Finite Group Databasebeta

← #21435225

Subgroups — #21435225

Order 7552.

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

normal subgroups only

This group has 404 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#16837798C211yesmaximal, characteristic, complemented
4#16837750C211yesmaximal, characteristic, complemented
5#16837749C211yesmaximal, characteristic, complemented
6#16837722C214yesmaximal, complemented
7#16837691C214yesmaximal, complemented
8#16837453C211yesmaximal, characteristic, complemented
9C59×#498C211yesmaximal, characteristic, complemented
10C2×#11745689—21no—
11C2×Q8×D59—21no—
12#14273727C2211yescharacteristic
13#14273726—21no—
14#14273726C2212yes—
15#14273725—21no—
16#14273724C2211yescharacteristic
17#14273716—21no—
18#14273702—21no—
19#14273702C2212yes—
20#14273689—21no—
21#14273689C2212yes—
22C4⋊C4×D59—21no—
23C4⋊C4×D59C2212yes—
24C42×D59C2211yescharacteristic
25#14273658C2214yes—
26Dic59⋊C8C2212yes—
27#14273649C2214yes—
28#14273647C2214yes—
29(C2×D59)⋊C8C2212yes—
30C4×C59⋊C8C2211yescharacteristic
31C4⋊Q8×C59C2211yescharacteristic
32C4⋊Q8×C59C2211yescharacteristic
33#106×C59C2214yes—
34C4×C8×C59C2211yescharacteristic
35C2×Q8×C59—21no—
36C2×Q8×C59—21no—
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—21no—
49C2×Dic118—22no—
50C2×C8×C59C2312yes—
51C4⋊C4×C59—21no—
52C4⋊C4×C59—21no—
53C4⋊C4×C59C2312yes—
54C4⋊C4×C59C2312yes—
55C42×C59C2311yescharacteristic
56(C2×D59)⋊C4—22no—
57(C2×D59)⋊C4—22no—
58C4⋊Dic59—21no—
59C4⋊Dic59C2312yes—
60Dic59⋊C4—22no—
61Dic59⋊C4—22no—
62C4×Dic59D411yescharacteristic
63C4×Dic59D411yescharacteristic
64C4×Dic59C2311yescharacteristic
65C2×C59⋊C8C2312yes—
66C22×D59—22no—
67C22×D59C2×D411yescharacteristic
68Q8×C59—42no—
69Q8×C59—42no—
70C2×C4×C59—22no—
71C2×C4×C59—22no—
72C2×C4×C59C2×D411yescharacteristic
73C2×C4×C59C2×D411yescharacteristic
74C2×C4×C59C2411yescharacteristic
75C2×Dic59—22no—
76C2×Dic59C2×D411yescharacteristic
77C2×Dic59C2×D411yescharacteristic
78C2×Dic59C2×D411yescharacteristic
79D236—24no—
80D236—44no—
81D236—42no—
82D236—42no—
83C4×D59—22no—
84C4×D59—22no—
85C4×D59—44no—
86C4×D59—44no—
87C4×D59SD1614yes—
88Dic118—44no—
89Dic118—42no—
90C8×C59—22no—
91C59⋊C8—22no—
92C22×C59C22×D411yescharacteristic
93C2×D59—21no—
94C2×D59—21no—
95C2×D59—42no—
96C2×D59—42no—
97C2×D59—42no—
98C2×D59C2×SD1612yes—
99C4×C59—42no—
100C4×C59—42no—
101C4×C59C2×SD1612yes—
102C4×C59AGL(1,ℤ/8)12yes—
103C4×C592−1+412yes—
104Dic59—21no—
105Dic59—21no—
106Dic59—42no—
107Dic59C2×SD1612yes—
108#3227—591nomaximal, Sylow
109C2×C59#53611yescharacteristic
110C2×C59C22×SD1611yescharacteristic
111C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
112D59—22no—
113D59—82no—
114#532—591no—
115C2×C4⋊Q8—591no—
116#498—591no—
117#498D5911yescharacteristic, complemented
118#486—591no—
119#486—591no—
120#481—594no—
121#476—594no—
122#431—591no—
123C59#322711yescharacteristic, complemented, Sylow
124C2×#41—1181no—
125C22×Q8—1181no—
126C4⋊Q8—592no—
127C4⋊Q8—591no—
128C4⋊Q8—1181no—
129C4⋊Q8—1181no—
130C4⋊Q8C2×D5911yescharacteristic
131C4⋊Q8C2×D5911yescharacteristic
132#127—591no—
133#124—1181no—
134#121—592no—
135#121—1181no—
136C4×D4—592no—
137C4×D4—1181no—
138C2×C4⋊C4—592no—
139C2×C4⋊C4—1181no—
140C2×C42—591no—
141#106—594no—
142#106C2×D5914yes—
143C4⋊C8—592no—
144#102—594no—
145#102—594no—
146C22⋊C8—592no—
147C4×C8—591no—
148C4×C8C2×D5911yescharacteristic
149#41—2364no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—1182no—
153C2×Q8—1181no—
154C2×Q8—2364no—
155C2×D4—592no—
156C2×D4—1182no—
157C2×D4—1181no—
158C2×D4—1181no—
159C22×C4—591no—
160C22×C4—591no—
161C22×C4—591no—
162C22×C4—1182no—
163C22×C4—1182no—
164C2×C8—592no—
165C2×C8C22×D5912yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—592no—
169C4⋊C4—1182no—
170C4⋊C4—1182no—
171C4⋊C4—1181no—
172C4⋊C4C22×D5912yes—
173C4⋊C4C22×D5912yes—
174C22⋊C4—1182no—
175C22⋊C4—1182no—
176C42—591no—
177C42—591no—
178C42—591no—
179C42C22×D5911yescharacteristic
180C23—591no—
181C23—1182no—
182Q8—42no—
183Q8—42no—
184Q8—2364no—
185Q8—2362no—
186D4—1184no—
187D4—2364no—
188D4—2362no—
189D4—2362no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—591no—
193C2×C4—594no—
194C2×C4—591no—
195C2×C4—591no—
196C2×C4—1182no—
197C2×C4—1182no—
198C2×C4—1182no—
199C2×C4—2364no—
200C2×C4—2364no—
201C2×C4D4×D5911yescharacteristic
202C2×C4D4×D5911yescharacteristic
203C2×C4C23×D5911yescharacteristic
204C8—22no—
205C8—1182no—
206C22—592no—
207C22—1181no—
208C22—1181no—
209C22—2362no—
210C22—2362no—
211C22—2362no—
212C22C2×D4×D5911yescharacteristic
213C4—42no—
214C4—42no—
215C4—592no—
216C4—1181no—
217C4—1181no—
218C4—2362no—
219C4SD16×D5912yes—
220C4#1427375412yes—
221C4#1427377512yes—
222C2—1182no—
223C2—4722no—
224C2#1683797711yescharacteristic
225C2C2×SD16×D5911yescharacteristic
226C2C2×#1427375411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#498C2
socleC22×C59C22×D4
hypercenter#498D59
hypocenterC59#3227
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D59C2semidirect
3. #16837798C2semidirect
4. #16837750C2semidirect
5. #16837749C2semidirect
6. #16837722C2semidirect
7. #16837691C2semidirect
8. #16837453C2semidirect
9. C59×#498C2semidirect
117. #498D59semidirect
123. C59#3227semidirect