Finite Group Databasebeta

← #28484598

Subgroups — #28484598

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427006C211yesmaximal, characteristic
2#21422660C211yesmaximal, characteristic, complemented
3#21422595C211yesmaximal, characteristic, complemented
4#21422487C211yesmaximal, characteristic
5#21421997C211yesmaximal, characteristic, complemented
6#21421956C211yesmaximal, characteristic, complemented
7C59×#3077C211yesmaximal, characteristic
8C2×D4×Dic59—21no—
9#16837455C2211yescharacteristic
10#16837455C2211yescharacteristic
11#16837301—21no—
12#16837277—22no—
13#16837268C2211yescharacteristic
14#16837262—21no—
15C4×C4⋊Dic59C2211yescharacteristic
16C2×C59⋊#95—21no—
17#16837139—21no—
18#16837139—21no—
19#16837139—22no—
20D4⋊(C59⋊C8)—22no—
21C59×#533C2211yescharacteristic
22C2×C4×D4×C59C2211yescharacteristic
23C59×#478—22no—
24C59×#476C412yescomplemented
25C59×#474C412yescomplemented
26C59×#428—21no—
27C59×#423C2211yescharacteristic
28C59×#418—21no—
29C23×Dic59—22no—
30D4×Dic59—42no—
31D4×Dic59—42no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—22no—
35C2×C22⋊Dic59—22no—
36C2×C4×Dic59—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C4×C59⋊C8—21no—
42C4×C59⋊C8—21no—
43C59⋊#95—22no—
44C59⋊#95—21no—
45C59⋊#95D411yescharacteristic
46C59⋊#95D411yescharacteristic
47C59⋊#95D412yes—
48C22×D4×C59D411yescharacteristic
49C23×C4×C59—21no—
50C2×M16×C59—21no—
51C4⋊Q8×C59C2×C411yescharacteristic, complemented
52C4⋊Q8×C59C2×C411yescharacteristic, complemented
53#125×C59—21no—
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59C2×C412yes—
61C4×D4×C59D412yes—
62#117×C59—21no—
63C2×C4⋊C4×C59D411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66#107×C59—22no—
67#106×C59—22no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59C2×C412yescomplemented
70#102×C59—22no—
71#102×C59—22no—
72C24×C59—21no—
73C2×Q8×C59—21no—
74C2×Q8×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59—21no—
78C2×D4×C59C22⋊C412yes—
79C2×D4×C59SD1612yes—
80C2×D4×C59Q1612yes—
81C22×C4×C59—21no—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—82no—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×Dic59—21no—
89C22×Dic59—22no—
90C22×Dic59—21no—
91C22×Dic59—44no—
92C22×Dic59—44no—
93C22×Dic59—82no—
94M16×C59—41no—
95M16×C59—41no—
96C2×C8×C59—22no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59C22⋊C412yes—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—41no—
108C42×C59—41no—
109C42×C59C22×C411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112C42×C59#4111yescharacteristic
113C22⋊Dic59—44no—
114C4⋊Dic59—42no—
115C4⋊Dic59—41no—
116C4⋊Dic59—41no—
117C4⋊Dic59—82no—
118C4×Dic59—41no—
119C4×Dic59—41no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C23×C59—22no—
123C23×C59—22no—
124C23×C59—22no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59#12011yescharacteristic
128Q8×C59—81no—
129Q8×C59—81no—
130D4×C59—22no—
131D4×C59—22no—
132D4×C59—22no—
133D4×C59#10314yes—
134C2×C4×C59—22no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—42no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—82no—
141C2×C4×C59—82no—
142C2×C4×C59C2×C22⋊C411yescharacteristic
143C2×C4×C59C4×D411yescharacteristic
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12311yescharacteristic
148C2×C4×C59C2×SD1611yescharacteristic
149C2×C4×C59C2×Q1611yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C59#13711yescharacteristic
152C2×Dic59—22no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159C2×Dic59—41no—
160C2×Dic59—42no—
161C2×Dic59—88no—
162C8×C59—42no—
163C59⋊C8—81no—
164C59⋊C8—81no—
165#9175—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—24no—
169C22×C59—44no—
170C22×C59—44no—
171C22×C59—42no—
172C22×C59—42no—
173C22×C59#38611yescharacteristic
174C22×C59#45011yescharacteristic
175C22×C59#45111yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59—81no—
182C4×C59—81no—
183C4×C59C2×#10311yescharacteristic
184C4×C59#41711yescharacteristic
185C4×C59#48311yescharacteristic
186C4×C59#48411yescharacteristic
187Dic59—42no—
188Dic59—81no—
189Dic59—82no—
190#3077D5911yescharacteristic
191#2349—591no—
192#1518—591no—
193#1503—591no—
194#1478—591no—
195#1303—591no—
196#1286—591no—
197C2×C59—21no—
198C2×C59—21no—
199C2×C59—44no—
200C2×C59#187211yescharacteristic
201C2×C59#190411yescharacteristic
202C2×C59#190611yescharacteristic
203#533C2×D5911yescharacteristic
204C2×C4×D4—1181no—
205C2×C4×D4C2×D5911yescharacteristic
206#478—22no—
207#476Dic5912yescomplemented
208#474Dic5912yescomplemented
209#431—591no—
210#431—591no—
211#428—21no—
212#423C2×D5911yescharacteristic
213#418—21no—
214#389—1181no—
215#385—1182no—
216#382—591no—
217#380—1181no—
218C4×C4⋊C4—591no—
219C2×#95—1181no—
220#328—1181no—
221#328—1181no—
222#328—1182no—
223D4⋊C8—1182no—
224C59#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D5911yescharacteristic
226C23×C4—21no—
227C23×C4—1182no—
228C2×M16—21no—
229C4⋊Q8C2×Dic5911yescharacteristic, complemented
230C4⋊Q8C2×Dic5911yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—2362no—
239C4×D4—2362no—
240C4×D4C2×Dic5912yes—
241C4×D4C22⋊D5912yes—
242#117—21no—
243C2×C4⋊C4—1182no—
244C2×C4⋊C4—1181no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4C22⋊D5911yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1182no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—1181no—
255C4⋊C8—1181no—
256C4⋊C8C2×Dic5912yescomplemented
257#102—22no—
258#102—22no—
259C22⋊C8—1181no—
260C22⋊C8—1181no—
261C4×C8—1181no—
262C4×C8—1181no—
263#95—591no—
264#95—591no—
265#95—592no—
266#95—1182no—
267#95—1181no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1174567112yes—
275C2×D4#1174567212yes—
276C2×D4C22⋊Dic5912yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1181no—
282C22×C4—1182no—
283C22×C4—1181no—
284C22×C4—4722no—
285C22×C4—2364no—
286C22×C4—2364no—
287C22×C4D4×D5911yescharacteristic
288C22×C4D4×D5911yescharacteristic
289C22×C4C2×C22⋊D5911yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—4722no—
303C4⋊C4—2362no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C4⋊C4C22⋊Dic5912yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—2364no—
311C42—41no—
312C42—2361no—
313C42—2361no—
314C42#1174568711yescharacteristic
315C42#1174568711yescharacteristic
316C42C22×Dic5911yescharacteristic
317C42C2×C22⋊D5911yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1427369511yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1427365114yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—1182no—
339C2×C4—4728no—
340C2×C4—2361no—
341C2×C4—2362no—
342C2×C4—2361no—
343C2×C4—2362no—
344C2×C4—2361no—
345C2×C4—2362no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4C2×C22⋊Dic5911yescharacteristic
349C2×C4D4×Dic5911yescharacteristic
350C2×C4D4×Dic5911yescharacteristic
351C2×C4#1427369911yescharacteristic
352C2×C4#1427369911yescharacteristic
353C2×C4#1427370811yescharacteristic
354C2×C4C2×#1174567111yescharacteristic
355C2×C4C2×#1174567211yescharacteristic
356C2×C4#1427375211yescharacteristic
357C2×C4#1427375711yescharacteristic
358C8—42no—
359C8—4721no—
360C8—4721no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1683728211yescharacteristic
369C22#1683755211yescharacteristic
370C22#1683755711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—4722no—
379C4—4721no—
380C4—2362no—
381C4C2×#1427365111yescharacteristic
382C4#1683740311yescharacteristic
383C4#1683773011yescharacteristic
384C4#1683773511yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2142440411yescharacteristic
389C2#2142458511yescharacteristic
390C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3077C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#3077D59
hypocenterC59#9175
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21422660C2semidirect
3. #21422595C2semidirect
5. #21421997C2semidirect
6. #21421956C2semidirect
24. C59×#476C4semidirect
25. C59×#474C4semidirect
51. C4⋊Q8×C59C2×C4semidirect
52. C4⋊Q8×C59C2×C4semidirect
69. C4⋊C8×C59C2×C4semidirect
207. #476Dic59semidirect
208. #474Dic59semidirect
224. C59#9175semidirect
229. C4⋊Q8C2×Dic59semidirect
230. C4⋊Q8C2×Dic59semidirect
256. C4⋊C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214270062
#214226602
#214225952
#214224872
#214219972
#214219562
C59×#30772
#917559