Finite Group Databasebeta

← #28499201

Subgroups — #28499201

Order 15104.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#21428383C211yesmaximal, characteristic
2C2×#16837171C211yesmaximal, characteristic
3#21422124C212yesmaximal, complemented
4#21422103C212yesmaximal, complemented
5C59×#1463C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273686C411yescharacteristic
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837332C411yescharacteristic
11#16837332C2211yescharacteristic
12#16837283—21no—
13#16837283—21no—
14#16837275—21no—
15#16837265—21no—
16#16837265C411yescharacteristic
17#16837171—21no—
18#16837171C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C2×#11745664—24no—
24C2×#11745664—22no—
25C2×#11745664—41no—
26C2×#11745664—42no—
27C22×C59⋊C8—41no—
28#14273686—41no—
29#14273686—41no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C4×Dic59—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C2×M16×C59—41no—
50C2×C4⋊C4×C59—41no—
51C2×C4⋊C4×C59—41no—
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C4⋊C4×C59C2311yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—42no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80M16×C59—82no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C22⋊Dic59—42no—
87C22⋊Dic59—41no—
88C22⋊Dic59—41no—
89C4⋊Dic59—41no—
90C4⋊Dic59—41no—
91Dic59⋊C4—42no—
92Dic59⋊C4—41no—
93Dic59⋊C4—41no—
94C4×Dic59—42no—
95#11745664—24no—
96#11745664—44no—
97#11745664—42no—
98C2×C59⋊C8—44no—
99C2×C59⋊C8—42no—
100C23×C59—22no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59#9911yescharacteristic
105C23×C59#9911yescharacteristic
106C23×C59C2×C22⋊C411yescharacteristic
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
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—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—81no—
127C2×C4×C59#10412yes—
128C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C59#13711yescharacteristic
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—42no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C8×C59—82no—
142C59⋊C8—44no—
143#11764—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59#35212yes—
154C22×C59C2×#9911yescharacteristic
155C22×C59#41711yescharacteristic
156C22×C59C2×#10411yescharacteristic
157C4×C59—21no—
158C4×C59—22no—
159C4×C59—21no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—41no—
164Dic59—81no—
165Dic59—81no—
166Dic59—81no—
167Dic59—81no—
168#2484—591no—
169C2×#343—591no—
170#1463D5911yescharacteristic
171#1353—592no—
172#1346—592no—
173C2×C59—21no—
174C2×C59—21no—
175C2×C59—22no—
176C2×C59—41no—
177C2×C59#150411yescharacteristic
178C2×C59C2×#35211yescharacteristic
179C2×C59#213311yescharacteristic
180C22×M16—1181no—
181C2×#117—591no—
182C22×C4⋊C4C2×D5911yescharacteristic
183#407—21no—
184#400—591no—
185#400—591no—
186#398—591no—
187#398—591no—
188#387—1181no—
189#387—1181no—
190#385—1181no—
191#381—591no—
192#381—1181no—
193#343—592no—
194#343—1181no—
195#324—21no—
196#324C2×D5912yes—
197C59#1176411yescharacteristic, complemented, Sylow
198C23×C4C4×D5911yescharacteristic
199C23×C4D23611yescharacteristic
200C23×C4C22⋊D5911yescharacteristic
201C2×M16—41no—
202C2×M16—1184no—
203C2×M16—1182no—
204C2×M16—2361no—
205C2×M16—2362no—
206C22×C8—2361no—
207#117—2361no—
208#117—2361no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—591no—
212C2×C4⋊C4—591no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4C4×D5911yescharacteristic
216C2×C4⋊C4D23611yescharacteristic
217C2×C4⋊C4C22⋊D5911yescharacteristic
218C2×C4⋊C4C22×D5911yescharacteristic
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C42—1181no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#95—591no—
225#95—591no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233C24(C2×D59)⋊C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×C4×D5911yescharacteristic
253C22×C4C2×D23611yescharacteristic
254C22×C4C2×C22⋊D5911yescharacteristic
255M16—82no—
256M16—1184no—
257M16—2364no—
258M16—2362no—
259C2×C8—42no—
260C2×C8—2364no—
261C2×C8—2362no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—2361no—
267C4⋊C4—2361no—
268C4⋊C4—2362no—
269C4⋊C4—2361no—
270C4⋊C4—2361no—
271C22⋊C4—2362no—
272C22⋊C4—2361no—
273C22⋊C4—2361no—
274C42—2362no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1427364111yescharacteristic
280C23#1427364111yescharacteristic
281C23C2×(C2×D59)⋊C411yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2362no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4#1427365312yes—
314C2×C4#1427375511yescharacteristic
315C2×C4#1427376011yescharacteristic
316C8—82no—
317C8—2364no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1683719112yes—
328C22C2×#1427364111yescharacteristic
329C22#1683740411yescharacteristic
330C22C2×#1427365311yescharacteristic
331C4—21no—
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—4721no—
339C4—4721no—
340C4—4721no—
341C4—4721no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2142260111yescharacteristic
347C2C2×#1683719111yescharacteristic
348C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1463C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1463D59
hypocenterC59#11764
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #21422124C2semidirect
4. #21422103C2semidirect
197. C59#11764semidirect

Maximal subgroups

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

subgroupindex
#214283832
C2×#168371712
#214221242
#214221032
C59×#14632
#1176459