Finite Group Databasebeta

← #22596603

Subgroups — #22596603

Order 9088.

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#17985936C211yesmaximal, characteristic, complemented
2#17985934C213yesmaximal, complemented
3#17985923C211yesmaximal, characteristic, complemented
4#17985922C213yesmaximal, complemented
5#17985921C213yesmaximal, complemented
6#17985854C213yesmaximal
7#396×C71C211yesmaximal, characteristic, complemented
8C24×D71C2211yescharacteristic
9C2×C4⋊Dic71C2213yescomplemented
10C2×Dic71⋊C4C2213yescomplemented
11C2×Dic71⋊C4C2213yescomplemented
12C2×C22⋊Dic71C2213yes—
13C2×(C2×D71)⋊C4C2213yes—
14C2×(C2×D71)⋊C4C2213yes—
15C2×(C2×D71)⋊C4C2213yes—
16C2×(C2×D71)⋊C4C2213yes—
17C71⋊#95C2213yescomplemented
18C2×C4⋊C4×C71C2213yescomplemented
19C2×C22⋊C4×C71C2213yes—
20#95×C71C2211yescharacteristic, complemented
21C23×D71—43no—
22C23×D71—43no—
23C23×D71Q814yes—
24C23×D71C2311yescharacteristic
25C23×D71C2311yescharacteristic
26C22×Dic71C2313yes—
27C22×Dic71C2313yes—
28C4⋊Dic71—43no—
29Dic71⋊C4—43no—
30Dic71⋊C4—43no—
31C22⋊Dic71—43no—
32(C2×D71)⋊C4—43no—
33(C2×D71)⋊C4—43no—
34(C2×D71)⋊C4—43no—
35(C2×D71)⋊C4—43no—
36C24×C71C2311yescharacteristic
37C22×C4×C71C2313yes—
38C22×C4×C71C2313yes—
39C4⋊C4×C71—43no—
40C22⋊C4×C71—43no—
41C23×C71—23no—
42C23×C71—23no—
43C23×C71C2×Q812yes—
44C23×C71C2411yescharacteristic
45C22×D71—23no—
46C22×D71—23no—
47C22×D71—23no—
48C22×D71—23no—
49C22×D71—412no—
50C22×D71—412no—
51C22×D71—44no—
52C22×D71C2×Q812yes—
53C22×D71C2×Q812yes—
54C2×C4×C71—23no—
55C2×C4×C71—23no—
56C2×C4×C71—23no—
57C2×C4×C71—23no—
58C2×C4×C71—23no—
59C2×C4×C71—23no—
60C2×Dic71—23no—
61C2×Dic71—23no—
62C2×Dic71—23no—
63C2×Dic71—23no—
64C2×Dic71—23no—
65C2×Dic71—23no—
66C22×C71—26no—
67C22×C71—43no—
68C22×C71—41no—
69C22×C71C22×Q811yescharacteristic
70C22×C71H(5,𝔽2)13yes—
71C22×C71H(5,𝔽2)13yes—
72C2×D71—26no—
73C2×D71—26no—
74C2×D71—416no—
75C2×D71—43no—
76C2×D71—43no—
77C2×D71—41no—
78C2×D71—41no—
79C4×C71—43no—
80C4×C71—43no—
81Dic71—43no—
82Dic71—43no—
83C2×C71—42no—
84C2×C71#54313yes—
85C2×C71#56013yes—
86C2×C71#56011yescharacteristic
87D71—42no—
88D71—42no—
89#2842—711nomaximal, Sylow
90C71#284211yescharacteristic, complemented, Sylow
91#399—713no—
92#399—711no—
93#396—713no—
94#396—713no—
95#396—711no—
96#396D7111yescharacteristic, complemented
97#379—713no—
98C25—711no—
99C2×C4⋊C4—713no—
100C2×C4⋊C4—713no—
101C2×C4⋊C4—713no—
102C2×C4⋊C4C2×D7113yescomplemented
103C2×C22⋊C4—713no—
104C2×C22⋊C4—713no—
105C2×C22⋊C4—713no—
106C2×C22⋊C4—713no—
107C2×C22⋊C4—713no—
108C2×C22⋊C4C2×D7113yes—
109#95—713no—
110#95C2×D7111yescharacteristic, complemented
111C24—2843no—
112C24—2843no—
113C24—714no—
114C24—711no—
115C24—711no—
116C24C22×D7111yescharacteristic
117C22×C4—713no—
118C22×C4—713no—
119C22×C4C22×D7113yes—
120C22×C4C22×D7113yes—
121C4⋊C4—43no—
122C4⋊C4—2843no—
123C4⋊C4—2843no—
124C4⋊C4—2843no—
125C22⋊C4—43no—
126C22⋊C4—2843no—
127C22⋊C4—2843no—
128C22⋊C4—2843no—
129C22⋊C4—2843no—
130C22⋊C4—2843no—
131C23—23no—
132C23—23no—
133C23—28412no—
134C23—28412no—
135C23—2844no—
136C23—712no—
137C23—712no—
138C23—1423no—
139C23—1423no—
140C23—1423no—
141C23—1423no—
142C23Q8×D7112yes—
143C23C23×D7111yescharacteristic
144C2×C4—23no—
145C2×C4—23no—
146C2×C4—23no—
147C2×C4—23no—
148C2×C4—23no—
149C2×C4—23no—
150C2×C4—1423no—
151C2×C4—1423no—
152C2×C4—1423no—
153C2×C4—1423no—
154C2×C4—1423no—
155C2×C4—1423no—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—28416no—
160C22—2843no—
161C22—2843no—
162C22—2841no—
163C22—2841no—
164C22—1426no—
165C22—1426no—
166C22C2×Q8×D7111yescharacteristic
167C22#1455617313yes—
168C22#1455617413yes—
169C4—43no—
170C4—43no—
171C4—2843no—
172C4—2843no—
173C2—42no—
174C2—2842no—
175C2—2842no—
176C2#1798662413yes—
177C2#1798673713yes—
178C2#1798674011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#396×C71C2
hypercenter#396D71
hypocenterC71#2842
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #17985936C2semidirect
2. #17985934C2semidirect
3. #17985923C2semidirect
4. #17985922C2semidirect
5. #17985921C2semidirect
7. #396×C71C2semidirect
9. C2×C4⋊Dic71C22semidirect
10. C2×Dic71⋊C4C22semidirect
11. C2×Dic71⋊C4C22semidirect
17. C71⋊#95C22semidirect
18. C2×C4⋊C4×C71C22semidirect
20. #95×C71C22semidirect
90. C71#2842semidirect
96. #396D71semidirect
102. C2×C4⋊C4C2×D71semidirect
110. #95C2×D71semidirect