Finite Group Databasebeta

← #21432159

Subgroups — #21432159

Order 7552.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14273720C211yesmaximal, characteristic
3#16837342C211yesmaximal, characteristic, complemented
4#16837340C211yesmaximal, characteristic, complemented
5#16837338C211yesmaximal, characteristic
6#16837335C212yesmaximal, complemented
7#16837330C212yesmaximal
8#16837322C212yesmaximal, complemented
9#16837314C212yesmaximal
10#16837269C211yesmaximal, characteristic, complemented
11C59×#399C211yesmaximal, characteristic
12C22×Dic118C2212yes—
13#14273726—24no—
14#14273720—24no—
15C2×C4⋊Dic59C2211yescharacteristic
16C2×C4⋊Dic59C2211yescharacteristic
17C2×Dic59⋊C4C2211yescharacteristic
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×C4×Dic59C2211yescharacteristic
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C59⋊#95C2211yescharacteristic
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
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×Dic59C2312yes—
43C22×Dic59C2311yescharacteristic
44C22×Dic59C2311yescharacteristic
45C22×Dic59C2312yes—
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C2×Dic118—24no—
49C2×Dic118—42no—
50C2×Dic118—42no—
51C4⋊C4×C59—22no—
52C4⋊C4×C59—42no—
53C22⋊C4×C59—22no—
54C22⋊C4×C59—42no—
55C22⋊Dic59—22no—
56C22⋊Dic59—22no—
57C22⋊Dic59—42no—
58C4⋊Dic59—22no—
59C4⋊Dic59—22no—
60Dic59⋊C4—22no—
61Dic59⋊C4—22no—
62Dic59⋊C4—24no—
63Dic59⋊C4—42no—
64C4×Dic59D414yes—
65C23×C59—21no—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—21no—
69C23×C59—21no—
70C23×C59C2411yescharacteristic
71C2×C4×C59—22no—
72C2×C4×C59—21no—
73C2×C4×C59—22no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59C2×D412yes—
81C2×Dic59—22no—
82C2×Dic59—21no—
83C2×Dic59—21no—
84C2×Dic59—21no—
85C2×Dic59—21no—
86C2×Dic59—22no—
87C2×Dic59—22no—
88C2×Dic59—22no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—21no—
92C2×Dic59—22no—
93C2×Dic59—21no—
94C2×Dic59—22no—
95C2×Dic59C2×D412yes—
96C2×Dic59C2×D412yes—
97Dic118—48no—
98C22×C59—41no—
99C22×C59—41no—
100C22×C59—41no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59C22×D411yescharacteristic
104C22×C59H(5,𝔽2)12yes—
105C22×C592−1+412yes—
106C22×C592−1+411yescharacteristic
107C22×C592−1+411yescharacteristic
108C4×C59—22no—
109C4×C59—42no—
110C4×C59—41no—
111C4×C59—42no—
112Dic59—22no—
113Dic59—22no—
114Dic59—42no—
115Dic59—41no—
116Dic59—41no—
117Dic59—42no—
118#2821—591nomaximal, Sylow
119C2×C59—81no—
120C2×C59#53511yescharacteristic
121C2×C59#53611yescharacteristic
122C2×C59#53611yescharacteristic
123C2×C59#56312yes—
124C2×C59#56312yes—
125C2×C4⋊Q8—591no—
126C2×#126—591no—
127#400—591no—
128#400—591no—
129#399—591no—
130#399D5911yescharacteristic
131#398—592no—
132#397—592no—
133#395—592no—
134#393—592no—
135#383—591no—
136C59#282111yescharacteristic, complemented, Sylow
137C22×Q8—592no—
138C4⋊Q8—1184no—
139#126—1184no—
140C2×C4⋊C4—591no—
141C2×C4⋊C4—591no—
142C2×C4⋊C4—591no—
143C2×C4⋊C4—592no—
144C2×C4⋊C4—591no—
145C2×C4⋊C4—592no—
146C2×C4⋊C4C2×D5911yescharacteristic
147C2×C4⋊C4C2×D5912yes—
148C2×C22⋊C4—591no—
149C2×C22⋊C4—591no—
150C2×C22⋊C4—592no—
151C2×C22⋊C4C2×D5911yescharacteristic
152C2×C22⋊C4C2×D5912yes—
153C2×C42—591no—
154#95—592no—
155#95—592no—
156#95—591no—
157#95—592no—
158#95—592no—
159#95—592no—
160#95—592no—
161#95C2×D5911yescharacteristic
162C24C22×D5911yescharacteristic
163C2×Q8—1184no—
164C2×Q8—2362no—
165C2×Q8—2362no—
166C22×C4—591no—
167C22×C4—591no—
168C22×C4—592no—
169C22×C4—591no—
170C22×C4—591no—
171C22×C4—592no—
172C22×C4C22×D5912yes—
173C22×C4C22×D5912yes—
174C22×C4C22×D5911yescharacteristic
175C22×C4C22×D5911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—1182no—
179C4⋊C4—1182no—
180C4⋊C4—1184no—
181C4⋊C4—1182no—
182C4⋊C4—1182no—
183C4⋊C4—2362no—
184C22⋊C4—22no—
185C22⋊C4—42no—
186C22⋊C4—1182no—
187C22⋊C4—1182no—
188C22⋊C4—2362no—
189C42—594no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C23×D5911yescharacteristic
196Q8—2368no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—592no—
207C2×C4—592no—
208C2×C4—1182no—
209C2×C4—1181no—
210C2×C4—1181no—
211C2×C4—1181no—
212C2×C4—1181no—
213C2×C4—1182no—
214C2×C4—1182no—
215C2×C4—1182no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4—1181no—
220C2×C4—1182no—
221C2×C4—1181no—
222C2×C4D4×D5912yes—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22C2×D4×D5911yescharacteristic
229C22#1427377212yes—
230C22#1427377412yes—
231C22#1427377411yescharacteristic
232C22#1427377511yescharacteristic
233C4—22no—
234C4—42no—
235C4—42no—
236C4—41no—
237C4—1182no—
238C4—1182no—
239C4—2362no—
240C4—2362no—
241C4—2361no—
242C4—2361no—
243C2—81no—
244C2#1683796511yescharacteristic
245C2#1683797411yescharacteristic
246C2#1683797511yescharacteristic
247C2#1683815312yes—
248C2#1683815512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2821
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. #16837335C2semidirect
8. #16837322C2semidirect
10. #16837269C2semidirect
136. C59#2821semidirect