Finite Group Databasebeta

← #28515705

Subgroups — #28515705

Order 15104.

1590 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
1C2×#16837339C211yesmaximal, characteristic
2C2×#16837298C213yesmaximal, complemented
3C2×#16837291C213yesmaximal
4#21422365C213yesmaximal, complemented
5#21422338C213yesmaximal
6#95×Dic59C211yesmaximal, characteristic, complemented
7C59×#1444C211yesmaximal, characteristic, complemented
8C22×C4⋊Dic59C2213yes—
9C22×Dic59⋊C4C2213yes—
10C22×(C2×D59)⋊C4C2211yescharacteristic
11C22×(C2×D59)⋊C4C2213yes—
12C22×(C2×D59)⋊C4C2213yes—
13C22×C4×Dic59C2213yes—
14#16837339C418yescomplemented
15#16837298—212no—
16#16837291—212no—
17C2×C59⋊#95C2216yes—
18C2×C59⋊#95C2213yes—
19C2×C59⋊#95C2213yes—
20C22×C4⋊C4×C59C2213yes—
21C2×#95×C59C2213yes—
22C2×#95×C59C2211yescharacteristic
23C24×D59C2311yescharacteristic
24C23×Dic59C2313yes—
25C23×Dic59C2313yes—
26C23×Dic59C2311yescharacteristic
27C2×C4⋊Dic59—26no—
28C2×C4⋊Dic59—26no—
29C2×C4⋊Dic59C2×C4112yes—
30C2×Dic59⋊C4—26no—
31C2×Dic59⋊C4—212no—
32C2×(C2×D59)⋊C4—26no—
33C2×(C2×D59)⋊C4—26no—
34C2×(C2×D59)⋊C4—26no—
35C2×(C2×D59)⋊C4—26no—
36C2×(C2×D59)⋊C4—46no—
37C2×(C2×D59)⋊C4C2×C4112yes—
38C2×C4×Dic59—46no—
39C2×C4×Dic59D4112yes—
40C59⋊#95—26no—
41C59⋊#95—26no—
42C59⋊#95—212no—
43C23×C4×C59C2311yescharacteristic
44C23×C4×C59C2313yes—
45C23×C4×C59C2313yes—
46C2×C4⋊C4×C59—26no—
47C2×C4⋊C4×C59—46no—
48#95×C59—26no—
49#95×C59C2×C414yes—
50C24×C59C2411yescharacteristic
51C23×D59—23no—
52C23×D59—23no—
53C23×D59—23no—
54C23×D59—21no—
55C23×D59—21no—
56C23×D59—23no—
57C23×D59C22×C412yes—
58C22×C4×C59—26no—
59C22×C4×C59—26no—
60C22×C4×C59—23no—
61C22×C4×C59—23no—
62C22×C4×C59—26no—
63C22×C4×C59—23no—
64C22×C4×C59—26no—
65C22×C4×C59—23no—
66C22×C4×C59—21no—
67C22×C4×C59—23no—
68C22×C4×C59C22×C416yes—
69C22×C4×C59C2×D416yes—
70C22×C4×C59#4116yes—
71C22×Dic59—23no—
72C22×Dic59—21no—
73C22×Dic59—23no—
74C22×Dic59—26no—
75C22×Dic59—26no—
76C22×Dic59—23no—
77C22×Dic59—26no—
78C22×Dic59—26no—
79C22×Dic59—23no—
80C22×Dic59—23no—
81C22×Dic59C22×C416yes—
82C22×Dic59C2×D416yes—
83C22×Dic59C2×D416yes—
84C4⋊C4×C59—412no—
85(C2×D59)⋊C4—224no—
86(C2×D59)⋊C4—412no—
87(C2×D59)⋊C4—44no—
88C4⋊Dic59—224no—
89Dic59⋊C4—412no—
90C4×Dic59—412no—
91C23×C59C23×C411yescharacteristic
92C23×C59C22×D413yes—
93C23×C59C2×#4113yes—
94C23×C59H(5,𝔽2)13yes—
95C23×C592−1+411yescharacteristic
96C23×C592−1+413yes—
97C23×C592−1+411yescharacteristic
98C22×D59—26no—
99C22×D59—22no—
100C22×D59—26no—
101C22×D59—41no—
102C22×D59—43no—
103C22×D59—43no—
104C22×D59—43no—
105C22×D59—43no—
106C22×D59—43no—
107C22×D59—46no—
108C22×D59—43no—
109C22×D59—43no—
110C2×C4×C59—212no—
111C2×C4×C59—26no—
112C2×C4×C59—26no—
113C2×C4×C59—212no—
114C2×C4×C59—43no—
115C2×C4×C59—41no—
116C2×C4×C59—43no—
117C2×C4×C59—46no—
118C2×C4×C59—43no—
119C2×C4×C59—43no—
120C2×C4×C59—43no—
121C2×C4×C59—46no—
122C2×C4×C59C4×D4112yes—
123C2×Dic59—26no—
124C2×Dic59—26no—
125C2×Dic59—212no—
126C2×Dic59—26no—
127C2×Dic59—26no—
128C2×Dic59—26no—
129C2×Dic59—26no—
130C2×Dic59—43no—
131C2×Dic59—43no—
132C2×Dic59—46no—
133C2×Dic59—46no—
134C2×Dic59—46no—
135C2×Dic59#12714yes—
136#14322—591nomaximal, Sylow
137C22×C59C2×C4×D413yes—
138C22×C59#51911yescharacteristic
139C22×C59#52013yes—
140C22×C59C2×#12711yescharacteristic
141C22×C59#53213yes—
142C22×C59#53413yes—
143C22×C59#53613yes—
144C22×C59#53613yes—
145C22×C59#53713yes—
146C22×C59#53916yes—
147C22×C59#54113yes—
148C22×C59#54213yes—
149C2×D59—42no—
150C2×D59—46no—
151C2×D59—46no—
152C2×D59—81no—
153C2×D59—81no—
154C2×D59—83no—
155C2×D59—83no—
156C4×C59—212no—
157C4×C59—83no—
158C4×C59—81no—
159Dic59—24no—
160Dic59—46no—
161Dic59—46no—
162C2×#399—591no—
163C2×#389—593no—
164C2×#388—593no—
165#1445—593no—
166#1444D5911yescharacteristic, complemented
167#1439—593no—
168C4×#95—591no—
169C2×C59C4×#12711yescharacteristic
170C2×C59#232213yes—
171C2×C59#232313yes—
172C2×C59#265611yescharacteristic
173C2×C59#265811yescharacteristic
174C2×C59#266313yes—
175C2×C59#266413yes—
176D59—82no—
177C22×C4⋊C4—593no—
178C22×C4⋊C4—593no—
179C22×C4⋊C4C2×D5913yes—
180C22×C22⋊C4—593no—
181C22×C22⋊C4—591no—
182C22×C22⋊C4—593no—
183C22×C42—593no—
184#399—598no—
185#389—11812no—
186#388—11812no—
187C2×#95—593no—
188C2×#95—596no—
189C2×#95—593no—
190C2×#95C2×D5913yes—
191C2×#95C2×D5911yescharacteristic
192C59#1432211yescharacteristic, complemented, Sylow
193C25—591no—
194C23×C4—591no—
195C23×C4—593no—
196C23×C4—593no—
197C23×C4C22×D5913yes—
198C23×C4C22×D5911yescharacteristic
199C23×C4C22×D5913yes—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—46no—
202C2×C4⋊C4—5912no—
203C2×C4⋊C4—1186no—
204C2×C4⋊C4—1186no—
205C2×C4⋊C4—1186no—
206C2×C4⋊C4—11812no—
207C2×C22⋊C4—5912no—
208C2×C22⋊C4—1186no—
209C2×C22⋊C4—1186no—
210C2×C22⋊C4—1186no—
211C2×C22⋊C4—1186no—
212C2×C22⋊C4—2366no—
213C2×C42—5912no—
214C2×C42—2366no—
215#95—26no—
216#95—1186no—
217#95—11812no—
218#95—1186no—
219#95C4×D5914yes—
220C24—592no—
221C24—1183no—
222C24—1183no—
223C24—1183no—
224C24—1181no—
225C24—1183no—
226C24—1181no—
227C24C23×D5911yescharacteristic
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—21no—
238C22×C4—596no—
239C22×C4—596no—
240C22×C4—596no—
241C22×C4—1183no—
242C22×C4—1181no—
243C22×C4—1183no—
244C22×C4—1186no—
245C22×C4—1186no—
246C22×C4—1183no—
247C22×C4—1186no—
248C22×C4—1186no—
249C22×C4—1183no—
250C22×C4—1183no—
251C22×C4C2×C4×D5916yes—
252C22×C4D4×D5916yes—
253C22×C4#1174568916yes—
254C4⋊C4—412no—
255C4⋊C4—11824no—
256C4⋊C4—23612no—
257C22⋊C4—11824no—
258C22⋊C4—2364no—
259C22⋊C4—23612no—
260C42—23612no—
261C23—1186no—
262C23—1182no—
263C23—1186no—
264C23—2361no—
265C23—2363no—
266C23—2363no—
267C23—2363no—
268C23—2363no—
269C23—2363no—
270C23—2366no—
271C23—2363no—
272C23—2363no—
273C23C22×C4×D5911yescharacteristic
274C23C2×D4×D5913yes—
275C23C2×#1174568913yes—
276C23#1427377213yes—
277C23#1427377413yes—
278C23#1427377411yescharacteristic
279C23#1427377511yescharacteristic
280C2×C4—26no—
281C2×C4—26no—
282C2×C4—212no—
283C2×C4—212no—
284C2×C4—41no—
285C2×C4—43no—
286C2×C4—43no—
287C2×C4—46no—
288C2×C4—43no—
289C2×C4—43no—
290C2×C4—43no—
291C2×C4—46no—
292C2×C4—594no—
293C2×C4—1186no—
294C2×C4—1186no—
295C2×C4—1186no—
296C2×C4—1186no—
297C2×C4—1186no—
298C2×C4—1186no—
299C2×C4—11812no—
300C2×C4—2363no—
301C2×C4—2363no—
302C2×C4—2366no—
303C2×C4—2366no—
304C2×C4—2366no—
305C2×C4#14273689112yes—
306C22—4721no—
307C22—4721no—
308C22—4723no—
309C22—4723no—
310C22—2366no—
311C22—2366no—
312C22—2362no—
313C22C2×#1427368913yes—
314C22#1683787811yescharacteristic
315C22#1683788113yes—
316C22#127×D5911yescharacteristic
317C22#1683795313yes—
318C22#1683796213yes—
319C22#1683797413yes—
320C22#1683797513yes—
321C22#1683798113yes—
322C22#1683800116yes—
323C22#1683801513yes—
324C22#1683802613yes—
325C4—212no—
326C4—83no—
327C4—81no—
328C4—1184no—
329C4—2366no—
330C4—2366no—
331C2—4722no—
332C2#2142665811yescharacteristic
333C2#2142676413yes—
334C2#2142677613yes—
335C2#2143019711yescharacteristic
336C2#2143020511yescharacteristic
337C2#2143025113yes—
338C2#2143026213yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1444C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1444D59
hypocenterC59#14322
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×#16837298C2semidirect
4. #21422365C2semidirect
6. #95×Dic59C2semidirect
7. C59×#1444C2semidirect
14. #16837339C4semidirect
166. #1444D59semidirect
192. C59#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#168373392
C2×#168372982
C2×#168372912
#214223652
#214223382
#95×Dic592
C59×#14442
#1432259