Finite Group Databasebeta

← #21432425

Subgroups — #21432425

Order 7552.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14273710C211yesmaximal, characteristic, complemented
2C2×#14273699C211yesmaximal, characteristic
3C22×C4×Dic59C211yesmaximal, characteristic, complemented
4#16837302C212yesmaximal, complemented
5#16837297C212yesmaximal, complemented
6#16837296C212yesmaximal, complemented
7#16837292C212yesmaximal, complemented
8#16837282C211yesmaximal, characteristic, complemented
9#16837279C211yesmaximal, characteristic, complemented
10#16837260C211yesmaximal, characteristic, complemented
11C2×#123×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C22⋊D59C2211yescharacteristic
14C23×Dic59C2211yescharacteristic
15C23×Dic59C2211yescharacteristic
16#14273710—24no—
17#14273699—24no—
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×C4×Dic59—24no—
28C2×C4×Dic59C2212yes—
29C2×C4×Dic59C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C22×D4×C59C2211yescharacteristic
34C23×C4×C59C2211yescharacteristic
35#123×C59—24no—
36C2×C4⋊C4×C59C2212yes—
37C2×C22⋊C4×C59C2211yescharacteristic
38C2×C22⋊C4×C59C2212yes—
39C24×C59C2311yescharacteristic
40C24×C59C2311yescharacteristic
41C23×D59C2311yescharacteristic
42C2×D4×C59—22no—
43C2×D4×C59—22no—
44C2×D4×C59—22no—
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59D414yescomplemented
48C22×C4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C22×C4×C59C2312yes—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—22no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—22no—
55C2×C22⋊D59—24no—
56C22×Dic59—22no—
57C22×Dic59—24no—
58C22×Dic59—24no—
59C22×Dic59D414yes—
60C22×Dic59C2312yes—
61C22×Dic59C2311yescharacteristic
62C22×Dic59C2311yescharacteristic
63C22×Dic59C2312yes—
64C22×Dic59C2311yescharacteristic
65C4⋊C4×C59—24no—
66C22⋊C4×C59—22no—
67C22⋊C4×C59—24no—
68C22⋊Dic59—22no—
69C22⋊Dic59—22no—
70C22⋊Dic59—24no—
71(C2×D59)⋊C4—24no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—24no—
74Dic59⋊C4—24no—
75Dic59⋊C4—22no—
76Dic59⋊C4—24no—
77C4×Dic59—24no—
78C4×Dic59—24no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—21no—
82C23×C59—21no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59—21no—
86C23×C59C2×D412yes—
87C23×C59C2×D412yes—
88C23×C59#4112yes—
89C23×C59C2411yescharacteristic
90C22×D59—21no—
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—22no—
94C22×D59—21no—
95D4×C59—44no—
96C2×C4×C59—28no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59C2×D414yes—
106C22⋊D59—44no—
107C22⋊D59—44no—
108C2×Dic59—28no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—28no—
113C2×Dic59—21no—
114C2×Dic59—22no—
115C2×Dic59—21no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—21no—
119C2×Dic59C2×D412yes—
120C2×Dic59C2×D412yes—
121C2×Dic59#4114yes—
122C2×Dic59#4114yes—
123C22×C59—22no—
124C22×C59—22no—
125C22×C59—24no—
126C22×C59—24no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—42no—
131C22×C59—42no—
132C22×C59#12114yes—
133C22×C59C22×D411yescharacteristic
134C22×C59C22×D411yescharacteristic
135C22×C59C2×#4112yes—
136C22×C59C2×#4112yes—
137C22×C59C2×#4111yescharacteristic
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—41no—
141C2×D59—42no—
142C2×D59—42no—
143C4×C59—24no—
144C4×C59—41no—
145C4×C59—42no—
146Dic59—22no—
147Dic59—24no—
148Dic59—41no—
149Dic59—42no—
150Dic59#12314yescomplemented
151#2854—591nomaximal, Sylow
152C2×C59—24no—
153C2×C59—81no—
154C2×C59C2×#12111yescharacteristic
155C2×C59C2×#12311yescharacteristic
156C2×C59#52511yescharacteristic
157C2×C59#52912yes—
158C2×C59#53212yes—
159D59—81no—
160C2×#123—591no—
161C2×#123D5911yescharacteristic, complemented
162C2×#121—591no—
163C22×C42—591no—
164#390—592no—
165#389—592no—
166#388—592no—
167#388—592no—
168#386—591no—
169#386—591no—
170#380—591no—
171C59#285411yescharacteristic, complemented, Sylow
172C22×D4—591no—
173C22×D4—591no—
174C22×D4C2×D5911yescharacteristic
175C23×C4—591no—
176C23×C4—591no—
177C23×C4C2×D5911yescharacteristic
178#123—24no—
179#123—1184no—
180#121—1184no—
181C2×C4⋊C4—591no—
182C2×C4⋊C4—592no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4C2×D5912yes—
185C2×C22⋊C4—591no—
186C2×C22⋊C4—591no—
187C2×C22⋊C4—592no—
188C2×C22⋊C4—591no—
189C2×C22⋊C4—592no—
190C2×C22⋊C4—592no—
191C2×C22⋊C4C2×D5911yescharacteristic
192C2×C22⋊C4C2×D5912yes—
193C2×C42—592no—
194C2×C42—592no—
195C2×C42—1184no—
196#95—592no—
197#95—592no—
198#95—592no—
199C24—591no—
200C24C22×D5911yescharacteristic
201C24C22×D5911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—1182no—
206C2×D4—1184no—
207C2×D4—1182no—
208C2×D4—1182no—
209C2×D4—1182no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—591no—
213C22×C4—594no—
214C22×C4—591no—
215C22×C4—592no—
216C22×C4—592no—
217C22×C4—591no—
218C22×C4—1184no—
219C22×C4—1184no—
220C22×C4—1182no—
221C22×C4C22⋊D5914yescomplemented
222C22×C4C22×D5912yes—
223C22×C4C22×D5911yescharacteristic
224C22×C4C22×D5912yes—
225C4⋊C4—24no—
226C4⋊C4—1182no—
227C4⋊C4—1184no—
228C4⋊C4—1184no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—1182no—
232C22⋊C4—1184no—
233C22⋊C4—1182no—
234C22⋊C4—1184no—
235C22⋊C4—1182no—
236C22⋊C4—1184no—
237C42—1184no—
238C42—1184no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—1181no—
247C23—1181no—
248C23—1182no—
249C23—1182no—
250C23—1181no—
251C23D4×D5912yes—
252C23#1174568712yes—
253C23C2×C22⋊D5912yes—
254C23C23×D5911yescharacteristic
255D4—44no—
256D4—2364no—
257D4—2364no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—592no—
268C2×C4—594no—
269C2×C4—592no—
270C2×C4—594no—
271C2×C4—1188no—
272C2×C4—1188no—
273C2×C4—1182no—
274C2×C4—1182no—
275C2×C4—1182no—
276C2×C4—1181no—
277C2×C4—1182no—
278C2×C4—1181no—
279C2×C4—1182no—
280C2×C4—1182no—
281C2×C4—1181no—
282C2×C4C2×C22⋊D5914yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—2361no—
293C22—2361no—
294C22—2361no—
295C22—2362no—
296C22—2362no—
297C22#1427369914yes—
298C22C2×D4×D5911yescharacteristic
299C22C22×C22⋊D5911yescharacteristic
300C22#41×D5912yes—
301C22#41×D5912yes—
302C22C2×#1174568711yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—594no—
307C4—1182no—
308C4—1184no—
309C4—2361no—
310C4—2362no—
311C2—24no—
312C2—81no—
313C2—4721no—
314C2C2×#1427369911yescharacteristic
315C2#123×D5911yescharacteristic
316C2#1683791011yescharacteristic
317C2#1683792912yes—
318C2#1683794812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2854
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14273710C2semidirect
3. C22×C4×Dic59C2semidirect
4. #16837302C2semidirect
5. #16837297C2semidirect
6. #16837296C2semidirect
7. #16837292C2semidirect
8. #16837282C2semidirect
9. #16837279C2semidirect
10. #16837260C2semidirect
11. C2×#123×C59C2semidirect
47. C22×C4×C59D4semidirect
150. Dic59#123semidirect
161. C2×#123D59semidirect
171. C59#2854semidirect
221. C22×C4C22⋊D59semidirect