Finite Group Databasebeta

← #28481841

Subgroups — #28481841

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837480C211yesmaximal, characteristic
2#21424233C211yesmaximal, characteristic, complemented
3#21422842C211yesmaximal, characteristic, complemented
4#21422521C211yesmaximal, characteristic
5#21421992C211yesmaximal, characteristic, complemented
6#21421991C211yesmaximal, characteristic, complemented
7C59×#1486C211yesmaximal, characteristic
8C22×#11745664—21no—
9#16837480—21no—
10#16837480—21no—
11#16837480C412yes—
12#16837480C412yes—
13#16837455C2211yescharacteristic
14#16837426C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×C22⋊(C59⋊C8)—21no—
17C2×C4×C59⋊C8C2211yescharacteristic
18#16837166—21no—
19#16837166—21no—
20#16837154—21no—
21#16837152—21no—
22D4⋊(C59⋊C8)—22no—
23C2×C4×D4×C59C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×D4⋊C8—22no—
27C2×#11745664—22no—
28C2×#11745664—22no—
29C2×#11745664—21no—
30C2×#11745664—42no—
31C2×#11745664—42no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8D411yescharacteristic
34C22×C59⋊C8D411yescharacteristic
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)C2×C412yes—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—21no—
43C4×C59⋊C8—21no—
44C4×C59⋊C8C2×C411yescharacteristic
45C4×C59⋊C8C2×C411yescharacteristic
46C4×C59⋊C8D411yescharacteristic
47C4×C59⋊C8D411yescharacteristic
48C22×D4×C59Q811yescharacteristic
49C23×C4×C59—21no—
50C2×M16×C59—21no—
51C4×D4×C59—21no—
52C4×D4×C59—21no—
53C4×D4×C59C2×C412yes—
54C4×D4×C59D412yes—
55C2×C4⋊C4×C59Q811yescharacteristic
56C2×C22⋊C4×C59—21no—
57C2×C42×C59C2311yescharacteristic
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C22⋊C8×C59—21no—
62C4×C8×C59—21no—
63C24×C59—21no—
64C2×D4×C59—21no—
65C2×D4×C59—42no—
66C2×D4×C59C4⋊C412yes—
67C22×C4×C59—21no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×Q811yescharacteristic
76C22×C4×C59#4111yescharacteristic
77M16×C59—41no—
78M16×C59—41no—
79C2×C8×C59—41no—
80C2×C8×C59—41no—
81C4⋊C4×C59—21no—
82C4⋊C4×C59C4⋊C412yes—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—22no—
85C42×C59C22×C411yescharacteristic
86C42×C59C2×D411yescharacteristic
87C42×C59C2×D411yescharacteristic
88C42×C59#4111yescharacteristic
89#11745664—44no—
90#11745664—44no—
91#11745664—41no—
92#11745664—41no—
93C2×C59⋊C8—21no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—21no—
97C2×C59⋊C8—22no—
98C2×C59⋊C8—41no—
99C2×C59⋊C8—41no—
100C2×C59⋊C8—41no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8C22⋊C411yescharacteristic
103C2×C59⋊C8C22⋊C411yescharacteristic
104C2×C59⋊C8C22⋊C411yescharacteristic
105C2×C59⋊C8C22⋊C411yescharacteristic
106C23×C59—22no—
107C23×C59—42no—
108C23×C59—42no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59C22⋊Q811yescharacteristic
112D4×C59—24no—
113D4×C59—42no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59C2×C22⋊C411yescharacteristic
124C2×C4×C59C2×C4⋊C411yescharacteristic
125C2×C4×C59#11711yescharacteristic
126C2×C4×C59#12111yescharacteristic
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59C22⋊Q811yescharacteristic
129C2×C4×C59#13511yescharacteristic
130C2×C4×C59#13511yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C8×C59—81no—
134C8×C59—81no—
135C59⋊C8—21no—
136C59⋊C8—21no—
137C59⋊C8—21no—
138C59⋊C8—21no—
139C59⋊C8—42no—
140C59⋊C8—81no—
141C59⋊C8—81no—
142#8664—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—82no—
149C22×C59—82no—
150C22×C59#38011yescharacteristic
151C22×C59#47811yescharacteristic
152C22×C59#47811yescharacteristic
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—42no—
157C4×C59#41611yescharacteristic
158C4×C59#41811yescharacteristic
159C4×C59#43012yes—
160C4×C59#46311yescharacteristic
161C4×C59#46311yescharacteristic
162C2×#436—591no—
163#1837—591no—
164#1564—591no—
165#1486—591no—
166#1486D5911yescharacteristic
167#1301—591no—
168#1301—591no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—82no—
172C2×C59#180411yescharacteristic
173C2×C59#181111yescharacteristic
174C2×C59#181411yescharacteristic
175C22×M16—1181no—
176C2×C4×D4C2×D5911yescharacteristic
177#436—592no—
178#436—592no—
179#436—1181no—
180#436—1181no—
181#431—591no—
182#431C2×D5911yescharacteristic
183#423—591no—
184#423C2×D5911yescharacteristic
185C2×C4⋊C8—591no—
186C2×C22⋊C8—1181no—
187C2×C4×C8—591no—
188#341—1181no—
189#341—1181no—
190#335—1181no—
191#334—1181no—
192D4⋊C8—22no—
193D4⋊C8—1182no—
194C59#866411yescharacteristic, complemented, Sylow
195C22×D4Dic11811yescharacteristic
196C23×C4—21no—
197C2×M16—21no—
198C2×M16—1182no—
199C2×M16—1182no—
200C2×M16—1181no—
201C2×M16—2362no—
202C2×M16—2362no—
203C22×C8—591no—
204C22×C8—591no—
205C22×C8—1181no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C4×D5912yes—
209C4×D4C22⋊D5912yes—
210C2×C4⋊C4Dic11811yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D5911yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—592no—
217C4⋊C8—1181no—
218C4⋊C8—1181no—
219C4⋊C8—1181no—
220C4⋊C8—1181no—
221C22⋊C8—21no—
222C22⋊C8—1182no—
223C22⋊C8—1182no—
224C22⋊C8—1181no—
225C4×C8—21no—
226C4×C8—591no—
227C4×C8—591no—
228C4×C8—591no—
229C4×C8—591no—
230C4×C8—1181no—
231C24—21no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4Dic59⋊C412yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C2×Dic11811yescharacteristic
243C22×C4D4×D5911yescharacteristic
244C22×C4#1174568711yescharacteristic
245M16—41no—
246M16—41no—
247M16—2364no—
248M16—2364no—
249M16—2361no—
250M16—2361no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—591no—
254C2×C8—591no—
255C2×C8—591no—
256C2×C8—591no—
257C2×C8—1181no—
258C2×C8—1181no—
259C2×C8—1181no—
260C2×C8—1181no—
261C2×C8—1182no—
262C2×C8—2361no—
263C2×C8—2361no—
264C2×C8—2361no—
265C2×C8—2361no—
266C4⋊C4—21no—
267C4⋊C4Dic59⋊C412yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42C2×C4×D5911yescharacteristic
271C42D4×D5911yescharacteristic
272C42#1174568711yescharacteristic
273C42C2×C22⋊D5911yescharacteristic
274C23—22no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23#1427370411yescharacteristic
280D4—24no—
281D4—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C22⋊C4×D5911yescharacteristic
292C2×C4C2×Dic59⋊C411yescharacteristic
293C2×C4#1427368311yescharacteristic
294C2×C4#1427369911yescharacteristic
295C2×C4#1427369911yescharacteristic
296C2×C4#1427370411yescharacteristic
297C2×C4#1427374611yescharacteristic
298C2×C4#1427374811yescharacteristic
299C2×C4#1427375111yescharacteristic
300C2×C4#1427375411yescharacteristic
301C8—81no—
302C8—81no—
303C8—1181no—
304C8—1181no—
305C8—1181no—
306C8—1181no—
307C8—4721no—
308C8—4721no—
309C8—2362no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—82no—
316C22—82no—
317C22#1683726011yescharacteristic
318C22#1683770011yescharacteristic
319C22#1683770411yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4#1683739511yescharacteristic
325C4#1683740511yescharacteristic
326C4#1683745012yes—
327C4#1683761511yescharacteristic
328C4#1683761911yescharacteristic
329C2—21no—
330C2—21no—
331C2—82no—
332C2#2142410411yescharacteristic
333C2#2142413611yescharacteristic
334C2#2142414811yescharacteristic
subgroupG/H
centerC2×C4#14273699
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1486C2
socleC22×C59#380
prime radicalC22×D4×C59Q8
cosocleC2×C42×C59C23
hypercenter#1486D59
hypocenterC59#8664
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21424233C2semidirect
3. #21422842C2semidirect
5. #21421992C2semidirect
6. #21421991C2semidirect
194. C59#8664semidirect

Maximal subgroups

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

subgroupindex
C2×#168374802
#214242332
#214228422
#214225212
#214219922
#214219912
C59×#14862
#866459