Finite Group Databasebeta

← #21432359

Subgroups — #21432359

Order 7552.

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

normal subgroups only

This group has 548 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#16837339C211yesmaximal, characteristic, complemented
2#16837337C213yesmaximal, complemented
3#16837326C211yesmaximal, characteristic, complemented
4#16837325C213yesmaximal, complemented
5#16837324C213yesmaximal, complemented
6#16837257C213yesmaximal
7C59×#396C211yesmaximal, characteristic, complemented
8C24×D59C2211yescharacteristic
9C2×C4⋊Dic59C2213yescomplemented
10C2×Dic59⋊C4C2213yescomplemented
11C2×Dic59⋊C4C2213yescomplemented
12C2×C22⋊Dic59C2213yes—
13C2×(C2×D59)⋊C4C2213yes—
14C2×(C2×D59)⋊C4C2213yes—
15C2×(C2×D59)⋊C4C2213yes—
16C2×(C2×D59)⋊C4C2213yes—
17C59⋊#95C2213yescomplemented
18C2×C4⋊C4×C59C2213yescomplemented
19C2×C22⋊C4×C59C2213yes—
20#95×C59C2211yescharacteristic, complemented
21C24×C59C2311yescharacteristic
22C23×D59—43no—
23C23×D59—43no—
24C23×D59Q814yes—
25C23×D59C2311yescharacteristic
26C23×D59C2311yescharacteristic
27C22×C4×C59C2313yes—
28C22×C4×C59C2313yes—
29C22×Dic59C2313yes—
30C22×Dic59C2313yes—
31C4⋊C4×C59—43no—
32C22⋊C4×C59—43no—
33C22⋊Dic59—43no—
34(C2×D59)⋊C4—43no—
35(C2×D59)⋊C4—43no—
36(C2×D59)⋊C4—43no—
37(C2×D59)⋊C4—43no—
38C4⋊Dic59—43no—
39Dic59⋊C4—43no—
40Dic59⋊C4—43no—
41C23×C59—23no—
42C23×C59—23no—
43C23×C59C2×Q812yes—
44C23×C59C2411yescharacteristic
45C22×D59—23no—
46C22×D59—23no—
47C22×D59—23no—
48C22×D59—23no—
49C22×D59—412no—
50C22×D59—412no—
51C22×D59—44no—
52C22×D59C2×Q812yes—
53C22×D59C2×Q812yes—
54C2×C4×C59—23no—
55C2×C4×C59—23no—
56C2×C4×C59—23no—
57C2×C4×C59—23no—
58C2×C4×C59—23no—
59C2×C4×C59—23no—
60C2×Dic59—23no—
61C2×Dic59—23no—
62C2×Dic59—23no—
63C2×Dic59—23no—
64C2×Dic59—23no—
65C2×Dic59—23no—
66C22×C59—26no—
67C22×C59—43no—
68C22×C59—41no—
69C22×C59C22×Q811yescharacteristic
70C22×C59H(5,𝔽2)13yes—
71C22×C59H(5,𝔽2)13yes—
72C2×D59—26no—
73C2×D59—26no—
74C2×D59—416no—
75C2×D59—43no—
76C2×D59—43no—
77C2×D59—41no—
78C2×D59—41no—
79C4×C59—43no—
80C4×C59—43no—
81Dic59—43no—
82Dic59—43no—
83#2842—591nomaximal, Sylow
84C2×C59—42no—
85C2×C59#54313yes—
86C2×C59#56013yes—
87C2×C59#56011yescharacteristic
88D59—42no—
89D59—42no—
90#399—593no—
91#399—591no—
92#396—593no—
93#396—593no—
94#396—591no—
95#396D5911yescharacteristic, complemented
96#379—593no—
97C59#284211yescharacteristic, complemented, Sylow
98C25—591no—
99C2×C4⋊C4—593no—
100C2×C4⋊C4—593no—
101C2×C4⋊C4—593no—
102C2×C4⋊C4C2×D5913yescomplemented
103C2×C22⋊C4—593no—
104C2×C22⋊C4—593no—
105C2×C22⋊C4—593no—
106C2×C22⋊C4—593no—
107C2×C22⋊C4—593no—
108C2×C22⋊C4C2×D5913yes—
109#95—593no—
110#95C2×D5911yescharacteristic, complemented
111C24—591no—
112C24—591no—
113C24—594no—
114C24—2363no—
115C24—2363no—
116C24C22×D5911yescharacteristic
117C22×C4—593no—
118C22×C4—593no—
119C22×C4C22×D5913yes—
120C22×C4C22×D5913yes—
121C4⋊C4—43no—
122C4⋊C4—2363no—
123C4⋊C4—2363no—
124C4⋊C4—2363no—
125C22⋊C4—43no—
126C22⋊C4—2363no—
127C22⋊C4—2363no—
128C22⋊C4—2363no—
129C22⋊C4—2363no—
130C22⋊C4—2363no—
131C23—23no—
132C23—23no—
133C23—592no—
134C23—592no—
135C23—1183no—
136C23—1183no—
137C23—1183no—
138C23—1183no—
139C23—23612no—
140C23—23612no—
141C23—2364no—
142C23Q8×D5912yes—
143C23C23×D5911yescharacteristic
144C2×C4—23no—
145C2×C4—23no—
146C2×C4—23no—
147C2×C4—23no—
148C2×C4—23no—
149C2×C4—23no—
150C2×C4—1183no—
151C2×C4—1183no—
152C2×C4—1183no—
153C2×C4—1183no—
154C2×C4—1183no—
155C2×C4—1183no—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—1186no—
160C22—1186no—
161C22—23616no—
162C22—2363no—
163C22—2363no—
164C22—2361no—
165C22—2361no—
166C22C2×Q8×D5911yescharacteristic
167C22#1427377213yes—
168C22#1427377313yes—
169C4—43no—
170C4—43no—
171C4—2363no—
172C4—2363no—
173C2—42no—
174C2—2362no—
175C2—2362no—
176C2#1683802713yes—
177C2#1683814013yes—
178C2#1683814311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#396C2
hypercenter#396D59
hypocenterC59#2842
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #16837339C2semidirect
2. #16837337C2semidirect
3. #16837326C2semidirect
4. #16837325C2semidirect
5. #16837324C2semidirect
7. C59×#396C2semidirect
9. C2×C4⋊Dic59C22semidirect
10. C2×Dic59⋊C4C22semidirect
11. C2×Dic59⋊C4C22semidirect
17. C59⋊#95C22semidirect
18. C2×C4⋊C4×C59C22semidirect
20. #95×C59C22semidirect
95. #396D59semidirect
97. C59#2842semidirect
102. C2×C4⋊C4C2×D59semidirect
110. #95C2×D59semidirect