Finite Group Databasebeta

← #21429910

Subgroups — #21429910

Order 7552.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14273707C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic
3#16837340C211yesmaximal, characteristic, complemented
4#16837336C212yesmaximal, complemented
5#16837334C212yesmaximal, complemented
6#16837323C211yesmaximal, characteristic, complemented
7#16837322C211yesmaximal, characteristic, complemented
8#16837276C212yesmaximal
9#16837264C212yesmaximal
10#16837261C211yesmaximal, characteristic
11C59×#400C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16#14273707—24no—
17C2×C4⋊Dic59C2212yes—
18C2×C4⋊Dic59C2211yescharacteristic
19C2×C4⋊Dic59C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×Dic59⋊C4C2212yes—
23C2×Dic59⋊C4C2212yes—
24C4⋊C4×D59—24no—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C59⋊#95C2211yescharacteristic
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34#95×C59C2211yescharacteristic
35#95×C59C2212yes—
36#95×C59C2212yes—
37C23×D59C2311yescharacteristic
38C22×C4×C59C2312yes—
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2311yescharacteristic
43C22×Dic59C2312yes—
44C22×Dic59C2312yes—
45C22×Dic59C2311yescharacteristic
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C2×C4×D59—22no—
49C2×C4×D59—22no—
50C2×C4×D59—22no—
51C2×C4×D59—42no—
52C2×C4×D59—42no—
53C2×C4×D59D414yes—
54C2×C4×D59Q814yescomplemented
55C2×Dic118—22no—
56C2×Dic118—42no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—22no—
59(C2×D59)⋊C4—24no—
60(C2×D59)⋊C4—24no—
61C4⋊Dic59—22no—
62C4⋊Dic59—24no—
63C4⋊Dic59—42no—
64Dic59⋊C4—22no—
65Dic59⋊C4—22no—
66Dic59⋊C4—42no—
67Dic59⋊C4—42no—
68C23×C59C2411yescharacteristic
69C22×D59—22no—
70C22×D59—22no—
71C22×D59C2×D412yes—
72C22×D59C2×Q812yes—
73C22×D59#4112yes—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—24no—
83C2×C4×C59—22no—
84C2×C4×C59C2×D412yes—
85C2×C4×C59C2×Q812yes—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—22no—
89C2×Dic59—21no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59C2×D412yes—
98C2×Dic59C2×Q812yes—
99C4×D59—28no—
100C4×D59—44no—
101C4×D59—44no—
102Dic118—44no—
103C22×C59C22×D411yescharacteristic
104C22×C59C22×Q811yescharacteristic
105C22×C59C2×#4111yescharacteristic
106C22×C59H(5,𝔽2)12yes—
107C22×C592−1+412yes—
108C2×D59—22no—
109C2×D59—22no—
110C2×D59—24no—
111C2×D59—24no—
112C2×D59C22⋊Q814yes—
113C4×C59—22no—
114C4×C59—22no—
115C4×C59—41no—
116C4×C59—42no—
117C4×C59—42no—
118Dic59—22no—
119Dic59—22no—
120Dic59—42no—
121Dic59—41no—
122Dic59—42no—
123#2627—591nomaximal, Sylow
124C2×C59C2×C22⋊Q811yescharacteristic
125C2×C59#53411yescharacteristic
126C2×C59#53611yescharacteristic
127C2×C59#53912yes—
128C2×C59#54412yes—
129D59—24no—
130C2×C22⋊Q8—591no—
131C22×C4⋊C4—591no—
132#400—591no—
133#400D5911yescharacteristic, complemented
134#398—592no—
135#398—592no—
136#395—591no—
137#395—591no—
138#385—592no—
139#381—592no—
140#380—591no—
141C59#262711yescharacteristic, complemented, Sylow
142C22×Q8—591no—
143C23×C4—591no—
144C23×C4—591no—
145C23×C4—591no—
146C22⋊Q8—1184no—
147C2×C4⋊C4—591no—
148C2×C4⋊C4—591no—
149C2×C4⋊C4—591no—
150C2×C4⋊C4—592no—
151C2×C4⋊C4—592no—
152C2×C4⋊C4—592no—
153C2×C4⋊C4—592no—
154C2×C4⋊C4—1184no—
155C2×C4⋊C4C2×D5911yescharacteristic
156C2×C4⋊C4C2×D5911yescharacteristic
157C2×C22⋊C4—592no—
158C2×C22⋊C4—592no—
159#95—591no—
160#95—592no—
161#95—592no—
162#95—592no—
163#95—592no—
164#95C2×D5911yescharacteristic
165#95C2×D5912yes—
166#95C2×D5912yes—
167C24—591no—
168C2×Q8—1182no—
169C2×Q8—2362no—
170C22×C4—591no—
171C22×C4—591no—
172C22×C4—594no—
173C22×C4—594no—
174C22×C4—592no—
175C22×C4—592no—
176C22×C4—591no—
177C22×C4—1182no—
178C22×C4—1182no—
179C22×C4—1182no—
180C22×C4—2362no—
181C22×C4—2362no—
182C22×C4C22×D5912yes—
183C22×C4C22×D5912yes—
184C22×C4C22×D5911yescharacteristic
185C22×C4C22×D5911yescharacteristic
186C22×C4C22×D5911yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—1182no—
190C4⋊C4—1182no—
191C4⋊C4—1184no—
192C4⋊C4—1182no—
193C4⋊C4—2362no—
194C4⋊C4—2362no—
195C4⋊C4—2362no—
196C22⋊C4—1184no—
197C22⋊C4—1184no—
198C23—592no—
199C23—592no—
200C23—592no—
201C23—1182no—
202C23—1182no—
203C23C23×D5911yescharacteristic
204Q8—2364no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—592no—
216C2×C4—592no—
217C2×C4—1188no—
218C2×C4—1181no—
219C2×C4—1181no—
220C2×C4—1182no—
221C2×C4—1182no—
222C2×C4—1181no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—1182no—
226C2×C4—1182no—
227C2×C4—1182no—
228C2×C4—1182no—
229C2×C4—2364no—
230C2×C4—2364no—
231C2×C4D4×D5912yes—
232C2×C4Q8×D5912yes—
233C22—594no—
234C22—1182no—
235C22—1182no—
236C22—1184no—
237C22—1184no—
238C22C2×D4×D5911yescharacteristic
239C22C2×Q8×D5911yescharacteristic
240C22#41×D5911yescharacteristic
241C22#1427377312yes—
242C22#1427377412yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—1182no—
249C4—1182no—
250C4—2361no—
251C4—2362no—
252C4—2362no—
253C2—1184no—
254C2C22⋊Q8×D5911yescharacteristic
255C2#1683796311yescharacteristic
256C2#1683797211yescharacteristic
257C2#1683799312yes—
258C2#1683803212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#400C2
hypercenter#400D59
hypocenterC59#2627
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16837340C2semidirect
4. #16837336C2semidirect
5. #16837334C2semidirect
6. #16837323C2semidirect
7. #16837322C2semidirect
11. C59×#400C2semidirect
54. C2×C4×D59Q8semidirect
133. #400D59semidirect
141. C59#2627semidirect