Finite Group Databasebeta

← #28476914

Subgroups — #28476914

Order 15104.

594 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16837300C211yesmaximal, characteristic
2#21423858C211yesmaximal, characteristic
3#21421918C214yesmaximal, complemented
4C59×#1749C211yesmaximal, characteristic
5C22×C4⋊Dic59C411yescharacteristic
6C22×C4⋊Dic59C411yescharacteristic
7C22×Dic59⋊C4—21no—
8C2×C22⋊(C59⋊C8)—21no—
9C2×#14273638—21no—
10#16837300—21no—
11#16837300—21no—
12#16837300C412yes—
13#16837300C2212yes—
14C2×C59⋊#95—21no—
15#16837157C2212yes—
16C22×C42×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C2×#97×C59—21no—
19C59×#336C2212yes—
20C23×Dic59—21no—
21C23×Dic59—21no—
22C22×C59⋊C8—22no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—24no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—42no—
27C2×C4⋊Dic59C2×C412yes—
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—44no—
32C22⋊(C59⋊C8)—24no—
33#14273638—42no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C22×C8×C59—22no—
40C2×C42×C59—42no—
41C2×C42×C59C2×C411yescharacteristic
42C2×C42×C59D411yescharacteristic
43C2×C42×C59D411yescharacteristic
44C2×C42×C59C2311yescharacteristic
45C22⋊C8×C59—24no—
46#97×C59—42no—
47C24×C59C22⋊C411yescharacteristic
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59—41no—
56C22×C4×C59—41no—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—24no—
64C22×Dic59—22no—
65C22×Dic59—22no—
66C22×Dic59—42no—
67C22×Dic59—42no—
68C22×Dic59—42no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C2×C8×C59—24no—
72C2×C8×C59—42no—
73C2×C8×C59—42no—
74C42×C59—22no—
75C42×C59—22no—
76C42×C59—41no—
77C42×C59—41no—
78C4⋊Dic59—24no—
79C4⋊Dic59—22no—
80C4⋊Dic59—42no—
81C4⋊Dic59—42no—
82C4⋊Dic59M1614yes—
83Dic59⋊C4—44no—
84Dic59⋊C4—44no—
85C2×C59⋊C8—24no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59#10112yes—
93C23×C59#10412yes—
94C23×C59#10412yes—
95C23×C59C2×C22⋊C411yescharacteristic
96C2×C4×C59—28no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—44no—
112C2×C4×C59—44no—
113C2×C4×C59#9912yes—
114C2×C4×C59C2×M1612yes—
115C2×C4×C59AGL(1,ℤ/8)12yes—
116C2×C4×C59#13712yes—
117C2×Dic59—24no—
118C2×Dic59—48no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—84no—
125C8×C59—44no—
126C59⋊C8—44no—
127#7627—591nomaximal, Sylow
128C22×C59—22no—
129C22×C59—22no—
130C22×C59—22no—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—22no—
134C22×C59#33014yes—
135C22×C59#40711yescharacteristic
136C22×C59C2×#9911yescharacteristic
137C22×C59C2×#10111yescharacteristic
138C22×C59#41811yescharacteristic
139C22×C59#41911yescharacteristic
140C22×C59C2×#10411yescharacteristic
141C22×C59C2×#10411yescharacteristic
142C4×C59—22no—
143C4×C59—22no—
144C4×C59—22no—
145C4×C59—22no—
146C4×C59—41no—
147C4×C59—41no—
148Dic59—44no—
149Dic59—82no—
150C2×#389—591no—
151#1749—591no—
152#1749D5911yescharacteristic
153#1271—594no—
154C2×C59—24no—
155C2×C59#146311yescharacteristic
156C2×C59#149011yescharacteristic
157C2×C59#149111yescharacteristic
158C2×C59#150711yescharacteristic
159C2×C59#150811yescharacteristic
160C2×C59C2×#33011yescharacteristic
161C2×C59#155611yescharacteristic
162C22×C4⋊C4—591no—
163C22×C4⋊C4—591no—
164C22×C4⋊C4—1181no—
165C22×C42C2×D5911yescharacteristic
166C2×C22⋊C8—21no—
167C2×C22⋊C8—1181no—
168C2×#97—21no—
169C2×#97—1181no—
170#389—592no—
171#389—592no—
172#389—1181no—
173#389—1181no—
174C2×#95—1181no—
175#336—592no—
176#336C2×D5912yes—
177C59#762711yescharacteristic, complemented, Sylow
178C23×C4—1181no—
179C23×C4—1181no—
180C23×C4C4×D5911yescharacteristic
181C23×C4D23611yescharacteristic
182C23×C4C22⋊D5911yescharacteristic
183C22×C8—22no—
184C22×C8—1182no—
185C2×C4⋊C4—592no—
186C2×C4⋊C4—592no—
187C2×C4⋊C4—1184no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4—1182no—
191C2×C4⋊C4—1182no—
192C2×C4⋊C4—2364no—
193C2×C4⋊C4—2362no—
194C2×C42—42no—
195C2×C42C4×D5911yescharacteristic
196C2×C42D23611yescharacteristic
197C2×C42C22⋊D5911yescharacteristic
198C2×C42C22×D5911yescharacteristic
199C22⋊C8—24no—
200C22⋊C8—1184no—
201#97—42no—
202#97—2362no—
203#95—1182no—
204#95—1182no—
205C24(C2×D59)⋊C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—1184no—
216C22×C4—1182no—
217C22×C4—1182no—
218C22×C4—2362no—
219C22×C4—2362no—
220C22×C4—2362no—
221C22×C4—2362no—
222C22×C4—2362no—
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4C2×C4×D5911yescharacteristic
227C22×C4C2×D23611yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—1184no—
233C2×C8—2362no—
234C2×C8—2362no—
235C4⋊C4—594no—
236C4⋊C4—1184no—
237C4⋊C4—1182no—
238C4⋊C4—2362no—
239C4⋊C4—2364no—
240C4⋊C4—2362no—
241C4⋊C4—2364no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1427364512yes—
251C23#1427365312yes—
252C23#1427365512yes—
253C23C2×(C2×D59)⋊C411yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—1184no—
272C2×C4—4724no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4—2362no—
276C2×C4—2368no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4#1427364112yes—
280C2×C4M16×D5912yes—
281C2×C4#1427375512yes—
282C2×C4#1427375712yes—
283C8—44no—
284C8—2364no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1683714314yes—
292C22#1683736511yescharacteristic
293C22C2×#1427364111yescharacteristic
294C22C2×#1427364511yescharacteristic
295C22#1683740811yescharacteristic
296C22#1683741011yescharacteristic
297C22C2×#1427365311yescharacteristic
298C22C2×#1427365511yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—4722no—
306C4—2364no—
307C2—24no—
308C2#2142242511yescharacteristic
309C2#2142254011yescharacteristic
310C2#2142254211yescharacteristic
311C2#2142261311yescharacteristic
312C2#2142261511yescharacteristic
313C2C2×#1683714311yescharacteristic
314C2#2142281211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1749C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1749D59
hypocenterC59#7627
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #21421918C2semidirect
177. C59#7627semidirect

Maximal subgroups

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

subgroupindex
C2×#168373002
#214238582
#214219182
C59×#17492
#762759