Finite Group Databasebeta

← #28489840

Subgroups — #28489840

Order 15104.

638 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic
2#21429050C211yesmaximal, characteristic
3#21425045C211yesmaximal, characteristic, complemented
4C2×#16837148C211yesmaximal, characteristic, complemented
5C2×#16837131C211yesmaximal, characteristic
6#21421936C211yesmaximal, characteristic, complemented
7C2×C59×#487C211yesmaximal, characteristic, complemented
8C2×#14273703—22no—
9C2×C4×D236C2211yescharacteristic
10#16837689—22no—
11#16837689—22no—
12C2×#14273658—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273647—21no—
16#16837337—22no—
17#16837303—21no—
18#16837271—21no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837164—22no—
21#16837148—22no—
22#16837131—22no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×C4⋊Q8×C59C2211yescharacteristic
25C59×#487—22no—
26C59×#487—22no—
27C2×#103×C59—22no—
28C2×C4×C8×C59C2211yescharacteristic
29C22×D236D411yescharacteristic
30C22×C4×D59—21no—
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33#14273703—44no—
34#14273703—44no—
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59D411yescharacteristic
40C2×C4⋊Dic59D412yes—
41C2×Dic59⋊C4—24no—
42C2×(C2×D59)⋊C4—24no—
43C2×(C2×D59)⋊C4—21no—
44C2×C4×Dic59—21no—
45#14273658—24no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)D412yes—
50#14273647—24no—
51C59⋊#95—21no—
52C22×Q8×C59—21no—
53C22×Q8×C59—21no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59D412yes—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59D412yes—
67C2×C42×C59C2311yescharacteristic
68#103×C59—28no—
69C4×C8×C59—21no—
70C4×C8×C59D412yes—
71C23×D59—21no—
72C2×Q8×C59—22no—
73C2×Q8×C59—44no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—42no—
78C2×Q8×C59—42no—
79C22×C4×C59—22no—
80C22×C4×C59—41no—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×Dic59—22no—
85C22×Dic59—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236—21no—
89C2×D236—21no—
90C2×D236—21no—
91C2×D236Q1612yes—
92C2×C4×D59—41no—
93C2×C4×D59—41no—
94C2×C4×D59—42no—
95C2×C4×D59—42no—
96C2×C8×C59—22no—
97C2×C8×C59—42no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59#4111yescharacteristic
110(C2×D59)⋊C4—48no—
111(C2×D59)⋊C4—41no—
112(C2×D59)⋊C4—41no—
113C4⋊Dic59—22no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59SD1614yes—
117Dic59⋊C4—48no—
118C4×Dic59—41no—
119C4×Dic59—41no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8—42no—
125C23×C59#12011yescharacteristic
126C22×D59—22no—
127C22×D59—41no—
128C22×D59—42no—
129C22×D59—41no—
130C22×D59—42no—
131Q8×C59—44no—
132Q8×C59—82no—
133Q8×C59—82no—
134Q8×C59—81no—
135Q8×C59—81no—
136C2×C4×C59—22no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—44no—
140C2×C4×C59—82no—
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59#12711yescharacteristic
147C2×C4×C59C2×SD1612yes—
148C2×C4×C59C2×Q1611yescharacteristic
149C2×C4×C59#13511yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)12yes—
151C2×C4×C59#13711yescharacteristic
152C2×C4×C59#13711yescharacteristic
153C2×Dic59—24no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—41no—
157C2×Dic59—42no—
158D236—22no—
159D236—22no—
160D236—41no—
161D236—41no—
162C4×D59—82no—
163C4×D59—81no—
164C4×D59—81no—
165C8×C59—42no—
166C59⋊C8—42no—
167C59⋊C8—81no—
168#10060—591nomaximal, Sylow
169C22×C59#39211yescharacteristic
170C22×C59#45012yes—
171C22×C59#45111yescharacteristic
172C22×C59#45211yescharacteristic
173C22×C59#45412yes—
174C2×D59—44no—
175C2×D59—42no—
176C2×D59—82no—
177C2×D59—81no—
178C2×D59—81no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—82no—
182C4×C59—82no—
183C4×C59#45111yescharacteristic
184C4×C59#45211yescharacteristic
185C4×C59#46012yes—
186C4×C59#46711yescharacteristic
187C4×C59#47011yescharacteristic
188C4×C59#49211yescharacteristic
189C4×C59#49611yescharacteristic
190Dic59—44no—
191Dic59—81no—
192C2×#487D5911yescharacteristic, complemented
193C2×#476—591no—
194#2542—591no—
195#1987—591no—
196C2×#332—591no—
197C2×D4⋊C8—591no—
198#1279—591no—
199C2×C59#163512yes—
200C2×C59#163712yes—
201C2×C59#200111yescharacteristic
202C2×C59#200311yescharacteristic
203C2×C59#201411yescharacteristic
204D59—82no—
205C2×C4⋊Q8C2×D5911yescharacteristic
206C2×C4⋊Q8C2×D5911yescharacteristic
207C2×C22⋊Q8—1182no—
208C2×C4×D4—591no—
209#487—22no—
210#487—22no—
211#476—1182no—
212#476—1182no—
213C2×#106—1181no—
214C2×C4⋊C8—591no—
215C2×C4⋊C8—591no—
216C2×#103—22no—
217C2×#102—1181no—
218C2×C4×C8C2×D5911yescharacteristic
219#399—1182no—
220#390—1181no—
221#384—1181no—
222C4×C4⋊C4—591no—
223#340—1182no—
224#332—1182no—
225D4⋊C8—1182no—
226C59#1006011yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—591no—
230C23×C4—1181no—
231C22×C8—21no—
232C22×C8—1181no—
233C22×C8—1181no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C22⋊D5912yes—
242C22⋊Q8—2364no—
243C22⋊Q8—2364no—
244C4×D4—1181no—
245C4×D4—1181no—
246C4×D4—1181no—
247C4×D4—1181no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—592no—
252C2×C4⋊C4—591no—
253C2×C4⋊C4—1184no—
254C2×C4⋊C4C22⋊D5912yes—
255C2×C22⋊C4—1184no—
256C2×C22⋊C4—1181no—
257C2×C42—1181no—
258C2×C42C22×D5911yescharacteristic
259#106—1184no—
260C4⋊C8—592no—
261C4⋊C8—1181no—
262C4⋊C8—1181no—
263C4⋊C8—1181no—
264#103—28no—
265#102—1184no—
266C4×C8—21no—
267C4×C8C22⋊D5912yes—
268#95—1181no—
269C24—1181no—
270C2×Q8—22no—
271C2×Q8—44no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—592no—
278C2×D4—1181no—
279C2×D4—1181no—
280C2×D4—1181no—
281C2×D4—1181no—
282C2×D4—1181no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—1182no—
286C22×C4—1181no—
287C22×C4—2361no—
288C22×C4—2361no—
289C22×C4—2362no—
290C22×C4—2362no—
291C22×C4D4×D5911yescharacteristic
292C22×C4D4×D5911yescharacteristic
293C22×C4C2×C22⋊D5911yescharacteristic
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—1182no—
297C2×C8—2361no—
298C2×C8—2362no—
299C2×C8—2361no—
300C2×C8—2361no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—594no—
310C4⋊C4—1182no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4—2368no—
314C22⋊C4—2368no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C42—2361no—
318C42—2361no—
319C42D4×D5911yescharacteristic
320C42#1174568711yescharacteristic
321C42C2×C22⋊D5911yescharacteristic
322C42C2×C22⋊D5911yescharacteristic
323C23—1182no—
324C23—2362no—
325C23—2361no—
326C23—2361no—
327C23—2362no—
328C23#1427369511yescharacteristic
329Q8—44no—
330Q8—82no—
331Q8—81no—
332Q8—81no—
333Q8—82no—
334D4—1182no—
335D4—1182no—
336D4—2361no—
337D4—2361no—
338C2×C4—22no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—1184no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4—4722no—
347C2×C4—2362no—
348C2×C4—2361no—
349C2×C4—2362no—
350C2×C4—2362no—
351C2×C4#1427369511yescharacteristic
352C2×C4#1427369611yescharacteristic
353C2×C4#1427369811yescharacteristic
354C2×C4#1427369911yescharacteristic
355C2×C4#1427369911yescharacteristic
356C2×C4#1427372311yescharacteristic
357C2×C4SD16×D5912yes—
358C2×C4Q16×D5911yescharacteristic
359C2×C4#1427375011yescharacteristic
360C2×C4#1427375412yes—
361C2×C4#1427375911yescharacteristic
362C2×C4#1427375911yescharacteristic
363C8—42no—
364C8—4721no—
365C8—2362no—
366C22—4721no—
367C22—4722no—
368C22—4721no—
369C22—2364no—
370C22—2362no—
371C22#1683731011yescharacteristic
372C22#1683755112yes—
373C22#1683755611yescharacteristic
374C22#1683756311yescharacteristic
375C22#1683757412yes—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—82no—
380C4—4721no—
381C4—2364no—
382C4#1683755611yescharacteristic
383C4#1683756311yescharacteristic
384C4#1683760012yes—
385C4#1683764011yescharacteristic
386C4#1683765711yescharacteristic
387C4#1683777311yescharacteristic
388C4#1683778811yescharacteristic
389C2—4722no—
390C2#2142324412yes—
391C2#2142325812yes—
392C2#2142510311yescharacteristic
393C2#2142511811yescharacteristic
394C2#2142518211yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#487C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenterC2×#487D59
hypocenterC59#10060
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #21425045C2semidirect
4. C2×#16837148C2semidirect
6. #21421936C2semidirect
7. C2×C59×#487C2semidirect
192. C2×#487D59semidirect
226. C59#10060semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
#214290502
#214250452
C2×#168371482
C2×#168371312
#214219362
C2×C59×#4872
#1006059