Finite Group Databasebeta

← #28483924

Subgroups — #28483924

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#21425040C211yesmaximal, characteristic, complemented
3#21422591C211yesmaximal, characteristic
4#21422498C211yesmaximal, characteristic, complemented
5#21421935C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7C59×#2935C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837798C411yescharacteristic, complemented
10#16837774C411yescharacteristic, complemented
11#16837754C411yescharacteristic, complemented
12#16837748C411yescharacteristic, complemented
13#16837629—22no—
14C4×#11745670—22no—
15#16837455C2211yescharacteristic
16C2×C4×C59⋊C8C2211yescharacteristic
17C4×C4⋊Dic59C2211yescharacteristic
18#16837163—21no—
19#16837163—21no—
20#16837136—22no—
21#16837132—21no—
22#16837132—21no—
23C2×#131×C59—21no—
24C59×#532C2211yescharacteristic
25C59×#436—21no—
26C59×#435—22no—
27C8×D4×C59—21no—
28C2×C4⋊C8×C59C2211yescharacteristic
29C59×#408—21no—
30C4×M16×C59C2211yescharacteristic
31C2×#11745670—22no—
32C22×C59⋊C8—21no—
33#14273727C2×C411yescharacteristic
34#14273727C2×C411yescharacteristic
35#14273719—21no—
36#14273706—21no—
37#14273706—21no—
38C4×Dic118—21no—
39C4×Dic118—21no—
40#14273686—21no—
41C2×C4⋊Dic59—21no—
42C2×C4×Dic59—21no—
43#14273658—22no—
44C4⋊(C59⋊C8)—21no—
45#14273650—22no—
46#14273648—22no—
47C22⋊(C59⋊C8)—21no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8C2×C411yescharacteristic
51C4×C59⋊C8C2×C411yescharacteristic
52C59⋊#95—21no—
53C2×#41×C59—21no—
54#131×C59—42no—
55#131×C59—42no—
56C2×M16×C59—22no—
57C2×M16×C59D411yescharacteristic
58C2×M16×C59D411yescharacteristic
59C22×C8×C59—22no—
60C22×C8×C59—21no—
61C4⋊Q8×C59C2×C411yescharacteristic
62#127×C59C2×C411yescharacteristic
63#124×C59—21no—
64#121×C59—21no—
65#121×C59D412yes—
66C4×D4×C59—21no—
67C4×D4×C59D412yes—
68C2×C42×C59C2311yescharacteristic
69C4⋊C8×C59—21no—
70C4⋊C8×C59D411yescharacteristic
71C4⋊C8×C59D411yescharacteristic
72C22⋊C8×C59—22no—
73C22⋊C8×C59—22no—
74#97×C59—21no—
75C4×C8×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×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—41no—
90C2×Dic118—21no—
91C2×Dic118—21no—
92M16×C59—21no—
93M16×C59—21no—
94M16×C59—21no—
95M16×C59—44no—
96M16×C59SD1612yescomplemented
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59—21no—
100C2×C8×C59—21no—
101C2×C8×C59—44no—
102C2×C8×C59—41no—
103C2×C8×C59—41no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59C22⋊C412yes—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—22no—
108C42×C59C22×C411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C22⋊Dic59—41no—
113#11745670—44no—
114C4⋊Dic59—22no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—41no—
120Dic59⋊C4—41no—
121Dic59⋊C4—41no—
122C4×Dic59—41no—
123C4×Dic59—41no—
124C4×Dic59—41no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8—42no—
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×C59C2×C22⋊C411yescharacteristic
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59#12011yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12311yescharacteristic
152C2×C4×C59C2×SD1611yescharacteristic
153C2×C4×C59#13511yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C59#13711yescharacteristic
156C2×Dic59—42no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—81no—
160C2×Dic59—81no—
161Dic118—42no—
162Dic118—41no—
163Dic118—41no—
164C8×C59—22no—
165C8×C59—41no—
166C8×C59—41no—
167C8×C59—41no—
168C59⋊C8—22no—
169C59⋊C8—41no—
170C59⋊C8—81no—
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×C59C4×SD1611yescharacteristic
185C4×C59#44011yescharacteristic
186C4×C59#44311yescharacteristic
187C4×C59#44411yescharacteristic
188C4×C59#45011yescharacteristic
189C4×C59#45211yescharacteristic
190Dic59—82no—
191Dic59—81no—
192Dic59—81no—
193#3167—591no—
194#2935D5911yescharacteristic
195#1986—591no—
196#1502—591no—
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—21no—
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—21no—
216#435—22no—
217C8×D4—21no—
218#431—591no—
219C2×C4⋊C8C2×D5911yescharacteristic
220#408—21no—
221C4×M16C2×D5911yescharacteristic
222C2×C4×C8—591no—
223C4×C4⋊C4—591no—
224#339—1181no—
225#339—1181no—
226#327—1182no—
227Q8⋊C8—1181no—
228Q8⋊C8—1181no—
229C59#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—1182no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16C22⋊D5911yescharacteristic
236C2×M16C22⋊D5911yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
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#121D23612yes—
250C4×Q8—1181no—
251C4×Q8—1181no—
252C4×D4—21no—
253C4×D4C22⋊D5912yes—
254#117—1181no—
255C2×C4⋊C4—1181no—
256C2×C42—1181no—
257C2×C42C22×D5911yescharacteristic
258#106—1182no—
259C4⋊C8—21no—
260C4⋊C8—1181no—
261C4⋊C8C22⋊D5911yescharacteristic
262C4⋊C8C22⋊D5911yescharacteristic
263#103—1182no—
264#102—1182no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—1181no—
268#97—21no—
269C4×C8—21no—
270C4×C8—591no—
271C4×C8—591no—
272C4×C8—1181no—
273C4×C8—1181no—
274#95—1181no—
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—2361no—
288C22×C4C2×D23611yescharacteristic
289C22×C4#1174568511yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291SD16—2364no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16#1174567012yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—1181no—
305C2×C8—1181no—
306C2×C8—2362no—
307C2×C8—2361no—
308C4⋊C4—21no—
309C4⋊C4—1182no—
310C4⋊C4—1181no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4—1181no—
314C4⋊C4—2361no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4(C2×D59)⋊C412yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—2361no—
321C42—2361no—
322C42—2361no—
323C42—2361no—
324C42C2×C4×D5911yescharacteristic
325C42#1174568511yescharacteristic
326C42C2×C22⋊D5911yescharacteristic
327C42C2×C22⋊D5911yescharacteristic
328C23—22no—
329C23#1427370111yescharacteristic
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—4721no—
349C2×C4—4721no—
350C2×C4—2362no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4C2×(C2×D59)⋊C411yescharacteristic
354C2×C4C4×C22⋊D5911yescharacteristic
355C2×C4C4×C22⋊D5911yescharacteristic
356C2×C4#1427369611yescharacteristic
357C2×C4#1427370111yescharacteristic
358C2×C4#1427371011yescharacteristic
359C2×C4C2×#1174567011yescharacteristic
360C2×C4#1427374711yescharacteristic
361C2×C4#1427375311yescharacteristic
362C2×C4#1427375711yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8—1182no—
368C8—4721no—
369C8—2361no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1683728111yescharacteristic
376C22#1683760711yescharacteristic
377C22#1683761711yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—4722no—
383C4—4721no—
384C4—4721no—
385C4C4×#1174567011yescharacteristic
386C4#1683750211yescharacteristic
387C4#1683751711yescharacteristic
388C4#1683752211yescharacteristic
389C4#1683755311yescharacteristic
390C4#1683756411yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2142442311yescharacteristic
395C2#2142443011yescharacteristic
396C2#2142447911yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2935C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#2935D59
hypocenterC59#9055
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #21425040C2semidirect
4. #21422498C2semidirect
5. #21421935C2semidirect
6. #21421934C2semidirect
9. #16837798C4semidirect
10. #16837774C4semidirect
11. #16837754C4semidirect
12. #16837748C4semidirect
96. M16×C59SD16semidirect
229. C59#9055semidirect
296. M16#11745670semidirect

Maximal subgroups

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

subgroupindex
#214346512
#214250402
#214225912
#214224982
#214219352
#214219342
C59×#29352
#905559