Finite Group Databasebeta

← #28479507

Subgroups — #28479507

Order 15104.

554 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
1#21423963C211yesmaximal, characteristic, complemented
2C4×#14273648C211yesmaximal, characteristic
3C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic
4C2×C8×C59⋊C8C211yesmaximal, characteristic, complemented
5#21421935C211yesmaximal, characteristic, complemented
6#21421930C211yesmaximal, characteristic, complemented
7C2×C59×#436C211yesmaximal, characteristic
8C2×C4⋊(C59⋊C8)C2211yescharacteristic
9C2×#14273648C412yes—
10C2×#14273637—21no—
11C2×C4×C59⋊C8C2211yescharacteristic
12C2×C4×C59⋊C8C2211yescharacteristic
13C4×C4⋊Dic59C2211yescharacteristic
14#16837164C412yes—
15#16837156—21no—
16D4⋊(C59⋊C8)C414yescomplemented
17C8×C59⋊C8—22no—
18C22×M16×C59—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#436—22no—
21C59×#436C414yescomplemented
22C2×C4⋊C8×C59C2211yescharacteristic
23C2×C22⋊C8×C59—21no—
24C2×C4×C8×C59C2211yescharacteristic
25C22×C59⋊C8—21no—
26C22×C59⋊C8—21no—
27C22×C59⋊C8C2×C412yes—
28C2×C4⋊Dic59C2×C412yes—
29C2×C4×Dic59—21no—
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)C2×C412yes—
32#14273648—24no—
33#14273637—42no—
34C4×C59⋊C8—21no—
35C4×C59⋊C8—21no—
36C4×C59⋊C8—21no—
37C4×C59⋊C8C2×C412yes—
38C59⋊#95—21no—
39C22×D4×C59C2×C411yescharacteristic
40C23×C4×C59—21no—
41C2×M16×C59—24no—
42C2×M16×C59—42no—
43C2×M16×C59—42no—
44C22×C8×C59—21no—
45C22×C8×C59D411yescharacteristic
46C22×C8×C59D411yescharacteristic
47C4×D4×C59—21no—
48C4×D4×C59—21no—
49C4×D4×C59C2×C412yes—
50C4×D4×C59C2×C412yes—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59—21no—
53C2×C42×C59C2311yescharacteristic
54C4⋊C8×C59—21no—
55C4⋊C8×C59C2×C412yes—
56C22⋊C8×C59—24no—
57C4×C8×C59C2×C412yes—
58C4×C8×C59D412yes—
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—
75M16×C59—48no—
76C2×C8×C59—21no—
77C2×C8×C59—21no—
78C2×C8×C59—22no—
79C2×C8×C59—41no—
80C2×C8×C59—41no—
81C2×C8×C59C22⋊C412yes—
82C2×C8×C59C22⋊C412yes—
83C2×C8×C59D812yes—
84C2×C8×C59SD1612yes—
85C4⋊C4×C59—21no—
86C4⋊C4×C59C4212yes—
87C22⋊C4×C59—22no—
88C22⋊C4×C59—22no—
89C42×C59C22×C411yescharacteristic
90C42×C59C22×C411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59#4111yescharacteristic
93C4⋊Dic59—22no—
94C4⋊Dic59—22no—
95C4×Dic59—41no—
96C4×Dic59—41no—
97C2×C59⋊C8—24no—
98C2×C59⋊C8—22no—
99C2×C59⋊C8—22no—
100C2×C59⋊C8—22no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8M1614yescomplemented
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×C59C2×D811yescharacteristic
131C2×C4×C59C2×SD1611yescharacteristic
132C2×C4×C59#13511yescharacteristic
133C2×C4×C59#13511yescharacteristic
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C8×C59—22no—
138C8×C59—42no—
139C8×C59#10214yescomplemented
140C59⋊C8—24no—
141C59⋊C8—24no—
142C59⋊C8—42no—
143#8206—591nomaximal, Sylow
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—82no—
148C22×C59—82no—
149C22×C59C4×C22⋊C411yescharacteristic
150C22×C59#43612yes—
151C22×C59C4×D811yescharacteristic
152C22×C59C4×SD1611yescharacteristic
153C22×C59#44312yes—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—42no—
157C4×C59C4×M1611yescharacteristic
158C4×C59#40511yescharacteristic
159C4×C59C2×#10211yescharacteristic
160C4×C59#41611yescharacteristic
161C4×C59#43512yes—
162C4×C59C4×D811yescharacteristic
163C4×C59C4×SD1611yescharacteristic
164Dic59—82no—
165C2×#436D5911yescharacteristic
166#1770—591no—
167C4×#102—591no—
168C2×D4⋊C8—591no—
169C2×C82—591no—
170#1279—591no—
171#1277—591no—
172C2×C59—82no—
173C2×C59#158912yes—
174C2×C59#159012yes—
175C2×C59#175311yescharacteristic
176C2×C59C4×#10211yescharacteristic
177C2×C59#176411yescharacteristic
178C22×M16—21no—
179C2×C4×D4C2×D5911yescharacteristic
180#436—22no—
181#436Dic5914yescomplemented
182C2×C4⋊C8—591no—
183C2×C4⋊C8C2×D5911yescharacteristic
184C2×#102—592no—
185C2×C22⋊C8—21no—
186C2×#97—1181no—
187C2×C4×C8—591no—
188C2×C4×C8—591no—
189C2×C4×C8C2×D5911yescharacteristic
190C4×C4⋊C4—591no—
191#340—592no—
192#336—1181no—
193D4⋊C8—594no—
194C82—1182no—
195C59#820611yescharacteristic, complemented, Sylow
196C22×D4C4×D5911yescharacteristic
197C23×C4—21no—
198C2×M16—24no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—21no—
202C22×C8—592no—
203C22×C8—1181no—
204C22×C8—1181no—
205C22×C8C22⋊D5911yescharacteristic
206C22×C8C22⋊D5911yescharacteristic
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—24no—
222#97—2362no—
223C4×C8—592no—
224C4×C8—1181no—
225C4×C8—1181no—
226C4×C8—1181no—
227C4×C8C2×Dic5912yes—
228C4×C8C22⋊D5912yes—
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×C4#1174568511yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246M16—48no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—594no—
253C2×C8—1184no—
254C2×C8—1182no—
255C2×C8—1182no—
256C2×C8—1182no—
257C2×C8—1182no—
258C2×C8—2361no—
259C2×C8—2361no—
260C2×C8#1174566912yes—
261C2×C8#1174567012yes—
262C2×C8C22⋊Dic5912yes—
263C2×C8C22⋊Dic5912yes—
264C4⋊C4—21no—
265C4⋊C4—1182no—
266C4⋊C4—1182no—
267C4⋊C4C4×Dic5912yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42—2361no—
271C42—2361no—
272C42C2×C4×D5911yescharacteristic
273C42#1174568511yescharacteristic
274C42C22×Dic5911yescharacteristic
275C42C2×C22⋊D5911yescharacteristic
276C23—22no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23C4×C22⋊D5911yescharacteristic
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—2362no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4C2×C4×Dic5911yescharacteristic
297C2×C4C2×C22⋊Dic5911yescharacteristic
298C2×C4#1427368611yescharacteristic
299C2×C4C4×C22⋊D5911yescharacteristic
300C2×C4C4×C22⋊D5911yescharacteristic
301C2×C4C4×C22⋊D5911yescharacteristic
302C2×C4M16×D5912yes—
303C2×C4#1427373512yes—
304C2×C4C2×#1174566911yescharacteristic
305C2×C4C2×#1174567011yescharacteristic
306C2×C4#1427374711yescharacteristic
307C2×C4#1427374711yescharacteristic
308C8—22no—
309C8—42no—
310C8—1184no—
311C8—1184no—
312C8—2362no—
313C8#1427364814yescomplemented
314C22—42no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—82no—
319C22C4×C22⋊Dic5911yescharacteristic
320C22#1683747812yes—
321C22C4×#1174566911yescharacteristic
322C22C4×#1174567011yescharacteristic
323C22#1683751712yes—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—4722no—
328C4M16×Dic5911yescharacteristic
329C4#1683736011yescharacteristic
330C4C2×#1427364811yescharacteristic
331C4#1683739811yescharacteristic
332C4#1683747112yes—
333C4C4×#1174566911yescharacteristic
334C4C4×#1174567011yescharacteristic
335C2—82no—
336C2#2142297912yes—
337C2#2142298412yes—
338C2#2142387711yescharacteristic
339C2C4×#1427364811yescharacteristic
340C2#2142393111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC2×C4×C59C2×C42
FrattiniC2×C42C22×D59
FittingC2×C59×#436C2
socleC23×C59C4×D4
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenterC2×#436D59
hypocenterC59#8206
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. #21423963C2semidirect
4. C2×C8×C59⋊C8C2semidirect
5. #21421935C2semidirect
6. #21421930C2semidirect
16. D4⋊(C59⋊C8)C4semidirect
21. C59×#436C4semidirect
104. C2×C59⋊C8M16semidirect
139. C8×C59#102semidirect
181. #436Dic59semidirect
195. C59#8206semidirect
313. C8#14273648semidirect

Maximal subgroups

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

subgroupindex
#214239632
C4×#142736482
C2×D4⋊(C59⋊C8)2
C2×C8×C59⋊C82
#214219352
#214219302
C2×C59×#4362
#820659