Finite Group Databasebeta

← #28571199

Subgroups — #28571199

Order 15104.

770 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434092C212yesmaximal, complemented
2C2×#16837481C211yesmaximal, characteristic, complemented
3#21423582C212yesmaximal, complemented
4#21423572C212yesmaximal, complemented
5#21423558C214yesmaximal, complemented
6#21422844C211yesmaximal, characteristic, complemented
7#21422745C212yesmaximal, complemented
8C59×#3147C211yesmaximal, characteristic, complemented
9C22×#11745659—21no—
10C2×C4×D236C2211yescharacteristic
11#16837713—22no—
12#16837679C2214yes—
13#16837594C2214yes—
14#16837535—22no—
15C4×#11745669—24no—
16#16837481—21no—
17#16837481—21no—
18#16837481C2212yes—
19#16837481C2212yes—
20#16837455C2212yes—
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×(C2×D59)⋊C8—21no—
23#16837155C2212yescomplemented
24#16837153C2212yescomplemented
25#16837148C2214yescomplemented
26#16837146C2214yescomplemented
27C2×#135×C59—21no—
28C2×#135×C59—21no—
29C59×#532C2212yes—
30C59×#495—21no—
31C59×#495—21no—
32C59×#494C2211yescharacteristic, complemented
33C59×#493C2211yescharacteristic, complemented
34C59×#492C2212yescomplemented
35C2×C4×C8×C59C2211yescharacteristic
36C22×D236D411yescharacteristic
37C22×C4×D59—21no—
38C2×#11745669—24no—
39C2×#11745659—22no—
40C2×#11745659—22no—
41C2×#11745659—44no—
42C22×C59⋊C8—21no—
43C4×D236—21no—
44C4×D236—21no—
45C4×D236D412yes—
46C4×D236C2312yes—
47C2×C4⋊Dic59D411yescharacteristic
48C2×(C2×D59)⋊C4—21no—
49#14273660—24no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)C2312yes—
52C4⋊(C59⋊C8)C2314yes—
53#14273648—24no—
54#14273647—24no—
55C22⋊(C59⋊C8)—22no—
56(C2×D59)⋊C8—22no—
57(C2×D59)⋊C8—22no—
58C4×C59⋊C8—22no—
59C2×#41×C59—22no—
60#135×C59—44no—
61#135×C59—44no—
62C2×Q16×C59—21no—
63C2×Q16×C59—21no—
64C2×SD16×C59—22no—
65C2×SD16×C59—22no—
66C2×D8×C59—21no—
67C2×D8×C59—21no—
68C22×C8×C59D411yescharacteristic
69C22×C8×C59D411yescharacteristic
70C4⋊Q8×C59C2312yes—
71#127×C59C2312yes—
72#124×C59—22no—
73#121×C59—22no—
74#121×C59—22no—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77C2×C42×C59C2311yescharacteristic
78C4×C8×C59D411yescharacteristic
79C4×C8×C59D411yescharacteristic
80C4×C8×C59C2311yescharacteristic
81C4×C8×C59C2311yescharacteristic
82C23×D59—21no—
83#41×C59—44no—
84#41×C59—44no—
85C2×Q8×C59—22no—
86C2×D4×C59—22no—
87C2×D4×C59—22no—
88C2×D4×C59—22no—
89C2×D4×C59—42no—
90C22×C4×C59—42no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×Dic59—21no—
95C2×D236—21no—
96C2×D236—42no—
97C2×D236C2×D412yes—
98C2×C4×D59—22no—
99C2×C4×D59—22no—
100C2×C4×D59—42no—
101C2×C4×D59—42no—
102Q16×C59—42no—
103Q16×C59—42no—
104SD16×C59—44no—
105SD16×C59—44no—
106D8×C59—42no—
107D8×C59—42no—
108C2×C8×C59—22no—
109C2×C8×C59D814yescomplemented
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59C2×D411yescharacteristic
113C2×C8×C59C2×D411yescharacteristic
114C4⋊C4×C59—22no—
115C4⋊C4×C59—22no—
116C22⋊C4×C59—42no—
117C22⋊C4×C59—42no—
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59#4111yescharacteristic
121C42×C59C2411yescharacteristic
122#11745669—48no—
123(C2×D59)⋊C4—22no—
124(C2×D59)⋊C4—22no—
125C4⋊Dic59—21no—
126C4⋊Dic59C2×D412yes—
127C2×C59⋊C8—24no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—42no—
130#11745659—44no—
131#11745659—44no—
132C23×C59—42no—
133C23×C59#12111yescharacteristic
134C22×D59—22no—
135C22×D59—42no—
136C22×D59—42no—
137C22×D59—42no—
138Q8×C59—42no—
139Q8×C59—42no—
140D4×C59—42no—
141D4×C59—42no—
142D4×C59—44no—
143D4×C59—82no—
144D4×C59—84no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—84no—
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59C2×D812yes—
153C2×C4×C59AGL(1,ℤ/8)12yes—
154C2×C4×C59C22×D411yescharacteristic
155C2×C4×C59C22×D411yescharacteristic
156C2×C4×C59C2×#4111yescharacteristic
157C2×Dic59—22no—
158C2×Dic59—42no—
159D236—24no—
160D236—42no—
161C4×D59—42no—
162C4×D59—42no—
163C4×D59—44no—
164C8×C59—21no—
165C8×C59—21no—
166C8×C59C2×D812yes—
167C8×C59C2×D812yes—
168C59⋊C8—42no—
169C59⋊C8—44no—
170#19133—591nomaximal, Sylow
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—82no—
174C22×C59—82no—
175C22×C59—82no—
176C22×C59#45912yes—
177C22×C59C2×#12111yescharacteristic
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—42no—
181C2×D59—84no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—82no—
186C4×C59#45912yes—
187C4×C59C22×D811yescharacteristic
188C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
189C4×C59#57512yes—
190Dic59—42no—
191#3147D5911yescharacteristic, complemented
192#3100—592no—
193C2×#436—591no—
194#1689—592no—
195#1687—592no—
196#1685—594no—
197#1564—591no—
198#1540—592no—
199C2×C59—22no—
200C2×C59—162no—
201C2×C59C2×#45911yescharacteristic
202C2×C59#305411yescharacteristic
203C2×C59#306711yescharacteristic
204D59—82no—
205C2×#135—21no—
206C2×#135—21no—
207C22×M16—1181no—
208#532C2×D5912yes—
209C2×C4×D4—591no—
210#495—21no—
211#495—21no—
212#494C2×D5911yescharacteristic, complemented
213#493C2×D5911yescharacteristic, complemented
214#492C2×D5912yescomplemented
215#480—1182no—
216#474—594no—
217#459—594no—
218#447—1182no—
219C4×D8—1184no—
220#436—592no—
221#436—592no—
222#436—1181no—
223#436—1181no—
224#431—592no—
225C2×C4⋊C8—591no—
226C2×C22⋊C8—1181no—
227C2×C4×C8C2×D5911yescharacteristic
228#335—592no—
229#334—592no—
230#332—594no—
231#331—594no—
232C59#1913311yescharacteristic, complemented, Sylow
233C2×#41—22no—
234C22×D4—591no—
235C23×C4—1181no—
236#135—44no—
237#135—44no—
238C2×Q16—21no—
239C2×Q16—21no—
240C2×SD16—22no—
241C2×SD16—22no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—1184no—
245C2×M16—1182no—
246C2×M16—1182no—
247C2×M16—2364no—
248C22×C8—1181no—
249C22×C8C22⋊D5911yescharacteristic
250C22×C8C22⋊D5911yescharacteristic
251C4⋊Q8C22×D5912yes—
252#127C22×D5912yes—
253#124—22no—
254#121—22no—
255#121—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—592no—
259C4×D4—592no—
260C4×D4—1181no—
261C4×D4—1181no—
262C2×C4⋊C4—591no—
263C2×C22⋊C4—1181no—
264C2×C42C22×D5911yescharacteristic
265#107—1184no—
266C4⋊C8—592no—
267C4⋊C8—594no—
268C4⋊C8—1181no—
269#102—1184no—
270#102—1184no—
271C22⋊C8—1182no—
272C22⋊C8—1182no—
273C22⋊C8—1182no—
274C4×C8—1182no—
275C4×C8C22⋊D5911yescharacteristic
276C4×C8C22⋊D5911yescharacteristic
277C4×C8C22×D5911yescharacteristic
278C4×C8C22×D5911yescharacteristic
279C24—1181no—
280#41—44no—
281#41—44no—
282C2×Q8—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—592no—
288C2×D4—1181no—
289C2×D4—2362no—
290C22×C4—42no—
291C22×C4—1181no—
292C22×C4—1182no—
293C22×C4—1182no—
294C22×C4—2362no—
295C22×C4—2362no—
296C22×C4D4×D5911yescharacteristic
297C22×C4#1174568711yescharacteristic
298C22×C4C2×C22⋊D5911yescharacteristic
299Q16—42no—
300Q16—42no—
301SD16—44no—
302SD16—44no—
303D8—42no—
304D8—42no—
305D8—2368no—
306M16—2364no—
307M16—2364no—
308C2×C8—22no—
309C2×C8—1184no—
310C2×C8—1182no—
311C2×C8—2362no—
312C2×C8#1174566914yescomplemented
313C2×C8C2×C22⋊D5911yescharacteristic
314C2×C8C2×C22⋊D5911yescharacteristic
315C2×C8C2×C22⋊D5911yescharacteristic
316C2×C8C2×C22⋊D5911yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—592no—
320C4⋊C4—1181no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—1182no—
324C22⋊C4—1182no—
325C42D4×D5911yescharacteristic
326C42#1174568711yescharacteristic
327C42C2×C22⋊D5911yescharacteristic
328C42C23×D5911yescharacteristic
329C23—42no—
330C23—1182no—
331C23—2362no—
332C23—2362no—
333C23—2362no—
334C23#1427369811yescharacteristic
335Q8—42no—
336Q8—42no—
337D4—42no—
338D4—42no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—1184no—
343D4—2362no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—1182no—
349C2×C4—2362no—
350C2×C4—2362no—
351C2×C4—2362no—
352C2×C4—2364no—
353C2×C4#1427369811yescharacteristic
354C2×C4#1427369811yescharacteristic
355C2×C4#1427369811yescharacteristic
356C2×C4C2×#1174566912yes—
357C2×C4#1427375312yes—
358C2×C4C2×D4×D5911yescharacteristic
359C2×C4C22×C22⋊D5911yescharacteristic
360C2×C4C2×#1174568711yescharacteristic
361C8—21no—
362C8—21no—
363C8—2362no—
364C8—2364no—
365C8C2×#1174566912yes—
366C8C2×#1174566912yes—
367C22—21no—
368C22—21no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—4724no—
373C22—2362no—
374C22—2362no—
375C22—2362no—
376C22#1683759412yes—
377C22C2×#1427369811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—2362no—
383C4#1683759412yes—
384C4C22×#1174566911yescharacteristic
385C4C2×#1427375311yescharacteristic
386C4#1683819912yes—
387C2—22no—
388C2—162no—
389C2—4722no—
390C2C2×#1683759411yescharacteristic
391C2#2143370111yescharacteristic
392C2#2143381211yescharacteristic
subgroupG/H
centerC2×C4#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3147C2
socleC22×C59C2×#121
hypercenter#3147D59
hypocenterC59#19133
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #21434092C2semidirect
2. C2×#16837481C2semidirect
3. #21423582C2semidirect
4. #21423572C2semidirect
5. #21423558C2semidirect
6. #21422844C2semidirect
7. #21422745C2semidirect
8. C59×#3147C2semidirect
23. #16837155C22semidirect
24. #16837153C22semidirect
25. #16837148C22semidirect
26. #16837146C22semidirect
32. C59×#494C22semidirect
33. C59×#493C22semidirect
34. C59×#492C22semidirect
109. C2×C8×C59D8semidirect
191. #3147D59semidirect
212. #494C2×D59semidirect
213. #493C2×D59semidirect
214. #492C2×D59semidirect
232. C59#19133semidirect
312. C2×C8#11745669semidirect

Maximal subgroups

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

subgroupindex
#214340922
C2×#168374812
#214235822
#214235722
#214235582
#214228442
#214227452
C59×#31472
#1913359