Finite Group Databasebeta

← #21429922

Subgroups — #21429922

Order 7552.

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

normal subgroups only

This group has 488 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×#14273726C211yesmaximal, characteristic, complemented
2C2×#14273683C211yesmaximal, characteristic
3#16837342C211yesmaximal, characteristic, complemented
4#16837340C211yesmaximal, characteristic, complemented
5#16837338C211yesmaximal, characteristic
6#16837322C212yesmaximal, complemented
7#16837314C212yesmaximal
8#16837297C212yesmaximal, complemented
9#16837266C211yesmaximal, characteristic, complemented
10#16837260C212yesmaximal
11C59×#399C211yesmaximal, characteristic
12C22×Dic118C2212yes—
13C23×Dic59C2211yescharacteristic
14#14273726—24no—
15#14273683—24no—
16C2×C4⋊Dic59C2211yescharacteristic
17C2×Dic59⋊C4C2212yes—
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2211yescharacteristic
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2212yes—
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×C4×Dic59C2211yescharacteristic
25C2×C4×Dic59C2211yescharacteristic
26C59⋊#95C2212yes—
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2212yes—
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C22×C4×C59C2312yes—
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×Dic59—22no—
43C22×Dic59—22no—
44C22×Dic59Q814yescomplemented
45C22×Dic59C2311yescharacteristic
46C22×Dic59C2312yes—
47C22×Dic59C2312yes—
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2311yescharacteristic
50C22×Dic59C2311yescharacteristic
51C2×Dic118—24no—
52C2×Dic118—42no—
53C2×Dic118—42no—
54C4⋊C4×C59—22no—
55C4⋊C4×C59—42no—
56C22⋊C4×C59—22no—
57C22⋊C4×C59—42no—
58C22⋊Dic59—22no—
59C22⋊Dic59—42no—
60C4⋊Dic59—22no—
61Dic59⋊C4—22no—
62Dic59⋊C4—22no—
63Dic59⋊C4—24no—
64Dic59⋊C4—24no—
65Dic59⋊C4—42no—
66C4×Dic59—22no—
67C4×Dic59D414yes—
68C23×C59—21no—
69C23×C59—21no—
70C23×C59—22no—
71C23×C59—22no—
72C23×C59C2×Q812yes—
73C23×C59C2411yescharacteristic
74C2×C4×C59—22no—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59C2×D412yes—
84C2×Dic59—21no—
85C2×Dic59—21no—
86C2×Dic59—22no—
87C2×Dic59—28no—
88C2×Dic59—21no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59C2×D412yes—
97C2×Dic59C2×D412yes—
98C2×Dic59C2×Q812yes—
99C2×Dic59C2×Q812yes—
100C2×Dic59#4112yes—
101Dic118—48no—
102C22×C59—22no—
103C22×C59—24no—
104C22×C59—41no—
105C22×C59—41no—
106C22×C59—42no—
107C22×C59C22×D411yescharacteristic
108C22×C59C22×Q811yescharacteristic
109C22×C59C2×#4111yescharacteristic
110C22×C59H(5,𝔽2)12yes—
111C22×C592−1+412yes—
112C4×C59—22no—
113C4×C59—42no—
114C4×C59—41no—
115C4×C59—42no—
116Dic59—22no—
117Dic59—22no—
118Dic59—42no—
119Dic59—41no—
120Dic59—42no—
121Dic59C22⋊Q814yes—
122#2628—591nomaximal, Sylow
123C2×C59—42no—
124C2×C59C2×C22⋊Q811yescharacteristic
125C2×C59#53511yescharacteristic
126C2×C59#53611yescharacteristic
127C2×C59#54212yes—
128C2×C59#54412yes—
129C2×C4⋊Q8—591no—
130C2×#117—591no—
131#400—591no—
132#400—591no—
133#399—591no—
134#399D5911yescharacteristic
135#395—592no—
136#393—592no—
137#389—592no—
138#382—591no—
139#380—592no—
140C59#262811yescharacteristic, complemented, Sylow
141C22×Q8—592no—
142C23×C4—591no—
143C4⋊Q8—1184no—
144#117—1184no—
145C2×C4⋊C4—591no—
146C2×C4⋊C4—591no—
147C2×C4⋊C4—592no—
148C2×C4⋊C4—591no—
149C2×C4⋊C4—592no—
150C2×C4⋊C4—592no—
151C2×C4⋊C4C2×D5911yescharacteristic
152C2×C4⋊C4C2×D5912yes—
153C2×C22⋊C4—591no—
154C2×C22⋊C4—592no—
155C2×C22⋊C4C2×D5911yescharacteristic
156C2×C22⋊C4C2×D5912yes—
157C2×C42—591no—
158C2×C42—591no—
159#95—592no—
160#95—592no—
161#95—592no—
162#95—591no—
163#95—592no—
164#95—592no—
165#95C2×D5911yescharacteristic
166C24C22×D5911yescharacteristic
167C2×Q8—1184no—
168C2×Q8—2362no—
169C2×Q8—2362no—
170C22×C4—591no—
171C22×C4—594no—
172C22×C4—591no—
173C22×C4—591no—
174C22×C4—591no—
175C22×C4—592no—
176C22×C4—592no—
177C22×C4—1182no—
178C22×C4—1182no—
179C22×C4C22×D5912yes—
180C22×C4C22×D5912yes—
181C22×C4C22×D5911yescharacteristic
182C22×C4C22×D5911yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—1182no—
186C4⋊C4—1184no—
187C4⋊C4—1184no—
188C4⋊C4—1182no—
189C4⋊C4—1182no—
190C4⋊C4—2362no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—1182no—
194C22⋊C4—2362no—
195C42—594no—
196C42—1182no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23Q8×D5912yes—
202C23C23×D5911yescharacteristic
203Q8—2368no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—592no—
214C2×C4—592no—
215C2×C4—592no—
216C2×C4—592no—
217C2×C4—592no—
218C2×C4—1188no—
219C2×C4—1181no—
220C2×C4—1181no—
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—1182no—
230C2×C4D4×D5912yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22C2×D4×D5911yescharacteristic
237C22C2×Q8×D5911yescharacteristic
238C22#41×D5911yescharacteristic
239C22#1427377212yes—
240C22#1427377412yes—
241C4—22no—
242C4—42no—
243C4—41no—
244C4—42no—
245C4—594no—
246C4—1182no—
247C4—1182no—
248C4—2362no—
249C4—2361no—
250C4—2362no—
251C2—42no—
252C2C22⋊Q8×D5911yescharacteristic
253C2#1683796511yescharacteristic
254C2#1683797411yescharacteristic
255C2#1683802012yes—
256C2#1683803112yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2628
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273726C2semidirect
3. #16837342C2semidirect
4. #16837340C2semidirect
6. #16837322C2semidirect
8. #16837297C2semidirect
9. #16837266C2semidirect
44. C22×Dic59Q8semidirect
140. C59#2628semidirect