Finite Group Databasebeta

← #28482006

Subgroups — #28482006

Order 15104.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433072C211yesmaximal, characteristic
2C4×#14273660C211yesmaximal, characteristic, complemented
3#21422857C211yesmaximal, characteristic, complemented
4#21422621C211yesmaximal, characteristic
5#21421952C211yesmaximal, characteristic, complemented
6#21421951C211yesmaximal, characteristic, complemented
7C59×#1509C211yesmaximal, characteristic
8#16837533C411yescharacteristic, complemented
9Q8×C59⋊C8C411yescharacteristic, complemented
10#16837480C411yescharacteristic, complemented
11#16837474—21no—
12D4×C59⋊C8—21no—
13D4×C59⋊C8C411yescharacteristic, complemented
14#16837463—21no—
15#16837459C2211yescharacteristic
16#16837459C2211yescharacteristic
17C2×#14273660—21no—
18#16837358—21no—
19C2×C4×C59⋊C8C2211yescharacteristic
20C4×C4⋊Dic59C2211yescharacteristic
21#16837164—21no—
22#16837154—21no—
23#16837152—21no—
24#16837139—21no—
25#16837136—21no—
26C59×#529C2211yescharacteristic
27C59×#432C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C59×#328—21no—
30C59×#327—21no—
31C2×#11745664—21no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8D411yescharacteristic
34C22×C59⋊C8D411yescharacteristic
35C2×C4⋊Dic59—21no—
36C2×C4×Dic59—21no—
37#14273660—24no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)C2×C411yescharacteristic
42C4⋊(C59⋊C8)C2×C411yescharacteristic
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—21no—
47#14273638—21no—
48#14273638—21no—
49#14273638—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8C2×C411yescharacteristic
52C4×C59⋊C8C2×C411yescharacteristic
53C4×C59⋊C8D411yescharacteristic
54C4×C59⋊C8D411yescharacteristic
55C59⋊#95—21no—
56#126×C59—21no—
57#125×C59Q811yescharacteristic
58#124×C59Q811yescharacteristic
59#123×C59—21no—
60C22⋊Q8×C59D411yescharacteristic
61#121×C59D411yescharacteristic
62C4×Q8×C59C2×C411yescharacteristic
63C4×D4×C59—21no—
64C4×D4×C59C2×C411yescharacteristic
65#117×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C42×C59—21no—
68C2×C42×C59C2311yescharacteristic
69C4⋊C8×C59—21no—
70C22⋊C8×C59—21no—
71#97×C59—21no—
72#95×C59—21no—
73C2×Q8×C59C4⋊C411yescharacteristic
74C2×D4×C59—21no—
75C2×D4×C59C4⋊C411yescharacteristic
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×Dic59—41no—
82C2×C8×C59—41no—
83C4⋊C4×C59—21no—
84C4⋊C4×C59—21no—
85C4⋊C4×C59—21no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59C4⋊C411yescharacteristic
90C4⋊C4×C59C4⋊C411yescharacteristic
91C22⋊C4×C59—21no—
92C22⋊C4×C59—21no—
93C22⋊C4×C59—21no—
94C22⋊C4×C59—21no—
95C22⋊C4×C59—21no—
96C42×C59—21no—
97C42×C59—81no—
98C42×C59C22×C411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59C2×Q811yescharacteristic
101C42×C59#4111yescharacteristic
102C4⋊Dic59—22no—
103C4⋊Dic59—41no—
104C4×Dic59—81no—
105#11745664—42no—
106C2×C59⋊C8—22no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—42no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8C22⋊C411yescharacteristic
113C2×C59⋊C8C22⋊C411yescharacteristic
114C2×C59⋊C8C22⋊C411yescharacteristic
115C2×C59⋊C8C22⋊C411yescharacteristic
116C23×C59—21no—
117C23×C59#12111yescharacteristic
118Q8×C59—41no—
119D4×C59—41no—
120D4×C59—41no—
121D4×C59—41no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59C2×C22⋊C411yescharacteristic
136C2×C4×C59C2×C4⋊C411yescharacteristic
137C2×C4×C59#11711yescharacteristic
138C2×C4×C59#12111yescharacteristic
139C2×C4×C59C22⋊Q811yescharacteristic
140C2×C4×C59C22⋊Q811yescharacteristic
141C2×Dic59—42no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144C8×C59—81no—
145C59⋊C8—21no—
146C59⋊C8—21no—
147C59⋊C8—21no—
148C59⋊C8—41no—
149C59⋊C8—41no—
150C59⋊C8—81no—
151C59⋊C8#10412yes—
152#8697—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59#38011yescharacteristic
159C22×C59#47211yescharacteristic
160C22×C59#47311yescharacteristic
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—21no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—82no—
170C4×C59C2×#10411yescharacteristic
171C4×C59#42111yescharacteristic
172Dic59—82no—
173#2966—591no—
174C4×#107—591no—
175#1568—591no—
176#1509—591no—
177#1509D5911yescharacteristic
178#1285—591no—
179#1285—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—81no—
183C2×C59#180611yescharacteristic
184C2×C59#181211yescharacteristic
185C2×C59#181611yescharacteristic
186#529C2×D5911yescharacteristic
187#446—591no—
188C8×Q8—591no—
189#436—591no—
190#435—1181no—
191C8×D4—591no—
192C8×D4—1181no—
193#433—1181no—
194#432—591no—
195#432—591no—
196#432C2×D5911yescharacteristic
197C2×#107—1181no—
198#405—1181no—
199C2×C4×C8—591no—
200C4×C4⋊C4—591no—
201C4×C4⋊C4C2×D5911yescharacteristic
202#340—1181no—
203#335—1181no—
204#334—1181no—
205#328—21no—
206#328—1181no—
207#327—21no—
208#327—1181no—
209C59#869711yescharacteristic, complemented, Sylow
210C2×M16—1181no—
211C22×C8—591no—
212C22×C8—591no—
213C22×C8—1181no—
214#126—21no—
215#125Dic11811yescharacteristic
216#124Dic11811yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D5911yescharacteristic
219#121C22⋊D5911yescharacteristic
220C4×Q8C4×D5911yescharacteristic
221C4×D4—21no—
222C4×D4C4×D5911yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1181no—
226C2×C42—21no—
227C2×C42—1181no—
228C2×C42C22×D5911yescharacteristic
229#107—1184no—
230C4⋊C8—21no—
231C4⋊C8—591no—
232C4⋊C8—591no—
233C4⋊C8—1181no—
234C4⋊C8—1181no—
235C4⋊C8—1181no—
236C22⋊C8—21no—
237C22⋊C8—1181no—
238C22⋊C8—1181no—
239C22⋊C8—1181no—
240C22⋊C8—1181no—
241#97—21no—
242#97—1181no—
243#97—1181no—
244#97—1181no—
245C4×C8—591no—
246C4×C8—591no—
247C4×C8—591no—
248C4×C8—591no—
249C4×C8—1181no—
250#95—21no—
251#95—1181no—
252C2×Q8Dic59⋊C411yescharacteristic
253C2×D4—21no—
254C2×D4Dic59⋊C411yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—2361no—
258C22×C4D4×D5911yescharacteristic
259C22×C4#1174568711yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261M16—2362no—
262C2×C8—41no—
263C2×C8—591no—
264C2×C8—591no—
265C2×C8—591no—
266C2×C8—591no—
267C2×C8—1182no—
268C2×C8—1181no—
269C2×C8—1181no—
270C2×C8—2361no—
271C2×C8—2362no—
272C2×C8—2361no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—1182no—
280C4⋊C4—2361no—
281C4⋊C4Dic59⋊C411yescharacteristic
282C4⋊C4Dic59⋊C411yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—4721no—
291C42C2×Dic11811yescharacteristic
292C42C2×C4×D5911yescharacteristic
293C42D4×D5911yescharacteristic
294C42#1174568711yescharacteristic
295C23—21no—
296C23#1427369911yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—4721no—
315C2×C4—4721no—
316C2×C4—2362no—
317C2×C4C22⋊C4×D5911yescharacteristic
318C2×C4C2×Dic59⋊C411yescharacteristic
319C2×C4#1427368311yescharacteristic
320C2×C4#1427369911yescharacteristic
321C2×C4#1427370411yescharacteristic
322C2×C4#1427370411yescharacteristic
323C8—81no—
324C8—592no—
325C8—1181no—
326C8—1181no—
327C8—1181no—
328C8—4721no—
329C8—2361no—
330C8—2361no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1683726011yescharacteristic
337C22#1683767011yescharacteristic
338C22#1683767511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—4722no—
349C4#104×D5911yescharacteristic
350C4#1683741711yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2142411411yescharacteristic
355C2#2142414011yescharacteristic
356C2#2142415811yescharacteristic
subgroupG/H
centerC22#16837260
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1509C2
socleC22×C59#380
prime radical#121×C59D4
cosocleC2×C42×C59C23
hypercenter#1509D59
hypocenterC59#8697
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#14273660C2semidirect
3. #21422857C2semidirect
5. #21421952C2semidirect
6. #21421951C2semidirect
8. #16837533C4semidirect
9. Q8×C59⋊C8C4semidirect
10. #16837480C4semidirect
13. D4×C59⋊C8C4semidirect
209. C59#8697semidirect

Maximal subgroups

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

subgroupindex
#214330722
C4×#142736602
#214228572
#214226212
#214219522
#214219512
C59×#15092
#869759