Finite Group Databasebeta

← #28505199

Subgroups — #28505199

Order 15104.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837229C211yesmaximal, characteristic, complemented
2C2×#16837227C211yesmaximal, characteristic, complemented
3C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic
4#21424125C211yesmaximal, characteristic
5#21422187C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#427C211yesmaximal, characteristic
8C22×C59⋊C16C2211yescharacteristic
9C22×C59⋊C16C2211yescharacteristic
10C2×#14273661C2211yescharacteristic
11C2×#14273661C2211yescharacteristic
12#16837262—21no—
13#16837229—22no—
14#16837227—22no—
15C22⋊(C59⋊C16)C414yescomplemented
16#16837165—21no—
17C23×C8×C59C2211yescharacteristic
18C2×#117×C59—21no—
19C59×#427—22no—
20C59×#427—22no—
21C2×#107×C59C2211yescharacteristic
22C2×#107×C59C2211yescharacteristic
23C2×#106×C59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C22⋊Dic59—41no—
27C2×C59⋊C16—21no—
28C2×C59⋊C16—21no—
29C2×C59⋊C16—21no—
30C2×C59⋊C16—21no—
31C2×C59⋊C16C2×C412yes—
32C2×C59⋊C16C2×C412yes—
33#14273661—21no—
34#14273661—21no—
35#14273661—21no—
36#14273661—21no—
37C59⋊#95—41no—
38C23×C4×C59D411yescharacteristic
39C22×C8×C59—21no—
40C22×C8×C59—21no—
41C22×C8×C59—21no—
42C22×C8×C59—21no—
43C22×C8×C59C2×C412yes—
44C22×C8×C59D411yescharacteristic
45C22×C8×C59Q812yes—
46C22×C8×C59C2311yescharacteristic
47#117×C59—24no—
48#117×C59—42no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C22⋊C4×C59—41no—
52C2×C42×C59—41no—
53#107×C59—21no—
54#107×C59—21no—
55#107×C59D412yes—
56#107×C59D412yes—
57#106×C59—24no—
58C24×C59D811yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59C4⋊C412yes—
66C22×C4×C59D811yescharacteristic
67C22×C4×C59Q1612yes—
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—41no—
70C22×Dic59—41no—
71C2×C8×C59—21no—
72C2×C8×C59—21no—
73C2×C8×C59—22no—
74C2×C8×C59—22no—
75C2×C8×C59—22no—
76C2×C8×C59—22no—
77C2×C8×C59C4⋊C412yes—
78C2×C8×C59C22×C411yescharacteristic
79C2×C8×C59C2×D411yescharacteristic
80C2×C8×C59C2×D411yescharacteristic
81C2×C8×C59C2×Q811yescharacteristic
82C2×C8×C59#4111yescharacteristic
83C2×C8×C59#4111yescharacteristic
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C22⋊C4×C59—44no—
89C42×C59—44no—
90C22⋊Dic59—82no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59—41no—
94C4⋊Dic59—41no—
95C59⋊C16—22no—
96C59⋊C16—22no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59#10712yes—
104C23×C59C2×D811yescharacteristic
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—81no—
114C2×C4×C59—81no—
115C2×C4×C59—81no—
116C2×C4×C59—81no—
117C2×C4×C59#10712yes—
118C2×C4×C59C2×C4⋊C411yescharacteristic
119C2×C4×C59#12011yescharacteristic
120C2×C4×C59#12311yescharacteristic
121C2×C4×C59C2×Q1611yescharacteristic
122C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C59#13711yescharacteristic
124C2×Dic59—81no—
125C2×Dic59—81no—
126C2×Dic59—81no—
127C2×Dic59—81no—
128C2×Dic59—81no—
129C2×Dic59—81no—
130C8×C59—22no—
131C8×C59C4×D411yescharacteristic
132C8×C59C4×D411yescharacteristic
133C8×C59C22⋊Q811yescharacteristic
134C8×C59C22⋊Q811yescharacteristic
135#12921—591nomaximal, Sylow
136C22×C59—22no—
137C22×C59—22no—
138C22×C59—22no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59C2×#10711yescharacteristic
144C22×C59#44711yescharacteristic
145C22×C59#48011yescharacteristic
146C22×C59#50811yescharacteristic
147C22×C59#50811yescharacteristic
148C22×C59#50911yescharacteristic
149C22×C59#51011yescharacteristic
150C4×C59—22no—
151C4×C59—82no—
152C4×C59—82no—
153C4×C59#38511yescharacteristic
154C4×C59#42811yescharacteristic
155C4×C59#45111yescharacteristic
156C4×C59#48411yescharacteristic
157Dic59—161no—
158Dic59—161no—
159C2×#427D5911yescharacteristic
160C2×#367—591no—
161C2×#366—591no—
162C2×C22⋊C16—591no—
163#1808—591no—
164#1380—591no—
165#1380—591no—
166C2×C59—42no—
167C2×C59#185211yescharacteristic
168C2×C59#215811yescharacteristic
169C2×C59#215911yescharacteristic
170C2×C59#218611yescharacteristic
171C2×C59#218911yescharacteristic
172C2×C59#223411yescharacteristic
173C2×C59#223711yescharacteristic
174C23×C8C2×D5911yescharacteristic
175C2×#117—21no—
176C22×C16—591no—
177C22×C16—591no—
178#427—22no—
179#427—22no—
180C2×#107—591no—
181C2×#107—591no—
182C2×#107C2×D5911yescharacteristic
183C2×#107C2×D5911yescharacteristic
184C2×#106—21no—
185#380—1181no—
186#367—1182no—
187#366—1182no—
188C22⋊C16—594no—
189#340—1181no—
190C59#1292111yescharacteristic, complemented, Sylow
191C23×C4C22⋊D5911yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8Dic11812yes—
197C22×C8C4×D5912yes—
198C22×C8C22⋊D5911yescharacteristic
199C22×C8C22×D5911yescharacteristic
200#117—24no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—2361no—
208C2×C42—41no—
209C2×C16—592no—
210C2×C16—592no—
211C2×C16—1181no—
212C2×C16—1181no—
213C2×C16—1181no—
214C2×C16—1181no—
215#107—21no—
216#107—21no—
217#107—1181no—
218#107—1181no—
219#107—1181no—
220#107—1181no—
221#107C22⋊D5912yes—
222#107C22⋊D5912yes—
223#106—24no—
224#95—2361no—
225C24#1174566911yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—2361no—
233C22×C4—2361no—
234C22×C4Dic59⋊C412yes—
235C22×C4#1174566911yescharacteristic
236C22×C4#1174567212yes—
237C22×C4C2×C22⋊D5911yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8Dic59⋊C412yes—
245C2×C8C2×Dic11811yescharacteristic
246C2×C8C2×C4×D5911yescharacteristic
247C2×C8#1174568511yescharacteristic
248C2×C8#1174568511yescharacteristic
249C2×C8C2×C22⋊D5911yescharacteristic
250C2×C8C2×C22⋊D5911yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2361no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C4⋊C4—2361no—
259C22⋊C4—44no—
260C22⋊C4—4722no—
261C42—44no—
262C16—1182no—
263C16—1182no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#1427366012yes—
271C23C2×#1174566911yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—4721no—
285C2×C4—4721no—
286C2×C4—4721no—
287C2×C4—4721no—
288C2×C4—4721no—
289C2×C4—4721no—
290C2×C4#1427366012yes—
291C2×C4C2×Dic59⋊C411yescharacteristic
292C2×C4#1427369611yescharacteristic
293C2×C4#1427371011yescharacteristic
294C2×C4C2×#1174567211yescharacteristic
295C2×C4#1427375311yescharacteristic
296C2×C4#1427375811yescharacteristic
297C8—22no—
298C8C4×C22⋊D5911yescharacteristic
299C8C4×C22⋊D5911yescharacteristic
300C8#1427370611yescharacteristic
301C8#1427370611yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×#1427366011yescharacteristic
310C22#1683753611yescharacteristic
311C22#1683771511yescharacteristic
312C22#1683783211yescharacteristic
313C22#1683783211yescharacteristic
314C22#1683783711yescharacteristic
315C22#1683784211yescharacteristic
316C4—22no—
317C4—82no—
318C4—82no—
319C4—9441no—
320C4—9441no—
321C4#1683727511yescharacteristic
322C4#1683744211yescharacteristic
323C4#1683755811yescharacteristic
324C4#1683773711yescharacteristic
325C2—42no—
326C2#2142428411yescharacteristic
327C2#2142587811yescharacteristic
328C2#2142588511yescharacteristic
329C2#2142601911yescharacteristic
330C2#2142603611yescharacteristic
331C2#2142628711yescharacteristic
332C2#2142630411yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#427C2
socleC23×C59C2×D8
prime radicalC24×C59D8
cosocleC22×C8×C59C23
hypercenterC2×#427D59
hypocenterC59#12921
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
1. C2×#16837229C2semidirect
2. C2×#16837227C2semidirect
5. #21422187C2semidirect
6. #21422187C2semidirect
15. C22⋊(C59⋊C16)C4semidirect
190. C59#12921semidirect

Maximal subgroups

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

subgroupindex
C2×#168372292
C2×#168372272
C2×C22⋊(C59⋊C16)2
#214241252
#214221872
C2×C59×#4272
#1292159