Finite Group Databasebeta

← #28574234

Subgroups — #28574234

Order 15104.

684 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837679C211yesmaximal, characteristic, complemented
2#21433556C211yesmaximal, characteristic
3#21433136C211yesmaximal, characteristic, complemented
4#21423640C214yesmaximal, complemented
5#21423558C214yesmaximal, complemented
6#21422871C211yesmaximal, characteristic, complemented
7#21422781C211yesmaximal, characteristic, complemented
8#21422745C211yesmaximal, characteristic, complemented
9C59×#3206C211yesmaximal, characteristic, complemented
10C2×#14273752—21no—
11C2×C4×D236C2211yescharacteristic
12#16837679—22no—
13#16837679C2214yes—
14#16837614—22no—
15#16837604C2212yes—
16#16837594C2212yes—
17#16837481C2214yes—
18#16837479—22no—
19#16837455C2211yescharacteristic
20C2×#14273660—21no—
21#16837426C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23#16837403—21no—
24C2×#14273647—21no—
25#16837364—22no—
26#16837155C2214yescomplemented
27#16837148C2212yescomplemented
28#16837148C2214yescomplemented
29#16837146C2212yescomplemented
30C59×#532C2211yescharacteristic
31C2×C4⋊Q8×C59C2211yescharacteristic
32C59×#492C2212yescomplemented
33C59×#487C2212yescomplemented
34C59×#461—22no—
35C59×#448—22no—
36C59×#431C2211yescharacteristic
37C22×D236D411yescharacteristic
38C22×C4×D59—21no—
39#14273752—44no—
40C2×#11745671—22no—
41C2×#11745669—22no—
42C2×#11745664—21no—
43C2×#11745659—24no—
44C22×C59⋊C8—21no—
45C4×D236—21no—
46C4×D236—21no—
47C4×D236D412yes—
48C4×D236C2312yes—
49C2×C4⋊Dic59D411yescharacteristic
50C2×(C2×D59)⋊C4—21no—
51#14273660—24no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)C2312yes—
55C4⋊(C59⋊C8)C2312yes—
56C4⋊(C59⋊C8)C2312yes—
57#14273651—22no—
58#14273648—22no—
59#14273647—24no—
60C22⋊(C59⋊C8)—21no—
61(C2×D59)⋊C8—24no—
62C4×C59⋊C8—21no—
63C2×#41×C59—21no—
64C22×Q8×C59—21no—
65C2×SD16×C59—24no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59C2312yes—
70C4⋊Q8×C59C2311yescharacteristic
71#127×C59C2311yescharacteristic
72#124×C59—21no—
73#121×C59—21no—
74#121×C59—21no—
75C4×D4×C59—21no—
76C4×D4×C59—21no—
77C2×C4⋊C4×C59—21no—
78C2×C4⋊C4×C59—21no—
79C2×C42×C59C2311yescharacteristic
80C4⋊C8×C59D412yes—
81#103×C59—24no—
82C22⋊C8×C59D412yes—
83C4×C8×C59C2312yes—
84C23×D59—21no—
85#41×C59—42no—
86#41×C59—42no—
87C2×Q8×C59—22no—
88C2×Q8×C59—21no—
89C2×Q8×C59—41no—
90C2×Q8×C59—82no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C22×Dic59—21no—
101C2×D236—22no—
102C2×D236—22no—
103C2×D236—21no—
104C2×D236C2×D412yes—
105C2×C4×D59—22no—
106C2×C4×D59—22no—
107C2×C4×D59—82no—
108SD16×C59—84no—
109C2×C8×C59C2×D414yes—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59—22no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59#4111yescharacteristic
121C42×C59C2411yescharacteristic
122#11745671—44no—
123#11745669—44no—
124(C2×D59)⋊C4—22no—
125(C2×D59)⋊C4—22no—
126C4⋊Dic59—21no—
127C4⋊Dic59C2×D412yes—
128#11745664—42no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—42no—
133#11745659—84no—
134C23×C59—41no—
135C23×C59#12111yescharacteristic
136C22×D59—22no—
137C22×D59—42no—
138C22×D59—81no—
139C22×D59—81no—
140Q8×C59—41no—
141Q8×C59—41no—
142Q8×C59—44no—
143Q8×C59—82no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—42no—
147D4×C59—81no—
148D4×C59—82no—
149C2×C4×C59—22no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—82no—
154C2×C4×C59—82no—
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59AGL(1,ℤ/8)12yes—
159C2×C4×C59#13712yes—
160C2×C4×C59C22×D411yescharacteristic
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C2×#4111yescharacteristic
163C2×Dic59—22no—
164C2×Dic59—81no—
165D236—22no—
166D236—22no—
167D236—42no—
168C4×D59—44no—
169C4×D59—82no—
170C8×C59—24no—
171C59⋊C8—42no—
172C59⋊C8—42no—
173C59⋊C8—42no—
174#19421—591nomaximal, Sylow
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59#46812yes—
181C22×C59C2×#12111yescharacteristic
182C2×D59—42no—
183C2×D59—42no—
184C2×D59—42no—
185C2×D59—84no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—82no—
190C4×C59—81no—
191C4×C59#46812yes—
192C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C59C2×#13711yescharacteristic
194C4×C59#57612yes—
195Dic59—42no—
196#3206D5911yescharacteristic, complemented
197C2×#474—591no—
198#3036—591no—
199#2975—591no—
200#1699—594no—
201#1685—594no—
202#1571—591no—
203#1549—591no—
204#1540—591no—
205C2×C59—22no—
206C2×C59—161no—
207C2×C59#304111yescharacteristic
208C2×C59C2×#46811yescharacteristic
209C2×C59#306911yescharacteristic
210D59—82no—
211C2×AGL(1,ℤ/8)—1181no—
212#532C2×D5911yescharacteristic
213C2×C4⋊Q8C2×D5911yescharacteristic
214C2×C4×D4—591no—
215#492C2×D5912yescomplemented
216#487C2×D5912yescomplemented
217#474—594no—
218#474—1182no—
219#463—1182no—
220#461—22no—
221#461—592no—
222#459—592no—
223#448—22no—
224#436—594no—
225#436—1182no—
226#431—591no—
227#431C2×D5911yescharacteristic
228C2×#107—1181no—
229#423—591no—
230C2×C4⋊C8—591no—
231#417—1181no—
232C2×#102—1181no—
233#407—1182no—
234#335—594no—
235#332—592no—
236#332—594no—
237#331—592no—
238C59#1942111yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×Q8—21no—
241C22×D4—591no—
242C23×C4—1181no—
243AGL(1,ℤ/8)—2364no—
244C2×SD16—24no—
245C2×SD16—1182no—
246C2×D8—1182no—
247C2×M16—1181no—
248C2×M16—1184no—
249C22×C8—1181no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22×D5912yes—
254C4⋊Q8C22×D5911yescharacteristic
255#127C22×D5911yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—592no—
262C4×D4—592no—
263C4×D4—1181no—
264C4×D4—1181no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—591no—
268C2×C22⋊C4—1181no—
269C2×C42C22×D5911yescharacteristic
270#107—1184no—
271C4⋊C8—592no—
272C4⋊C8—592no—
273C4⋊C8—592no—
274C4⋊C8—1181no—
275C4⋊C8—1181no—
276C4⋊C8C22⋊D5912yes—
277#103—24no—
278#103—1182no—
279#102—1184no—
280#102—1182no—
281C22⋊C8—1181no—
282C22⋊C8—1184no—
283C22⋊C8C22⋊D5912yes—
284C4×C8—1181no—
285C4×C8C22×D5912yes—
286C24—1181no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—82no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—592no—
298C2×D4—1181no—
299C2×D4—1182no—
300C2×D4—1182no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—1181no—
304C22×C4—1182no—
305C22×C4—1182no—
306C22×C4—4722no—
307C22×C4D4×D5911yescharacteristic
308C22×C4#1174568711yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310SD16—84no—
311SD16—2364no—
312D8—2364no—
313M16—4724no—
314M16—2362no—
315C2×C8—1182no—
316C2×C8—1182no—
317C2×C8—1182no—
318C2×C8—2362no—
319C2×C8C2×C22⋊D5914yes—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—592no—
327C4⋊C4—1181no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1182no—
331C22⋊C4—1182no—
332C42D4×D5911yescharacteristic
333C42#1174568711yescharacteristic
334C42C2×C22⋊D5911yescharacteristic
335C42C23×D5911yescharacteristic
336C23—41no—
337C23—1182no—
338C23—4721no—
339C23—4721no—
340C23—2362no—
341C23#1427369811yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—44no—
345Q8—82no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—82no—
351D4—1182no—
352D4—1182no—
353D4—2362no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—1182no—
361C2×C4—4722no—
362C2×C4—4721no—
363C2×C4—2364no—
364C2×C4#1427369811yescharacteristic
365C2×C4#1427369811yescharacteristic
366C2×C4#1427369811yescharacteristic
367C2×C4#1427375412yes—
368C2×C4#1427375612yes—
369C2×C4C2×D4×D5911yescharacteristic
370C2×C4C22×C22⋊D5911yescharacteristic
371C2×C4C2×#1174568711yescharacteristic
372C8—24no—
373C8—2362no—
374C8—2362no—
375C8—2362no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—4724no—
382C22—2362no—
383C22—2362no—
384C22—2362no—
385C22#1683764512yes—
386C22C2×#1427369811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—82no—
392C4—2362no—
393C4#1683764512yes—
394C4C2×#1427375411yescharacteristic
395C4C2×#1427375611yescharacteristic
396C4#1683820512yes—
397C2—22no—
398C2—161no—
399C2—4722no—
400C2#2143359511yescharacteristic
401C2C2×#1683764511yescharacteristic
402C2#2143382911yescharacteristic
subgroupG/H
centerC22C2×#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3206C2
socleC22×C59C2×#121
prime radical#21433556C2
hypercenter#3206D59
hypocenterC59#19421
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. C2×#16837679C2semidirect
3. #21433136C2semidirect
4. #21423640C2semidirect
5. #21423558C2semidirect
6. #21422871C2semidirect
7. #21422781C2semidirect
8. #21422745C2semidirect
9. C59×#3206C2semidirect
26. #16837155C22semidirect
27. #16837148C22semidirect
28. #16837148C22semidirect
29. #16837146C22semidirect
32. C59×#492C22semidirect
33. C59×#487C22semidirect
196. #3206D59semidirect
215. #492C2×D59semidirect
216. #487C2×D59semidirect
238. C59#19421semidirect

Maximal subgroups

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

subgroupindex
C2×#168376792
#214335562
#214331362
#214236402
#214235582
#214228712
#214227812
#214227452
C59×#32062
#1942159