Finite Group Databasebeta

← #28499266

Subgroups — #28499266

Order 15104.

446 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429001C211yesmaximal, characteristic
2#21422440C211yesmaximal, characteristic, complemented
3#21422101C212yesmaximal, complemented
4#21422098C212yesmaximal
5C4×#101×C59C211yesmaximal, characteristic, complemented
6C2×#14273703—21no—
7C2×C4×D236C411yescharacteristic
8#16837459—21no—
9#16837332C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837319C411yescharacteristic
12#16837319C2211yescharacteristic
13#16837293—21no—
14#16837293—21no—
15#16837268C411yescharacteristic
16#16837259—21no—
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C2×#101×C59—21no—
21C4×M16×C59—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C4×D59—21no—
25#14273703—44no—
26C4×D236—41no—
27C4×D236—41no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C4⋊(C59⋊C8)—41no—
38C22⋊(C59⋊C8)—22no—
39#14273638—41no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×Q8×C59D411yescharacteristic
46C2×M16×C59—22no—
47C4×Q8×C59—41no—
48C4×Q8×C59—41no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55#101×C59—24no—
56#97×C59—41no—
57C4×C8×C59—41no—
58C23×D59—21no—
59C2×Q8×C59—22no—
60C2×Q8×C59—22no—
61C2×Q8×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×D236—41no—
73C2×D236—42no—
74C2×C4×D59—42no—
75C2×C4×D59—42no—
76C2×C4×D59—42no—
77M16×C59—24no—
78M16×C59—42no—
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—42no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—41no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—42no—
95C4⋊Dic59—42no—
96C4⋊Dic59—41no—
97Dic59⋊C4—42no—
98Dic59⋊C4—42no—
99C2×C59⋊C8—42no—
100C23×C59C2×C22⋊C411yescharacteristic
101C22×D59—22no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105C22×D59—41no—
106C22×D59—41no—
107C22×D59—41no—
108Q8×C59—44no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#10412yes—
126C2×C4×C59AGL(1,ℤ/8)12yes—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136D236—42no—
137D236—42no—
138C4×D59—84no—
139C8×C59—44no—
140C59⋊C8—82no—
141#11778—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59C2×#9911yescharacteristic
145C22×C59#41811yescharacteristic
146C22×C59C2×#10411yescharacteristic
147C2×D59—42no—
148C2×D59—42no—
149C2×D59—42no—
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—81no—
153C2×D59—81no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—22no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162Dic59—81no—
163Dic59—81no—
164Dic59—81no—
165#2536—591no—
166C4×#101D5911yescharacteristic, complemented
167#1467—591no—
168#1346—592no—
169#1345—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59#150711yescharacteristic
173C2×C59#212011yescharacteristic
174C2×C59#213311yescharacteristic
175D59—82no—
176C2×C22⋊Q8—1181no—
177C2×C4×Q8C2×D5911yescharacteristic
178C2×C4×D4—591no—
179#432—1181no—
180C2×#101—21no—
181C4×M16—21no—
182#398—591no—
183#398—591no—
184#394—591no—
185#394—591no—
186#388—1181no—
187#388—1181no—
188#382—591no—
189#380—1181no—
190#343C2×D5912yes—
191#324—592no—
192#324—1181no—
193C59#1177811yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D5911yescharacteristic
195C22×D4—591no—
196C23×C4—1181no—
197C2×M16—22no—
198C22⋊Q8—2364no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—2361no—
202C4×D4—2361no—
203C2×C4⋊C4—591no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4C4×D5911yescharacteristic
208C2×C4⋊C4D23611yescharacteristic
209C2×C4⋊C4C22×D5911yescharacteristic
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C42C4×D5911yescharacteristic
216C2×C42D23611yescharacteristic
217C2×C42C22⋊D5911yescharacteristic
218C4⋊C8—2361no—
219#101—24no—
220C22⋊C8—1182no—
221#97—41no—
222#97—2361no—
223C4×C8—41no—
224#95—591no—
225#95—591no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229C24—1181no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—2361no—
234C2×D4—2362no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4—1181no—
238C22×C4—2362no—
239C22×C4—2362no—
240C22×C4—2362no—
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4C2×D23611yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248M16—24no—
249M16—42no—
250C2×C8—42no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2361no—
256C4⋊C4—2362no—
257C4⋊C4—2362no—
258C4⋊C4—2362no—
259C22⋊C4—2362no—
260C22⋊C4—2362no—
261C22⋊C4—2362no—
262C22⋊C4—2361no—
263C22⋊C4—2361no—
264C22⋊C4—2362no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—1182no—
272C23—2361no—
273C23—2361no—
274C23—2361no—
275C23—2361no—
276C23—2361no—
277C23—2361no—
278C23C2×(C2×D59)⋊C411yescharacteristic
279Q8—44no—
280D4—2362no—
281D4—2362no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—4724no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4#1427364111yescharacteristic
307C2×C4#1427364111yescharacteristic
308C2×C4#1427365512yes—
309C2×C4#1427375212yes—
310C8—44no—
311C8—4722no—
312C22—21no—
313C22—21no—
314C22—4721no—
315C22—4721no—
316C22—4721no—
317C22—4721no—
318C22—2362no—
319C22—2362no—
320C22—2362no—
321C22C2×#1427364111yescharacteristic
322C22#1683740611yescharacteristic
323C22C2×#1427365511yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—4721no—
333C4—4721no—
334C4—4721no—
335C2—21no—
336C2—21no—
337C2—4722no—
338C2#2142261111yescharacteristic
339C2#2142573811yescharacteristic
340C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC4×#101×C59C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenterC4×#101D59
hypocenterC59#11778
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #21422440C2semidirect
3. #21422101C2semidirect
5. C4×#101×C59C2semidirect
166. C4×#101D59semidirect
193. C59#11778semidirect

Maximal subgroups

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

subgroupindex
#214290012
#214224402
#214221012
#214220982
C4×#101×C592
#1177859