Finite Group Databasebeta

← #21430255

Subgroups — #21430255

Order 7552.

434 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14273721C211yesmaximal, characteristic
2C2×#14273719C211yesmaximal, characteristic, complemented
3#16837342C211yesmaximal, characteristic, complemented
4#16837339C211yesmaximal, characteristic
5#16837334C211yesmaximal, characteristic, complemented
6#16837332C212yesmaximal, complemented
7#16837329C211yesmaximal, characteristic
8#16837324C212yesmaximal
9#16837300C212yesmaximal, complemented
10#16837293C212yesmaximal
11C4×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12#14273721—24no—
13#14273719—24no—
14C2×C4⋊Dic59C2211yescharacteristic
15C2×C4⋊Dic59C2211yescharacteristic
16C2×C4⋊Dic59C2212yes—
17C2×C4⋊Dic59C2212yes—
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2212yes—
33C2×C42×C59C2212yes—
34C2×C42×C59C2211yescharacteristic
35#95×C59C2211yescharacteristic
36#95×C59C2211yescharacteristic
37C23×D59C2311yescharacteristic
38C22×C4×C59C2312yes—
39C22×C4×C59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2312yes—
43C22×Dic59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C4⋊C4×C59—24no—
49C42×C59—42no—
50C42×C59D414yes—
51(C2×D59)⋊C4—22no—
52(C2×D59)⋊C4—22no—
53(C2×D59)⋊C4—22no—
54(C2×D59)⋊C4—24no—
55(C2×D59)⋊C4—42no—
56C4⋊Dic59—22no—
57C4⋊Dic59—22no—
58C4⋊Dic59—24no—
59C4⋊Dic59—42no—
60Dic59⋊C4—24no—
61Dic59⋊C4—24no—
62Dic59⋊C4—22no—
63Dic59⋊C4—22no—
64Dic59⋊C4—22no—
65C23×C59C2411yescharacteristic
66C22×D59—21no—
67C22×D59—22no—
68C22×D59—21no—
69C22×D59—22no—
70C22×D59—21no—
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×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59C2×D412yes—
81C2×C4×C59C2×D412yes—
82C2×C4×C59C2×D412yes—
83C2×C4×C59#4114yes—
84C2×Dic59—22no—
85C2×Dic59—22no—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—21no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—21no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59—21no—
97C22×C59C22×D411yescharacteristic
98C22×C59C2×#4112yes—
99C22×C59H(5,𝔽2)11yescharacteristic
100C22×C592−1+412yes—
101C22×C592−1+411yescharacteristic
102C2×D59—41no—
103C2×D59—41no—
104C2×D59—41no—
105C2×D59—42no—
106C2×D59—42no—
107C4×C59—22no—
108C4×C59—22no—
109C4×C59—22no—
110C4×C59—24no—
111C4×C59—42no—
112Dic59—42no—
113Dic59—41no—
114Dic59—41no—
115Dic59—42no—
116Dic59—41no—
117#2664—591nomaximal, Sylow
118C2×C59#53211yescharacteristic
119C2×C59#53611yescharacteristic
120C2×C59#53711yescharacteristic
121C2×C59#53912yes—
122C2×C59#54112yes—
123D59—81no—
124C2×#126—591no—
125C2×#125—591no—
126#400—591no—
127#399—591no—
128#398—592no—
129#398—591no—
130#397—591no—
131#396—592no—
132#389—592no—
133#388—592no—
134C4×C4⋊C4D5911yescharacteristic, complemented
135C59#266411yescharacteristic, complemented, Sylow
136#126—1184no—
137#125—1184no—
138C2×C4⋊C4—592no—
139C2×C4⋊C4—592no—
140C2×C4⋊C4—591no—
141C2×C4⋊C4—591no—
142C2×C4⋊C4—591no—
143C2×C4⋊C4—591no—
144C2×C4⋊C4—591no—
145C2×C4⋊C4—592no—
146C2×C4⋊C4—592no—
147C2×C4⋊C4C2×D5912yes—
148C2×C22⋊C4—591no—
149C2×C22⋊C4—591no—
150C2×C22⋊C4—591no—
151C2×C22⋊C4—592no—
152C2×C22⋊C4—592no—
153C2×C42C2×D5912yes—
154C2×C42C2×D5911yescharacteristic
155#95—592no—
156#95—592no—
157#95—592no—
158#95—592no—
159#95C2×D5911yescharacteristic
160#95C2×D5911yescharacteristic
161C24—591no—
162C22×C4—591no—
163C22×C4—591no—
164C22×C4—591no—
165C22×C4—592no—
166C22×C4—592no—
167C22×C4C22×D5912yes—
168C22×C4C22×D5911yescharacteristic
169C22×C4C22×D5911yescharacteristic
170C22×C4C22×D5911yescharacteristic
171C22×C4C22×D5912yes—
172C4⋊C4—24no—
173C4⋊C4—1184no—
174C4⋊C4—1184no—
175C4⋊C4—1182no—
176C4⋊C4—1182no—
177C4⋊C4—1182no—
178C4⋊C4—1182no—
179C4⋊C4—1182no—
180C4⋊C4—1184no—
181C4⋊C4—2362no—
182C22⋊C4—1182no—
183C22⋊C4—1182no—
184C22⋊C4—1182no—
185C22⋊C4—1184no—
186C22⋊C4—2362no—
187C42—42no—
188C42D23614yes—
189C23—1181no—
190C23—1182no—
191C23—1181no—
192C23—1182no—
193C23—1181no—
194C23C23×D5911yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—1181no—
205C2×C4—1181no—
206C2×C4—1181no—
207C2×C4—1182no—
208C2×C4—1182no—
209C2×C4—1182no—
210C2×C4—1182no—
211C2×C4—1182no—
212C2×C4—1182no—
213C2×C4—1181no—
214C2×C4—1182no—
215C2×C4—1181no—
216C2×C4—1182no—
217C2×C4C2×D23612yes—
218C2×C4C2×D23612yes—
219C2×C4C2×D23612yes—
220C2×C4#1174568514yes—
221C22—2361no—
222C22—2361no—
223C22—2361no—
224C22—2362no—
225C22—2362no—
226C22C22×D23611yescharacteristic
227C22C2×#1174568512yes—
228C22#1427377211yescharacteristic
229C22#1427377412yes—
230C22#1427377511yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—24no—
235C4—42no—
236C4—2361no—
237C4—2362no—
238C4—2361no—
239C4—2361no—
240C4—2362no—
241C2—4721no—
242C2#1683795211yescharacteristic
243C2#1683797811yescharacteristic
244C2#1683798311yescharacteristic
245C2#1683799412yes—
246C2#1683801012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC4×C4⋊C4×C59C2
hypercenterC4×C4⋊C4D59
hypocenterC59#2664
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14273719C2semidirect
3. #16837342C2semidirect
5. #16837334C2semidirect
6. #16837332C2semidirect
9. #16837300C2semidirect
11. C4×C4⋊C4×C59C2semidirect
134. C4×C4⋊C4D59semidirect
135. C59#2664semidirect