Finite Group Databasebeta

← #21429836

Subgroups — #21429836

Order 7552.

824 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14273697C211yesmaximal, characteristic, complemented
3C2×#14273695C211yesmaximal, characteristic, complemented
4C22×(C2×D59)⋊C4C211yesmaximal, characteristic, complemented
5#16837331C212yesmaximal, complemented
6#16837320C212yesmaximal, complemented
7#16837317C212yesmaximal, complemented
8#16837311C212yesmaximal, complemented
9#16837281C211yesmaximal, characteristic, complemented
10#16837260C211yesmaximal, characteristic, complemented
11C59×#386C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×D236C2212yes—
14C22×C22⋊D59C2211yescharacteristic
15C22×C22⋊D59C2212yes—
16C22×C22⋊D59C2211yescharacteristic, complemented
17C23×Dic59C2211yescharacteristic
18#14273699—24no—
19#14273697—24no—
20#14273695—24no—
21C2×Dic59⋊C4C2211yescharacteristic, complemented
22C2×Dic59⋊C4C2212yescomplemented
23C2×C22⋊Dic59C2211yescharacteristic
24C2×C22⋊Dic59C2211yescharacteristic, complemented
25C2×(C2×D59)⋊C4—24no—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×(C2×D59)⋊C4C2212yes—
30C2×(C2×D59)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D59)⋊C4C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C59⋊#95C2212yescomplemented
34C22×D4×C59C2211yescharacteristic
35C23×C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2211yescharacteristic
37C2×C22⋊C4×C59C2212yescomplemented
38#95×C59C2212yescomplemented
39C24×C59C2311yescharacteristic
40C24×C59C2311yescharacteristic
41C23×D59—22no—
42C23×D59—22no—
43C23×D59—42no—
44C23×D59—42no—
45C23×D59D414yescomplemented
46C23×D59C2311yescharacteristic
47C23×D59C2312yes—
48C2×D4×C59—22no—
49C2×D4×C59—22no—
50C2×D4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—42no—
53C22×C4×C59C2311yescharacteristic
54C22×C4×C59C2312yes—
55C22×C4×C59C2312yes—
56C2×C22⋊D59—22no—
57C2×C22⋊D59—22no—
58C2×C22⋊D59—22no—
59C2×C22⋊D59—24no—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—42no—
62C2×C22⋊D59—42no—
63C2×C22⋊D59—42no—
64C22×Dic59—42no—
65C22×Dic59D414yescomplemented
66C22×Dic59C2311yescharacteristic
67C22×Dic59C2312yes—
68C22×Dic59C2311yescharacteristic
69C22×Dic59C2311yescharacteristic
70C2×D236—24no—
71C2×D236—42no—
72C2×D236—42no—
73C22⋊C4×C59—22no—
74C22⋊C4×C59—42no—
75C22⋊Dic59—22no—
76C22⋊Dic59—22no—
77(C2×D59)⋊C4—24no—
78(C2×D59)⋊C4—24no—
79(C2×D59)⋊C4—24no—
80(C2×D59)⋊C4—24no—
81(C2×D59)⋊C4—22no—
82(C2×D59)⋊C4—42no—
83Dic59⋊C4—22no—
84Dic59⋊C4—42no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59C2×D412yes—
93C23×C59C2×D412yes—
94C23×C59#4112yes—
95C23×C59C2411yescharacteristic
96C22×D59—28no—
97C22×D59—22no—
98C22×D59—22no—
99C22×D59—22no—
100C22×D59—22no—
101C22×D59—22no—
102C22×D59—22no—
103C22×D59—21no—
104C22×D59—21no—
105C22×D59—21no—
106C22×D59—22no—
107C22×D59—22no—
108C22×D59—48no—
109C22×D59—44no—
110C22×D59—44no—
111C22×D59—48no—
112C22×D59C2×D414yes—
113D4×C59—44no—
114C2×C4×C59—21no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—44no—
124C22⋊D59—44no—
125C22⋊D59—48no—
126C22⋊D59—44no—
127C2×Dic59—22no—
128C2×Dic59—22no—
129C2×Dic59—21no—
130C2×Dic59—21no—
131C2×Dic59—22no—
132C2×Dic59—21no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—44no—
136C2×Dic59C2×D412yes—
137C2×Dic59C2×D412yes—
138D236—48no—
139C22×C59—22no—
140C22×C59—24no—
141C22×C59—22no—
142C22×C59—24no—
143C22×C59—41no—
144C22×C59—42no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—42no—
148C22×C59#12114yes—
149C22×C59C22×D411yescharacteristic
150C22×C59C22×D411yescharacteristic
151C22×C59C2×#4111yescharacteristic
152C22×C59H(5,𝔽2)12yes—
153C22×C59H(5,𝔽2)12yes—
154C2×D59—24no—
155C2×D59—24no—
156C2×D59—24no—
157C2×D59—416no—
158C2×D59—42no—
159C2×D59—42no—
160C2×D59—42no—
161C2×D59—42no—
162C2×D59—41no—
163C2×D59—42no—
164C2×D59—41no—
165C2×D59—41no—
166C2×D59—42no—
167C4×C59—41no—
168C4×C59—42no—
169C4×C59—42no—
170Dic59—22no—
171Dic59—22no—
172Dic59—41no—
173Dic59—42no—
174#2617—591nomaximal, Sylow
175C2×C59—24no—
176C2×C59—81no—
177C2×C59C2×#12111yescharacteristic
178C2×C59#53412yes—
179C2×C59#53411yescharacteristic
180C2×C59#53511yescharacteristic
181C2×C59#53812yes—
182D59—44no—
183D59—81no—
184C2×#121—591no—
185C2×#120—591no—
186C2×#120—591no—
187C22×C22⋊C4—591no—
188#397—592no—
189#394—592no—
190#394—592no—
191#392—592no—
192#386—591no—
193#386D5911yescharacteristic, complemented
194#380—591no—
195C59#261711yescharacteristic, complemented, Sylow
196C25—591no—
197C22×D4—591no—
198C22×D4—592no—
199C22×D4—591no—
200C22×D4—592no—
201C22×D4C2×D5911yescharacteristic
202C23×C4—591no—
203C23×C4C2×D5911yescharacteristic
204#121—1184no—
205#120—1184no—
206#120—1184no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—592no—
209C2×C22⋊C4—591no—
210C2×C22⋊C4—591no—
211C2×C22⋊C4—592no—
212C2×C22⋊C4—592no—
213C2×C22⋊C4—592no—
214C2×C22⋊C4—592no—
215C2×C22⋊C4—591no—
216C2×C22⋊C4—592no—
217C2×C22⋊C4—1184no—
218C2×C22⋊C4C2×D5911yescharacteristic
219C2×C22⋊C4C2×D5912yescomplemented
220#95—592no—
221#95—592no—
222#95C2×D5912yescomplemented
223C24—592no—
224C24—594no—
225C24—591no—
226C24—1182no—
227C24—1182no—
228C24—2362no—
229C24—2362no—
230C24C22×D5911yescharacteristic
231C24C22×D5911yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—1184no—
236C2×D4—1182no—
237C2×D4—1184no—
238C2×D4—1182no—
239C2×D4—1182no—
240C2×D4—1182no—
241C2×D4—2362no—
242C2×D4—2362no—
243C2×D4—2362no—
244C2×D4—2362no—
245C2×D4—2362no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—591no—
249C22×C4—591no—
250C22×C4—594no—
251C22×C4—591no—
252C22×C4—592no—
253C22×C4—2362no—
254C22×C4C22×D5912yes—
255C22×C4C22×D5912yes—
256C22×C4C22×D5911yescharacteristic
257C4⋊C4—1182no—
258C4⋊C4—2362no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—1184no—
262C22⋊C4—1184no—
263C22⋊C4—1182no—
264C22⋊C4—1184no—
265C22⋊C4—1184no—
266C22⋊C4—1182no—
267C22⋊C4—1182no—
268C22⋊C4—2362no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—594no—
277C23—1188no—
278C23—1182no—
279C23—1182no—
280C23—1182no—
281C23—1182no—
282C23—1182no—
283C23—1182no—
284C23—1181no—
285C23—1181no—
286C23—1181no—
287C23—1182no—
288C23—1182no—
289C23—2368no—
290C23—2364no—
291C23—2364no—
292C23—2368no—
293C23#1174568512yes—
294C23D4×D5912yes—
295C23D4×D5912yes—
296C23C23×D5911yescharacteristic
297D4—44no—
298D4—2368no—
299D4—2364no—
300D4—2368no—
301D4—2364no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—592no—
313C2×C4—592no—
314C2×C4—1182no—
315C2×C4—1182no—
316C2×C4—1181no—
317C2×C4—1182no—
318C2×C4—1181no—
319C2×C4—1181no—
320C2×C4—1182no—
321C2×C4—1182no—
322C2×C4—2364no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—41no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—1184no—
333C22—1184no—
334C22—1184no—
335C22—23616no—
336C22—2361no—
337C22—2362no—
338C22—2361no—
339C22—2361no—
340C22—2362no—
341C22—2362no—
342C22—2362no—
343C22—2362no—
344C22—2362no—
345C22#1427370014yes—
346C22C2×D4×D5911yescharacteristic
347C22C2×D4×D5911yescharacteristic
348C22C2×#1174568511yescharacteristic
349C22#1427377212yes—
350C22#1427377312yes—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—1182no—
355C4—1182no—
356C4—2361no—
357C4—2362no—
358C2—24no—
359C2—81no—
360C2—4721no—
361C2—2364no—
362C2C2×#1427370011yescharacteristic
363C2#1683796012yes—
364C2#1683796311yescharacteristic
365C2#1683796711yescharacteristic
366C2#1683798512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#386C2
hypercenter#386D59
hypocenterC59#2617
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14273699C2semidirect
2. C2×#14273697C2semidirect
3. C2×#14273695C2semidirect
4. C22×(C2×D59)⋊C4C2semidirect
5. #16837331C2semidirect
6. #16837320C2semidirect
7. #16837317C2semidirect
8. #16837311C2semidirect
9. #16837281C2semidirect
10. #16837260C2semidirect
11. C59×#386C2semidirect
16. C22×C22⋊D59C22semidirect
21. C2×Dic59⋊C4C22semidirect
22. C2×Dic59⋊C4C22semidirect
24. C2×C22⋊Dic59C22semidirect
30. C2×(C2×D59)⋊C4C22semidirect
31. C2×(C2×D59)⋊C4C22semidirect
32. C59⋊#95C22semidirect
33. C59⋊#95C22semidirect
37. C2×C22⋊C4×C59C22semidirect
38. #95×C59C22semidirect
45. C23×D59D4semidirect
65. C22×Dic59D4semidirect
193. #386D59semidirect
195. C59#2617semidirect
219. C2×C22⋊C4C2×D59semidirect
222. #95C2×D59semidirect