Finite Group Databasebeta

← #28483887

Subgroups — #28483887

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434100C211yesmaximal, characteristic
2#21433037C211yesmaximal, characteristic, complemented
3#21424893C211yesmaximal, characteristic
4#21422522C211yesmaximal, characteristic
5#21421993C211yesmaximal, characteristic, complemented
6#21421940C211yesmaximal, characteristic, complemented
7C59×#1542C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9#16837730C412yes—
10#16837689—21no—
11#16837604—21no—
12#16837552C412yes—
13#16837533—22no—
14#16837533—22no—
15C4×#11745671—22no—
16#16837455C2211yescharacteristic
17#16837429—21no—
18#16837426C2211yescharacteristic
19C2×(C2×D59)⋊C8—21no—
20C2×(C2×D59)⋊C8—21no—
21#16837358—21no—
22C2×#14273638C2211yescharacteristic
23#16837131—22no—
24C59×#533C2211yescharacteristic
25C59×#431C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#333—21no—
28C59×#332—21no—
29C22×D236C2×C411yescharacteristic
30C22×C4×D59—21no—
31C2×#11745671—22no—
32C2×#11745664—21no—
33C22×C59⋊C8D411yescharacteristic
34C22×C59⋊C8D411yescharacteristic
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×(C2×D59)⋊C4—21no—
41#14273658—22no—
42C4⋊(C59⋊C8)—24no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D412yes—
46#14273651—22no—
47#14273647—22no—
48C22⋊(C59⋊C8)C2×C412yes—
49(C2×D59)⋊C8—42no—
50(C2×D59)⋊C8—42no—
51#14273638—22no—
52#14273638—21no—
53#14273638—21no—
54C4×C59⋊C8—22no—
55C4×C59⋊C8—21no—
56C22×C8×C59—21no—
57C4⋊Q8×C59D411yescharacteristic
58C4⋊Q8×C59D411yescharacteristic
59#125×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59C2×C412yes—
62C4×Q8×C59—21no—
63C4×Q8×C59D412yes—
64#117×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C22⋊C8×C59—21no—
70C4×C8×C59—21no—
71C23×D59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59C22⋊C412yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59#4111yescharacteristic
77C22×Dic59—21no—
78C2×D236—21no—
79C2×D236—21no—
80C2×D236—42no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59—41no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C4⋊C4×C59—22no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59C22⋊C412yes—
97C22⋊C4×C59—22no—
98C42×C59—22no—
99C42×C59C2×D411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59#4111yescharacteristic
103#11745671—44no—
104(C2×D59)⋊C4—41no—
105(C2×D59)⋊C4—41no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108#11745664—41no—
109#11745664—41no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—21no—
115C2×C59⋊C8—21no—
116C2×C59⋊C8—21no—
117C2×C59⋊C8SD1612yescomplemented
118C23×C59C4×D411yescharacteristic
119C22×D59—22no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126Q8×C59—42no—
127Q8×C59—81no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59C2×C22⋊C411yescharacteristic
133C2×C4×C59C4×D411yescharacteristic
134C2×C4×C59#12011yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59#12311yescharacteristic
138C2×C4×C59C2×SD1611yescharacteristic
139C2×C4×C59#13511yescharacteristic
140C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C59#13711yescharacteristic
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145D236—22no—
146D236—22no—
147D236—41no—
148D236—41no—
149C4×D59—81no—
150C4×D59—81no—
151C4×D59—81no—
152C4×D59—81no—
153C8×C59—81no—
154C8×C59—81no—
155C59⋊C8—22no—
156C59⋊C8—42no—
157C59⋊C8—41no—
158C59⋊C8—42no—
159#9050—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59#38611yescharacteristic
163C22×C59C4×SD1611yescharacteristic
164C22×C59#44011yescharacteristic
165C2×D59—42no—
166C2×D59—42no—
167C2×D59—42no—
168C2×D59—81no—
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—81no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—42no—
177C4×C59—42no—
178C4×C59#44811yescharacteristic
179C4×C59#44911yescharacteristic
180C4×C59#46011yescharacteristic
181C4×C59#46411yescharacteristic
182Dic59—81no—
183#3101—591no—
184#2961—591no—
185#1955—591no—
186#1542D5911yescharacteristic, complemented
187#1486—591no—
188#1301—591no—
189#1281—591no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59#188011yescharacteristic
193C2×C59#188111yescharacteristic
194C2×C59#189111yescharacteristic
195D59—82no—
196#533C2×D5911yescharacteristic
197C2×C4×D4—591no—
198#483—592no—
199#476—1181no—
200#461—1181no—
201#450—592no—
202#446—1182no—
203#446—1182no—
204C4×SD16—1182no—
205#431—591no—
206#431C2×D5911yescharacteristic
207#424—1181no—
208#423—591no—
209C2×C4⋊C8C2×D5911yescharacteristic
210C2×C22⋊C8—1181no—
211C2×C22⋊C8—1181no—
212#405—1181no—
213C2×#97—591no—
214#333—21no—
215#332—21no—
216D4⋊C8—1182no—
217C59#905011yescharacteristic, complemented, Sylow
218C22×D4—591no—
219C23×C4—1181no—
220C2×SD16—1182no—
221C2×M16—1181no—
222C22×C8—21no—
223C22×C8—591no—
224C22×C8—591no—
225C4⋊Q8C22⋊D5911yescharacteristic
226C4⋊Q8C22⋊D5911yescharacteristic
227#125—21no—
228C22⋊Q8—21no—
229C22⋊Q8C4×D5912yes—
230C4×Q8—21no—
231C4×Q8D23612yes—
232C4×D4—1181no—
233C4×D4—1181no—
234C4×D4—1181no—
235C4×D4—1181no—
236#117—21no—
237C2×C4⋊C4—591no—
238C2×C22⋊C4—1181no—
239C2×C42C22×D5911yescharacteristic
240#106—1182no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—592no—
245C4⋊C8—1184no—
246C4⋊C8—1181no—
247C4⋊C8—1181no—
248#103—1182no—
249#102—1182no—
250C22⋊C8—21no—
251C22⋊C8—592no—
252C22⋊C8—2362no—
253C22⋊C8—2362no—
254#97—1182no—
255#97—1181no—
256#97—1181no—
257C4×C8—21no—
258C4×C8—1182no—
259C4×C8—1181no—
260C24—1181no—
261C2×Q8—21no—
262C2×Q8(C2×D59)⋊C412yes—
263C2×D4—1181no—
264C2×D4—1181no—
265C2×D4—2362no—
266C22×C4—1181no—
267C22×C4—2361no—
268C22×C4—2361no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4—2361no—
272C22×C4—2361no—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4D4×D5911yescharacteristic
275C22×C4#1174568711yescharacteristic
276SD16—2364no—
277M16—2361no—
278M16—2361no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—592no—
284C2×C8—1181no—
285C2×C8—1181no—
286C2×C8—1182no—
287C2×C8—1182no—
288C2×C8—1181no—
289C2×C8—1181no—
290C2×C8—1181no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4(C2×D59)⋊C412yes—
299C22⋊C4—22no—
300C22⋊C4—2361no—
301C22⋊C4—2361no—
302C42—22no—
303C42C2×D23611yescharacteristic
304C42D4×D5911yescharacteristic
305C42#1174568711yescharacteristic
306C42C2×C22⋊D5911yescharacteristic
307C23—1182no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23—2361no—
312C23—2361no—
313C23—2361no—
314C23(C22⋊D59)⋊C411yescharacteristic
315Q8—42no—
316Q8—81no—
317D4—1182no—
318D4—1182no—
319D4—2361no—
320D4—2361no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—4721no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4C2×(C2×D59)⋊C411yescharacteristic
333C2×C4(C22⋊D59)⋊C411yescharacteristic
334C2×C4#1427369711yescharacteristic
335C2×C4#1427369911yescharacteristic
336C2×C4#1427369911yescharacteristic
337C2×C4#1427371111yescharacteristic
338C2×C4SD16×D5911yescharacteristic
339C2×C4#1427375011yescharacteristic
340C2×C4#1427375411yescharacteristic
341C2×C4#1427375911yescharacteristic
342C8—81no—
343C8—81no—
344C8—1182no—
345C8—2362no—
346C8—2361no—
347C8—2362no—
348C22—21no—
349C22—21no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22—4721no—
354C22—2362no—
355C22—2362no—
356C22—2362no—
357C22#1683727911yescharacteristic
358C22#1683748711yescharacteristic
359C22#1683750511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—4721no—
367C4#1683754211yescharacteristic
368C4#1683755011yescharacteristic
369C4#1683760011yescharacteristic
370C4#1683762611yescharacteristic
371C2—21no—
372C2—21no—
373C2—4722no—
374C2#2142444911yescharacteristic
375C2#2142445611yescharacteristic
376C2#2142451211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1542C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1542D59
hypocenterC59#9050
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. #21433037C2semidirect
5. #21421993C2semidirect
6. #21421940C2semidirect
7. C59×#1542C2semidirect
117. C2×C59⋊C8SD16semidirect
186. #1542D59semidirect
217. C59#9050semidirect

Maximal subgroups

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

subgroupindex
#214341002
#214330372
#214248932
#214225222
#214219932
#214219402
C59×#15422
#905059