Finite Group Databasebeta

← #22597984

Subgroups — #22597984

Order 9088.

524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14556134C211yesmaximal, characteristic, complemented
2#17986566C212yesmaximal, complemented
3#17986249C214yesmaximal, complemented
4#17986232C214yesmaximal, complemented
5#17986037C211yesmaximal, characteristic, complemented
6#17986004C212yesmaximal, complemented
7C22×D8×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556134—24no—
10C2×#11765849C2211yescharacteristic
11C2×#11765848C2212yescomplemented
12C2×C8×D71C2212yescomplemented
13#14556124—22no—
14#14556115—22no—
15#14556099C2218yescomplemented
16#14556097—24no—
17#14556087C2212yes—
18#14556062C2212yescomplemented
19#14556060C2212yescomplemented
20#14556049C2218yescomplemented
21C22×D4×C71C2212yes—
22C2×D8×C71—24no—
23C2×D8×C71C2212yes—
24C2×D8×C71C2212yes—
25C22×C8×C71C2211yescharacteristic
26#11765865—22no—
27#11765865—22no—
28C2×Dic142D411yescharacteristic
29C2×D284D411yescharacteristic
30C2×C22⋊D71—28no—
31C2×C22⋊D71D412yes—
32C2×C4×D71C2312yes—
33#11765849—22no—
34#11765848—24no—
35C2×C71⋊C8C2312yes—
36C8×D71—24no—
37C4⋊Dic71C2314yes—
38C22⋊Dic71—22no—
39C22⋊Dic71—28no—
40C4×Dic71—22no—
41C24×C71—22no—
42C2×D4×C71—24no—
43C2×D4×C71—22no—
44C2×D4×C71—24no—
45C2×D4×C71C2314yes—
46C22×C4×C71C2311yescharacteristic
47D8×C71—24no—
48D8×C71—24no—
49C2×C8×C71D414yescomplemented
50C2×C8×C71C2311yescharacteristic
51C2×C8×C71C2311yescharacteristic
52C23×C71—24no—
53C23×C71—44no—
54C23×C71—84no—
55C23×C71C2×D411yescharacteristic
56C22×D71C2×D412yes—
57D4×C71—24no—
58D4×C71—24no—
59D4×C71—44no—
60C2×C4×C71C2×D411yescharacteristic
61C2×C4×C71C2×D412yes—
62C2×C4×C71#4112yes—
63C2×C4×C71C2411yescharacteristic
64C22⋊D71—24no—
65C22⋊D71—88no—
66C2×Dic71—24no—
67C2×Dic71C2×D412yes—
68D284—22no—
69C4×D71—22no—
70C4×D71—22no—
71Dic142—22no—
72C8×C71C2×D412yes—
73C8×C71C2×D412yes—
74C71⋊C8—22no—
75C22×C71—22no—
76C22×C71—44no—
77C22×C71—48no—
78C22×C71—88no—
79C22×C71#12112yes—
80C22×C71C22×D411yescharacteristic
81C2×D71—22no—
82C2×D71—24no—
83C4×C71#12112yes—
84C4×C71C22×D411yescharacteristic
85C4×C71C2×#4111yescharacteristic
86Dic71—22no—
87Dic71—44no—
88C2×C71—22no—
89C2×C71—84no—
90C2×C71C2×#12111yescharacteristic
91C2×C71#57612yes—
92D71—42no—
93#3058—711nomaximal, Sylow
94C71#305811yescharacteristic, complemented, Sylow
95C22×D8D7111yescharacteristic, complemented
96C2×#131—711no—
97#535—712no—
98#469—714no—
99#466—714no—
100#427—711no—
101#418—712no—
102C2×#41—711no—
103C22×D4C2×D7112yes—
104C2×D8—24no—
105C2×D8C2×D7112yes—
106C2×D8C2×D7112yes—
107#131—1424no—
108C2×M16—711no—
109C2×M16—712no—
110C22×C8—712no—
111C22×C8C2×D7111yescharacteristic
112#127—1422no—
113#124—1422no—
114#121—718no—
115#120—1424no—
116#117—712no—
117#107—712no—
118#106—712no—
119#102—718no—
120C24—22no—
121#41—1422no—
122#41—1422no—
123C2×Q8—711no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—712no—
128C2×D4—711no—
129C2×D4—1428no—
130C2×D4C22×D7114yes—
131C22×C4—712no—
132C22×C4C22×D7111yescharacteristic
133D8—24no—
134D8—24no—
135M16—1422no—
136M16—1424no—
137C2×C8—712no—
138C2×C8—1424no—
139C2×C8C22⋊D7114yescomplemented
140C2×C8C22×D7111yescharacteristic
141C2×C8C22×D7111yescharacteristic
142C4⋊C4—714no—
143C22⋊C4—1428no—
144C22⋊C4—1422no—
145C42—1422no—
146C23—24no—
147C23—44no—
148C23—84no—
149C23—712no—
150C23D4×D7111yescharacteristic
151Q8—1422no—
152D4—24no—
153D4—24no—
154D4—44no—
155D4—5688no—
156D4—1424no—
157D4—1422no—
158C2×C4—712no—
159C2×C4—1424no—
160C2×C4—1422no—
161C2×C4—1422no—
162C2×C4D4×D7111yescharacteristic
163C2×C4C2×C22⋊D7112yes—
164C2×C4#1176586412yes—
165C2×C4C23×D7111yescharacteristic
166C8—1422no—
167C8C2×C22⋊D7112yes—
168C8C2×C22⋊D7112yes—
169C22—22no—
170C22—44no—
171C22—48no—
172C22—88no—
173C22—1422no—
174C22—1424no—
175C22#1455609912yes—
176C22C2×D4×D7111yescharacteristic
177C4—2844no—
178C4—1422no—
179C4#1455609912yes—
180C4C22×C22⋊D7111yescharacteristic
181C4C2×#1176586411yescharacteristic
182C2—22no—
183C2—84no—
184C2—2842no—
185C2C2×#1455609911yescharacteristic
186C2#1798680012yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC22×D8×C71C2
socleC22×C71C22×D4
hypercenterC22×D8D71
hypocenterC71#3058
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#14556134C2semidirect
2. #17986566C2semidirect
3. #17986249C2semidirect
4. #17986232C2semidirect
5. #17986037C2semidirect
6. #17986004C2semidirect
7. C22×D8×C71C2semidirect
11. C2×#11765848C22semidirect
12. C2×C8×D71C22semidirect
15. #14556099C22semidirect
18. #14556062C22semidirect
19. #14556060C22semidirect
20. #14556049C22semidirect
49. C2×C8×C71D4semidirect
94. C71#3058semidirect
95. C22×D8D71semidirect
139. C2×C8C22⋊D71semidirect