Finite Group Databasebeta

← #28572666

Subgroups — #28572666

Order 15104.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433905C211yesmaximal, characteristic
2#21433905C211yesmaximal, characteristic
3C8×#11745685C211yesmaximal, characteristic
4#21423551C214yesmaximal, complemented
5#21423544C214yesmaximal, complemented
6#21422855C211yesmaximal, characteristic, complemented
7#21422781C211yesmaximal, characteristic, complemented
8#21422777C211yesmaximal, characteristic, complemented
9C2×C59×#487C211yesmaximal, characteristic, complemented
10C4×#11745685C2211yescharacteristic
11#16837694—21no—
12#16837694C2212yes—
13#16837689—21no—
14#16837689C2212yes—
15#16837684—21no—
16#16837684C2212yes—
17#16837679—21no—
18#16837679C2212yes—
19C8×Dic118C2212yes—
20C8×D236C2212yes—
21C8×C22⋊D59—22no—
22#16837455C2211yescharacteristic
23#16837453—21no—
24#16837442—21no—
25#16837442—21no—
26#16837426C2211yescharacteristic
27#16837426C2211yescharacteristic
28#16837401—21no—
29#16837401—21no—
30C4×C8×D59—21no—
31#16837155C2214yes—
32#16837151C2214yes—
33#16837148C2214yes—
34C2×C4⋊Q8×C59C2211yescharacteristic
35C2×C4⋊Q8×C59C2211yescharacteristic
36C59×#487—21no—
37C59×#487—21no—
38C59×#487C2214yes—
39C2×#103×C59—22no—
40C2×C4×C8×C59C2211yescharacteristic
41C2×#11745685D411yescharacteristic
42C2×#11745664—21no—
43C2×#11745664—21no—
44C2×C8×D59—22no—
45C4×Dic118D411yescharacteristic
46C4×Dic118C2311yescharacteristic
47C4×D236D411yescharacteristic
48C4×D236C2311yescharacteristic
49C4×C22⋊D59—21no—
50C4×C22⋊D59—21no—
51#14273686D411yescharacteristic
52#14273684—21no—
53C42×D59—21no—
54#14273660—22no—
55#14273660—22no—
56#14273658—22no—
57#14273658—22no—
58C4⋊(C59⋊C8)—21no—
59C4⋊(C59⋊C8)—21no—
60C4⋊(C59⋊C8)C2312yes—
61C4⋊(C59⋊C8)C2312yes—
62C4⋊(C59⋊C8)C2312yes—
63Dic59⋊C8—22no—
64#14273650—22no—
65#14273650—22no—
66#14273647—22no—
67#14273647—22no—
68C22⋊(C59⋊C8)—21no—
69(C2×D59)⋊C8—22no—
70C4×C59⋊C8—21no—
71C8×Dic59—22no—
72C22×Q8×C59—21no—
73C22×Q8×C59—21no—
74C22×C8×C59D412yes—
75C4⋊Q8×C59—21no—
76C4⋊Q8×C59—21no—
77C4⋊Q8×C59—21no—
78C4⋊Q8×C59—21no—
79C4⋊Q8×C59—21no—
80C4⋊Q8×C59—21no—
81C4⋊Q8×C59C2312yes—
82C4⋊Q8×C59C2312yes—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—21no—
87C2×C42×C59C2311yescharacteristic
88#103×C59—24no—
89#103×C59—24no—
90C4×C8×C59D411yescharacteristic
91C4×C8×C59D411yescharacteristic
92C4×C8×C59C2312yes—
93C2×Q8×C59—22no—
94C2×Q8×C59—22no—
95C2×Q8×C59—44no—
96C2×Q8×C59—41no—
97C2×Q8×C59—41no—
98C2×Q8×C59—82no—
99C22×C4×C59—41no—
100C22×C4×C59—41no—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59#4111yescharacteristic
104C2×C22⋊D59—21no—
105#11745685—22no—
106#11745685Q1614yes—
107C2×D236C2×D411yescharacteristic
108C2×C4×D59—21no—
109C2×C4×D59—21no—
110C2×C4×D59—21no—
111C2×Dic118C2×D411yescharacteristic
112C2×C8×C59—42no—
113C2×C8×C59C2×D412yes—
114C2×C8×C59C2×D412yes—
115C4⋊C4×C59—22no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59—22no—
118C4⋊C4×C59—22no—
119C4⋊C4×C59—41no—
120C4⋊C4×C59—41no—
121C4⋊C4×C59—41no—
122C4⋊C4×C59—41no—
123C42×C59C2×D411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59#4111yescharacteristic
126C42×C59C2411yescharacteristic
127C22⋊Dic59—21no—
128(C2×D59)⋊C4—21no—
129(C2×D59)⋊C4—21no—
130C4⋊Dic59C2×D411yescharacteristic
131C4⋊Dic59C2×D411yescharacteristic
132Dic59⋊C4—21no—
133Dic59⋊C4—21no—
134C4×Dic59—21no—
135C4×Dic59—21no—
136C4×Dic59—21no—
137#11745664—42no—
138#11745664—42no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8—22no—
141C2×C59⋊C8—22no—
142C8×D59—44no—
143C23×C59#12111yescharacteristic
144C22×D59—21no—
145Q8×C59—44no—
146Q8×C59—82no—
147Q8×C59—82no—
148Q8×C59—82no—
149C2×C4×C59—22no—
150C2×C4×C59—22no—
151C2×C4×C59—42no—
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×C59C2×Q1612yes—
159C2×C4×C59#13512yes—
160C2×C4×C59C22×D411yescharacteristic
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C2×#4111yescharacteristic
163C22⋊D59—24no—
164C2×Dic59—21no—
165C2×Dic59—21no—
166C2×Dic59—21no—
167D236—21no—
168D236C2×Q1612yes—
169C4×D59—24no—
170C4×D59—42no—
171C4×D59—42no—
172Dic118—21no—
173Dic118C2×Q1612yes—
174C8×C59—22no—
175C8×C59—22no—
176C59⋊C8—42no—
177C59⋊C8—42no—
178C59⋊C8—42no—
179#19277—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59#46712yes—
183C22×C59C2×#12111yescharacteristic
184C2×D59—22no—
185C2×D59—41no—
186C2×D59—41no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—82no—
190C4×C59—82no—
191C4×C59#46712yes—
192C4×C59#47112yes—
193C4×C59C22×Q1611yescharacteristic
194C4×C59C2×#13511yescharacteristic
195C4×C59#57512yes—
196Dic59—22no—
197Dic59—41no—
198Dic59—41no—
199C2×#487D5911yescharacteristic, complemented
200#3079—591no—
201#3079—591no—
202C8×#41—591no—
203#1684—594no—
204#1683—594no—
205#1567—591no—
206#1549—591no—
207#1548—591no—
208C2×C59—22no—
209C2×C59#304011yescharacteristic
210C2×C59C2×#46711yescharacteristic
211C2×C59C2×#47111yescharacteristic
212D59—42no—
213C2×C4⋊Q8C2×D5911yescharacteristic
214C2×C4⋊Q8C2×D5911yescharacteristic
215C4×#41—591no—
216#487—21no—
217#487—21no—
218#487C2×D5914yes—
219#477—592no—
220#477—1181no—
221#476—592no—
222#476—1181no—
223#475—592no—
224#475—1181no—
225#474—592no—
226#474—1181no—
227C8×Q8—592no—
228C8×D4—592no—
229C8×D4—1182no—
230#431—591no—
231#431—1181no—
232#428—1181no—
233#428—1181no—
234#423—591no—
235#423—591no—
236#417—1181no—
237#417—1181no—
238C2×#103—22no—
239C2×C4×C8—1181no—
240C2×C4×C8C2×D5911yescharacteristic
241#335—594no—
242#333—594no—
243#332—594no—
244C59#1927711yescharacteristic, complemented, Sylow
245C2×#41—591no—
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—1181no—
249C2×M16—1181no—
250C22×C8—1182no—
251C22×C8C22⋊D5912yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22×D5912yes—
259C4⋊Q8C22×D5912yes—
260C4×Q8—591no—
261C4×Q8—591no—
262C4×D4—591no—
263C4×D4—591no—
264C4×D4—1181no—
265C4×D4—1181no—
266#117—591no—
267#117—1181no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42—1181no—
273C2×C42C22×D5911yescharacteristic
274#107—1182no—
275#107—1182no—
276#106—1182no—
277#106—1182no—
278C4⋊C8—592no—
279C4⋊C8—592no—
280C4⋊C8—592no—
281C4⋊C8—1182no—
282C4⋊C8—1181no—
283C4⋊C8—1181no—
284#103—24no—
285#103—24no—
286#103—1182no—
287#103—1182no—
288#102—1182no—
289#102—1182no—
290C22⋊C8—1181no—
291C22⋊C8—1182no—
292C4×C8—1182no—
293C4×C8—1181no—
294C4×C8C22⋊D5911yescharacteristic
295C4×C8C22⋊D5911yescharacteristic
296C4×C8C22×D5912yes—
297#41—594no—
298#41—1182no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—591no—
306C2×D4—591no—
307C2×D4—1181no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1181no—
311C22×C4—1181no—
312C22×C4—1181no—
313C22×C4D4×D5911yescharacteristic
314C22×C4#1174568711yescharacteristic
315C22×C4C2×C22⋊D5911yescharacteristic
316M16—2362no—
317M16—2362no—
318C2×C8—42no—
319C2×C8—1182no—
320C2×C8—1182no—
321C2×C8—1182no—
322C2×C8—2364no—
323C2×C8C2×C22⋊D5912yes—
324C2×C8C2×C22⋊D5912yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—591no—
334C4⋊C4—591no—
335C4⋊C4—1181no—
336C4⋊C4—1181no—
337C22⋊C4—1181no—
338C22⋊C4—1181no—
339C22⋊C4—1181no—
340C42—1181no—
341C42—1181no—
342C42—1181no—
343C42D4×D5911yescharacteristic
344C42#1174568711yescharacteristic
345C42C2×C22⋊D5911yescharacteristic
346C42C23×D5911yescharacteristic
347C23—1181no—
348C23#1427369811yescharacteristic
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—592no—
354Q8—1181no—
355D4—592no—
356D4—1184no—
357D4—1181no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—1181no—
365C2×C4—1184no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4#1427369811yescharacteristic
371C2×C4#1427369811yescharacteristic
372C2×C4#1427369811yescharacteristic
373C2×C4Q16×D5912yes—
374C2×C4#1427375012yes—
375C2×C4C2×D4×D5911yescharacteristic
376C2×C4C22×C22⋊D5911yescharacteristic
377C2×C4C2×#1174568711yescharacteristic
378C8—22no—
379C8—22no—
380C8—2362no—
381C8—2362no—
382C8—2362no—
383C22—21no—
384C22—21no—
385C22—1182no—
386C22—2361no—
387C22—2361no—
388C22#1683764012yes—
389C22C2×#1427369811yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—1182no—
395C4—2361no—
396C4—2361no—
397C4#1683764012yes—
398C4#1683766212yes—
399C4C2×Q16×D5911yescharacteristic
400C4C2×#1427375011yescharacteristic
401C4#1683820012yes—
402C2—22no—
403C2—2362no—
404C2#2143358611yescharacteristic
405C2C2×#1683764011yescharacteristic
406C2C2×#1683766211yescharacteristic
subgroupG/H
centerC22C2×#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#487C2
socleC22×C59C2×#121
prime radicalC2×#11745685D4
hypercenterC2×#487D59
hypocenterC59#19277
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #21423551C2semidirect
5. #21423544C2semidirect
6. #21422855C2semidirect
7. #21422781C2semidirect
8. #21422777C2semidirect
9. C2×C59×#487C2semidirect
199. C2×#487D59semidirect
244. C59#19277semidirect

Maximal subgroups

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

subgroupindex
#214339052
C8×#117456852
#214235512
#214235442
#214228552
#214227812
#214227772
C2×C59×#4872
#1927759