Finite Group Databasebeta

← #28483922

Subgroups — #28483922

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434651C211yesmaximal, characteristic
2#21432840C211yesmaximal, characteristic
3#21425039C211yesmaximal, characteristic, complemented
4#21422500C211yesmaximal, characteristic, complemented
5#21421937C211yesmaximal, characteristic, complemented
6#21421932C211yesmaximal, characteristic, complemented
7C59×#1502C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837959C2211yescharacteristic
10#16837798C411yescharacteristic, complemented
11#16837774C411yescharacteristic, complemented
12#16837754C411yescharacteristic, complemented
13#16837748C411yescharacteristic, complemented
14#16837629—22no—
15C4×#11745670—22no—
16#16837480—21no—
17#16837474—22no—
18D4×C59⋊C8—21no—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20#16837369—21no—
21C4×#11745664C2211yescharacteristic
22C2×C4×C59⋊C8C2211yescharacteristic
23#16837162—21no—
24#16837162—21no—
25#16837134—21no—
26#16837134—21no—
27C59×#532C2211yescharacteristic
28C59×#431C2211yescharacteristic
29C4×C4⋊C4×C59C2211yescharacteristic
30C59×#327—22no—
31C2×#11745670—22no—
32#14273734—42no—
33#14273734—42no—
34C2×#11745664—22no—
35C2×#11745664D411yescharacteristic
36C2×#11745664D411yescharacteristic
37C22×C59⋊C8—22no—
38C22×C59⋊C8—21no—
39C22×C59⋊C8—21no—
40#14273727C2×C411yescharacteristic
41#14273727C2×C411yescharacteristic
42#14273719—21no—
43#14273706—21no—
44#14273706—21no—
45C4×Dic118—21no—
46C4×Dic118—21no—
47#14273686—21no—
48#14273658—22no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)D411yescharacteristic
51C4⋊(C59⋊C8)D411yescharacteristic
52#14273650—22no—
53#14273648—22no—
54C22⋊(C59⋊C8)—22no—
55C22⋊(C59⋊C8)—22no—
56#14273638—21no—
57C4×C59⋊C8—21no—
58C4×C59⋊C8—21no—
59C4×C59⋊C8C2×C411yescharacteristic
60C4×C59⋊C8C2×C411yescharacteristic
61C2×#41×C59—21no—
62C4⋊Q8×C59C2×C411yescharacteristic
63#127×C59C2×C411yescharacteristic
64#124×C59—21no—
65#121×C59—21no—
66#121×C59D412yes—
67C4×D4×C59—21no—
68C4×D4×C59D412yes—
69C2×C4⋊C4×C59—21no—
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72C4⋊C8×C59—21no—
73C22⋊C8×C59—21no—
74C4×C8×C59—21no—
75#95×C59—21no—
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79#41×C59—41no—
80C2×Q8×C59—21no—
81C2×D4×C59—22no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59C22⋊C412yes—
85C22×C4×C59—22no—
86C22×C4×C59—41no—
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C2×Dic118—21no—
91C2×Dic118—21no—
92C2×C8×C59—41no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59C22⋊C412yes—
97C22⋊C4×C59—22no—
98C22⋊C4×C59—22no—
99C42×C59—41no—
100C42×C59—41no—
101C42×C59C22×C411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59#4111yescharacteristic
105C22⋊Dic59—41no—
106#11745670—44no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111Dic59⋊C4—41no—
112Dic59⋊C4—41no—
113C4×Dic59—41no—
114#11745664—21no—
115#11745664—21no—
116#11745664—21no—
117#11745664—44no—
118#11745664SD1612yescomplemented
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—42no—
126C2×C59⋊C8—44no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—41no—
129C23×C59—22no—
130C23×C59#12111yescharacteristic
131Q8×C59—41no—
132Q8×C59—41no—
133D4×C59—42no—
134D4×C59—42no—
135D4×C59—42no—
136D4×C59—41no—
137D4×C59—41no—
138D4×C59—41no—
139D4×C59—41no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59C4×D411yescharacteristic
152C2×C4×C59#12011yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12311yescharacteristic
155C2×C4×C59C2×SD1611yescharacteristic
156C2×C4×C59#13511yescharacteristic
157C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C59#13711yescharacteristic
159C2×Dic59—41no—
160C2×Dic59—41no—
161Dic118—42no—
162Dic118—41no—
163Dic118—41no—
164C8×C59—81no—
165C59⋊C8—22no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—41no—
169C59⋊C8—41no—
170C59⋊C8—41no—
171#9055—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—42no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59#38611yescharacteristic
178C22×C59#46111yescharacteristic
179C22×C59#46311yescharacteristic
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—42no—
184C4×C59—82no—
185C4×C59C4×SD1611yescharacteristic
186C4×C59#44011yescharacteristic
187C4×C59#44311yescharacteristic
188C4×C59#44411yescharacteristic
189C4×C59#45011yescharacteristic
190C4×C59#45211yescharacteristic
191Dic59—81no—
192Dic59—81no—
193#3167—591no—
194#2935—591no—
195#1986—591no—
196#1502D5911yescharacteristic
197#1481—591no—
198#1279—591no—
199#1278—591no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—82no—
203C2×C59#187611yescharacteristic
204C2×C59#187711yescharacteristic
205C2×C59#188411yescharacteristic
206C2×#131—1181no—
207#533—591no—
208#532C2×D5911yescharacteristic
209#498—591no—
210#492—591no—
211#487—591no—
212#486—591no—
213#465—1182no—
214C4×SD16—1182no—
215#436—1181no—
216#435—1182no—
217C8×D4—1181no—
218#431C2×D5911yescharacteristic
219C2×C4⋊C8—591no—
220#408—1181no—
221C4×M16—591no—
222C2×C4×C8—591no—
223C4×C4⋊C4C2×D5911yescharacteristic
224#339—1181no—
225#339—1181no—
226#327—22no—
227Q8⋊C8—1181no—
228Q8⋊C8—1181no—
229C59#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—1182no—
232#131—2362no—
233#131—2362no—
234C2×M16—591no—
235C2×M16—591no—
236C2×M16—1182no—
237C22×C8—1182no—
238C22×C8—1181no—
239C22×C8—1181no—
240C4⋊Q8—591no—
241C4⋊Q8—591no—
242C4⋊Q8C4×D5911yescharacteristic
243#127C4×D5911yescharacteristic
244#125—1181no—
245#124—21no—
246C22⋊Q8—1181no—
247C22⋊Q8—1181no—
248#121—21no—
249#121C22⋊D5912yes—
250C4×Q8—1181no—
251C4×Q8—1181no—
252C4×D4—21no—
253C4×D4D23612yes—
254#117—1181no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C22×D5911yescharacteristic
258#106—1182no—
259C4⋊C8—21no—
260C4⋊C8—591no—
261C4⋊C8—591no—
262C4⋊C8—1181no—
263#103—1182no—
264#102—1182no—
265C22⋊C8—21no—
266C22⋊C8—1182no—
267C22⋊C8—1182no—
268#97—1181no—
269C4×C8—21no—
270C4×C8—591no—
271C4×C8—591no—
272C4×C8—1181no—
273C4×C8—1181no—
274#95—21no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—1181no—
281C2×Q8—1181no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4(C2×D59)⋊C412yes—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4D4×D5911yescharacteristic
289C22×C4#1174568711yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291SD16—2364no—
292M16—592no—
293M16—1181no—
294M16—1181no—
295M16—1181no—
296M16—2364no—
297C2×C8—41no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—1181no—
304C2×C8—2362no—
305C2×C8—2364no—
306C2×C8—2361no—
307C2×C8—2361no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4—1181no—
314C4⋊C4—1181no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4(C2×D59)⋊C412yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—2361no—
321C42—41no—
322C42—41no—
323C42—2361no—
324C42C2×C4×D5911yescharacteristic
325C42C2×D23611yescharacteristic
326C42D4×D5911yescharacteristic
327C42#1174568711yescharacteristic
328C23—22no—
329C23#1427369911yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—2362no—
333Q8—2361no—
334Q8—2361no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4C2×(C2×D59)⋊C411yescharacteristic
354C2×C4(C22⋊D59)⋊C411yescharacteristic
355C2×C4(C22⋊D59)⋊C411yescharacteristic
356C2×C4#1427369711yescharacteristic
357C2×C4#1427369911yescharacteristic
358C2×C4#1427371111yescharacteristic
359C2×C4SD16×D5911yescharacteristic
360C2×C4#1427374611yescharacteristic
361C2×C4#1427375111yescharacteristic
362C2×C4#1427375611yescharacteristic
363C8—81no—
364C8—1182no—
365C8—1182no—
366C8—2361no—
367C8—2361no—
368C8—2361no—
369C8—2361no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1683727911yescharacteristic
376C22#1683760511yescharacteristic
377C22#1683761511yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—4721no—
384C4—4721no—
385C4#1683749311yescharacteristic
386C4#1683749911yescharacteristic
387C4#1683751611yescharacteristic
388C4#1683752111yescharacteristic
389C4#1683755411yescharacteristic
390C4#1683756511yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2142442111yescharacteristic
395C2#2142442811yescharacteristic
396C2#2142447711yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1502C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#1502D59
hypocenterC59#9055
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
3. #21425039C2semidirect
4. #21422500C2semidirect
5. #21421937C2semidirect
6. #21421932C2semidirect
10. #16837798C4semidirect
11. #16837774C4semidirect
12. #16837754C4semidirect
13. #16837748C4semidirect
118. #11745664SD16semidirect
229. C59#9055semidirect

Maximal subgroups

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

subgroupindex
#214346512
#214328402
#214250392
#214225002
#214219372
#214219322
C59×#15022
#905559