Finite Group Databasebeta

← #21429470

Subgroups — #21429470

Order 7552.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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#16837333C211yesmaximal, characteristic, complemented
4#16837323C211yesmaximal, characteristic, complemented
5#16837322C211yesmaximal, characteristic, complemented
6#16837301C212yesmaximal, complemented
7#16837299C212yesmaximal, complemented
8#16837276C212yesmaximal
9#16837273C212yesmaximal
10#16837249C211yesmaximal, characteristic
11C59×#398C211yesmaximal, 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⋊C4C2212yes—
21C2×Dic59⋊C4C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×Dic59⋊C4C2212yes—
24C4⋊C4×D59—24no—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×C4×Dic59C2212yes—
28C59⋊#95C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2212yes—
34C2×C4⋊C4×C59C2211yescharacteristic
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C23×D59C2311yescharacteristic
38C22×C4×C59C2311yescharacteristic
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×Dic59C2312yes—
44C22×Dic59C2311yescharacteristic
45C22×Dic59C2311yescharacteristic
46C22×Dic59C2312yes—
47C22×Dic59C2311yescharacteristic
48C2×C4×D59—22no—
49C2×C4×D59—22no—
50C2×C4×D59—22no—
51C2×C4×D59—22no—
52C2×C4×D59—42no—
53C2×C4×D59—42no—
54C2×C4×D59D414yes—
55C2×Dic118—22no—
56C2×Dic118—42no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—24no—
60(C2×D59)⋊C4—24no—
61(C2×D59)⋊C4Q818yescomplemented
62C4⋊Dic59—22no—
63C4⋊Dic59—24no—
64C4⋊Dic59—42no—
65Dic59⋊C4—22no—
66Dic59⋊C4—22no—
67Dic59⋊C4—24no—
68Dic59⋊C4—42no—
69C4×Dic59—42no—
70C23×C59C2411yescharacteristic
71C22×D59—21no—
72C22×D59—21no—
73C22×D59—22no—
74C22×D59C2×D412yes—
75C22×D59C2×Q814yes—
76C2×C4×C59—21no—
77C2×C4×C59—22no—
78C2×C4×C59—21no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59C2×D412yes—
87C2×C4×C59C2×Q814yes—
88C2×C4×C59#4112yes—
89C2×Dic59—21no—
90C2×Dic59—22no—
91C2×Dic59—21no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—21no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59C2×D412yes—
100C2×Dic59C2×Q814yes—
101C2×Dic59#4112yes—
102C4×D59—28no—
103C4×D59—44no—
104C4×D59—44no—
105Dic118—44no—
106C22×C59C22×D411yescharacteristic
107C22×C59C22×Q812yes—
108C22×C59C2×#4111yescharacteristic
109C22×C59C2×#4111yescharacteristic
110C22×C59H(5,𝔽2)12yes—
111C2×D59—22no—
112C2×D59—22no—
113C2×D59—24no—
114C2×D59—24no—
115C2×D59C4⋊Q814yes—
116C4×C59—22no—
117C4×C59—22no—
118C4×C59—24no—
119C4×C59—42no—
120C4×C59—41no—
121Dic59—22no—
122Dic59—22no—
123Dic59—24no—
124Dic59—42no—
125Dic59—41no—
126#2583—591nomaximal, Sylow
127C2×C59#52511yescharacteristic
128C2×C59C2×C4⋊Q811yescharacteristic
129C2×C59#53411yescharacteristic
130C2×C59#55412yes—
131C2×C59#55412yes—
132D59—24no—
133C2×C22⋊Q8—591no—
134C22×C4⋊C4—591no—
135#398—591no—
136#398D5911yescharacteristic, complemented
137#395—591no—
138#395—591no—
139#389—592no—
140#389—592no—
141#385—592no—
142#385—592no—
143C4×C22⋊C4—591no—
144C59#258311yescharacteristic, complemented, Sylow
145C22×Q8—591no—
146C23×C4—591no—
147C23×C4—591no—
148C23×C4—591no—
149C22⋊Q8—1184no—
150C2×C4⋊C4—591no—
151C2×C4⋊C4—591no—
152C2×C4⋊C4—591no—
153C2×C4⋊C4—592no—
154C2×C4⋊C4—592no—
155C2×C4⋊C4—592no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—1184no—
158C2×C4⋊C4C2×D5911yescharacteristic
159C2×C4⋊C4C2×D5911yescharacteristic
160C2×C4⋊C4C2×D5912yes—
161C2×C22⋊C4—592no—
162C2×C22⋊C4—592no—
163C2×C42—592no—
164#95—591no—
165#95—592no—
166#95—592no—
167#95—592no—
168#95C2×D5911yescharacteristic
169#95C2×D5912yes—
170C24—591no—
171C2×Q8—1182no—
172C2×Q8—2362no—
173C22×C4—592no—
174C22×C4—591no—
175C22×C4—591no—
176C22×C4—594no—
177C22×C4—592no—
178C22×C4—591no—
179C22×C4—1182no—
180C22×C4—1182no—
181C22×C4—1182no—
182C22×C4—1182no—
183C22×C4—2362no—
184C22×C4—2362no—
185C22×C4C22×D5912yes—
186C22×C4C22×D5911yescharacteristic
187C22×C4C22×D5911yescharacteristic
188C22×C4C22×D5911yescharacteristic
189C22×C4C22×D5912yes—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—1182no—
194C4⋊C4—1182no—
195C4⋊C4—1184no—
196C4⋊C4—1184no—
197C4⋊C4—1182no—
198C4⋊C4—2362no—
199C4⋊C4—2362no—
200C22⋊C4—598no—
201C22⋊C4—1184no—
202C42—2362no—
203C23—592no—
204C23—594no—
205C23—1181no—
206C23—1181no—
207C23—1182no—
208C23C23×D5911yescharacteristic
209Q8—2364no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—592no—
221C2×C4—594no—
222C2×C4—592no—
223C2×C4—1188no—
224C2×C4—1181no—
225C2×C4—1182no—
226C2×C4—1181no—
227C2×C4—1182no—
228C2×C4—1182no—
229C2×C4—1182no—
230C2×C4—1181no—
231C2×C4—1182no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—2364no—
235C2×C4—2364no—
236C2×C4D4×D5912yes—
237C2×C4#1174568712yes—
238C2×C4Q8×D5914yes—
239C22—594no—
240C22—1182no—
241C22—1182no—
242C22—1184no—
243C22—1184no—
244C22C2×D4×D5911yescharacteristic
245C22C2×Q8×D5912yes—
246C22#41×D5911yescharacteristic
247C22C2×#1174568711yescharacteristic
248C22#1427377312yes—
249C4—22no—
250C4—22no—
251C4—24no—
252C4—42no—
253C4—41no—
254C4—1182no—
255C4—1182no—
256C4—1184no—
257C4—2361no—
258C4—2362no—
259C2—1184no—
260C2#1683791211yescharacteristic
261C2C4⋊Q8×D5911yescharacteristic
262C2#1683796311yescharacteristic
263C2#1683810812yes—
264C2#1683811012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#398C2
hypercenter#398D59
hypocenterC59#2583
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #16837333C2semidirect
4. #16837323C2semidirect
5. #16837322C2semidirect
6. #16837301C2semidirect
7. #16837299C2semidirect
11. C59×#398C2semidirect
61. (C2×D59)⋊C4Q8semidirect
136. #398D59semidirect
144. C59#2583semidirect