Finite Group Databasebeta

← #21435022

Subgroups — #21435022

Order 7552.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#16837953C211yesmaximal, characteristic
2C4⋊Q8×D59C211yesmaximal, characteristic, complemented
3#16837776C211yesmaximal, characteristic, complemented
4#16837775C211yesmaximal, characteristic, complemented
5#16837754C211yesmaximal, characteristic, complemented
6#16837606C214yesmaximal, complemented
7#16837542C214yesmaximal, complemented
8#16837453C211yesmaximal, characteristic, complemented
9C59×#487C211yesmaximal, characteristic, complemented
10C2×#11745689—21no—
11C2×Q8×D59C2212yes—
12C2×#11745660C2214yes—
13C2×#11745671C2214yes—
14#14273727C2211yescharacteristic
15#14273726—21no—
16#14273726—21no—
17#14273725C2212yes—
18#14273724C2211yescharacteristic
19#14273716—21no—
20#14273702—21no—
21#14273702C2212yes—
22#14273689—21no—
23#14273689C2212yes—
24C4⋊C4×D59—21no—
25C4⋊C4×D59—21no—
26C42×D59C2211yescharacteristic
27Dic59⋊C8C2212yes—
28#14273650C2214yes—
29(C2×D59)⋊C8C2212yes—
30C4×C59⋊C8C2211yescharacteristic
31C4⋊Q8×C59C2211yescharacteristic
32C4⋊Q8×C59C2211yescharacteristic
33#103×C59C2214yes—
34C4×C8×C59C2211yescharacteristic
35C2×Q8×C59—21no—
36C2×Q8×C59C2312yes—
37#11745689—44no—
38Q8×D59—44no—
39C2×D236—22no—
40C2×D236—21no—
41C2×D236—21no—
42C2×D236C2312yes—
43C2×C4×D59—22no—
44C2×C4×D59—22no—
45C2×C4×D59D411yescharacteristic
46C2×C4×D59D411yescharacteristic
47C2×C4×D59C2311yescharacteristic
48C2×Dic118—22no—
49C2×Dic118C2312yes—
50C2×C8×C59C2312yes—
51C4⋊C4×C59—21no—
52C4⋊C4×C59—21no—
53C4⋊C4×C59—21no—
54C4⋊C4×C59C2312yes—
55C42×C59C2311yescharacteristic
56#11745671—44no—
57(C2×D59)⋊C4—22no—
58(C2×D59)⋊C4—22no—
59C4⋊Dic59—21no—
60C4⋊Dic59—21no—
61Dic59⋊C4—22no—
62Dic59⋊C4—22no—
63C4×Dic59D411yescharacteristic
64C4×Dic59D411yescharacteristic
65C4×Dic59C2311yescharacteristic
66C2×C59⋊C8C2312yes—
67#11745660—44no—
68C22×D59—22no—
69C22×D59C2×D411yescharacteristic
70Q8×C59—24no—
71Q8×C59—42no—
72C2×C4×C59—22no—
73C2×C4×C59—22no—
74C2×C4×C59C2×D411yescharacteristic
75C2×C4×C59C2×D411yescharacteristic
76C2×C4×C59C2411yescharacteristic
77C2×Dic59—22no—
78C2×Dic59C2×D411yescharacteristic
79C2×Dic59C2×D411yescharacteristic
80C2×Dic59C2×D411yescharacteristic
81D236—24no—
82D236—44no—
83D236—42no—
84D236—42no—
85C4×D59—22no—
86C4×D59—22no—
87C4×D59—44no—
88C4×D59—44no—
89C4×D59SD1614yescomplemented
90Dic118—24no—
91Dic118—44no—
92C8×C59—22no—
93C59⋊C8—22no—
94C22×C59C22×D411yescharacteristic
95C2×D59—21no—
96C2×D59—21no—
97C2×D59—42no—
98C2×D59—42no—
99C2×D59—42no—
100C2×D59C2×SD1612yes—
101C4×C59—42no—
102C4×C59—42no—
103C4×C59C2×SD1612yes—
104C4×C59#13712yes—
105C4×C59H(5,𝔽2)12yes—
106Dic59—21no—
107Dic59—21no—
108Dic59—42no—
109Dic59C2×SD1612yes—
110#3206—591nomaximal, Sylow
111C2×C59#53511yescharacteristic
112C2×C59C22×SD1611yescharacteristic
113C2×C59C2×#13711yescharacteristic
114D59—22no—
115D59—82no—
116#532—591no—
117C2×C4⋊Q8—591no—
118#492—591no—
119#492—591no—
120#487—591no—
121#487D5911yescharacteristic, complemented
122#461—594no—
123#448—594no—
124#431—591no—
125C59#320611yescharacteristic, complemented, Sylow
126C2×#41—1181no—
127C22×Q8—592no—
128C2×SD16—594no—
129C2×SD16—594no—
130C4⋊Q8—592no—
131C4⋊Q8—591no—
132C4⋊Q8—1181no—
133C4⋊Q8—1181no—
134C4⋊Q8C2×D5911yescharacteristic
135C4⋊Q8C2×D5911yescharacteristic
136#127—591no—
137#124—1181no—
138#121—592no—
139#121—1181no—
140C4×D4—592no—
141C4×D4—1181no—
142C2×C4⋊C4—1181no—
143C2×C4⋊C4—1181no—
144C2×C42—591no—
145C4⋊C8—592no—
146#103—594no—
147#103C2×D5914yes—
148C22⋊C8—592no—
149C4×C8—591no—
150C4×C8C2×D5911yescharacteristic
151#41—2364no—
152C2×Q8—21no—
153C2×Q8—592no—
154C2×Q8—1182no—
155C2×Q8—2364no—
156C2×Q8C22×D5912yes—
157C2×D4—592no—
158C2×D4—1182no—
159C2×D4—1181no—
160C2×D4—1181no—
161C22×C4—591no—
162C22×C4—591no—
163C22×C4—591no—
164C22×C4—1182no—
165C22×C4—1182no—
166SD16—2364no—
167SD16—2364no—
168C2×C8—592no—
169C2×C8C22×D5912yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—1182no—
174C4⋊C4—1182no—
175C4⋊C4—1181no—
176C4⋊C4—1181no—
177C4⋊C4C22×D5912yes—
178C22⋊C4—1182no—
179C22⋊C4—1182no—
180C42—591no—
181C42—591no—
182C42—591no—
183C42C22×D5911yescharacteristic
184C23—591no—
185C23—1182no—
186Q8—24no—
187Q8—42no—
188Q8—1184no—
189Q8—2364no—
190D4—1184no—
191D4—2364no—
192D4—2362no—
193D4—2362no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—591no—
197C2×C4—594no—
198C2×C4—591no—
199C2×C4—591no—
200C2×C4—1182no—
201C2×C4—1182no—
202C2×C4—1182no—
203C2×C4—2364no—
204C2×C4—2364no—
205C2×C4D4×D5911yescharacteristic
206C2×C4D4×D5911yescharacteristic
207C2×C4C23×D5911yescharacteristic
208C8—22no—
209C8—1182no—
210C22—592no—
211C22—1181no—
212C22—1181no—
213C22—2362no—
214C22—2362no—
215C22—2362no—
216C22C2×D4×D5911yescharacteristic
217C4—42no—
218C4—42no—
219C4—592no—
220C4—1181no—
221C4—1181no—
222C4—2362no—
223C4SD16×D5912yes—
224C4#1427375912yes—
225C4#1427377312yes—
226C2—1182no—
227C2—4722no—
228C2#1683797011yescharacteristic
229C2C2×SD16×D5911yescharacteristic
230C2C2×#1427375911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#487C2
socleC22×C59C22×D4
hypercenter#487D59
hypocenterC59#3206
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D59C2semidirect
3. #16837776C2semidirect
4. #16837775C2semidirect
5. #16837754C2semidirect
6. #16837606C2semidirect
7. #16837542C2semidirect
8. #16837453C2semidirect
9. C59×#487C2semidirect
89. C4×D59SD16semidirect
121. #487D59semidirect
125. C59#3206semidirect