Finite Group Databasebeta

← #28499064

Subgroups — #28499064

Order 15104.

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

normal subgroups only

This group has 440 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#21427355C211yesmaximal, characteristic
2C4×#14273644C211yesmaximal, characteristic
3#21422124C212yesmaximal, complemented
4#21422103C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic
7C2×#14273644—21no—
8C4×#11745664—21no—
9#16837341C2211yescharacteristic
10#16837338C411yescharacteristic
11#16837332C2211yescharacteristic
12#16837330—21no—
13#16837327C411yescharacteristic
14#16837316—21no—
15C4×Dic59⋊C4—21no—
16C4×C22⋊Dic59—21no—
17C4×C22⋊Dic59C411yescharacteristic
18#16837171C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C2×#11745664—22no—
24#14273686—41no—
25#14273686—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36#14273644—24no—
37#14273638—41no—
38C4×C59⋊C8—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×D4×C59D411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C22×C8×C59—41no—
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C22⋊C8×C59—22no—
56C22⋊C8×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×C8×C59—42no—
78C2×C8×C59—82no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C4⋊Dic59—41no—
89C4⋊Dic59—41no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92C4×Dic59—42no—
93C4×Dic59—41no—
94C4×Dic59—41no—
95#11745664—24no—
96#11745664—42no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#10412yes—
108C23×C59C2×C22⋊C411yescharacteristic
109D4×C59—44no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—81no—
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#13512yes—
126C2×Dic59—22no—
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—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C8×C59—82no—
139C59⋊C8—44no—
140#11735—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—22no—
144C22×C59—42no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59C2×#9911yescharacteristic
154C22×C59#41611yescharacteristic
155C22×C59C2×#10411yescharacteristic
156C4×C59—21no—
157C4×C59—22no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59#35212yes—
162Dic59—42no—
163Dic59—81no—
164Dic59—81no—
165Dic59—81no—
166#2395—591no—
167C4×#100—591no—
168#1459D5911yescharacteristic
169#1353—592no—
170#1346—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—42no—
174C2×C59—81no—
175C2×C59#150111yescharacteristic
176C2×C59C2×#35211yescharacteristic
177C2×C59#213211yescharacteristic
178C2×C4×D4C2×D5911yescharacteristic
179C2×#117—591no—
180C2×#100—1181no—
181C2×C22⋊C8—21no—
182C4×M16—1181no—
183#400—591no—
184#399—591no—
185#398—591no—
186#397—591no—
187#397—1181no—
188#394—1181no—
189C4×C4⋊C4—1181no—
190C4×C22⋊C4—591no—
191C4×C22⋊C4—1181no—
192#343—592no—
193#324C2×D5912yes—
194#323—21no—
195C59#1173511yescharacteristic, complemented, Sylow
196C22×D4D23611yescharacteristic
197C23×C4C4×D5911yescharacteristic
198C23×C4C22⋊D5911yescharacteristic
199C2×M16—1182no—
200C22×C8—41no—
201C4×D4—41no—
202C4×D4—41no—
203#117—2361no—
204#117—2361no—
205C2×C4⋊C4—591no—
206C2×C4⋊C4—591no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4C22×D5911yescharacteristic
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4C4×D5911yescharacteristic
215C2×C22⋊C4C22⋊D5911yescharacteristic
216C2×C42—1181no—
217C2×C42—1181no—
218C2×C42D23611yescharacteristic
219#100—1184no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222#97—2361no—
223C4×C8—2361no—
224#95—591no—
225#95—591no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230C24(C2×D59)⋊C411yescharacteristic
231C24(C2×D59)⋊C411yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4C2×C4×D5911yescharacteristic
248C22×C4C2×D23611yescharacteristic
249C22×C4C2×C22⋊D5911yescharacteristic
250M16—1184no—
251M16—2362no—
252C2×C8—42no—
253C2×C8—82no—
254C2×C8—2362no—
255C4⋊C4—41no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C4⋊C4—2362no—
259C4⋊C4—2362no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C22⋊C4—2362no—
265C22⋊C4—2362no—
266C42—21no—
267C42—21no—
268C42—2362no—
269C42—2361no—
270C42—2361no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1427365312yes—
281C23C2×(C2×D59)⋊C411yescharacteristic
282D4—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—1182no—
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—2361no—
307C2×C4—2361no—
308C2×C4#1427364111yescharacteristic
309C2×C4#1427364111yescharacteristic
310C2×C4#1427374912yes—
311C8—82no—
312C8—2364no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#1427364111yescharacteristic
326C22#1683739911yescharacteristic
327C22C2×#1427365311yescharacteristic
328C4—21no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4721no—
334C4—4721no—
335C4—4721no—
336C4—2362no—
337C4#1683719112yes—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2142258811yescharacteristic
343C2C2×#1683719111yescharacteristic
344C2#2142577911yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11735
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21422124C2semidirect
4. #21422103C2semidirect
195. C59#11735semidirect

Maximal subgroups

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

subgroupindex
#214273552
C4×#142736442
#214221242
#214221032
C59×#14592
#1173559