Finite Group Databasebeta

← #21431797

Subgroups — #21431797

Order 7552.

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

normal subgroups only

This group has 472 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
1#16837341C211yesmaximal, characteristic, complemented
2#16837325C212yesmaximal, complemented
3#16837322C211yesmaximal, characteristic, complemented
4#16837314C212yesmaximal, complemented
5#16837286C212yesmaximal, complemented
6#16837282C212yesmaximal
7#16837274C212yesmaximal, complemented
8#16837260C212yesmaximal, complemented
9C59×#390C211yesmaximal, characteristic
10C22×Dic118C2211yescharacteristic
11C23×Dic59C2212yes—
12C2×C4⋊Dic59C2211yescharacteristic
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×Dic59⋊C4C2212yes—
16C2×C22⋊Dic59C2212yes—
17C2×C22⋊Dic59C2212yes—
18C2×C22⋊Dic59C2212yes—
19C59⋊#95C2211yescharacteristic
20C59⋊#95C2212yes—
21C59⋊#95C2211yescharacteristic
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C22×D4×C59C2211yescharacteristic
27C2×C4⋊C4×C59C2211yescharacteristic
28C2×C22⋊C4×C59C2212yes—
29C2×C22⋊C4×C59C2212yes—
30C2×C42×C59C2211yescharacteristic
31C24×C59C2312yes—
32C2×D4×C59—41no—
33C2×D4×C59—41no—
34C2×D4×C59Q814yescomplemented
35C22×C4×C59C2312yes—
36C22×C4×C59C2311yescharacteristic
37C22×C4×C59C2311yescharacteristic
38C22×C4×C59C2311yescharacteristic
39C22×Dic59—24no—
40C22×Dic59—42no—
41C22×Dic59—42no—
42C22×Dic59C2312yes—
43C22×Dic59C2312yes—
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C2×Dic118—42no—
47C2×Dic118—41no—
48C4⋊C4×C59—41no—
49C22⋊C4×C59—42no—
50C22⋊C4×C59—42no—
51C42×C59—41no—
52C22⋊Dic59—24no—
53C22⋊Dic59—42no—
54C22⋊Dic59—42no—
55C4⋊Dic59—41no—
56Dic59⋊C4—24no—
57Dic59⋊C4—42no—
58Dic59⋊C4—42no—
59C23×C59—22no—
60C23×C59—22no—
61C23×C59—22no—
62C23×C59—22no—
63C23×C59—22no—
64C23×C59—22no—
65C23×C59C2×Q814yes—
66C23×C59C2411yescharacteristic
67D4×C59—44no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—22no—
71C2×C4×C59—22no—
72C2×C4×C59—22no—
73C2×C4×C59—21no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—21no—
77C2×C4×C59C2×Q812yes—
78C2×Dic59—22no—
79C2×Dic59—22no—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—22no—
83C2×Dic59—22no—
84C2×Dic59—22no—
85C2×Dic59—22no—
86C2×Dic59—22no—
87C2×Dic59—22no—
88C2×Dic59—22no—
89C2×Dic59—48no—
90C2×Dic59#4114yes—
91Dic118—44no—
92C22×C59—24no—
93C22×C59—24no—
94C22×C59—24no—
95C22×C59—42no—
96C22×C59—42no—
97C22×C59—42no—
98C22×C59—42no—
99C22×C59C22×Q811yescharacteristic
100C22×C59C2×#4112yes—
101C22×C59H(5,𝔽2)11yescharacteristic
102C22×C59H(5,𝔽2)12yes—
103C22×C59H(5,𝔽2)11yescharacteristic
104C4×C59—22no—
105C4×C59—42no—
106C4×C59—41no—
107C4×C59—41no—
108Dic59—24no—
109Dic59—42no—
110Dic59—42no—
111Dic59—42no—
112#2784—591nomaximal, Sylow
113C2×C59—44no—
114C2×C59#53812yes—
115C2×C59#54311yescharacteristic
116C2×C59#55412yes—
117C2×C59#55511yescharacteristic
118C2×C59#56011yescharacteristic
119#400—591no—
120#396—592no—
121#395—591no—
122#393—592no—
123#390D5911yescharacteristic
124#387—592no—
125#386—592no—
126#385—592no—
127#380—592no—
128C59#278411yescharacteristic, complemented, Sylow
129C22×Q8—591no—
130C22×D4C2×D5911yescharacteristic
131C23×C4—592no—
132C2×C4⋊C4—592no—
133C2×C4⋊C4—592no—
134C2×C4⋊C4—592no—
135C2×C4⋊C4—591no—
136C2×C4⋊C4C2×D5911yescharacteristic
137C2×C22⋊C4—592no—
138C2×C22⋊C4—592no—
139C2×C22⋊C4—592no—
140C2×C22⋊C4C2×D5912yes—
141C2×C22⋊C4C2×D5912yes—
142C2×C42C2×D5911yescharacteristic
143#95—591no—
144#95—592no—
145#95—591no—
146#95—592no—
147#95—592no—
148#95—592no—
149#95—592no—
150C24C22×D5912yes—
151C2×Q8—2362no—
152C2×Q8—2361no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic11814yescomplemented
156C22×C4—592no—
157C22×C4—592no—
158C22×C4—592no—
159C22×C4—592no—
160C22×C4—1184no—
161C22×C4—2362no—
162C22×C4—2362no—
163C22×C4C22×D5911yescharacteristic
164C22×C4C22×D5912yes—
165C22×C4C22×D5911yescharacteristic
166C22×C4C22×D5911yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—1184no—
169C4⋊C4—2362no—
170C4⋊C4—2362no—
171C4⋊C4—2361no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—1184no—
175C22⋊C4—2362no—
176C22⋊C4—2362no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic11814yes—
185C23C23×D5911yescharacteristic
186Q8—2364no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—594no—
198C2×C4—1182no—
199C2×C4—1182no—
200C2×C4—1182no—
201C2×C4—1182no—
202C2×C4—1182no—
203C2×C4—1182no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4—1182no—
207C2×C4—1182no—
208C2×C4—1182no—
209C2×C4—2368no—
210C2×C4C2×Dic11812yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic11811yescharacteristic
219C22#41×D5912yes—
220C22#1427377211yescharacteristic
221C22#1427377212yes—
222C22#1427377211yescharacteristic
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—1184no—
228C4—2362no—
229C4—2362no—
230C4—2362no—
231C2—44no—
232C2#1683798412yes—
233C2#1683802811yescharacteristic
234C2#1683811112yes—
235C2#1683811411yescharacteristic
236C2#1683813911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#390C2
hypercenter#390D59
hypocenterC59#2784
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16837341C2semidirect
2. #16837325C2semidirect
3. #16837322C2semidirect
4. #16837314C2semidirect
5. #16837286C2semidirect
7. #16837274C2semidirect
8. #16837260C2semidirect
34. C2×D4×C59Q8semidirect
128. C59#2784semidirect
155. C2×D4Dic118semidirect