Finite Group Databasebeta

← #21429834

Subgroups — #21429834

Order 7552.

824 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273699C211yesmaximal, characteristic, complemented
2C2×#14273696C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic59C211yesmaximal, characteristic, complemented
4#16837330C212yesmaximal, complemented
5#16837317C212yesmaximal, complemented
6#16837316C212yesmaximal, complemented
7#16837310C212yesmaximal, complemented
8#16837282C211yesmaximal, characteristic, complemented
9#16837279C211yesmaximal, characteristic, complemented
10#16837260C211yesmaximal, characteristic, complemented
11C2×#120×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14C22×C22⋊D59C2211yescharacteristic, complemented
15C23×Dic59C2211yescharacteristic
16C23×Dic59C2211yescharacteristic
17#14273699—24no—
18#14273696—24no—
19C2×Dic59⋊C4C2211yescharacteristic, complemented
20C2×Dic59⋊C4C2212yescomplemented
21C2×C22⋊Dic59—24no—
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×C22⋊Dic59C2212yes—
25C2×C22⋊Dic59C2212yes—
26C2×C22⋊Dic59C2211yescharacteristic, complemented
27C2×C22⋊Dic59C2212yescomplemented
28C2×(C2×D59)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D59)⋊C4C2212yescomplemented
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C25×C59C2211yescharacteristic
34C22×D4×C59C2211yescharacteristic
35C22×D4×C59C2212yes—
36#120×C59—24no—
37C2×C22⋊C4×C59C2211yescharacteristic
38C2×C22⋊C4×C59C2212yes—
39C24×C59—22no—
40C24×C59—22no—
41C24×C59—42no—
42C24×C59—42no—
43C24×C59D414yescomplemented
44C24×C59C2311yescharacteristic
45C24×C59C2312yes—
46C24×C59C2311yescharacteristic
47C23×D59C2311yescharacteristic
48C2×D4×C59—24no—
49C2×D4×C59—22no—
50C2×D4×C59—22no—
51C2×D4×C59—22no—
52C2×D4×C59—42no—
53C2×D4×C59—42no—
54C22×C4×C59C2311yescharacteristic
55C22×C4×C59C2312yes—
56C2×C22⋊D59—24no—
57C2×C22⋊D59—22no—
58C2×C22⋊D59—22no—
59C2×C22⋊D59—22no—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—42no—
62C2×C22⋊D59—42no—
63C2×C22⋊D59—42no—
64C22×Dic59—22no—
65C22×Dic59—42no—
66C22×Dic59—42no—
67C22×Dic59D414yescomplemented
68C22×Dic59C2312yes—
69C22×Dic59C2311yescharacteristic
70C22×Dic59C2312yes—
71C22×Dic59C2311yescharacteristic
72C22×Dic59C2311yescharacteristic
73C22⋊C4×C59—24no—
74C22⋊C4×C59—22no—
75C22⋊Dic59—24no—
76C22⋊Dic59—22no—
77C22⋊Dic59—24no—
78C22⋊Dic59—24no—
79C22⋊Dic59—22no—
80C22⋊Dic59—42no—
81(C2×D59)⋊C4—22no—
82(C2×D59)⋊C4—42no—
83Dic59⋊C4—22no—
84Dic59⋊C4—42no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—28no—
88C23×C59—22no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—48no—
100C23×C59—44no—
101C23×C59—44no—
102C23×C59—48no—
103C23×C59C2×D412yes—
104C23×C59C2×D414yes—
105C23×C59C2×D412yes—
106C23×C59#4112yes—
107C23×C59C2411yescharacteristic
108C22×D59—21no—
109C22×D59—21no—
110C22×D59—21no—
111C22×D59—22no—
112C22×D59—22no—
113D4×C59—48no—
114D4×C59—44no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C22⋊D59—44no—
122C22⋊D59—48no—
123C22⋊D59—44no—
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—21no—
127C2×Dic59—22no—
128C2×Dic59—21no—
129C2×Dic59—22no—
130C2×Dic59—21no—
131C2×Dic59—22no—
132C2×Dic59—22no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—44no—
136C2×Dic59—44no—
137C2×Dic59C2×D412yes—
138C2×Dic59C2×D412yes—
139C22×C59—22no—
140C22×C59—24no—
141C22×C59—22no—
142C22×C59—24no—
143C22×C59—24no—
144C22×C59—24no—
145C22×C59—24no—
146C22×C59—416no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—42no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—42no—
154C22×C59—41no—
155C22×C59—42no—
156C22×C59#12114yes—
157C22×C59C22×D411yescharacteristic
158C22×C59C22×D411yescharacteristic
159C22×C59C2×#4111yescharacteristic
160C22×C59H(5,𝔽2)12yes—
161C22×C59H(5,𝔽2)12yes—
162C2×D59—41no—
163C2×D59—42no—
164C2×D59—41no—
165C2×D59—41no—
166C2×D59—42no—
167C4×C59—41no—
168C4×C59—42no—
169Dic59—22no—
170Dic59—22no—
171Dic59—41no—
172Dic59—42no—
173Dic59—42no—
174#2617—591nomaximal, Sylow
175C2×C59—24no—
176C2×C59—44no—
177C2×C59—81no—
178C2×C59C2×#12111yescharacteristic
179C2×C59#53412yes—
180C2×C59#53411yescharacteristic
181C2×C59#53511yescharacteristic
182C2×C59#53812yes—
183D59—81no—
184C2×#121—591no—
185C2×#120—591no—
186C2×#120D5911yescharacteristic, complemented
187C22×C22⋊C4—591no—
188#397—592no—
189#394—592no—
190#394—592no—
191#392—592no—
192#386—591no—
193#386—591no—
194#380—591no—
195C59#261711yescharacteristic, complemented, Sylow
196C25C2×D5911yescharacteristic
197C22×D4—591no—
198C22×D4—592no—
199C22×D4—591no—
200C22×D4C2×D5911yescharacteristic
201C22×D4C2×D5912yes—
202C23×C4—591no—
203C23×C4—591no—
204#121—1184no—
205#120—24no—
206#120—1184no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—592no—
209C2×C22⋊C4—592no—
210C2×C22⋊C4—592no—
211C2×C22⋊C4—591no—
212C2×C22⋊C4—592no—
213C2×C22⋊C4—591no—
214C2×C22⋊C4—591no—
215C2×C22⋊C4—592no—
216C2×C22⋊C4—592no—
217C2×C22⋊C4—1184no—
218C2×C22⋊C4C2×D5911yescharacteristic
219C2×C22⋊C4C2×D5912yes—
220#95—592no—
221#95—592no—
222#95—592no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24—591no—
228C24C22⋊D5914yescomplemented
229C24C22×D5911yescharacteristic
230C24C22×D5912yes—
231C24C22×D5911yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—1182no—
239C2×D4—1182no—
240C2×D4—1182no—
241C2×D4—1184no—
242C2×D4—1182no—
243C2×D4—2362no—
244C2×D4—2362no—
245C2×D4—2362no—
246C22×C4—591no—
247C22×C4—591no—
248C22×C4—594no—
249C22×C4—592no—
250C22×C4—591no—
251C22×C4—592no—
252C22×C4—1182no—
253C22×C4—2362no—
254C22×C4—2362no—
255C22×C4C22×D5911yescharacteristic
256C22×C4C22×D5912yes—
257C4⋊C4—1182no—
258C4⋊C4—2362no—
259C22⋊C4—24no—
260C22⋊C4—22no—
261C22⋊C4—1182no—
262C22⋊C4—1184no—
263C22⋊C4—1184no—
264C22⋊C4—1182no—
265C22⋊C4—1184no—
266C22⋊C4—1182no—
267C22⋊C4—2362no—
268C22⋊C4—2362no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—1181no—
288C23—1181no—
289C23—1181no—
290C23—1182no—
291C23—1182no—
292C23D4×D5912yes—
293C23#1174568712yes—
294C23C2×C22⋊D5914yes—
295C23C2×C22⋊D5912yes—
296C23C23×D5911yescharacteristic
297D4—48no—
298D4—44no—
299D4—2364no—
300D4—2368no—
301D4—2364no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—592no—
309C2×C4—592no—
310C2×C4—1182no—
311C2×C4—1182no—
312C2×C4—1182no—
313C2×C4—1181no—
314C2×C4—1181no—
315C2×C4—1182no—
316C2×C4—1181no—
317C2×C4—1182no—
318C2×C4—1182no—
319C2×C4—1182no—
320C2×C4—1182no—
321C2×C4—2364no—
322C2×C4—2364no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—24no—
328C22—24no—
329C22—24no—
330C22—416no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—2361no—
341C22—2362no—
342C22—2361no—
343C22—2361no—
344C22—2362no—
345C22#1427369914yes—
346C22C2×D4×D5911yescharacteristic
347C22C22×C22⋊D5911yescharacteristic
348C22C2×#1174568711yescharacteristic
349C22#1427377212yes—
350C22#1427377212yes—
351C4—41no—
352C4—42no—
353C4—1182no—
354C4—1182no—
355C4—2361no—
356C4—2362no—
357C4—2362no—
358C2—24no—
359C2—44no—
360C2—81no—
361C2—4721no—
362C2C2×#1427369911yescharacteristic
363C2#1683796112yes—
364C2#1683796211yescharacteristic
365C2#1683796711yescharacteristic
366C2#1683798712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#120×C59C2
hypercenterC2×#120D59
hypocenterC59#2617
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14273699C2semidirect
2. C2×#14273696C2semidirect
3. C22×C22⋊Dic59C2semidirect
4. #16837330C2semidirect
5. #16837317C2semidirect
6. #16837316C2semidirect
7. #16837310C2semidirect
8. #16837282C2semidirect
9. #16837279C2semidirect
10. #16837260C2semidirect
11. C2×#120×C59C2semidirect
14. C22×C22⋊D59C22semidirect
19. C2×Dic59⋊C4C22semidirect
20. C2×Dic59⋊C4C22semidirect
26. C2×C22⋊Dic59C22semidirect
27. C2×C22⋊Dic59C22semidirect
28. C2×(C2×D59)⋊C4C22semidirect
29. C2×(C2×D59)⋊C4C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
32. C59⋊#95C22semidirect
43. C24×C59D4semidirect
67. C22×Dic59D4semidirect
186. C2×#120D59semidirect
195. C59#2617semidirect
228. C24C22⋊D59semidirect