Finite Group Databasebeta

← #28479554

Subgroups — #28479554

Order 15104.

550 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837480C211yesmaximal, characteristic
2#21423961C211yesmaximal, characteristic, complemented
3C4×#14273648C211yesmaximal, characteristic
4C2×#16837123C211yesmaximal, characteristic, complemented
5#21421936C211yesmaximal, characteristic, complemented
6#21421930C211yesmaximal, characteristic, complemented
7C2×C59×D4⋊C8C211yesmaximal, characteristic
8C22×#11745664—21no—
9#16837480—22no—
10#16837480C414yescomplemented
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×#14273648C412yes—
13C2×C22⋊(C59⋊C8)—21no—
14C2×C4×C59⋊C8C2211yescharacteristic
15C2×C4×C59⋊C8C2211yescharacteristic
16C2×C8×Dic59—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837164C412yes—
19#16837156—21no—
20#16837123—22no—
21C2×C4×D4×C59C2211yescharacteristic
22C2×C4⋊C8×C59C2211yescharacteristic
23C2×C4×C8×C59C2211yescharacteristic
24C59×D4⋊C8C414yescomplemented
25C2×#11745664—24no—
26C2×#11745664—42no—
27C2×#11745664—42no—
28C22×C59⋊C8—21no—
29C22×C59⋊C8C2×C412yes—
30C22×C59⋊C8D411yescharacteristic
31C22×C59⋊C8D411yescharacteristic
32C2×C4⋊Dic59C2×C412yes—
33C2×C4×Dic59—21no—
34C4⋊(C59⋊C8)—21no—
35C4⋊(C59⋊C8)C2×C412yes—
36#14273648—24no—
37C22⋊(C59⋊C8)—24no—
38C4×C59⋊C8—21no—
39C4×C59⋊C8—21no—
40C4×C59⋊C8C2×C412yes—
41C4×C59⋊C8D412yes—
42C8×Dic59—42no—
43C59⋊#95—21no—
44C22×D4×C59C2×C411yescharacteristic
45C23×C4×C59—21no—
46C22×C8×C59—21no—
47C22×C8×C59—21no—
48C4×D4×C59—21no—
49C4×D4×C59—21no—
50C4×D4×C59C2×C412yes—
51C4×D4×C59C2×C412yes—
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59—21no—
54C2×C42×C59C2311yescharacteristic
55C4⋊C8×C59—21no—
56C4⋊C8×C59C2×C412yes—
57C4×C8×C59—21no—
58C4×C8×C59C2×C412yes—
59C24×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59C4212yes—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59#4111yescharacteristic
74C22×Dic59—22no—
75C2×C8×C59—24no—
76C2×C8×C59—22no—
77C2×C8×C59—22no—
78C2×C8×C59—41no—
79C2×C8×C59—41no—
80C4⋊C4×C59—21no—
81C4⋊C4×C59C4212yes—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C42×C59C22×C411yescharacteristic
85C42×C59C22×C411yescharacteristic
86C42×C59C2×D411yescharacteristic
87C42×C59#4111yescharacteristic
88C4⋊Dic59—22no—
89C4⋊Dic59M1614yescomplemented
90C4×Dic59—41no—
91C4×Dic59—41no—
92#11745664—48no—
93C2×C59⋊C8—21no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—21no—
97C2×C59⋊C8—22no—
98C2×C59⋊C8—22no—
99C2×C59⋊C8—22no—
100C2×C59⋊C8—22no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8C22⋊C412yes—
104C2×C59⋊C8C22⋊C412yes—
105C23×C59—22no—
106C23×C59—42no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—42no—
110C23×C59C4×D411yescharacteristic
111D4×C59—22no—
112D4×C59—22no—
113D4×C59—42no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59C2×C4211yescharacteristic
123C2×C4×C59C2×C22⋊C411yescharacteristic
124C2×C4×C59#11711yescharacteristic
125C2×C4×C59C4×D411yescharacteristic
126C2×C4×C59C4×D411yescharacteristic
127C2×C4×C59C4×D411yescharacteristic
128C2×C4×C59C2×M1612yes—
129C2×C4×C59#13112yes—
130C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C59#13711yescharacteristic
134C2×Dic59—24no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C8×C59—24no—
138C8×C59—42no—
139C59⋊C8—22no—
140C59⋊C8—22no—
141C59⋊C8—42no—
142C59⋊C8—42no—
143#8212—591nomaximal, Sylow
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—82no—
148C22×C59—82no—
149C22×C59C4×C22⋊C411yescharacteristic
150C22×C59#43512yes—
151C22×C59#44011yescharacteristic
152C22×C59#44211yescharacteristic
153C22×C59#44312yes—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—42no—
157C4×C59C4×M1611yescharacteristic
158C4×C59#40511yescharacteristic
159C4×C59#41711yescharacteristic
160C4×C59#41811yescharacteristic
161C4×C59#43612yes—
162C4×C59#44011yescharacteristic
163C4×C59#44211yescharacteristic
164Dic59—44no—
165C2×#436—591no—
166C8×C4⋊C4—591no—
167C4×#102—591no—
168C2×D4⋊C8D5911yescharacteristic
169C2×#322—591no—
170#1279—591no—
171#1277—591no—
172C2×C59—82no—
173C2×C59#158612yes—
174C2×C59#158812yes—
175C2×C59#175311yescharacteristic
176C2×C59#176211yescharacteristic
177C2×C59#176411yescharacteristic
178C22×M16—1181no—
179C2×C4×D4C2×D5911yescharacteristic
180#436—594no—
181#436—1182no—
182C2×C4⋊C8—591no—
183C2×C4⋊C8C2×D5911yescharacteristic
184C2×#102—592no—
185C2×C22⋊C8—1181no—
186C2×C4×C8—591no—
187C2×C4×C8—591no—
188C2×C4×C8—1181no—
189C2×C4×C8C2×D5911yescharacteristic
190C4×C4⋊C4—591no—
191#340—592no—
192#336—1181no—
193D4⋊C8Dic5914yescomplemented
194#322—1182no—
195C59#821211yescharacteristic, complemented, Sylow
196C22×D4C4×D5911yescharacteristic
197C23×C4—21no—
198C2×M16—1184no—
199C2×M16—2362no—
200C2×M16—2362no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—591no—
204C22×C8—591no—
205C22×C8—592no—
206C22×C8—1181no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C4×D5912yes—
210C4×D4C2×Dic5912yes—
211C2×C4⋊C4—592no—
212C2×C4⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4—21no—
214C2×C42—1181no—
215C2×C42C22×D5911yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—592no—
218C4⋊C8—1181no—
219C4⋊C8C2×Dic5912yes—
220#102—1184no—
221C22⋊C8—1184no—
222C4×C8—21no—
223C4×C8—592no—
224C4×C8—592no—
225C4×C8—1181no—
226C4×C8—1181no—
227C4×C8—2362no—
228C4×C8C2×Dic5912yes—
229#95—1181no—
230C24—21no—
231C2×D4—21no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4C4×Dic5912yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1182no—
243C22×C4C2×C4×D5911yescharacteristic
244C22×C4D4×D5911yescharacteristic
245C22×C4#1174568711yescharacteristic
246M16—2368no—
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—592no—
253C2×C8—592no—
254C2×C8—1181no—
255C2×C8—1181no—
256C2×C8—1181no—
257C2×C8—1181no—
258C2×C8—1182no—
259C2×C8—1182no—
260C2×C8—1182no—
261C2×C8—1182no—
262C2×C8—2361no—
263C2×C8—2361no—
264C4⋊C4—21no—
265C4⋊C4—594no—
266C4⋊C4—1182no—
267C4⋊C4C4×Dic5912yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42—2361no—
271C42—2361no—
272C42C2×C4×D5911yescharacteristic
273C42D4×D5911yescharacteristic
274C42#1174568711yescharacteristic
275C42C22×Dic5911yescharacteristic
276C23—22no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23(C22⋊D59)⋊C411yescharacteristic
282D4—22no—
283D4—22no—
284D4—42no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1184no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4C2×C4×Dic5911yescharacteristic
297C2×C4C22⋊C4×D5911yescharacteristic
298C2×C4#1427368311yescharacteristic
299C2×C4(C22⋊D59)⋊C411yescharacteristic
300C2×C4D4×Dic5911yescharacteristic
301C2×C4D4×Dic5911yescharacteristic
302C2×C4M16×D5912yes—
303C2×C4#1427373312yes—
304C2×C4#1427375111yescharacteristic
305C2×C4#1427375111yescharacteristic
306C2×C4#1427375411yescharacteristic
307C2×C4#1427375611yescharacteristic
308C8—24no—
309C8—42no—
310C8—1182no—
311C8—1182no—
312C8—2362no—
313C8—2362no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—82no—
319C22C22⋊C4×Dic5911yescharacteristic
320C22#1683747612yes—
321C22#1683749911yescharacteristic
322C22#1683751111yescharacteristic
323C22#1683751612yes—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—2364no—
328C4M16×Dic5911yescharacteristic
329C4#1683735711yescharacteristic
330C4#1683740011yescharacteristic
331C4#1683740511yescharacteristic
332C4#1683747712yes—
333C4#1683750311yescharacteristic
334C4#1683751411yescharacteristic
335C2—82no—
336C2#2142296312yes—
337C2#2142297712yes—
338C2#2142388211yescharacteristic
339C2#2142392311yescharacteristic
340C2#2142393411yescharacteristic
subgroupG/H
centerC22×C4D4×D59
commutatorC2×C4×C59C2×C42
FrattiniC2×C42C22×D59
FittingC2×C59×D4⋊C8C2
socleC23×C59C4×D4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenterC2×D4⋊C8D59
hypocenterC59#8212
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #21423961C2semidirect
4. C2×#16837123C2semidirect
5. #21421936C2semidirect
6. #21421930C2semidirect
10. #16837480C4semidirect
24. C59×D4⋊C8C4semidirect
89. C4⋊Dic59M16semidirect
193. D4⋊C8Dic59semidirect
195. C59#8212semidirect

Maximal subgroups

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

subgroupindex
C2×#168374802
#214239612
C4×#142736482
C2×#168371232
#214219362
#214219302
C2×C59×D4⋊C82
#821259