Finite Group Databasebeta

← #28480187

Subgroups — #28480187

Order 15104.

802 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837959C211yesmaximal, characteristic
2#21424076C211yesmaximal, characteristic
3C2×#16837150C212yesmaximal, complemented
4C2×#16837149C212yesmaximal, complemented
5C59×#1797C211yesmaximal, characteristic
6#16837959—24no—
7#16837959—24no—
8C2×#14273727C2211yescharacteristic
9C2×#14273727C2211yescharacteristic
10C2×#14273719—21no—
11C2×#14273706—21no—
12C2×#14273706C412yes—
13C2×C4×Dic118—21no—
14C2×C4×Dic118C412yes—
15C2×#14273686—21no—
16C2×C4⋊(C59⋊C8)C2212yes—
17C2×C22⋊(C59⋊C8)—21no—
18#16837157—21no—
19#16837150—24no—
20#16837149—24no—
21C22×C42×C59C2211yescharacteristic
22C2×C4⋊C8×C59C2212yes—
23C2×C22⋊C8×C59—21no—
24C59×#336—21no—
25C22×Dic118—21no—
26C22×Dic118C2×C412yes—
27C22×C59⋊C8—22no—
28#14273727—22no—
29#14273727—22no—
30#14273727D414yes—
31#14273727D414yes—
32#14273719—28no—
33#14273706—28no—
34#14273706—24no—
35#14273706—24no—
36C4×Dic118—28no—
37C4×Dic118—24no—
38C4×Dic118—24no—
39#14273686—28no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59C2×C412yes—
44C2×Dic59⋊C4—22no—
45C2×Dic59⋊C4—22no—
46C2×C22⋊Dic59—22no—
47C2×C4×Dic59—22no—
48C4⋊(C59⋊C8)—22no—
49C4⋊(C59⋊C8)—22no—
50C22⋊(C59⋊C8)—42no—
51C23×C4×C59C2×C411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C22×C8×C59—22no—
55C2×C42×C59—22no—
56C2×C42×C59—22no—
57C2×C42×C59C2×C411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—22no—
62C4⋊C8×C59—22no—
63C22⋊C8×C59—42no—
64C24×C59C22⋊C411yescharacteristic
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59SD1612yes—
79C22×C4×C59Q1612yes—
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×Dic59—22no—
84C22×Dic59—22no—
85C2×Dic118—22no—
86C2×Dic118—24no—
87C2×Dic118—42no—
88C2×Dic118—41no—
89C2×Dic118—41no—
90C2×Dic118—42no—
91C2×Dic118—42no—
92C2×Dic118C22⋊C414yes—
93C2×C8×C59—42no—
94C2×C8×C59—42no—
95C2×C8×C59—42no—
96C42×C59—21no—
97C42×C59—21no—
98C42×C59—22no—
99C42×C59—22no—
100C42×C59C2×D412yes—
101C42×C59C2×D412yes—
102C22⋊Dic59—28no—
103C4⋊Dic59—22no—
104C4⋊Dic59—24no—
105C4⋊Dic59—24no—
106C4⋊Dic59—24no—
107C4⋊Dic59C22⋊C414yes—
108Dic59⋊C4—28no—
109Dic59⋊C4—28no—
110C4×Dic59—28no—
111C2×C59⋊C8—42no—
112C2×C59⋊C8—42no—
113C2×C59⋊C8—42no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59#10312yes—
121C23×C59C2×C22⋊C411yescharacteristic
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—24no—
127C2×C4×C59—24no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59#10312yes—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C2×C22⋊C411yescharacteristic
141C2×C4×C59#12012yes—
142C2×C4×C59#12012yes—
143C2×C4×C59C2×SD1611yescharacteristic
144C2×C4×C59C2×Q1611yescharacteristic
145C2×C4×C59#13512yes—
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×C4×C59#13712yes—
149C2×Dic59—24no—
150C2×Dic59—24no—
151C2×Dic59—42no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155Dic118—44no—
156Dic118—44no—
157Dic118—44no—
158Dic118—44no—
159C8×C59—82no—
160C59⋊C8—82no—
161#8343—591nomaximal, Sylow
162C22×C59—22no—
163C22×C59—22no—
164C22×C59—22no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59#37911yescharacteristic
170C22×C59#41311yescharacteristic
171C22×C59#41311yescharacteristic
172C22×C59C2×#10311yescharacteristic
173C22×C59#41611yescharacteristic
174C22×C59#41711yescharacteristic
175C22×C59#41911yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—22no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59#44812yes—
182C4×C59#44912yes—
183C4×C59#45112yes—
184C4×C59#45212yes—
185Dic59—44no—
186Dic59—44no—
187C2×#533—591no—
188#1797—591no—
189#1797D5911yescharacteristic
190C2×#333—592no—
191C2×#332—592no—
192C2×C59—42no—
193C2×C59#154211yescharacteristic
194C2×C59#155011yescharacteristic
195C2×C59#155411yescharacteristic
196C2×C59#155911yescharacteristic
197C2×C59#178511yescharacteristic
198C2×C59#178711yescharacteristic
199C2×C59#178811yescharacteristic
200#533—1184no—
201#533—1184no—
202C2×C4⋊Q8—591no—
203C2×C4⋊Q8—591no—
204C2×#125—1181no—
205C2×C22⋊Q8—592no—
206C2×C22⋊Q8—1181no—
207C2×C4×Q8—592no—
208C2×C4×Q8—1181no—
209C2×#117—1181no—
210C22×C42C2×D5911yescharacteristic
211C2×C4⋊C8—592no—
212C2×C4⋊C8C2×D5912yes—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—1181no—
215#336—21no—
216#336—1181no—
217#333—1184no—
218#332—1184no—
219C59#834311yescharacteristic, complemented, Sylow
220C22×Q8—592no—
221C22×Q8—1181no—
222C23×C4C4×D5911yescharacteristic
223C23×C4D23611yescharacteristic
224C23×C4C22⋊D5911yescharacteristic
225C22×C8—22no—
226C22×C8—1182no—
227C4⋊Q8—594no—
228C4⋊Q8—594no—
229C4⋊Q8—1182no—
230C4⋊Q8—1182no—
231#125—1188no—
232C22⋊Q8—1184no—
233C22⋊Q8—1184no—
234C22⋊Q8—1188no—
235C4×Q8—1184no—
236C4×Q8—1184no—
237C4×Q8—1188no—
238#117—1188no—
239C2×C4⋊C4—592no—
240C2×C4⋊C4—1182no—
241C2×C4⋊C4—1182no—
242C2×C4⋊C4—1181no—
243C2×C4⋊C4—1181no—
244C2×C4⋊C4—1181no—
245C2×C22⋊C4—1182no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—1182no—
249C2×C42C4×D5911yescharacteristic
250C2×C42D23611yescharacteristic
251C2×C42C22⋊D5911yescharacteristic
252C2×C42C22×D5911yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—1182no—
256C4⋊C8—1182no—
257C22⋊C8—42no—
258C22⋊C8—2362no—
259C24(C2×D59)⋊C411yescharacteristic
260C2×Q8—594no—
261C2×Q8—1182no—
262C2×Q8—1184no—
263C2×Q8—2362no—
264C2×Q8—2362no—
265C2×Q8—2362no—
266C2×Q8—2361no—
267C2×Q8—2361no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—1182no—
279C22×C4—1182no—
280C22×C4#1174566012yes—
281C22×C4Dic23612yes—
282C22×C4(C2×D59)⋊C411yescharacteristic
283C22×C4(C2×D59)⋊C411yescharacteristic
284C22×C4(C2×D59)⋊C411yescharacteristic
285C22×C4C2×C4×D5911yescharacteristic
286C22×C4C2×D23611yescharacteristic
287C22×C4C2×C22⋊D5911yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—2362no—
292C2×C8—2362no—
293C2×C8—2362no—
294C4⋊C4—594no—
295C4⋊C4—1188no—
296C4⋊C4—1188no—
297C4⋊C4—1182no—
298C4⋊C4—1184no—
299C4⋊C4—1184no—
300C4⋊C4—1184no—
301C22⋊C4—1188no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—1188no—
307C42D4×D5912yes—
308C42D4×D5912yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#1427365212yes—
316C23C2×(C2×D59)⋊C411yescharacteristic
317Q8—2364no—
318Q8—2364no—
319Q8—2364no—
320Q8—2364no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—1184no—
338C2×C4—1184no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4#1427365212yes—
344C2×C4C22⋊C4×D5911yescharacteristic
345C2×C4C22⋊C4×D5911yescharacteristic
346C2×C4#1427369512yes—
347C2×C4#1427369712yes—
348C2×C4C2×#1174566011yescharacteristic
349C2×C4C2×Dic23611yescharacteristic
350C2×C4#1427374712yes—
351C2×C4#1427375311yescharacteristic
352C2×C4#1427375811yescharacteristic
353C2×C4#1427376012yes—
354C8—82no—
355C8—4722no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22#1683725711yescharacteristic
364C22#1683738411yescharacteristic
365C22#1683738411yescharacteristic
366C22C2×#1427365211yescharacteristic
367C22#1683739611yescharacteristic
368C22#1683740111yescharacteristic
369C22#1683741211yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—2364no—
376C4—2364no—
377C4#1683754312yes—
378C4#1683754512yes—
379C4#1683756012yes—
380C4#1683756212yes—
381C2—42no—
382C2#2142275111yescharacteristic
383C2#2142278411yescharacteristic
384C2#2142280711yescharacteristic
385C2#2142282411yescharacteristic
386C2#2142402211yescharacteristic
387C2#2142403211yescharacteristic
388C2#2142403411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1797C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1797D59
hypocenterC59#8343
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. C2×#16837150C2semidirect
4. C2×#16837149C2semidirect
219. C59#8343semidirect

Maximal subgroups

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

subgroupindex
C2×#168379592
#214240762
C2×#168371502
C2×#168371492
C59×#17972
#834359