Finite Group Databasebeta

← #28498993

Subgroups — #28498993

Order 15104.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#21427364C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422122C212yesmaximal, complemented
5C59×#1742C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7#16837455—21no—
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837335—21no—
11#16837325—21no—
12#16837322C2211yescharacteristic
13#16837314C411yescharacteristic
14C4×Dic59⋊C4—21no—
15C4×C4⋊Dic59C411yescharacteristic
16C4×C22⋊Dic59—21no—
17#16837128C2212yes—
18#16837126—21no—
19C22×M16×C59—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37C4×C59⋊C8—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C2×M16×C59—24no—
46C2×M16×C59—22no—
47C2×M16×C59—41no—
48C2×M16×C59—42no—
49C22×C8×C59—41no—
50#117×C59—41no—
51#117×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59SD1612yes—
66C22×C4×C59Q1612yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×Dic118—41no—
75C2×Dic118—42no—
76M16×C59—24no—
77M16×C59—44no—
78M16×C59—42no—
79C2×C8×C59—42no—
80C2×C8×C59—44no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C4⋊Dic59—42no—
92C4⋊Dic59—41no—
93Dic59⋊C4—42no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96C4×Dic59—41no—
97C4×Dic59—41no—
98C4×Dic59—42no—
99C2×C59⋊C8—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59#10312yes—
105C23×C59C2×C22⋊C411yescharacteristic
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#10312yes—
123C2×C4×C59#10412yes—
124C2×C4×C59C2×SD1611yescharacteristic
125C2×C4×C59C2×Q1611yescharacteristic
126C2×Dic59—22no—
127C2×Dic59—22no—
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—
138Dic118—42no—
139Dic118—42no—
140C8×C59—44no—
141C59⋊C8—82no—
142#11721—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—42no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—81no—
150C22×C59C2×#9911yescharacteristic
151C22×C59C2×#10311yescharacteristic
152C22×C59C2×#10411yescharacteristic
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59#32812yes—
160C4×C59#32812yes—
161Dic59—42no—
162Dic59—42no—
163Dic59—81no—
164Dic59—81no—
165#2396—591no—
166#1742D5911yescharacteristic
167#1458—591no—
168#1354—592no—
169#1353—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—42no—
173C2×C59C2×#32811yescharacteristic
174C2×C59#212011yescharacteristic
175C2×C59#213211yescharacteristic
176C22×M16—21no—
177C2×C4×Q8—591no—
178C2×#117C2×D5911yescharacteristic
179#431—1181no—
180#400—591no—
181#399—591no—
182#398—1181no—
183#396—1181no—
184#395—591no—
185#393—591no—
186C4×C4⋊C4—591no—
187C4×C4⋊C4—1181no—
188C4×C22⋊C4—1181no—
189#343—21no—
190#343C2×D5912yes—
191#324—592no—
192#323—1181no—
193C59#1172111yescharacteristic, complemented, Sylow
194C22×Q8—591no—
195C23×C4C22⋊D5911yescharacteristic
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C22×C8—41no—
201C4×Q8—2361no—
202C4×Q8—2361no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—591no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4C22⋊D5911yescharacteristic
211C2×C4⋊C4C22×D5911yescharacteristic
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4C4×D5911yescharacteristic
215C2×C22⋊C4D23611yescharacteristic
216C2×C42—1181no—
217C2×C42—1181no—
218C2×C42C4×D5911yescharacteristic
219C2×C42D23611yescharacteristic
220C4⋊C8—2361no—
221C22⋊C8—1182no—
222C4×C8—2361no—
223#95—591no—
224#95—591no—
225#95—1181no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229C24(C2×D59)⋊C411yescharacteristic
230C2×Q8—2361no—
231C2×Q8—2362no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4—1181no—
238C22×C4—1181no—
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4#1174567012yes—
243C22×C4#1174567212yes—
244C22×C4C2×C4×D5911yescharacteristic
245C22×C4C2×D23611yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247M16—24no—
248M16—44no—
249M16—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—2362no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2361no—
256C4⋊C4—2362no—
257C4⋊C4—2362no—
258C4⋊C4—2362no—
259C4⋊C4—2362no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—2361no—
269C42—2361no—
270C42—2362no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1427365012yes—
276C23C2×(C2×D59)⋊C411yescharacteristic
277Q8—2362no—
278Q8—2362no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—2361no—
296C2×C4—2361no—
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#1427364111yescharacteristic
306C2×C4#1427364111yescharacteristic
307C2×C4#1427365012yes—
308C2×C4#1427365512yes—
309C2×C4C2×#1174567011yescharacteristic
310C2×C4C2×#1174567211yescharacteristic
311C8—44no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1427364111yescharacteristic
321C22C2×#1427365011yescharacteristic
322C22C2×#1427365511yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4721no—
330C4—4721no—
331C4—2362no—
332C4—2362no—
333C4#1683713812yes—
334C4#1683713812yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1683713811yescharacteristic
339C2#2142573811yescharacteristic
340C2#2142577911yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1742C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1742D59
hypocenterC59#11721
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422122C2semidirect
193. C59#11721semidirect

Maximal subgroups

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

subgroupindex
#214273642
#214224102
#214221252
#214221222
C59×#17422
#1172159