Finite Group Databasebeta

← #21428408

Subgroups — #21428408

Order 7552.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14273689C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
3#16837341C211yesmaximal, characteristic, complemented
4#16837339C211yesmaximal, characteristic, complemented
5#16837337C211yesmaximal, characteristic, complemented
6#16837299C212yesmaximal, complemented
7#16837280C212yesmaximal
8#16837276C212yesmaximal, complemented
9#16837261C212yesmaximal, complemented
10#16837261C211yesmaximal, characteristic, complemented
11C59×#400C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C22×C4×D59C2212yes—
16#14273689—24no—
17C2×C4⋊Dic59C2211yescharacteristic
18C2×C4⋊Dic59C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C2×Dic59⋊C4C2212yes—
22C4⋊C4×D59—24no—
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×C4×Dic59C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34#95×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C23×D59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C2×D236—22no—
49C2×D236—22no—
50C2×D236Q814yes—
51C2×C4×D59—24no—
52C2×C4×D59—24no—
53C2×C4×D59—22no—
54C2×C4×D59—24no—
55C2×C4×D59—42no—
56C2×C4×D59—42no—
57C2×C4×D59D414yes—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—22no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—24no—
62(C2×D59)⋊C4—22no—
63(C2×D59)⋊C4—42no—
64(C2×D59)⋊C4—42no—
65C4⋊Dic59—22no—
66C4⋊Dic59—42no—
67Dic59⋊C4—24no—
68Dic59⋊C4—24no—
69Dic59⋊C4—42no—
70C4×Dic59—22no—
71C23×C59C2411yescharacteristic
72C22×D59—22no—
73C22×D59—22no—
74C22×D59—21no—
75C22×D59—22no—
76C22×D59—21no—
77C22×D59—22no—
78C22×D59C2×D412yes—
79C22×D59C2×Q812yes—
80C22×D59C2×Q812yes—
81C22×D59#4112yes—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59C2×D412yes—
94C2×C4×C59C2×Q812yes—
95C2×Dic59—21no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59C2×D412yes—
104C2×Dic59#4114yes—
105D236—28no—
106C4×D59—28no—
107C4×D59—48no—
108C4×D59—44no—
109C22×C59C22×D411yescharacteristic
110C22×C59C22×Q811yescharacteristic
111C22×C59C2×#4111yescharacteristic
112C22×C59C2×#4112yes—
113C22×C59H(5,𝔽2)12yes—
114C2×D59—22no—
115C2×D59—22no—
116C2×D59—24no—
117C2×D59—24no—
118C2×D59—22no—
119C2×D59—24no—
120C2×D59—41no—
121C2×D59—41no—
122C2×D59—42no—
123C2×D59C22⋊Q814yes—
124C4×C59—22no—
125C4×C59—22no—
126C4×C59—41no—
127C4×C59—42no—
128C4×C59—42no—
129Dic59—42no—
130Dic59—41no—
131Dic59—42no—
132Dic59#12314yes—
133#2487—591nomaximal, Sylow
134C2×C59C2×C22⋊Q811yescharacteristic
135C2×C59C2×#12311yescharacteristic
136C2×C59#53511yescharacteristic
137C2×C59#55312yes—
138C2×C59#55412yes—
139D59—24no—
140D59—42no—
141C2×C4×D4—591no—
142C22×C4⋊C4—591no—
143#400—591no—
144#400D5911yescharacteristic, complemented
145#399—591no—
146#399—591no—
147#389—592no—
148#386—592no—
149#385—592no—
150#380—592no—
151#380—591no—
152C59#248711yescharacteristic, complemented, Sylow
153C22×D4—591no—
154C23×C4—591no—
155C23×C4—591no—
156C23×C4—592no—
157C4×D4—1184no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—591no—
160C2×C4⋊C4—592no—
161C2×C4⋊C4—592no—
162C2×C4⋊C4—592no—
163C2×C4⋊C4—1184no—
164C2×C4⋊C4C2×D5911yescharacteristic
165C2×C4⋊C4C2×D5911yescharacteristic
166C2×C22⋊C4—591no—
167C2×C22⋊C4—591no—
168C2×C22⋊C4—592no—
169C2×C22⋊C4—592no—
170C2×C22⋊C4—592no—
171C2×C42—591no—
172#95—592no—
173#95—591no—
174#95—592no—
175#95C2×D5912yes—
176#95C2×D5911yescharacteristic
177#95C2×D5912yes—
178C24—591no—
179C24—591no—
180C2×D4—594no—
181C2×D4—1182no—
182C2×D4—1182no—
183C22×C4—591no—
184C22×C4—594no—
185C22×C4—592no—
186C22×C4—592no—
187C22×C4—591no—
188C22×C4—1184no—
189C22×C4—1182no—
190C22×C4—1184no—
191C22×C4—1184no—
192C22×C4—2362no—
193C22×C4—2362no—
194C22×C4C22×D5911yescharacteristic
195C22×C4C22×D5912yes—
196C22×C4C22×D5912yes—
197C22×C4C22×D5911yescharacteristic
198C22×C4C22×D5911yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—1184no—
202C4⋊C4—1184no—
203C4⋊C4—1182no—
204C4⋊C4—2362no—
205C4⋊C4—2362no—
206C22⋊C4—1182no—
207C22⋊C4—1184no—
208C22⋊C4—1182no—
209C22⋊C4—2362no—
210C22⋊C4—2362no—
211C42—1182no—
212C23—592no—
213C23—592no—
214C23—592no—
215C23—592no—
216C23—1182no—
217C23—1182no—
218C23—1181no—
219C23—1182no—
220C23—1181no—
221C23—1182no—
222C23C23×D5911yescharacteristic
223D4—1188no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—592no—
236C2×C4—594no—
237C2×C4—1188no—
238C2×C4—1181no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—1182no—
244C2×C4—1182no—
245C2×C4—1182no—
246C2×C4—2364no—
247C2×C4—2368no—
248C2×C4D4×D5912yes—
249C2×C4Q8×D5912yes—
250C22—594no—
251C22—1182no—
252C22—1182no—
253C22—1184no—
254C22—1184no—
255C22—1182no—
256C22—1184no—
257C22—2361no—
258C22—2361no—
259C22—2362no—
260C22C2×D4×D5911yescharacteristic
261C22C2×Q8×D5911yescharacteristic
262C22#41×D5911yescharacteristic
263C22#41×D5912yes—
264C22#1427377312yes—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—42no—
269C4—42no—
270C4—594no—
271C4—2362no—
272C4—2361no—
273C4—2362no—
274C2—1184no—
275C2—2362no—
276C2C22⋊Q8×D5911yescharacteristic
277C2#123×D5911yescharacteristic
278C2#1683797011yescharacteristic
279C2#1683810312yes—
280C2#1683810812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#400C2
hypercenter#400D59
hypocenterC59#2487
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D59C2semidirect
3. #16837341C2semidirect
4. #16837339C2semidirect
5. #16837337C2semidirect
6. #16837299C2semidirect
8. #16837276C2semidirect
9. #16837261C2semidirect
10. #16837261C2semidirect
11. C59×#400C2semidirect
144. #400D59semidirect
152. C59#2487semidirect