Finite Group Databasebeta

← #21430246

Subgroups — #21430246

Order 7552.

498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14273706C211yesmaximal, characteristic, complemented
2#16837341C211yesmaximal, characteristic, complemented
3#16837338C211yesmaximal, characteristic, complemented
4#16837330C212yesmaximal, complemented
5#16837322C211yesmaximal, characteristic, complemented
6#16837316C212yesmaximal
7#16837314C211yesmaximal, characteristic, complemented
8#16837297C212yesmaximal, complemented
9#16837296C212yesmaximal, complemented
10C4×C22⋊Dic59C211yesmaximal, characteristic, complemented
11C2×#123×C59C211yesmaximal, characteristic
12C22×Dic118C2211yescharacteristic
13#14273706—24no—
14C2×C4⋊Dic59C2211yescharacteristic
15C2×Dic59⋊C4C2212yes—
16C2×Dic59⋊C4C2212yes—
17C2×Dic59⋊C4C2212yes—
18C2×C22⋊Dic59C2212yes—
19C2×C22⋊Dic59C2212yes—
20C2×C22⋊Dic59C2212yes—
21C2×C22⋊Dic59C2212yes—
22C2×C4×Dic59C2212yes—
23C59⋊#95C2211yescharacteristic
24C59⋊#95C2211yescharacteristic
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29C22×D4×C59C2211yescharacteristic
30C23×C4×C59C2211yescharacteristic
31#123×C59—24no—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C24×C59C2311yescharacteristic
39C2×D4×C59—22no—
40C2×D4×C59—42no—
41C22×C4×C59—42no—
42C22×C4×C59D414yes—
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2311yescharacteristic
48C22×Dic59C2312yes—
49C22×Dic59C2312yes—
50C22×Dic59C2312yes—
51C22×Dic59C2312yes—
52C2×Dic118—22no—
53C2×Dic118—42no—
54C4⋊C4×C59—22no—
55C4⋊C4×C59—22no—
56C22⋊C4×C59—22no—
57C22⋊C4×C59—22no—
58C22⋊C4×C59—22no—
59C22⋊Dic59—24no—
60C22⋊Dic59—24no—
61C22⋊Dic59—24no—
62C22⋊Dic59—42no—
63C4⋊Dic59—22no—
64Dic59⋊C4—24no—
65Dic59⋊C4—24no—
66Dic59⋊C4—42no—
67C4×Dic59—42no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59—21no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—21no—
76C23×C59—21no—
77C23×C59C2×D412yes—
78C23×C59C2411yescharacteristic
79D4×C59—44no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—44no—
90C2×C4×C59C2×D412yes—
91C2×C4×C59C2×D412yes—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59#4114yes—
104Dic118—44no—
105C22×C59—22no—
106C22×C59—22no—
107C22×C59—22no—
108C22×C59—42no—
109C22×C59—42no—
110C22×C59—41no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59—42no—
114C22×C59—42no—
115C22×C59C22×D411yescharacteristic
116C22×C59C2×#4112yes—
117C22×C59H(5,𝔽2)12yes—
118C22×C592−1+411yescharacteristic
119C22×C592−1+411yescharacteristic
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—41no—
123C4×C59—41no—
124C4×C59—41no—
125Dic59—24no—
126Dic59—42no—
127Dic59—42no—
128Dic59—42no—
129#2663—591nomaximal, Sylow
130C2×C59—42no—
131C2×C59—81no—
132C2×C59#53211yescharacteristic
133C2×C59#53411yescharacteristic
134C2×C59#53611yescharacteristic
135C2×C59#53912yes—
136C2×C59#54212yes—
137C2×#123D5911yescharacteristic
138C2×C22⋊Q8—591no—
139#400—591no—
140#399—591no—
141#397—592no—
142#395—591no—
143#394—592no—
144#393—591no—
145#389—592no—
146#388—592no—
147C4×C22⋊C4—591no—
148C59#266311yescharacteristic, complemented, Sylow
149C22×Q8—591no—
150C22×D4C2×D5911yescharacteristic
151C23×C4C2×D5911yescharacteristic
152#123—24no—
153C22⋊Q8—1184no—
154C2×C4⋊C4—592no—
155C2×C4⋊C4—591no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4C2×D5911yescharacteristic
159C2×C4⋊C4C2×D5911yescharacteristic
160C2×C22⋊C4—592no—
161C2×C22⋊C4—592no—
162C2×C22⋊C4—592no—
163C2×C22⋊C4—592no—
164C2×C22⋊C4C2×D5911yescharacteristic
165C2×C22⋊C4C2×D5911yescharacteristic
166C2×C22⋊C4C2×D5911yescharacteristic
167C2×C42—592no—
168#95—591no—
169#95—591no—
170#95—592no—
171#95—592no—
172#95—592no—
173#95—592no—
174C24C22×D5911yescharacteristic
175C24C22×D5911yescharacteristic
176C2×Q8—1182no—
177C2×Q8—2362no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—42no—
181C22×C4—592no—
182C22×C4—592no—
183C22×C4—592no—
184C22×C4—592no—
185C22×C4C22⋊D5914yes—
186C22×C4C22×D5911yescharacteristic
187C22×C4C22×D5911yescharacteristic
188C22×C4C22×D5911yescharacteristic
189C22×C4C22×D5911yescharacteristic
190C22×C4C22×D5911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—1184no—
194C4⋊C4—1184no—
195C4⋊C4—1182no—
196C4⋊C4—2362no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—1184no—
201C22⋊C4—1184no—
202C22⋊C4—1184no—
203C22⋊C4—2362no—
204C42—2362no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23C2×C22⋊D5912yes—
215C23C23×D5911yescharacteristic
216Q8—2364no—
217D4—44no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—594no—
229C2×C4—1182no—
230C2×C4—1182no—
231C2×C4—1182no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—1182no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4C2×C22⋊D5912yes—
241C2×C4C2×C22⋊D5912yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22C22×C22⋊D5911yescharacteristic
253C22#41×D5912yes—
254C22#1427377212yes—
255C22#1427377411yescharacteristic
256C22#1427377411yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—1184no—
263C4—2362no—
264C4—2362no—
265C4—2362no—
266C2—42no—
267C2—81no—
268C2#1683794811yescharacteristic
269C2#1683796111yescharacteristic
270C2#1683797311yescharacteristic
271C2#1683799512yes—
272C2#1683802012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2663
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14273706C2semidirect
2. #16837341C2semidirect
3. #16837338C2semidirect
4. #16837330C2semidirect
5. #16837322C2semidirect
7. #16837314C2semidirect
8. #16837297C2semidirect
9. #16837296C2semidirect
10. C4×C22⋊Dic59C2semidirect
148. C59#2663semidirect