Finite Group Databasebeta

← #21432039

Subgroups — #21432039

Order 7552.

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

normal subgroups only

This group has 480 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#16837344C212yesmaximal, complemented
2#16837314C212yesmaximal, complemented
3#16837314C212yesmaximal, complemented
4#16837306C212yesmaximal, complemented
5#16837305C211yesmaximal, characteristic
6#16837282C211yesmaximal, characteristic
7#16837275C212yesmaximal, complemented
8#16837274C212yesmaximal, complemented
9C59×#386C211yesmaximal, characteristic
10C22×Dic118C2212yes—
11C23×Dic59C2211yescharacteristic
12C2×C4⋊Dic59C2211yescharacteristic
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×C22⋊Dic59C2212yes—
16C2×C22⋊Dic59C2212yes—
17C2×C22⋊Dic59C2212yes—
18C2×C22⋊Dic59C2211yescharacteristic
19C2×C4×Dic59C2211yescharacteristic
20C59⋊#95C2212yes—
21C59⋊#95C2212yes—
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C22×D4×C59C2211yescharacteristic
27C23×C4×C59C2211yescharacteristic
28C2×C22⋊C4×C59C2212yes—
29C2×C22⋊C4×C59C2211yescharacteristic
30#95×C59C2212yes—
31C24×C59C2311yescharacteristic
32C24×C59C2311yescharacteristic
33C2×D4×C59—41no—
34C2×D4×C59—41no—
35C2×D4×C59Q814yescomplemented
36C22×C4×C59—41no—
37C22×C4×C59—42no—
38C22×C4×C59C2312yes—
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×Dic59—41no—
42C22×Dic59—42no—
43C22×Dic59C2312yes—
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2312yes—
47C2×Dic118—24no—
48C2×Dic118—42no—
49C2×Dic118—42no—
50C22⋊C4×C59—41no—
51C22⋊C4×C59—42no—
52C22⋊Dic59—24no—
53C22⋊Dic59—24no—
54C22⋊Dic59—41no—
55C22⋊Dic59—42no—
56C4⋊Dic59—41no—
57Dic59⋊C4—42no—
58Dic59⋊C4—42no—
59C4×Dic59—41no—
60C23×C59—22no—
61C23×C59—22no—
62C23×C59—22no—
63C23×C59—21no—
64C23×C59—22no—
65C23×C59—21no—
66C23×C59—21no—
67C23×C59—21no—
68C23×C59C2×Q812yes—
69C23×C59C2×Q812yes—
70C23×C59C2411yescharacteristic
71D4×C59—44no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
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×C59—44no—
81C2×C4×C59C2×Q812yes—
82C2×Dic59—22no—
83C2×Dic59—22no—
84C2×Dic59—22no—
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—44no—
94C2×Dic59#4114yes—
95Dic118—48no—
96C22×C59—22no—
97C22×C59—22no—
98C22×C59—24no—
99C22×C59—24no—
100C22×C59—42no—
101C22×C59—42no—
102C22×C59—41no—
103C22×C59—41no—
104C22×C59—41no—
105C22×C59—41no—
106C22×C59C22×Q811yescharacteristic
107C22×C59C2×#4112yes—
108C22×C59H(5,𝔽2)11yescharacteristic
109C22×C59H(5,𝔽2)12yes—
110C22×C59H(5,𝔽2)11yescharacteristic
111C4×C59—22no—
112C4×C59—42no—
113C4×C59—42no—
114Dic59—24no—
115Dic59—42no—
116Dic59—42no—
117Dic59—42no—
118#2804—591nomaximal, Sylow
119C2×C59—42no—
120C2×C59—42no—
121C2×C59#53812yes—
122C2×C59#54311yescharacteristic
123C2×C59#55412yes—
124C2×C59#55911yescharacteristic
125C2×C59H(3,𝔽4)11yescharacteristic
126#400—592no—
127#393—592no—
128#393—592no—
129#391—592no—
130#390—591no—
131#386—591no—
132#386D5911yescharacteristic
133#385—592no—
134#385—592no—
135C59#280411yescharacteristic, complemented, Sylow
136C22×Q8—592no—
137C22×D4C2×D5911yescharacteristic
138C23×C4—591no—
139C23×C4C2×D5911yescharacteristic
140C2×C4⋊C4—592no—
141C2×C4⋊C4—592no—
142C2×C4⋊C4—591no—
143C2×C22⋊C4—592no—
144C2×C22⋊C4—592no—
145C2×C22⋊C4—592no—
146C2×C22⋊C4—591no—
147C2×C22⋊C4C2×D5912yes—
148C2×C22⋊C4C2×D5911yescharacteristic
149C2×C42—591no—
150#95—592no—
151#95—592no—
152#95—592no—
153#95—592no—
154#95—592no—
155#95—592no—
156#95C2×D5912yes—
157C24C22×D5911yescharacteristic
158C24C22×D5911yescharacteristic
159C2×Q8—1184no—
160C2×Q8—2362no—
161C2×Q8—2362no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic11814yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—592no—
168C22×C4—592no—
169C22×C4—592no—
170C22×C4—592no—
171C22×C4—2362no—
172C22×C4—2361no—
173C22×C4C22×D5912yes—
174C22×C4C22×D5912yes—
175C22×C4C22×D5911yescharacteristic
176C4⋊C4—2362no—
177C4⋊C4—2362no—
178C4⋊C4—2361no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—1184no—
182C22⋊C4—1184no—
183C22⋊C4—2362no—
184C22⋊C4—2361no—
185C42—2361no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×Dic11812yes—
195C23C2×Dic11812yes—
196C23C23×D5911yescharacteristic
197Q8—2368no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
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×C4—2364no—
221C2×C4C2×Dic11812yes—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—24no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22C22×Dic11811yescharacteristic
233C22#41×D5912yes—
234C22#1427377212yes—
235C22#1427377211yescharacteristic
236C22#1427377211yescharacteristic
237C4—22no—
238C4—42no—
239C4—42no—
240C4—1184no—
241C4—2362no—
242C4—2362no—
243C4—2362no—
244C2—42no—
245C2—42no—
246C2#1683798412yes—
247C2#1683802811yescharacteristic
248C2#1683811112yes—
249C2#1683813611yescharacteristic
250C2#1683814511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#386C2
hypercenter#386D59
hypocenterC59#2804
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #16837344C2semidirect
2. #16837314C2semidirect
3. #16837314C2semidirect
4. #16837306C2semidirect
7. #16837275C2semidirect
8. #16837274C2semidirect
35. C2×D4×C59Q8semidirect
135. C59#2804semidirect
164. C2×D4Dic118semidirect