Finite Group Databasebeta

← #28619252

Subgroups — #28619252

Order 15104.

1322 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11745669C214yesmaximal, complemented
2C4×#14273724C211yesmaximal, characteristic, complemented
3#21425625C212yesmaximal, complemented
4#21425624C212yesmaximal, complemented
5#21425167C212yesmaximal, complemented
6#21425166C212yesmaximal, complemented
7#21423842C211yesmaximal, characteristic, complemented
8C4×#127×C59C211yesmaximal, characteristic, complemented
9C22×#11745669C2214yes—
10C2×#14273724C2211yescharacteristic
11C2×C4×D236—22no—
12C2×C4×D236C2212yes—
13C4×#11745669—216no—
14C2×#14273660C2214yes—
15C2×#14273648C2214yes—
16C2×#14273647C2214yes—
17C2×C4×C59⋊C8C2212yes—
18#16837304—22no—
19#16837304C2212yes—
20#16837272C2211yescharacteristic
21#16837157C2212yes—
22#16837157C2212yes—
23C2×#127×C59C2211yescharacteristic
24C2×C4×D4×C59—22no—
25C2×C4×D4×C59C2212yes—
26C59×#390—22no—
27C59×#390C2212yes—
28C59×#384C2211yescharacteristic
29C43×C59C2211yescharacteristic
30C22×D236—22no—
31C22×D236C2312yes—
32C22×C4×D59—22no—
33C2×#11745669—28no—
34C2×#11745669—28no—
35C2×#11745669—28no—
36C22×C59⋊C8C2314yes—
37#14273724—24no—
38C4×D236—28no—
39C4×D236—48no—
40C2×C4⋊Dic59—22no—
41C2×C4⋊Dic59C2312yes—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—24no—
44#14273660—28no—
45#14273648—28no—
46#14273647—28no—
47C4×C59⋊C8D418yescomplemented
48C22×D4×C59—22no—
49C22×D4×C59C2312yes—
50C23×C4×C59—22no—
51#127×C59—24no—
52C4×D4×C59—28no—
53C4×D4×C59—48no—
54C2×C4⋊C4×C59—22no—
55C2×C4⋊C4×C59C2312yes—
56C2×C22⋊C4×C59—22no—
57C2×C22⋊C4×C59—24no—
58C2×C42×C59D412yes—
59C2×C42×C59D412yes—
60C2×C42×C59C2311yescharacteristic
61C2×C42×C59C2311yescharacteristic
62C2×C42×C59C2311yescharacteristic
63C24×C59—22no—
64C23×D59—22no—
65C2×D4×C59—28no—
66C2×D4×C59—24no—
67C2×D4×C59—24no—
68C2×D4×C59—22no—
69C2×D4×C59—44no—
70C2×D4×C59—44no—
71C2×D4×C59#4114yes—
72C22×C4×C59—22no—
73C22×C4×C59—44no—
74C22×C4×C59—48no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D412yes—
77C22×C4×C59C2×D412yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2411yescharacteristic
80C22×Dic59—22no—
81C2×D236—24no—
82C2×D236—24no—
83C2×D236—28no—
84C2×D236—22no—
85C2×D236—44no—
86C2×D236—44no—
87C2×D236#4114yes—
88C2×C4×D59—44no—
89C2×C4×D59—48no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—44no—
92C22⋊C4×C59—44no—
93C22⋊C4×C59—48no—
94C42×C59—22no—
95C42×C59—22no—
96C42×C59—42no—
97C42×C59D818yescomplemented
98C42×C59C2×D414yes—
99#11745669—232no—
100(C2×D59)⋊C4—44no—
101(C2×D59)⋊C4—48no—
102C4⋊Dic59—24no—
103C4⋊Dic59—44no—
104C2×C59⋊C8—24no—
105C2×C59⋊C8—24no—
106C2×C59⋊C8C2×D418yes—
107C23×C59—24no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—44no—
111C23×C59—44no—
112C23×C59C22×D411yescharacteristic
113C22×D59—24no—
114C22×D59—42no—
115C22×D59—42no—
116C22×D59—44no—
117C22×D59—44no—
118D4×C59—28no—
119D4×C59—416no—
120D4×C59—44no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—22no—
125C2×C4×C59—22no—
126C2×C4×C59—42no—
127C2×C4×C59—44no—
128C2×C4×C59—88no—
129C2×C4×C59#12712yes—
130C2×C4×C59#12712yes—
131C2×C4×C59C2×D814yes—
132C2×C4×C59C2×D814yes—
133C2×C4×C59C2×D814yes—
134C2×C4×C59C22×D412yes—
135C2×C4×C59C2×#4112yes—
136C2×C4×C59C2×#4112yes—
137C2×Dic59—42no—
138C2×Dic59—44no—
139D236—28no—
140D236—416no—
141D236—44no—
142C4×D59—88no—
143C59⋊C8—28no—
144#23798—591nomaximal, Sylow
145C22×C59—48no—
146C22×C59—44no—
147C22×C59—84no—
148C22×C59—82no—
149C22×C59—82no—
150C22×C59#52511yescharacteristic
151C22×C59#52511yescharacteristic
152C22×C59C2×#12711yescharacteristic
153C22×C59C22×D812yes—
154C22×C59#57512yes—
155C2×D59—48no—
156C2×D59—44no—
157C2×D59—84no—
158C2×D59—82no—
159C2×D59—82no—
160C4×C59—24no—
161C4×C59—24no—
162C4×C59—22no—
163C4×C59—82no—
164C4×C59#49314yes—
165C4×C59#52514yes—
166Dic59—82no—
167C2×C4×D8—594no—
168C4×#127—591no—
169C4×#127D5911yescharacteristic, complemented
170#2093—592no—
171#2093—592no—
172#2011—592no—
173#2011—592no—
174#1744—591no—
175C2×C59—84no—
176C2×C59#239711yescharacteristic
177C2×C59#310012yes—
178C2×C59#310012yes—
179C2×C59C2×#49311yescharacteristic
180C2×C59#315011yescharacteristic
181D59—84no—
182C22×D8—594no—
183C2×#127—591no—
184C2×#127C2×D5911yescharacteristic
185C2×C4×D4—22no—
186C2×C4×D4—592no—
187C2×C4×D4—1182no—
188C2×C4×D4C2×D5912yes—
189C4×D8—11816no—
190C2×#107—594no—
191C2×#102—594no—
192C2×#102—594no—
193C2×C4×C8—592no—
194#390—22no—
195#390—592no—
196#390—1182no—
197#390C2×D5912yes—
198#384—591no—
199#384C2×D5911yescharacteristic
200C43C2×D5911yescharacteristic
201#336—592no—
202#336—592no—
203C59#2379811yescharacteristic, complemented, Sylow
204C22×D4—22no—
205C22×D4—592no—
206C22×D4—1182no—
207C22×D4C22×D5912yes—
208C23×C4—22no—
209C23×C4—1182no—
210C2×D8—1188no—
211C2×D8—1188no—
212C2×D8—1188no—
213C22×C8—594no—
214#127—24no—
215#127—1184no—
216C4×D4—28no—
217C4×D4—48no—
218C4×D4—1188no—
219C4×D4—2368no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—592no—
222C2×C4⋊C4—1182no—
223C2×C4⋊C4C22×D5912yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—1182no—
227C2×C22⋊C4—1184no—
228C2×C42C22⋊D5912yes—
229C2×C42C22⋊D5912yes—
230C2×C42C22×D5911yescharacteristic
231C2×C42C22×D5911yescharacteristic
232C2×C42C22×D5911yescharacteristic
233#107—1188no—
234#102—1188no—
235#102—1188no—
236C4×C8—598no—
237C24—22no—
238C24—1182no—
239C2×D4—28no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—594no—
246C2×D4—1188no—
247C2×D4—1184no—
248C2×D4—1182no—
249C2×D4—1184no—
250C2×D4—2364no—
251C2×D4—2364no—
252C2×D4#1174568514yes—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—48no—
256C22×C4—1182no—
257C22×C4—2364no—
258C22×C4—2368no—
259C22×C4C2×C22⋊D5911yescharacteristic
260C22×C4C2×C22⋊D5912yes—
261C22×C4C2×C22⋊D5912yes—
262C22×C4C2×C22⋊D5911yescharacteristic
263C22×C4C23×D5911yescharacteristic
264D8—11832no—
265C2×C8—598no—
266C2×C8—1184no—
267C2×C8—1184no—
268C4⋊C4—24no—
269C4⋊C4—44no—
270C4⋊C4—1184no—
271C4⋊C4—2364no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—2364no—
275C22⋊C4—2368no—
276C42—22no—
277C42—22no—
278C42—42no—
279C42#1174566918yescomplemented
280C42D4×D5914yes—
281C23—24no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—44no—
286C23—1184no—
287C23—2362no—
288C23—2364no—
289C23—2362no—
290C23—2364no—
291C23C22×C22⋊D5911yescharacteristic
292D4—28no—
293D4—416no—
294D4—44no—
295D4—1188no—
296D4—23616no—
297D4—2364no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—88no—
306C2×C4—4728no—
307C2×C4—2364no—
308C2×C4—2362no—
309C2×C4#1427372312yes—
310C2×C4#1427372312yes—
311C2×C4C2×#1174566914yes—
312C2×C4C2×#1174566914yes—
313C2×C4C2×#1174566914yes—
314C2×C4C2×D4×D5912yes—
315C2×C4#41×D5912yes—
316C2×C4C2×#1174568512yes—
317C8—1188no—
318C22—44no—
319C22—48no—
320C22—82no—
321C22—82no—
322C22—84no—
323C22—4722no—
324C22—4722no—
325C22—4724no—
326C22—2364no—
327C22—2368no—
328C22#1683790911yescharacteristic
329C22#1683791311yescharacteristic
330C22C2×#1427372311yescharacteristic
331C22C22×#1174566912yes—
332C22#1683820012yes—
333C4—22no—
334C4—24no—
335C4—24no—
336C4—82no—
337C4—4722no—
338C4#1683777814yes—
339C4#1683791114yes—
340C2—84no—
341C2—4724no—
342C2#2142737511yescharacteristic
343C2#2143409212yes—
344C2#2143409612yes—
345C2C2×#1683777811yescharacteristic
346C2#2143454111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#127×C59C2
socleC23×C59C22×D4
hypercenterC4×#127D59
hypocenterC59#23798
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11745669C2semidirect
2. C4×#14273724C2semidirect
3. #21425625C2semidirect
4. #21425624C2semidirect
5. #21425167C2semidirect
6. #21425166C2semidirect
7. #21423842C2semidirect
8. C4×#127×C59C2semidirect
47. C4×C59⋊C8D4semidirect
97. C42×C59D8semidirect
169. C4×#127D59semidirect
203. C59#23798semidirect
279. C42#11745669semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117456692
C4×#142737242
#214256252
#214256242
#214251672
#214251662
#214238422
C4×#127×C592
#2379859