Finite Group Databasebeta

← #28483926

Subgroups — #28483926

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#21432840C211yesmaximal, characteristic
2#21425041C211yesmaximal, characteristic, complemented
3#21422591C211yesmaximal, characteristic
4#21422499C211yesmaximal, characteristic, complemented
5#21421936C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C59×#3167C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837480—21no—
10#16837474—22no—
11D4×C59⋊C8—21no—
12#16837455C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14#16837369—21no—
15C4×#11745664C2211yescharacteristic
16C4×C4⋊Dic59C2211yescharacteristic
17#16837161—21no—
18#16837161—21no—
19#16837136—22no—
20Q8⋊(C59⋊C8)—21no—
21Q8⋊(C59⋊C8)—21no—
22C59×#533C2211yescharacteristic
23C59×#532C2211yescharacteristic
24C59×#498C411yescharacteristic, complemented
25C59×#492C411yescharacteristic, complemented
26C59×#487C411yescharacteristic, complemented
27C59×#486C411yescharacteristic, complemented
28C59×#465—22no—
29C4×SD16×C59—22no—
30C2×C4×C8×C59C2211yescharacteristic
31#14273734—42no—
32#14273734—42no—
33C2×#11745664—22no—
34C2×#11745664D411yescharacteristic
35C2×#11745664D411yescharacteristic
36C22×C59⋊C8—22no—
37C22×C59⋊C8—21no—
38C2×C4⋊Dic59—21no—
39C2×C4×Dic59—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)D411yescharacteristic
43C4⋊(C59⋊C8)D411yescharacteristic
44C22⋊(C59⋊C8)—22no—
45C22⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—21no—
47#14273638—21no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C2×#41×C59—21no—
52C2×SD16×C59—22no—
53C22×C8×C59—21no—
54C4⋊Q8×C59C2×C411yescharacteristic
55C4⋊Q8×C59C2×C411yescharacteristic
56C4⋊Q8×C59C2×C411yescharacteristic
57#127×C59C2×C411yescharacteristic
58#125×C59—21no—
59#124×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—21no—
62#121×C59—21no—
63#121×C59D412yes—
64C4×Q8×C59—21no—
65C4×Q8×C59—21no—
66C4×D4×C59—21no—
67C4×D4×C59D412yes—
68#117×C59—21no—
69C2×C42×C59C2311yescharacteristic
70#106×C59—22no—
71#103×C59—22no—
72#102×C59—22no—
73C4×C8×C59—21no—
74C4×C8×C59C2×C411yescharacteristic
75C4×C8×C59C2×C411yescharacteristic
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79#41×C59—41no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—21no—
83C2×D4×C59—22no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59C22⋊C412yes—
87C22×C4×C59—22no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59#4111yescharacteristic
91C22×Dic59—41no—
92SD16×C59—44no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59—42no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59C22⋊C412yes—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—41no—
107C42×C59—41no—
108C42×C59C22×C411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C4⋊Dic59—22no—
113C4⋊Dic59—41no—
114C4×Dic59—41no—
115C4×Dic59—41no—
116#11745664—21no—
117#11745664—21no—
118#11745664—21no—
119#11745664—44no—
120#11745664SD1612yescomplemented
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—44no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—41no—
129C23×C59—22no—
130C23×C59#12111yescharacteristic
131Q8×C59—42no—
132Q8×C59—41no—
133Q8×C59—41no—
134Q8×C59—41no—
135Q8×C59—41no—
136D4×C59—42no—
137D4×C59—42no—
138D4×C59—42no—
139D4×C59—41no—
140D4×C59—41no—
141D4×C59—41no—
142D4×C59—41no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59C2×C22⋊C411yescharacteristic
152C2×C4×C59C4×D411yescharacteristic
153C2×C4×C59C4×D411yescharacteristic
154C2×C4×C59#12011yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12311yescharacteristic
157C2×C4×C59C2×SD1611yescharacteristic
158C2×C4×C59#13511yescharacteristic
159C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C59#13711yescharacteristic
161C2×Dic59—42no—
162C2×Dic59—81no—
163C2×Dic59—81no—
164C8×C59—22no—
165C8×C59—41no—
166C59⋊C8—22no—
167C59⋊C8—41no—
168C59⋊C8—41no—
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×C59—81no—
185C4×C59—81no—
186C4×C59C4×SD1611yescharacteristic
187C4×C59#44011yescharacteristic
188C4×C59#44311yescharacteristic
189C4×C59#44411yescharacteristic
190C4×C59#45011yescharacteristic
191C4×C59#45211yescharacteristic
192Dic59—82no—
193#3167D5911yescharacteristic
194#2935—591no—
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—1181no—
207#533C2×D5911yescharacteristic
208#532C2×D5911yescharacteristic
209#498Dic5911yescharacteristic, complemented
210#492Dic5911yescharacteristic, complemented
211#487Dic5911yescharacteristic, complemented
212#486Dic5911yescharacteristic, complemented
213#465—22no—
214C4×SD16—22no—
215#436—1181no—
216#435—1182no—
217C8×D4—1181no—
218#431—591no—
219C2×C4⋊C8—591no—
220#408—1181no—
221C4×M16—591no—
222C2×C4×C8C2×D5911yescharacteristic
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—22no—
232#131—2362no—
233#131—2362no—
234C2×M16—591no—
235C2×M16—591no—
236C2×M16—1182no—
237C22×C8—21no—
238C22×C8—1182no—
239C22×C8—1181no—
240C4⋊Q8C2×Dic5911yescharacteristic
241C4⋊Q8C2×Dic5911yescharacteristic
242C4⋊Q8C2×Dic5911yescharacteristic
243#127C2×Dic5911yescharacteristic
244#125—21no—
245#124—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121C22⋊D5912yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4C22⋊D5912yes—
254#117—21no—
255C2×C4⋊C4—1181no—
256C2×C42—1181no—
257C2×C42C22×D5911yescharacteristic
258#106—22no—
259C4⋊C8—591no—
260C4⋊C8—591no—
261C4⋊C8—1181no—
262C4⋊C8—1181no—
263#103—22no—
264#102—22no—
265C22⋊C8—1182no—
266C22⋊C8—1182no—
267C22⋊C8—1181no—
268#97—1181no—
269C4×C8—21no—
270C4×C8—1181no—
271C4×C8—1181no—
272C4×C8C2×Dic5911yescharacteristic
273C4×C8C2×Dic5911yescharacteristic
274#95—1181no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊Dic5912yes—
286C22×C4—22no—
287C22×C4—2361no—
288C22×C4D4×D5911yescharacteristic
289C22×C4#1174568711yescharacteristic
290C22×C4C2×C22⋊D5911yescharacteristic
291SD16—44no—
292M16—592no—
293M16—1181no—
294M16—1181no—
295M16—1181no—
296M16—2364no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—1181no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—1181no—
304C2×C8—2364no—
305C2×C8—2361no—
306C2×C8—2361no—
307C2×C8—2361no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1182no—
316C4⋊C4—2361no—
317C4⋊C4C22⋊Dic5912yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C42—41no—
322C42—2361no—
323C42—2361no—
324C42D4×D5911yescharacteristic
325C42#1174568711yescharacteristic
326C42C22×Dic5911yescharacteristic
327C42C2×C22⋊D5911yescharacteristic
328C23—22no—
329C23#1427369911yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
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—41no—
349C2×C4—41no—
350C2×C4—4721no—
351C2×C4—4721no—
352C2×C4—2362no—
353C2×C4C2×C22⋊Dic5911yescharacteristic
354C2×C4D4×Dic5911yescharacteristic
355C2×C4D4×Dic5911yescharacteristic
356C2×C4#1427369511yescharacteristic
357C2×C4#1427369911yescharacteristic
358C2×C4#1427370811yescharacteristic
359C2×C4SD16×D5911yescharacteristic
360C2×C4#1427374811yescharacteristic
361C2×C4#1427375411yescharacteristic
362C2×C4#1427375611yescharacteristic
363C8—22no—
364C8—41no—
365C8—1182no—
366C8—4721no—
367C8—2361no—
368C8—2361no—
369C8—2361no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1683728211yescharacteristic
376C22#1683760511yescharacteristic
377C22#1683761911yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4—4722no—
385C4SD16×Dic5911yescharacteristic
386C4#1683750311yescharacteristic
387C4#1683751911yescharacteristic
388C4#1683752411yescharacteristic
389C4#1683755111yescharacteristic
390C4#1683756111yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2142442511yescharacteristic
395C2#2142443211yescharacteristic
396C2#2142448011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3167C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3167D59
hypocenterC59#9055
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #21425041C2semidirect
4. #21422499C2semidirect
5. #21421936C2semidirect
6. #21421933C2semidirect
24. C59×#498C4semidirect
25. C59×#492C4semidirect
26. C59×#487C4semidirect
27. C59×#486C4semidirect
120. #11745664SD16semidirect
209. #498Dic59semidirect
210. #492Dic59semidirect
211. #487Dic59semidirect
212. #486Dic59semidirect
229. C59#9055semidirect

Maximal subgroups

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

subgroupindex
#214328402
#214250412
#214225912
#214224992
#214219362
#214219332
C59×#31672
#905559