Finite Group Databasebeta

← #28482813

Subgroups — #28482813

Order 15104.

360 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433232C211yesmaximal, characteristic
2#21422879C211yesmaximal, characteristic, complemented
3#21422580C211yesmaximal, characteristic
4#21422507C211yesmaximal, characteristic
5#21421989C211yesmaximal, characteristic, complemented
6#21421946C211yesmaximal, characteristic, complemented
7C59×#1832C211yesmaximal, characteristic, complemented
8#16837936C2211yescharacteristic
9#16837531—21no—
10#16837473—21no—
11#16837472—21no—
12C8×C22⋊D59—21no—
13#16837462—21no—
14#16837459C2211yescharacteristic
15#16837455C2211yescharacteristic
16#16837427—21no—
17C4×#11745664C2211yescharacteristic
18#16837154—21no—
19#16837152—21no—
20#16837138C411yescharacteristic
21#16837138C411yescharacteristic
22#16837135C411yescharacteristic
23#16837135C411yescharacteristic
24#16837132—21no—
25#16837129—21no—
26C2×C4⋊C8×C59C2211yescharacteristic
27C2×C4⋊C8×C59C2211yescharacteristic
28C59×#376C2211yescharacteristic
29C59×#336—21no—
30C59×#336—21no—
31C2×#11745664—21no—
32C2×#11745659—21no—
33C2×C8×D59—21no—
34#14273721—21no—
35#14273719Q811yescharacteristic
36#14273715Q811yescharacteristic
37#14273710—21no—
38#14273706C2×C411yescharacteristic
39#14273701C2×C411yescharacteristic
40C4×Dic118D411yescharacteristic
41C4×D236D411yescharacteristic
42C4×C22⋊D59—21no—
43#14273684—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D411yescharacteristic
47Dic59⋊C8—21no—
48Dic59⋊C8—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)C2×C411yescharacteristic
51C22⋊(C59⋊C8)C2×C411yescharacteristic
52(C2×D59)⋊C8—21no—
53(C2×D59)⋊C8—21no—
54#14273638—21no—
55#14273638—21no—
56#14273637—21no—
57C4×C59⋊C8—21no—
58C4×C59⋊C8—21no—
59C8×Dic59—21no—
60C22×C8×C59—21no—
61C22×C8×C59D411yescharacteristic
62C22×C8×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69#95×C59—21no—
70#95×C59C2×C411yescharacteristic
71#95×C59C2×C411yescharacteristic
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59#4111yescharacteristic
77C2×C22⋊D59—21no—
78C2×D236C4⋊C411yescharacteristic
79C2×C4×D59—21no—
80C2×Dic118C4⋊C411yescharacteristic
81C2×C8×C59—21no—
82C2×C8×C59—21no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59—42no—
86C2×C8×C59—41no—
87C42×C59—41no—
88C42×C59—41no—
89C42×C59C2×D411yescharacteristic
90C42×C59C2×D411yescharacteristic
91C42×C59C2×Q811yescharacteristic
92C42×C59#4111yescharacteristic
93C22⋊Dic59—21no—
94(C2×D59)⋊C4—21no—
95(C2×D59)⋊C4—21no—
96(C2×D59)⋊C4—21no—
97(C2×D59)⋊C4—21no—
98C4⋊Dic59C4⋊C411yescharacteristic
99C4⋊Dic59C4⋊C411yescharacteristic
100Dic59⋊C4—21no—
101Dic59⋊C4—21no—
102Dic59⋊C4—21no—
103Dic59⋊C4—21no—
104C4×Dic59—21no—
105#11745664—22no—
106#11745664—41no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—41no—
112#11745659—81no—
113C8×D59—81no—
114C23×C59C4×D411yescharacteristic
115C22×D59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59C2×C4⋊C411yescharacteristic
128C2×C4×C59C4×D411yescharacteristic
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59C22⋊Q811yescharacteristic
131C2×C4×C59C22⋊Q811yescharacteristic
132C2×C4×C59#12311yescharacteristic
133C22⋊D59—41no—
134C22⋊D59—41no—
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—21no—
138D236—41no—
139C4×D59—41no—
140C4×D59—41no—
141Dic118—41no—
142C8×C59—41no—
143C8×C59—41no—
144C8×C59—81no—
145C59⋊C8—41no—
146C59⋊C8—41no—
147C59⋊C8—42no—
148C59⋊C8—41no—
149C59⋊C8—41no—
150#8877—591nomaximal, Sylow
151C22×C59—21no—
152C22×C59—21no—
153C22×C59#38511yescharacteristic
154C22×C59#44311yescharacteristic
155C22×C59#44411yescharacteristic
156C2×D59—41no—
157C2×D59—41no—
158C2×D59—41no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59#45411yescharacteristic
167C4×C59#45511yescharacteristic
168Dic59—41no—
169Dic59—41no—
170Dic59—41no—
171#2992—591no—
172#1832D5911yescharacteristic, complemented
173#1573—591no—
174#1500—591no—
175#1483—591no—
176#1300—591no—
177#1283—591no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59#186011yescharacteristic
181C2×C59#186611yescharacteristic
182C2×C59#186811yescharacteristic
183D59—81no—
184#529—591no—
185#446—1181no—
186#435—1181no—
187#435—1181no—
188C8×D4—1181no—
189#433—1181no—
190#432—591no—
191#431—591no—
192#424—1181no—
193C2×C4⋊C8C2×D5911yescharacteristic
194C2×C4⋊C8C2×D5911yescharacteristic
195C4×M16—591no—
196#376C2×D5911yescharacteristic
197#336—21no—
198#336—21no—
199#335—1181no—
200#334—1181no—
201#328—591no—
202#328—591no—
203#327—591no—
204#327—591no—
205Q8⋊C8—1181no—
206D4⋊C8—1181no—
207C59#887711yescharacteristic, complemented, Sylow
208C2×M16—1181no—
209C2×M16—1181no—
210C22×C8—21no—
211C22×C8—1181no—
212C22×C8C22⋊D5911yescharacteristic
213C22×C8C22⋊D5911yescharacteristic
214#126—1181no—
215#125—591no—
216#124—591no—
217#123—1181no—
218C22⋊Q8—591no—
219#121—591no—
220C4×Q8—591no—
221C4×D4—591no—
222C4×D4—1181no—
223#117—1181no—
224C2×C42—21no—
225C2×C42C22×D5911yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—591no—
231C4⋊C8—591no—
232C4⋊C8—1181no—
233C4⋊C8—1181no—
234C4⋊C8—1181no—
235C22⋊C8—591no—
236C22⋊C8—591no—
237C22⋊C8—1181no—
238C22⋊C8—1181no—
239C22⋊C8—1181no—
240#97—1181no—
241#97—1181no—
242#97—1181no—
243C4×C8—1181no—
244C4×C8—1181no—
245C4×C8—1181no—
246#95—21no—
247#95C4×D5911yescharacteristic
248#95C4×D5911yescharacteristic
249C2×Q8—591no—
250C2×D4—591no—
251C2×D4—1181no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—1181no—
255C22×C4C2×C4×D5911yescharacteristic
256C22×C4#1174568511yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258M16—1182no—
259M16—4721no—
260M16—2361no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—1181no—
268C2×C8—1181no—
269C2×C8—1181no—
270C2×C8—1181no—
271C2×C8—4721no—
272C2×C8—2361no—
273C4⋊C4—591no—
274C4⋊C4—591no—
275C4⋊C4—1181no—
276C4⋊C4—1181no—
277C4⋊C4—1181no—
278C4⋊C4—1181no—
279C22⋊C4—1181no—
280C22⋊C4—1181no—
281C22⋊C4—1181no—
282C22⋊C4—1181no—
283C22⋊C4—1181no—
284C42—41no—
285C42—41no—
286C42—1181no—
287C42D4×D5911yescharacteristic
288C42D4×D5911yescharacteristic
289C42Q8×D5911yescharacteristic
290C42#1174568911yescharacteristic
291C23—1181no—
292C23C4×C22⋊D5911yescharacteristic
293Q8—2361no—
294D4—2361no—
295D4—2361no—
296D4—2361no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—1181no—
309C2×C4—1181no—
310C2×C4—1181no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4C4⋊C4×D5911yescharacteristic
314C2×C4#1427368911yescharacteristic
315C2×C4#1427369511yescharacteristic
316C2×C4#1427370311yescharacteristic
317C2×C4#1427370511yescharacteristic
318C2×C4#1427371211yescharacteristic
319C8—41no—
320C8—41no—
321C8—81no—
322C8—2361no—
323C8—2361no—
324C8—2362no—
325C8—2361no—
326C8—2361no—
327C22—21no—
328C22—21no—
329C22—2361no—
330C22—2361no—
331C22—2361no—
332C22#1683727311yescharacteristic
333C22#1683751711yescharacteristic
334C22#1683752211yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—81no—
342C4—2361no—
343C4—2361no—
344C4—2361no—
345C4#1683757411yescharacteristic
346C4#1683757911yescharacteristic
347C2—21no—
348C2—21no—
349C2—4721no—
350C2#2142432811yescharacteristic
351C2#2142437011yescharacteristic
352C2#2142438011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1832C2
socleC22×C59#385
prime radical#14273701C2×C4
cosocleC2×C42×C59C23
hypercenter#1832D59
hypocenterC59#8877
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
2. #21422879C2semidirect
5. #21421989C2semidirect
6. #21421946C2semidirect
7. C59×#1832C2semidirect
172. #1832D59semidirect
207. C59#8877semidirect

Maximal subgroups

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

subgroupindex
#214332322
#214228792
#214225802
#214225072
#214219892
#214219462
C59×#18322
#887759