Finite Group Databasebeta

← #28571767

Subgroups — #28571767

Order 15104.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435214C212yesmaximal
2C2×#16837782C211yesmaximal, characteristic, complemented
3#21423585C214yesmaximal, complemented
4#21423461C214yesmaximal, complemented
5#21422844C211yesmaximal, characteristic, complemented
6#21422769C212yesmaximal, complemented
7C59×#2931C211yesmaximal, characteristic
8C22×Dic236—21no—
9C22×Dic236—21no—
10C2×#14273727C2212yes—
11#16837805—22no—
12#16837782—21no—
13#16837782—21no—
14#16837782C2212yes—
15#16837782C2212yes—
16#16837748—22no—
17#16837713—24no—
18#16837682C2218yes—
19#16837455C2212yes—
20#16837155C2212yes—
21#16837153C2212yes—
22#16837149C2218yes—
23C2×#131×C59—21no—
24C59×#532C2211yescharacteristic
25C59×#436—21no—
26C59×#436C2212yes—
27C8×D4×C59—21no—
28C8×D4×C59C2212yes—
29C59×#423C2211yescharacteristic
30C59×#408—21no—
31C2×C4×C8×C59C2211yescharacteristic
32C22×Dic118—22no—
33C2×Dic236—22no—
34C2×Dic236—22no—
35C2×Dic236—22no—
36C2×Dic236—22no—
37C2×Dic236—44no—
38C2×Dic236—44no—
39#14273727—22no—
40#14273727—22no—
41#14273727—22no—
42#14273727C2314yes—
43C2×C4⋊Dic59—22no—
44C2×C4⋊Dic59—22no—
45#14273660—28no—
46C4⋊(C59⋊C8)C2314yes—
47#14273648—28no—
48C22⋊(C59⋊C8)—22no—
49C4×C59⋊C8—22no—
50C2×#41×C59—21no—
51#131×C59—44no—
52C2×M16×C59—21no—
53C2×M16×C59—22no—
54C22×C8×C59—22no—
55C22×C8×C59D411yescharacteristic
56C22×C8×C59D411yescharacteristic
57C4⋊Q8×C59D411yescharacteristic
58#127×C59D411yescharacteristic
59#124×C59—21no—
60#121×C59—21no—
61#121×C59D412yes—
62C4×D4×C59—21no—
63C4×D4×C59C2312yes—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59C2312yes—
67C22⋊C8×C59—22no—
68C22⋊C8×C59—22no—
69C4×C8×C59D411yescharacteristic
70C4×C8×C59D411yescharacteristic
71C4×C8×C59C2311yescharacteristic
72C4×C8×C59C2311yescharacteristic
73#41×C59—42no—
74#41×C59—42no—
75C2×Q8×C59—21no—
76C2×D4×C59—22no—
77C2×D4×C59—21no—
78C2×D4×C59—21no—
79C2×D4×C59C2×D412yes—
80C22×C4×C59—22no—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59#4111yescharacteristic
84C22×Dic59—42no—
85C2×Dic118—24no—
86C2×Dic118—44no—
87C2×Dic118—44no—
88C2×Dic118—42no—
89M16×C59—42no—
90M16×C59—44no—
91C2×C8×C59—22no—
92C2×C8×C59—22no—
93C2×C8×C59—44no—
94C2×C8×C59D814yes—
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C4⋊C4×C59—21no—
100C4⋊C4×C59C2×D412yes—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—22no—
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C42×C59C2411yescharacteristic
107C4⋊Dic59—24no—
108C4⋊Dic59—24no—
109C4⋊Dic59—42no—
110C4⋊Dic59—42no—
111C2×C59⋊C8—24no—
112Dic236—44no—
113Dic236—44no—
114Dic236—44no—
115Dic236—44no—
116C23×C59—22no—
117C23×C59#12111yescharacteristic
118Q8×C59—42no—
119D4×C59—24no—
120D4×C59—44no—
121D4×C59—42no—
122D4×C59—42no—
123C2×C4×C59—22no—
124C2×C4×C59—22no—
125C2×C4×C59—22no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59#12111yescharacteristic
129C2×C4×C59#12111yescharacteristic
130C2×C4×C59#12111yescharacteristic
131C2×C4×C59C2×D812yes—
132C2×C4×C59AGL(1,ℤ/8)12yes—
133C2×C4×C59C22×D411yescharacteristic
134C2×C4×C59C22×D411yescharacteristic
135C2×C4×C59C2×#4111yescharacteristic
136C2×Dic59—44no—
137C2×Dic59—84no—
138Dic118—44no—
139Dic118—44no—
140Dic118—84no—
141C8×C59—21no—
142C8×C59—21no—
143C8×C59—42no—
144C8×C59C2×D812yes—
145C8×C59C2×D812yes—
146C59⋊C8—44no—
147#19188—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—42no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59#45912yes—
154C22×C59C2×#12111yescharacteristic
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—42no—
158C4×C59#45912yes—
159C4×C59#47312yes—
160C4×C59C22×D811yescharacteristic
161C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
162C4×C59#57812yes—
163Dic59—84no—
164#3226—592no—
165C2×#494—591no—
166#2931D5911yescharacteristic
167#1689—594no—
168#1669—594no—
169#1564—591no—
170#1546—592no—
171C2×C59—22no—
172C2×C59—82no—
173C2×C59C2×#45911yescharacteristic
174C2×C59#305911yescharacteristic
175C2×C59C2×#47311yescharacteristic
176C22×Q16—1181no—
177C22×Q16—1181no—
178C2×#131—21no—
179#532C2×D5911yescharacteristic
180C2×C4⋊Q8—592no—
181#500—1182no—
182#494—592no—
183#494—592no—
184#494—1181no—
185#494—1181no—
186#486—1182no—
187#480—1184no—
188#474—598no—
189#436—21no—
190#436C2×D5912yes—
191C8×D4—21no—
192C8×D4C2×D5912yes—
193#431—592no—
194#423C2×D5911yescharacteristic
195#408—21no—
196C2×C4×C8C2×D5911yescharacteristic
197#335—592no—
198#334—592no—
199#332—598no—
200C59#1918811yescharacteristic, complemented, Sylow
201C2×#41—21no—
202C22×Q8—1182no—
203C2×Q16—1182no—
204C2×Q16—1182no—
205C2×Q16—1182no—
206C2×Q16—1182no—
207C2×Q16—2364no—
208C2×Q16—2364no—
209#131—44no—
210C2×M16—21no—
211C2×M16—22no—
212C22×C8—22no—
213C22×C8C22⋊D5911yescharacteristic
214C22×C8C22⋊D5911yescharacteristic
215C4⋊Q8—594no—
216C4⋊Q8—1182no—
217C4⋊Q8—1182no—
218C4⋊Q8—1182no—
219C4⋊Q8D23611yescharacteristic
220#127D23611yescharacteristic
221#124—21no—
222#121—21no—
223#121D23612yes—
224C4×D4—21no—
225C4×D4C22×D5912yes—
226C2×C4⋊C4—1182no—
227C2×C4⋊C4—1182no—
228C2×C42C22×D5911yescharacteristic
229#107—1188no—
230C4⋊C8—21no—
231C4⋊C8—594no—
232C4⋊C8C22×D5912yes—
233#102—1188no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—1182no—
237C4×C8—1182no—
238C4×C8C22⋊D5911yescharacteristic
239C4×C8C22⋊D5911yescharacteristic
240C4×C8C22×D5911yescharacteristic
241C4×C8C22×D5911yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×Q8—1184no—
246C2×Q8—2364no—
247C2×Q8—2364no—
248C2×Q8—2362no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C2×D23612yes—
253C22×C4—22no—
254C22×C4—2362no—
255C22×C4C2×D23611yescharacteristic
256C22×C4#1174568511yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258Q16—2364no—
259Q16—2364no—
260Q16—2364no—
261Q16—2364no—
262M16—42no—
263M16—44no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—44no—
267C2×C8—1184no—
268C2×C8#1174566914yes—
269C2×C8C2×C22⋊D5911yescharacteristic
270C2×C8C2×C22⋊D5911yescharacteristic
271C2×C8C2×C22⋊D5911yescharacteristic
272C2×C8C2×C22⋊D5911yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—1184no—
275C4⋊C4—1184no—
276C4⋊C4—2362no—
277C4⋊C4—2362no—
278C4⋊C4C2×D23612yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C2×D23611yescharacteristic
282C42#1174568511yescharacteristic
283C42C2×C22⋊D5911yescharacteristic
284C42C23×D5911yescharacteristic
285C23—22no—
286C23#1427370111yescharacteristic
287Q8—42no—
288Q8—4724no—
289Q8—2364no—
290Q8—2364no—
291D4—24no—
292D4—44no—
293D4—42no—
294D4—42no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—4724no—
301C2×C4—2364no—
302C2×C4#1427370111yescharacteristic
303C2×C4#1427370111yescharacteristic
304C2×C4#1427370111yescharacteristic
305C2×C4C2×#1174566912yes—
306C2×C4#1427375212yes—
307C2×C4C22×C22⋊D5911yescharacteristic
308C2×C4C22×D23611yescharacteristic
309C2×C4C2×#1174568511yescharacteristic
310C8—21no—
311C8—21no—
312C8—42no—
313C8—2364no—
314C8C2×#1174566912yes—
315C8C2×#1174566912yes—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22#1683759712yes—
322C22C2×#1427370111yescharacteristic
323C4—21no—
324C4—21no—
325C4—42no—
326C4—4724no—
327C4#1683759712yes—
328C4#1683767712yes—
329C4C22×#1174566911yescharacteristic
330C4C2×#1427375211yescharacteristic
331C4#1683821912yes—
332C2—22no—
333C2—82no—
334C2C2×#1683759711yescharacteristic
335C2#2143375111yescharacteristic
336C2C2×#1683767711yescharacteristic
subgroupG/H
centerC22C2×#14273701
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#2931C2
socleC22×C59C2×#121
prime radicalC59×#532C22
hypercenter#2931D59
hypocenterC59#19188
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931

Complements

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

NH ≅ G/Ntype
2. C2×#16837782C2semidirect
3. #21423585C2semidirect
4. #21423461C2semidirect
5. #21422844C2semidirect
6. #21422769C2semidirect
200. C59#19188semidirect

Maximal subgroups

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

subgroupindex
#214352142
C2×#168377822
#214235852
#214234612
#214228442
#214227692
C59×#29312
#1918859