D59)⋊C412yes—
314C2×C4C2×(C2×D59)⋊C412yes—
315C2×C4C2×C8×D5912yes—
316C2×C4C2×C8×D5912yes—
317C2×C4C2×#1174565912yes—
318C2×C4C2×#1174565912yes—
319C2×C4C22×C22⋊D5911yescharacteristic
320C2×C4C22×D23611yescharacteristic
321C2×C4#1427377414yes—
322C8—44no—
323C8—2364no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—22no—
328C22C2×(C2×D59)⋊C814yes—
329C22C22×(C2×D59)⋊C411yescharacteristic
330C22#1683787912yes—
331C22C22×C8×D5911yescharacteristic
332C22C22×#1174565911yescharacteristic
333C22#1683816812yes—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—1188no—
338C4—2364no—
339C4C2×(C2×D59)⋊C814yes—
340C4C2×(C2×D59)⋊C814yes—
341C4#1683797312yes—
342C4#1683797812yes—
343C2—24no—
344C2#2142668011yescharacteristic
345C2C22×(C2×D59)⋊C811yescharacteristic
346C2#2143254011yescharacteristic
347C2#2143317612yes—
348C2#2143317912yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C23×C8
FittingC2×C59×#431C2
socleC23×C59C23×C4
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenterC2×#431D59
hypocenterC59#21818
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. #21425032C2semidirect
4. #21425031C2semidirect
29. C22×Dic118C8semidirect
201. C59#21818semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1182
C2×#168374552
#214250322
#214250312
#214241622
C2×C59×#4312
#2181859