Finite Group Databasebeta

← #21431024

Subgroups — #21431024

Order 7552.

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

normal subgroups only

This group has 504 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×#14273708C211yesmaximal, characteristic
2#16837335C212yesmaximal, complemented
3#16837330C212yesmaximal, complemented
4#16837330C212yesmaximal, complemented
5#16837305C211yesmaximal, characteristic
6#16837297C212yesmaximal, complemented
7#16837275C212yesmaximal, complemented
8#16837274C212yesmaximal, complemented
9C2×#123×C59C211yesmaximal, characteristic
10C23×Dic59C2211yescharacteristic
11#14273708—24no—
12C2×C4⋊Dic59C2211yescharacteristic
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×Dic59⋊C4C2212yes—
16C2×Dic59⋊C4C2212yes—
17C2×C22⋊Dic59C2212yes—
18C2×C22⋊Dic59C2211yescharacteristic
19C2×C22⋊Dic59C2212yes—
20C2×C22⋊Dic59C2212yes—
21C2×C4×Dic59C2211yescharacteristic
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C22×D4×C59C2211yescharacteristic
28C23×C4×C59C2211yescharacteristic
29#123×C59—24no—
30C2×C4⋊C4×C59C2212yes—
31C2×C22⋊C4×C59C2212yes—
32C2×C22⋊C4×C59C2211yescharacteristic
33C24×C59C2311yescharacteristic
34C24×C59C2311yescharacteristic
35C2×D4×C59—41no—
36C2×D4×C59—41no—
37C2×D4×C59D414yes—
38C22×C4×C59—22no—
39C22×C4×C59—41no—
40C22×C4×C59—41no—
41C22×C4×C59C2312yes—
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C22×Dic59—22no—
45C22×Dic59—41no—
46C22×Dic59—41no—
47C22×Dic59C2312yes—
48C22×Dic59C2312yes—
49C22×Dic59C2312yes—
50C22×Dic59C2312yes—
51C4⋊C4×C59—24no—
52C22⋊C4×C59—22no—
53C22⋊C4×C59—24no—
54C22⋊Dic59—22no—
55C22⋊Dic59—24no—
56C22⋊Dic59—24no—
57C22⋊Dic59—24no—
58C4⋊Dic59—41no—
59Dic59⋊C4—24no—
60Dic59⋊C4—24no—
61Dic59⋊C4—42no—
62Dic59⋊C4—42no—
63C4×Dic59—41no—
64C23×C59—22no—
65C23×C59—22no—
66C23×C59—21no—
67C23×C59—21no—
68C23×C59—21no—
69C23×C59—21no—
70C23×C59—22no—
71C23×C59—22no—
72C23×C59C2×D412yes—
73C23×C59C2×D412yes—
74C23×C59C2411yescharacteristic
75D4×C59—44no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—44no—
85C2×C4×C59C2×D412yes—
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—22no—
97C2×Dic59—44no—
98C2×Dic59#4114yes—
99C22×C59—22no—
100C22×C59—22no—
101C22×C59—24no—
102C22×C59—24no—
103C22×C59—42no—
104C22×C59—42no—
105C22×C59—41no—
106C22×C59—41no—
107C22×C59—41no—
108C22×C59—41no—
109C22×C59C22×D411yescharacteristic
110C22×C59C2×#4112yes—
111C22×C59H(5,𝔽2)11yescharacteristic
112C22×C59H(5,𝔽2)11yescharacteristic
113C22×C592−1+412yes—
114C4×C59—22no—
115C4×C59—42no—
116C4×C59—42no—
117Dic59—24no—
118Dic59—42no—
119Dic59—42no—
120Dic59—42no—
121#2724—591nomaximal, Sylow
122C2×C59—42no—
123C2×C59—42no—
124C2×C59#53411yescharacteristic
125C2×C59#53912yes—
126C2×C59#54712yes—
127C2×C59#55911yescharacteristic
128C2×C59#56311yescharacteristic
129C2×#123—591no—
130C2×#123D5911yescharacteristic
131#398—592no—
132#397—592no—
133#397—592no—
134#390—591no—
135#389—592no—
136#385—592no—
137#385—592no—
138C59#272411yescharacteristic, complemented, Sylow
139C22×D4C2×D5911yescharacteristic
140C23×C4—591no—
141C23×C4C2×D5911yescharacteristic
142#123—24no—
143#123—1184no—
144C2×C4⋊C4—592no—
145C2×C4⋊C4—592no—
146C2×C4⋊C4—592no—
147C2×C4⋊C4—592no—
148C2×C4⋊C4—591no—
149C2×C4⋊C4C2×D5912yes—
150C2×C22⋊C4—592no—
151C2×C22⋊C4—591no—
152C2×C22⋊C4—592no—
153C2×C22⋊C4—592no—
154C2×C22⋊C4C2×D5912yes—
155C2×C22⋊C4C2×D5911yescharacteristic
156C2×C42—591no—
157#95—592no—
158#95—592no—
159#95—592no—
160#95—592no—
161#95—592no—
162C24C22×D5911yescharacteristic
163C24C22×D5911yescharacteristic
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C22⋊D5914yes—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—592no—
171C22×C4—592no—
172C22×C4—592no—
173C22×C4—592no—
174C22×C4—1182no—
175C22×C4—2361no—
176C22×C4—2361no—
177C22×C4C22×D5912yes—
178C22×C4C22×D5912yes—
179C22×C4C22×D5911yescharacteristic
180C4⋊C4—24no—
181C4⋊C4—1184no—
182C4⋊C4—1184no—
183C4⋊C4—2362no—
184C4⋊C4—2362no—
185C4⋊C4—2361no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—1184no—
189C22⋊C4—1184no—
190C22⋊C4—1182no—
191C22⋊C4—1184no—
192C42—2361no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23C2×C22⋊D5912yes—
202C23C2×C22⋊D5912yes—
203C23C23×D5911yescharacteristic
204D4—44no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—594no—
215C2×C4—1182no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4—1182no—
220C2×C4—1182no—
221C2×C4—1182no—
222C2×C4—1182no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—1182no—
226C2×C4—2364no—
227C2×C4C2×C22⋊D5912yes—
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22C22×C22⋊D5911yescharacteristic
239C22#41×D5912yes—
240C22#1427377211yescharacteristic
241C22#1427377211yescharacteristic
242C22#1427377412yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—1184no—
247C4—2362no—
248C4—2362no—
249C4—2362no—
250C2—42no—
251C2—42no—
252C2#1683796111yescharacteristic
253C2#1683799512yes—
254C2#1683805112yes—
255C2#1683813611yescharacteristic
256C2#1683815511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2724
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #16837335C2semidirect
3. #16837330C2semidirect
4. #16837330C2semidirect
6. #16837297C2semidirect
7. #16837275C2semidirect
8. #16837274C2semidirect
138. C59#2724semidirect