Finite Group Databasebeta

← #21430004

Subgroups — #21430004

Order 7552.

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

normal subgroups only

This group has 450 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×C4×Dic118C211yesmaximal, characteristic, complemented
2#16837335C212yesmaximal, complemented
3#16837332C211yesmaximal, characteristic, complemented
4#16837327C211yesmaximal, characteristic
5#16837325C212yesmaximal
6#16837322C211yesmaximal, characteristic, complemented
7#16837314C211yesmaximal, characteristic
8#16837290C212yesmaximal
9#16837283C212yesmaximal, complemented
10#16837268C211yesmaximal, characteristic, complemented
11C2×#126×C59C211yesmaximal, characteristic
12C22×Dic118C2211yescharacteristic
13C4×Dic118—24no—
14C2×C4⋊Dic59C2211yescharacteristic
15C2×C4⋊Dic59C2212yes—
16C2×Dic59⋊C4C2212yes—
17C2×Dic59⋊C4C2212yes—
18C2×Dic59⋊C4C2212yes—
19C2×C22⋊Dic59C2212yes—
20C2×C22⋊Dic59C2212yes—
21C2×C4×Dic59C2212yes—
22C59⋊#95C2212yes—
23C59⋊#95C2211yescharacteristic
24C59⋊#95C2211yescharacteristic
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29#126×C59—24no—
30C2×C4⋊C4×C59C2211yescharacteristic
31C2×C4⋊C4×C59C2211yescharacteristic
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2211yescharacteristic
36C2×C42×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C22×C4×C59C2311yescharacteristic
39C22×C4×C59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2312yes—
47C22×Dic59C2312yes—
48C2×Dic118—22no—
49C2×Dic118—24no—
50C4⋊C4×C59—22no—
51C4⋊C4×C59—22no—
52C4⋊C4×C59—22no—
53C22⋊C4×C59—22no—
54C22⋊C4×C59—22no—
55C22⋊C4×C59—22no—
56C42×C59D414yes—
57C22⋊Dic59—24no—
58C22⋊Dic59—42no—
59C4⋊Dic59—22no—
60C4⋊Dic59—42no—
61Dic59⋊C4—24no—
62Dic59⋊C4—24no—
63Dic59⋊C4—42no—
64C4×Dic59—24no—
65C23×C59—21no—
66C23×C59—22no—
67C23×C59—21no—
68C23×C59—22no—
69C23×C59—21no—
70C23×C59C2411yescharacteristic
71C2×C4×C59—22no—
72C2×C4×C59—21no—
73C2×C4×C59—22no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59—21no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
82C2×C4×C59C2×D412yes—
83C2×C4×C59C2×D412yes—
84C2×C4×C59C2×D412yes—
85C2×Dic59—22no—
86C2×Dic59—22no—
87C2×Dic59—22no—
88C2×Dic59—22no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59#4114yes—
97Dic118—28no—
98C22×C59—41no—
99C22×C59—41no—
100C22×C59—41no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59C22×D411yescharacteristic
104C22×C59C2×#4112yes—
105C22×C59H(5,𝔽2)11yescharacteristic
106C22×C592−1+412yes—
107C22×C592−1+411yescharacteristic
108C4×C59—22no—
109C4×C59—22no—
110C4×C59—22no—
111C4×C59—41no—
112C4×C59—41no—
113C4×C59—41no—
114Dic59—24no—
115Dic59—42no—
116Dic59—42no—
117Dic59—42no—
118#2637—591nomaximal, Sylow
119C2×C59—81no—
120C2×C59#52511yescharacteristic
121C2×C59#53611yescharacteristic
122C2×C59#53711yescharacteristic
123C2×C59#54112yes—
124C2×C59#54212yes—
125C2×#126D5911yescharacteristic
126C2×C4×Q8—591no—
127#398—591no—
128#398—592no—
129#397—591no—
130#396—592no—
131#395—591no—
132#393—591no—
133#388—592no—
134#387—592no—
135#382—591no—
136C59#263711yescharacteristic, complemented, Sylow
137C22×Q8—591no—
138#126—24no—
139C4×Q8—1184no—
140C2×C4⋊C4—592no—
141C2×C4⋊C4—591no—
142C2×C4⋊C4—592no—
143C2×C4⋊C4—592no—
144C2×C4⋊C4—592no—
145C2×C4⋊C4C2×D5911yescharacteristic
146C2×C4⋊C4C2×D5911yescharacteristic
147C2×C4⋊C4C2×D5911yescharacteristic
148C2×C22⋊C4—592no—
149C2×C22⋊C4—592no—
150C2×C22⋊C4C2×D5911yescharacteristic
151C2×C22⋊C4C2×D5911yescharacteristic
152C2×C22⋊C4C2×D5911yescharacteristic
153C2×C42—592no—
154C2×C42C2×D5911yescharacteristic
155#95—592no—
156#95—591no—
157#95—591no—
158#95—592no—
159#95—592no—
160#95—592no—
161#95—592no—
162C24C22×D5911yescharacteristic
163C2×Q8—1184no—
164C2×Q8—1182no—
165C22×C4—592no—
166C22×C4—592no—
167C22×C4—592no—
168C22×C4—592no—
169C22×C4C22×D5911yescharacteristic
170C22×C4C22×D5911yescharacteristic
171C22×C4C22×D5911yescharacteristic
172C22×C4C22×D5911yescharacteristic
173C22×C4C22×D5911yescharacteristic
174C22×C4C22×D5911yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—1184no—
179C4⋊C4—1184no—
180C4⋊C4—1182no—
181C4⋊C4—2362no—
182C4⋊C4—2362no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—1184no—
187C22⋊C4—2362no—
188C42—1184no—
189C42C22⋊D5914yes—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23C23×D5911yescharacteristic
196Q8—1188no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—594no—
209C2×C4—1182no—
210C2×C4—1182no—
211C2×C4—1182no—
212C2×C4—1182no—
213C2×C4—1182no—
214C2×C4—1182no—
215C2×C4—1182no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4—1182no—
220C2×C4C2×C22⋊D5912yes—
221C2×C4C2×C22⋊D5912yes—
222C2×C4C2×C22⋊D5912yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C22×C22⋊D5911yescharacteristic
229C22#41×D5912yes—
230C22#1427377311yescharacteristic
231C22#1427377412yes—
232C22#1427377411yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—1184no—
240C4—2362no—
241C4—2362no—
242C4—2362no—
243C2—81no—
244C2#1683790911yescharacteristic
245C2#1683797311yescharacteristic
246C2#1683798011yescharacteristic
247C2#1683800912yes—
248C2#1683801812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#126×C59C2
hypercenterC2×#126D59
hypocenterC59#2637
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic118C2semidirect
2. #16837335C2semidirect
3. #16837332C2semidirect
6. #16837322C2semidirect
9. #16837283C2semidirect
10. #16837268C2semidirect
136. C59#2637semidirect