Finite Group Databasebeta

← #28487608

Subgroups — #28487608

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11745660C211yesmaximal, characteristic
2#21422844C211yesmaximal, characteristic, complemented
3#21422490C211yesmaximal, characteristic, complemented
4#21422486C211yesmaximal, characteristic
5#21422419C211yesmaximal, characteristic, complemented
6#21422416C211yesmaximal, characteristic
7C59×#1450C211yesmaximal, characteristic, complemented
8C22×#11745660C411yescharacteristic
9C2×C4×Dic118C2211yescharacteristic
10C2×C4×D236C2211yescharacteristic
11C4×#11745660—21no—
12C4×#11745660—21no—
13C4×#11745660—21no—
14C4×#11745660—21no—
15#16837455C2211yescharacteristic
16#16837455C2211yescharacteristic
17C2×#14273659C411yescharacteristic
18C2×#14273652C411yescharacteristic
19C2×#14273649C411yescharacteristic
20#16837155—21no—
21#16837153—21no—
22#16837132—22no—
23#16837129—22no—
24#16837127—22no—
25#16837125—22no—
26C59×#431C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C2×C4×C8×C59C2211yescharacteristic
29C59×#322—21no—
30C82×C59—21no—
31C22×Dic118C2×C411yescharacteristic
32C22×D236C2×C411yescharacteristic
33C22×C4×D59—21no—
34C2×#11745660—21no—
35C2×#11745660—21no—
36C2×#11745660—22no—
37C2×#11745660C814yescomplemented
38C4×Dic118—21no—
39C4×Dic118—21no—
40C4×Dic118—21no—
41C4×Dic118—21no—
42C4×D236—21no—
43C4×D236—21no—
44C4×D236—21no—
45C4×D236—21no—
46C2×C4⋊Dic59C2×C411yescharacteristic
47C2×C4⋊Dic59C2×C411yescharacteristic
48C2×Dic59⋊C4—21no—
49C2×(C2×D59)⋊C4—21no—
50C2×C4×Dic59—21no—
51#14273659—21no—
52#14273659—21no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)—21no—
55#14273652—21no—
56#14273652—21no—
57#14273649—21no—
58#14273649—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C4×C59⋊C8—21no—
62C4×C59⋊C8—21no—
63C22×C8×C59C2×C411yescharacteristic
64C22×C8×C59C2×C411yescharacteristic
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59D412yes—
68C22⋊C8×C59—21no—
69C22⋊C8×C59D412yes—
70C4×C8×C59—21no—
71C4×C8×C59—21no—
72C4×C8×C59D411yescharacteristic
73C4×C8×C59D411yescharacteristic
74C4×C8×C59D411yescharacteristic
75C4×C8×C59D411yescharacteristic
76C23×D59—21no—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C2×D236—21no—
84C2×D236—21no—
85C2×D236—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236C2×C812yes—
89C2×C4×D59—41no—
90C2×C4×D59—41no—
91C2×C4×D59—82no—
92C2×Dic118—21no—
93C2×Dic118—21no—
94C2×Dic118—21no—
95C2×Dic118—21no—
96C2×Dic118—21no—
97C2×Dic118C2×C812yes—
98C2×C8×C59—21no—
99C2×C8×C59—21no—
100C2×C8×C59—22no—
101C2×C8×C59—22no—
102C2×C8×C59C22⋊C411yescharacteristic
103C2×C8×C59C22⋊C411yescharacteristic
104C2×C8×C59C22⋊C411yescharacteristic
105C2×C8×C59C22⋊C411yescharacteristic
106C2×C8×C59C2×C812yes—
107C2×C8×C59M1612yes—
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59#4111yescharacteristic
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118Dic59⋊C4—41no—
119Dic59⋊C4—41no—
120C4×Dic59—41no—
121C4×Dic59—41no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124#11745660—24no—
125#11745660—24no—
126C23×C59C4×D411yescharacteristic
127C22×D59—22no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—81no—
131C22×D59—81no—
132C2×C4×C59—22no—
133C2×C4×C59—41no—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C4×D411yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59#12111yescharacteristic
138C2×C4×C59#12411yescharacteristic
139C2×C4×C59#12711yescharacteristic
140C2×C4×C59C22×C811yescharacteristic
141C2×C4×C59C2×M1611yescharacteristic
142C2×C4×C59#13111yescharacteristic
143C2×C4×C59#13111yescharacteristic
144C2×Dic59—22no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—81no—
150C2×Dic59—81no—
151D236—22no—
152D236—22no—
153D236—41no—
154D236—41no—
155C4×D59—82no—
156C4×D59—82no—
157Dic118—22no—
158Dic118—22no—
159Dic118—41no—
160Dic118—41no—
161C8×C59—21no—
162C8×C59—21no—
163C8×C59—42no—
164C8×C59—42no—
165C8×C59C22⋊C812yes—
166C8×C59C22⋊C812yes—
167C59⋊C8—81no—
168C59⋊C8—81no—
169#9667—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59#39011yescharacteristic
173C22×C59C8×D411yescharacteristic
174C22×C59#43611yescharacteristic
175C2×D59—42no—
176C2×D59—82no—
177C2×D59—81no—
178C2×D59—81no—
179C2×D59—82no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59C2×C22⋊C811yescharacteristic
184C4×C59#40811yescharacteristic
185C4×C59C8×D411yescharacteristic
186C4×C59#43611yescharacteristic
187C4×C59#47112yes—
188Dic59—42no—
189Dic59—81no—
190Dic59—81no—
191C2×C4×SD16—591no—
192#1564—591no—
193#1479—591no—
194#1478—591no—
195#1461—591no—
196#1460—591no—
197#1450D5911yescharacteristic, complemented
198C2×C59—21no—
199C2×C59—21no—
200C2×C59#195411yescharacteristic
201C2×C59#197611yescharacteristic
202C2×C59#197711yescharacteristic
203D59—82no—
204C22×SD16—591no—
205C2×C4×Q8—591no—
206C2×C4×D4—591no—
207C4×SD16—1181no—
208C4×SD16—1181no—
209C4×SD16—1181no—
210C4×SD16—1181no—
211#431—591no—
212#431—591no—
213#431C2×D5911yescharacteristic
214#431C2×D5911yescharacteristic
215C2×#106—591no—
216C2×#103—591no—
217C2×#102—591no—
218C2×C4×C8C2×D5911yescharacteristic
219#335—1181no—
220#334—1181no—
221Q8⋊C8—1182no—
222D4⋊C8—1182no—
223#324—1182no—
224#323—1182no—
225#322—21no—
226C82—21no—
227C59#966711yescharacteristic, complemented, Sylow
228C22×Q8—591no—
229C22×D4—591no—
230C23×C4—1181no—
231C2×SD16—594no—
232C2×SD16—1182no—
233C2×SD16—1181no—
234C2×SD16—1181no—
235C22×C8C4×D5911yescharacteristic
236C22×C8C4×D5911yescharacteristic
237C4×Q8—1181no—
238C4×Q8—1181no—
239C4×Q8—1181no—
240C4×Q8—1181no—
241C4×D4—1181no—
242C4×D4—1181no—
243C4×D4—1181no—
244C4×D4—1181no—
245C2×C4⋊C4—591no—
246C2×C4⋊C4—591no—
247C2×C4⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C42—1181no—
250C2×C42C22×D5911yescharacteristic
251#106—1181no—
252#106—1181no—
253C4⋊C8—21no—
254C4⋊C8—1181no—
255C4⋊C8—1181no—
256C4⋊C8D23612yes—
257#103—1181no—
258#103—1181no—
259#102—1181no—
260#102—1181no—
261C22⋊C8—21no—
262C22⋊C8—1181no—
263C22⋊C8—1181no—
264C22⋊C8D23612yes—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—1181no—
268C4×C8—1181no—
269C4×C8D23611yescharacteristic
270C4×C8D23611yescharacteristic
271C4×C8C22⋊D5911yescharacteristic
272C4×C8C22⋊D5911yescharacteristic
273C24—1181no—
274C2×Q8—592no—
275C2×Q8—1181no—
276C2×Q8—1181no—
277C2×Q8—1181no—
278C2×Q8—1181no—
279C2×Q8—1181no—
280C2×D4—592no—
281C2×D4—1181no—
282C2×D4—1181no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—1181no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—4722no—
290C22×C4—2361no—
291C22×C4—2361no—
292C22×C4C2×C4×D5911yescharacteristic
293C22×C4C2×D23611yescharacteristic
294C22×C4#1174568511yescharacteristic
295SD16—1184no—
296SD16—1184no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—2361no—
302C2×C8—2361no—
303C2×C8C8×D5912yes—
304C2×C8#1174565912yes—
305C2×C8(C2×D59)⋊C411yescharacteristic
306C2×C8(C2×D59)⋊C411yescharacteristic
307C2×C8(C2×D59)⋊C411yescharacteristic
308C2×C8(C2×D59)⋊C411yescharacteristic
309C4⋊C4—1181no—
310C4⋊C4—1181no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4—2361no—
314C4⋊C4—2361no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C42—2361no—
318C42—2361no—
319C42C2×D23611yescharacteristic
320C42C2×D23611yescharacteristic
321C42#1174568511yescharacteristic
322C42C2×C22⋊D5911yescharacteristic
323C23—1182no—
324C23—4721no—
325C23—4721no—
326C23—2361no—
327C23—2361no—
328C23C4×D23611yescharacteristic
329Q8—1182no—
330Q8—1182no—
331Q8—2361no—
332Q8—2361no—
333D4—1182no—
334D4—1182no—
335D4—2361no—
336D4—2361no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—1182no—
340C2×C4—4722no—
341C2×C4—4722no—
342C2×C4—4721no—
343C2×C4—4721no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4C2×(C2×D59)⋊C411yescharacteristic
349C2×C4C4×D23611yescharacteristic
350C2×C4#1427370111yescharacteristic
351C2×C4#1427370111yescharacteristic
352C2×C4#1427371511yescharacteristic
353C2×C4#1427372411yescharacteristic
354C2×C4C2×C8×D5911yescharacteristic
355C2×C4C2×#1174565911yescharacteristic
356C2×C4#1427373311yescharacteristic
357C2×C4#1427373311yescharacteristic
358C8—21no—
359C8—21no—
360C8—42no—
361C8—42no—
362C8—4721no—
363C8—4721no—
364C8(C2×D59)⋊C812yes—
365C8(C2×D59)⋊C812yes—
366C22—21no—
367C22—21no—
368C22—4722no—
369C22—4722no—
370C22—4721no—
371C22—4721no—
372C22—2362no—
373C22#1683730411yescharacteristic
374C22C8×D23611yescharacteristic
375C22#1683748111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—4721no—
380C4—4721no—
381C4—2362no—
382C4C2×(C2×D59)⋊C811yescharacteristic
383C4#1683736811yescharacteristic
384C4C8×D23611yescharacteristic
385C4#1683748111yescharacteristic
386C4#1683766512yes—
387C2—21no—
388C2—21no—
389C2—4722no—
390C2#2142489011yescharacteristic
391C2#2142499411yescharacteristic
392C2#2142499911yescharacteristic
subgroupG/H
centerC2×C4#14273701
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1450C2
socleC22×C59#390
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1450D59
hypocenterC59#9667
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #21422844C2semidirect
3. #21422490C2semidirect
5. #21422419C2semidirect
7. C59×#1450C2semidirect
37. C2×#11745660C8semidirect
197. #1450D59semidirect
227. C59#9667semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117456602
#214228442
#214224902
#214224862
#214224192
#214224162
C59×#14502
#966759