Finite Group Databasebeta

← #22594056

Subgroups — #22594056

Order 9088.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×D284C211yesmaximal, characteristic, complemented
2C2×#14556096C212yesmaximal, complemented
3#17985916C214yesmaximal, complemented
4#17985908C214yesmaximal, complemented
5#17985862C211yesmaximal, characteristic, complemented
6#17985854C212yesmaximal, complemented
7C2×#123×C71C211yesmaximal, characteristic, complemented
8C24×D71C2212yes—
9C22×D284—28no—
10C22×D284—24no—
11C22×D284C2214yes—
12C22×C22⋊D71C2214yescomplemented
13#14556096—28no—
14C2×C22⋊Dic71C2212yes—
15C2×(C2×D71)⋊C4C2214yescomplemented
16C2×(C2×D71)⋊C4C2218yescomplemented
17C71⋊#95C2214yescomplemented
18C22×D4×C71C2211yescharacteristic
19C23×C4×C71C2211yescharacteristic
20#123×C71—24no—
21C2×C4⋊C4×C71C2212yescomplemented
22C2×C22⋊C4×C71C2211yescharacteristic
23C2×C22⋊C4×C71C2212yescomplemented
24C23×D71—24no—
25C23×D71—24no—
26C23×D71—48no—
27C23×D71D418yescomplemented
28C23×D71C2314yes—
29C2×D284—232no—
30C2×D284—216no—
31C2×D284—28no—
32C2×C22⋊D71—28no—
33C2×C22⋊D71—48no—
34C22×Dic71C2314yes—
35C22⋊Dic71—24no—
36(C2×D71)⋊C4—28no—
37(C2×D71)⋊C4—48no—
38C24×C71C2311yescharacteristic
39C24×C71C2311yescharacteristic
40C2×D4×C71—22no—
41C2×D4×C71—24no—
42C22×C4×C71—24no—
43C22×C4×C71D414yescomplemented
44C22×C4×C71C2312yes—
45C22×C4×C71C2311yescharacteristic
46C22×C4×C71C2312yes—
47C4⋊C4×C71—24no—
48C22⋊C4×C71—22no—
49C22⋊C4×C71—24no—
50C23×C71—24no—
51C23×C71—21no—
52C23×C71—22no—
53C23×C71—24no—
54C23×C71C2×D414yes—
55C23×C71C2×D412yes—
56C23×C71C2411yescharacteristic
57C22×D71—216no—
58C22×D71—24no—
59C22×D71—24no—
60C22×D71—28no—
61C22×D71—28no—
62C22×D71—416no—
63C22×D71—416no—
64C22×D71—416no—
65C22×D71C2×D418yes—
66D4×C71—44no—
67C2×C4×C71—28no—
68C2×C4×C71—24no—
69C2×C4×C71—22no—
70C2×C4×C71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—24no—
73C2×C4×C71C2×D414yes—
74C22⋊D71—416no—
75C2×Dic71—24no—
76C2×Dic71—28no—
77D284—232no—
78C22×C71—28no—
79C22×C71—24no—
80C22×C71—41no—
81C22×C71—44no—
82C22×C71—42no—
83C22×C71#12014yes—
84C22×C71C22×D412yes—
85C22×C71C22×D411yescharacteristic
86C22×C71H(5,𝔽2)14yes—
87C2×D71—28no—
88C2×D71—28no—
89C2×D71—28no—
90C2×D71—432no—
91C2×D71—48no—
92C2×D71—48no—
93C4×C71—24no—
94C4×C71—41no—
95C4×C71—42no—
96Dic71—44no—
97C2×C71—24no—
98C2×C71—81no—
99C2×C71C2×#12011yescharacteristic
100C2×C71#53414yes—
101C2×C71#53512yes—
102D71—48no—
103#2613—711nomaximal, Sylow
104C71#261311yescharacteristic, complemented, Sylow
105C23×D4—711no—
106C2×#123D7111yescharacteristic, complemented
107C2×#120—712no—
108#394—714no—
109#392—714no—
110#381—711no—
111#379—712no—
112C25—712no—
113C22×D4—714no—
114C22×D4—714no—
115C22×D4—1428no—
116C22×D4—1424no—
117C22×D4C2×D7111yescharacteristic
118C23×C4C2×D7111yescharacteristic
119#123—24no—
120#120—1428no—
121C2×C4⋊C4C2×D7112yescomplemented
122C2×C22⋊C4—718no—
123C2×C22⋊C4—712no—
124C2×C22⋊C4—714no—
125C2×C22⋊C4C2×D7111yescharacteristic
126C2×C22⋊C4C2×D7112yescomplemented
127#95—714no—
128C24—2848no—
129C24—718no—
130C24—714no—
131C24—1424no—
132C24—1424no—
133C24C22×D7111yescharacteristic
134C24C22×D7111yescharacteristic
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—2848no—
138C2×D4—14216no—
139C2×D4—14232no—
140C2×D4—1428no—
141C2×D4—1428no—
142C22×C4—24no—
143C22×C4—714no—
144C22×C4C22⋊D7114yescomplemented
145C22×C4C22×D7112yes—
146C22×C4C22×D7111yescharacteristic
147C22×C4C22×D7112yes—
148C4⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—24no—
151C22⋊C4—2848no—
152C22⋊C4—1428no—
153C22⋊C4—1424no—
154C23—24no—
155C23—22no—
156C23—21no—
157C23—24no—
158C23—28416no—
159C23—28416no—
160C23—28416no—
161C23—718no—
162C23—1424no—
163C23—1424no—
164C23—1428no—
165C23—1428no—
166C23—14216no—
167C23D4×D7114yes—
168C23C2×C22⋊D7112yes—
169C23C23×D7111yescharacteristic
170D4—44no—
171D4—28416no—
172D4—14232no—
173C2×C4—24no—
174C2×C4—28no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—1424no—
180C2×C4—1428no—
181C2×C4C2×C22⋊D7114yes—
182C22—28no—
183C22—24no—
184C22—44no—
185C22—42no—
186C22—41no—
187C22—28432no—
188C22—2848no—
189C22—2848no—
190C22—1428no—
191C22—1428no—
192C22—1428no—
193C22#1455609614yes—
194C22C2×D4×D7112yes—
195C22C22×C22⋊D7111yescharacteristic
196C22#1455617414yes—
197C4—24no—
198C4—42no—
199C4—41no—
200C4—2844no—
201C2—24no—
202C2—81no—
203C2—2848no—
204C2C2×#1455609611yescharacteristic
205C2#1798656014yes—
206C2#1798656312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#123×C71C2
hypercenterC2×#123D71
hypocenterC71#2613
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×D284C2semidirect
2. C2×#14556096C2semidirect
3. #17985916C2semidirect
4. #17985908C2semidirect
5. #17985862C2semidirect
6. #17985854C2semidirect
7. C2×#123×C71C2semidirect
12. C22×C22⋊D71C22semidirect
15. C2×(C2×D71)⋊C4C22semidirect
16. C2×(C2×D71)⋊C4C22semidirect
17. C71⋊#95C22semidirect
21. C2×C4⋊C4×C71C22semidirect
23. C2×C22⋊C4×C71C22semidirect
27. C23×D71D4semidirect
43. C22×C4×C71D4semidirect
104. C71#2613semidirect
106. C2×#123D71semidirect
121. C2×C4⋊C4C2×D71semidirect
126. C2×C22⋊C4C2×D71semidirect
144. C22×C4C22⋊D71semidirect