Finite Group Databasebeta

← #28498943

Subgroups — #28498943

Order 15104.

450 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×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
2C2×#16837167C211yesmaximal, characteristic, complemented
3#21422114C212yesmaximal, complemented
4#21422111C212yesmaximal, complemented
5C59×#3050C211yesmaximal, characteristic
6C22×#11745664—21no—
7C22×C59⋊C16—21no—
8C22⋊(C59⋊C16)—21no—
9C22⋊(C59⋊C16)C2212yes—
10#16837167—21no—
11#16837167C2212yes—
12C2×#135×C59C411yescharacteristic
13C23×C8×C59C2211yescharacteristic
14C59×#525—21no—
15C59×#467C411yescharacteristic
16C59×#467C2211yescharacteristic
17C59×#466C411yescharacteristic
18C59×#466C2211yescharacteristic
19C59×#465—21no—
20C59×#465—21no—
21C59×#427C411yescharacteristic
22C59×#416—21no—
23C2×#11745664—24no—
24C2×#11745664—22no—
25C2×#11745664—42no—
26C2×#11745664—41no—
27C22×C59⋊C8—41no—
28C2×C59⋊C16—22no—
29C2×C59⋊C16—42no—
30C2×#41×C59—21no—
31C23×C4×C59D411yescharacteristic
32#135×C59—42no—
33C2×Q16×C59C2×C411yescharacteristic
34C2×SD16×C59—21no—
35C2×D8×C59C2×C411yescharacteristic
36C22×C8×C59—42no—
37C22×C8×C59C2×C411yescharacteristic
38C22×C8×C59C2×C411yescharacteristic
39C22×C8×C59D411yescharacteristic
40C22×C8×C59D411yescharacteristic
41C22×C8×C59D411yescharacteristic
42C22×C8×C59C2311yescharacteristic
43#123×C59—41no—
44C22⋊Q8×C59—21no—
45C22⋊Q8×C59—21no—
46#121×C59—21no—
47#121×C59—21no—
48#120×C59—41no—
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51#117×C59—21no—
52#107×C59C2×C411yescharacteristic
53#107×C59C2×C411yescharacteristic
54#106×C59—21no—
55#103×C59—21no—
56#103×C59—21no—
57#102×C59—21no—
58#102×C59—21no—
59C24×C59SD1611yescharacteristic
60#41×C59—42no—
61#41×C59—42no—
62C2×Q8×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C2×D4×C59—41no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—41no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59D811yescharacteristic
73C22×C4×C59D811yescharacteristic
74C22×C4×C59SD1611yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76Q16×C59—41no—
77SD16×C59—42no—
78D8×C59—41no—
79C2×C8×C59—22no—
80C2×C8×C59—22no—
81C2×C8×C59—41no—
82C2×C8×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59C22⋊C411yescharacteristic
86C2×C8×C59C22⋊C411yescharacteristic
87C2×C8×C59C22×C411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C42×C59—41no—
99#11745664—24no—
100#11745664—44no—
101#11745664—42no—
102C2×C59⋊C8—44no—
103C2×C59⋊C8—42no—
104C59⋊C16—42no—
105C23×C59—22no—
106C23×C59—22no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59#10211yescharacteristic
110C23×C59#10211yescharacteristic
111C23×C59C2×SD1611yescharacteristic
112Q8×C59—41no—
113Q8×C59—41no—
114D4×C59—41no—
115D4×C59—41no—
116D4×C59—82no—
117D4×C59—82no—
118D4×C59—82no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—82no—
130C2×C4×C59#9912yes—
131C2×C4×C59#10211yescharacteristic
132C2×C4×C59#10211yescharacteristic
133C2×C4×C59#10412yes—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C2×D811yescharacteristic
136C8×C59—21no—
137C8×C59—21no—
138C8×C59—21no—
139C8×C59—21no—
140C59⋊C8—44no—
141#11709—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—44no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59#32712yes—
154C22×C59#35912yes—
155C22×C59C2×#10211yescharacteristic
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—81no—
159C4×C59—81no—
160C4×C59—81no—
161C4×C59#32712yes—
162C4×C59C2×#9911yescharacteristic
163C4×C59C2×#10411yescharacteristic
164#3050D5911yescharacteristic
165C2×C22⋊C16—591no—
166C2×#341—591no—
167#1350—592no—
168#1349—592no—
169C2×C59—22no—
170C2×C59—22no—
171C2×C59—41no—
172C2×C59—161no—
173C2×C59C2×#32711yescharacteristic
174C2×C59C2×#35911yescharacteristic
175C2×C59#214811yescharacteristic
176C2×#135Dic5911yescharacteristic
177C22×M16—1181no—
178C23×C8C2×D5911yescharacteristic
179#525—21no—
180C22×C16—1181no—
181#467Dic5911yescharacteristic
182#467C2×D5911yescharacteristic
183#466Dic5911yescharacteristic
184#466C2×D5911yescharacteristic
185#465—21no—
186#465—21no—
187#427Dic5911yescharacteristic
188#416—21no—
189C22⋊C16—592no—
190C22⋊C16—1181no—
191#341—592no—
192#341—1181no—
193C59#1170911yescharacteristic, complemented, Sylow
194C2×#41—21no—
195C23×C4C22⋊D5911yescharacteristic
196#135—42no—
197C2×Q16C2×Dic5911yescharacteristic
198C2×SD16—21no—
199C2×D8C2×Dic5911yescharacteristic
200C2×M16—1184no—
201C2×M16—1182no—
202C2×M16—2362no—
203C2×M16—2361no—
204C22×C8—42no—
205C22×C8—2361no—
206C22×C8C2×Dic5911yescharacteristic
207C22×C8C2×Dic5911yescharacteristic
208C22×C8C22⋊D5911yescharacteristic
209C22×C8C22⋊D5911yescharacteristic
210C22×C8C22⋊D5911yescharacteristic
211C22×C8C22×D5911yescharacteristic
212#123—41no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215#121—21no—
216#121—21no—
217#120—41no—
218C4×D4—41no—
219C4×D4—41no—
220#117—21no—
221C2×C16—1182no—
222C2×C16—2362no—
223#107C2×Dic5911yescharacteristic
224#107C2×Dic5911yescharacteristic
225#106—21no—
226#103—21no—
227#103—21no—
228#102—21no—
229#102—21no—
230C24#1174567011yescharacteristic
231#41—42no—
232#41—42no—
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4#1174566911yescharacteristic
242C22×C4#1174566911yescharacteristic
243C22×C4#1174567011yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247Q16—41no—
248SD16—42no—
249D8—41no—
250M16—1184no—
251M16—2364no—
252M16—2362no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—2364no—
260C2×C8—2362no—
261C2×C8C22⋊Dic5911yescharacteristic
262C2×C8C22⋊Dic5911yescharacteristic
263C2×C8C22×Dic5911yescharacteristic
264C2×C8C2×C22⋊D5911yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—41no—
275C16—2362no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23#1427364811yescharacteristic
281C23#1427364811yescharacteristic
282C23C2×#1174567011yescharacteristic
283Q8—41no—
284Q8—41no—
285D4—41no—
286D4—41no—
287D4—82no—
288D4—82no—
289D4—82no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4#1427364212yes—
302C2×C4#1427364811yescharacteristic
303C2×C4#1427364811yescharacteristic
304C2×C4#1427365412yes—
305C2×C4C2×C22⋊Dic5911yescharacteristic
306C2×C4C2×#1174566911yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—2364no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—44no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#1683713612yes—
324C22#1683720912yes—
325C22C2×#1427364811yescharacteristic
326C4—21no—
327C4—21no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4#1683713612yes—
332C4C2×#1427364211yescharacteristic
333C4C2×#1427365411yescharacteristic
334C2—22no—
335C2—22no—
336C2—41no—
337C2—161no—
338C2C2×#1683713611yescharacteristic
339C2C2×#1683720911yescharacteristic
340C2#2142584511yescharacteristic
subgroupG/H
centerC2×C4#14273648
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3050C2
socleC22×C59C2×#102
cosocleC22×C8×C59C23
hypercenter#3050D59
hypocenterC59#11709
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C59⋊C16)C2semidirect
2. C2×#16837167C2semidirect
3. #21422114C2semidirect
4. #21422111C2semidirect
193. C59#11709semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C59⋊C16)2
C2×#168371672
#214221142
#214221112
C59×#30502
#1170959