Finite Group Databasebeta

← #28593393

Subgroups — #28593393

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426570C211yesmaximal, characteristic
2#21425036C211yesmaximal, characteristic
3#21425035C211yesmaximal, characteristic
4#21424820C211yesmaximal, characteristic
5#21424663C212yesmaximal
6#21424657C212yesmaximal
7#21423955C212yesmaximal
8#21423951C211yesmaximal, characteristic
9#21423846C212yesmaximal
10#97×Dic59C211yesmaximal, characteristic
11C59×#1940C211yesmaximal, characteristic
12C2×C4×Dic118C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2212yes—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273638C2211yescharacteristic
16C2×#14273637C2211yescharacteristic
17C2×#14273637C2211yescharacteristic
18C2×#14273637C2212yes—
19C2×C8×Dic59C2211yescharacteristic
20C2×C8×Dic59C2211yescharacteristic
21#16837307C412yes—
22#16837298C412yes—
23#16837284C414yes—
24C4×Dic59⋊C4C2212yes—
25#16837248C2212yes—
26C42×Dic59C2211yescharacteristic
27#16837157C2212yes—
28#16837156C2212yes—
29#16837156C2212yes—
30#16837156C2212yes—
31#16837156C2212yes—
32#16837156C2212yes—
33C2×C4⋊C8×C59C2212yes—
34C2×C4⋊C8×C59C2211yescharacteristic
35C2×#97×C59C2211yescharacteristic
36C4×C4⋊C4×C59C2211yescharacteristic
37C59×#336C2212yes—
38C22×Dic118C2×C411yescharacteristic
39C22×C59⋊C8C2312yes—
40C22×C59⋊C8C2311yescharacteristic
41C22×C59⋊C8C2311yescharacteristic
42C4×Dic118—22no—
43C4×Dic118—22no—
44C2×C4⋊Dic59C2×C411yescharacteristic
45C2×Dic59⋊C4C2×C412yes—
46C2×Dic59⋊C4C2×C414yes—
47C2×C4×Dic59C2×C412yes—
48C2×C4×Dic59C2×C412yes—
49C2×C4×Dic59C2312yes—
50C2×C4×Dic59C2311yescharacteristic
51C2×C4×Dic59C2311yescharacteristic
52C4⋊(C59⋊C8)—22no—
53C4⋊(C59⋊C8)—24no—
54#14273638—22no—
55#14273637—22no—
56#14273637—24no—
57#14273637—22no—
58C8×Dic59—22no—
59C8×Dic59Q814yescomplemented
60C59⋊#95C2×C414yes—
61C59⋊#95C2×C414yes—
62C59⋊#95C2×C414yes—
63C22×C8×C59C2312yes—
64C22×C8×C59C2311yescharacteristic
65C22×C8×C59C2311yescharacteristic
66C2×C4⋊C4×C59C2×C412yes—
67C2×C42×C59C2312yes—
68C2×C42×C59C2311yescharacteristic
69C4⋊C8×C59—24no—
70C4⋊C8×C59—22no—
71#97×C59—22no—
72#95×C59C2×C412yes—
73C22×C4×C59C22×C414yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2411yescharacteristic
77C22×Dic59C22×C412yes—
78C22×Dic59C22×C412yes—
79C22×Dic59C22×C412yes—
80C22×Dic59C22×C412yes—
81C2×Dic118—22no—
82C2×Dic118—42no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—22no—
88C2×C8×C59—22no—
89C2×C8×C59—22no—
90C2×C8×C59C2×Q812yes—
91C2×C8×C59#4112yes—
92C4⋊C4×C59—42no—
93C42×C59—22no—
94C42×C59—22no—
95C42×C59#4112yes—
96C42×C59#4112yes—
97C4⋊Dic59—22no—
98Dic59⋊C4—24no—
99Dic59⋊C4—44no—
100C4×Dic59—21no—
101C4×Dic59—21no—
102C4×Dic59—22no—
103C4×Dic59—22no—
104C4×Dic59—42no—
105C4×Dic59—42no—
106C4×Dic59C2×Q812yes—
107C4×Dic59#4112yes—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8C2×Q812yes—
116C2×C59⋊C8#4112yes—
117C23×C59C23×C411yescharacteristic
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—24no—
121C2×C4×C59—24no—
122C2×C4×C59—24no—
123C2×C4×C59#11712yes—
124C2×C4×C59#11712yes—
125C2×C4×C59C22×Q811yescharacteristic
126C2×C4×C59C2×#4111yescharacteristic
127C2×C4×C59C2×#4111yescharacteristic
128C2×C4×C59C2×#4111yescharacteristic
129C2×C4×C59C2×#4111yescharacteristic
130C2×C4×C59C2×#4111yescharacteristic
131C2×Dic59—22no—
132C2×Dic59—22no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59—22no—
137C2×Dic59—22no—
138C2×Dic59—24no—
139C2×Dic59—24no—
140C2×Dic59C4×Q814yes—
141Dic118—44no—
142C8×C59—22no—
143C8×C59—22no—
144C8×C59—42no—
145C59⋊C8—22no—
146C59⋊C8—22no—
147C59⋊C8—42no—
148#21342—591nomaximal, Sylow
149C22×C59C2×#11711yescharacteristic
150C22×C59C2×C4×Q811yescharacteristic
151C22×C59C4×#4111yescharacteristic
152C22×C59#56812yes—
153C22×C59#56812yes—
154C4×C59—22no—
155C4×C59—22no—
156C4×C59—44no—
157C4×C59#52911yescharacteristic
158C4×C59#52911yescharacteristic
159C4×C59#53311yescharacteristic
160C4×C59#53311yescharacteristic
161Dic59—24no—
162Dic59—42no—
163Dic59—42no—
164Dic59—42no—
165#2295—591no—
166#1985—591no—
167#1985—591no—
168#1940—591no—
169#1940D5911yescharacteristic
170#1917—592no—
171#1917—592no—
172#1768—592no—
173#1767—591no—
174#1746—592no—
175C4×#97—591no—
176C2×C59#229511yescharacteristic
177C2×C59#292111yescharacteristic
178C2×C59#292311yescharacteristic
179C2×C59#296112yes—
180C2×C59#296712yes—
181C2×C4×Q8—591no—
182C2×C4⋊C8—592no—
183C2×C4⋊C8—591no—
184C2×C4⋊C8C2×D5912yes—
185C2×C4⋊C8C2×D5911yescharacteristic
186C2×#97—591no—
187C2×#97—591no—
188C2×#97—592no—
189C2×#97—591no—
190C2×#97C2×D5911yescharacteristic
191C2×C4×C8—591no—
192C2×C4×C8—591no—
193#391—592no—
194#389—592no—
195#387—594no—
196C4×C4⋊C4—592no—
197C4×C4⋊C4C2×D5911yescharacteristic
198#376—592no—
199C43—591no—
200#336—592no—
201#336—592no—
202#336—592no—
203#336—592no—
204#336—592no—
205#336—592no—
206#336C2×D5912yes—
207C59#2134211yescharacteristic, complemented, Sylow
208C22×Q8—591no—
209C22×C8—591no—
210C22×C8—591no—
211C22×C8—592no—
212C22×C8C22×D5912yes—
213C22×C8C22×D5911yescharacteristic
214C22×C8C22×D5911yescharacteristic
215C4×Q8—1182no—
216C4×Q8—1182no—
217C2×C4⋊C4—592no—
218C2×C4⋊C4—591no—
219C2×C4⋊C4—594no—
220C2×C4⋊C4C4×D5912yes—
221C2×C42—591no—
222C2×C42—592no—
223C2×C42—591no—
224C2×C42—592no—
225C2×C42—592no—
226C2×C42C22×D5912yes—
227C2×C42C22×D5911yescharacteristic
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—1184no—
231C4⋊C8—1182no—
232#97—22no—
233#97—1184no—
234#97—1182no—
235#97—1182no—
236#97—1182no—
237C4×C8—594no—
238C4×C8—1182no—
239#95—594no—
240#95—594no—
241#95—594no—
242#95C4×D5912yes—
243C2×Q8—1182no—
244C2×Q8—2362no—
245C22×C4—592no—
246C22×C4—592no—
247C22×C4—592no—
248C22×C4—592no—
249C22×C4C2×C4×D5914yes—
250C22×C4C2×C4×D5911yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C23×D5911yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—592no—
261C2×C8—592no—
262C2×C8—1181no—
263C2×C8—1181no—
264C2×C8—1181no—
265C2×C8—1181no—
266C2×C8—1182no—
267C2×C8—1182no—
268C2×C8—1182no—
269C2×C8#1174568712yes—
270C2×C8Q8×D5912yes—
271C4⋊C4—42no—
272C4⋊C4—1184no—
273C4⋊C4—1182no—
274C4⋊C4—2364no—
275C42—22no—
276C42—22no—
277C42—592no—
278C42—592no—
279C42—1181no—
280C42—1181no—
281C42—1182no—
282C42—1182no—
283C42—2362no—
284C42—2362no—
285C42#1174568712yes—
286C42#1174568912yes—
287C23C22×C4×D5911yescharacteristic
288Q8—2364no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—594no—
295C2×C4—1182no—
296C2×C4—1182no—
297C2×C4—1182no—
298C2×C4—1184no—
299C2×C4—1182no—
300C2×C4—1184no—
301C2×C4—1182no—
302C2×C4—1182no—
303C2×C4—1182no—
304C2×C4#1427368512yes—
305C2×C4#1427368512yes—
306C2×C4C2×Q8×D5911yescharacteristic
307C2×C4#41×D5911yescharacteristic
308C2×C4#41×D5911yescharacteristic
309C2×C4C2×#1174568711yescharacteristic
310C2×C4C2×#1174568711yescharacteristic
311C2×C4C2×#1174568911yescharacteristic
312C8—22no—
313C8—22no—
314C8—42no—
315C8—1182no—
316C8—1182no—
317C8—2362no—
318C22C2×#1427368511yescharacteristic
319C22C4×Q8×D5911yescharacteristic
320C22C4×#1174568711yescharacteristic
321C22#1683816712yes—
322C22#1683816812yes—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—1184no—
327C4—2362no—
328C4—2362no—
329C4—2362no—
330C4#1683793311yescharacteristic
331C4#1683793711yescharacteristic
332C4#1683795411yescharacteristic
333C4#1683795711yescharacteristic
334C2#2142656711yescharacteristic
335C2#2143273311yescharacteristic
336C2#2143274611yescharacteristic
337C2#2143303212yes—
338C2#2143307812yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutator, socle, prime radicalC23×C59C23×C4
FittingC59×#1940C2
cosocleC22×C4×C59C24
hypercenter#1940D59
hypocenterC59#21342
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
59. C8×Dic59Q8semidirect
207. C59#21342semidirect

Maximal subgroups

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

subgroupindex
#214265702
#214250362
#214250352
#214248202
#214246632
#214246572
#214239552
#214239512
#214238462
#97×Dic592
C59×#19402
#2134259