Finite Group Databasebeta

← #22598721

Subgroups — #22598721

Order 9088.

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

normal subgroups only

This group has 380 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
1#17986556C211yesmaximal, characteristic
2C2×#14556128C211yesmaximal, characteristic
3#17986368C214yesmaximal, complemented
4#17986350C214yesmaximal, complemented
5#17986009C214yesmaximal
6C4×M16×C71C211yesmaximal, characteristic
7C22×Dic142C2212yes—
8#14556128—22no—
9#14556128C2212yes—
10#14556128C2212yes—
11#14556128C2212yes—
12#14556120C2212yes—
13#14556107—22no—
14C4×Dic142—22no—
15#14556087C2212yes—
16C2×C4⋊Dic71—22no—
17#14556053C22116yes—
18C2×M16×C71C2212yes—
19C2×C42×C71C2211yescharacteristic
20#97×C71C2212yes—
21C4×C8×C71C2212yes—
22C2×Dic142—28no—
23C2×Dic142—22no—
24C2×Dic142—22no—
25C2×Dic142C2314yes—
26C22×Dic71—22no—
27C4⋊Dic71—24no—
28C4⋊Dic71—24no—
29C4⋊Dic71—22no—
30C4⋊Dic71C2314yes—
31Dic71⋊C4—24no—
32C22⋊Dic71—22no—
33C4×Dic71—22no—
34C22×C4×C71D412yes—
35C22×C4×C71C2311yescharacteristic
36M16×C71—24no—
37C2×C8×C71C2314yes—
38C42×C71D412yes—
39C42×C71C2311yescharacteristic
40C42×C71C2311yescharacteristic
41C23×C71C2×D411yescharacteristic
42C2×C4×C71—24no—
43C2×C4×C71C2×D412yes—
44C2×C4×C71C2×D411yescharacteristic
45C2×C4×C71C2×D412yes—
46C2×C4×C71#4114yes—
47C2×C4×C71C2411yescharacteristic
48C2×Dic71—24no—
49C2×Dic71—24no—
50C2×Dic71—44no—
51Dic142—28no—
52Dic142—44no—
53Dic142—44no—
54C8×C71—24no—
55C22×C71—22no—
56C22×C71#12412yes—
57C22×C71C22×D411yescharacteristic
58C4×C71—22no—
59C4×C71#12412yes—
60C4×C71#13714yes—
61C4×C71C2×#4112yes—
62Dic71—44no—
63Dic71—44no—
64C2×C71—22no—
65C2×C71C2×#12411yescharacteristic
66C2×C71C2×#13712yes—
67#3139—711nomaximal, Sylow
68C71#313911yescharacteristic, complemented, Sylow
69#533—711no—
70C2×C4⋊Q8—711no—
71#491—714no—
72#487—714no—
73#419—714no—
74C4×M16D7111yescharacteristic
75C22×Q8—712no—
76C2×M16C2×D7112yes—
77C4⋊Q8—712no—
78C4⋊Q8—712no—
79C4⋊Q8—712no—
80C4⋊Q8—1422no—
81#125—712no—
82C22⋊Q8—1422no—
83C4×Q8—1422no—
84#117—712no—
85C2×C4⋊C4—1422no—
86C2×C42C2×D7111yescharacteristic
87#103—7116no—
88#97C2×D7112yes—
89C4×C8C2×D7112yes—
90C2×Q8—714no—
91C2×Q8—1428no—
92C2×Q8—1422no—
93C2×Q8—1422no—
94C22×C4—1422no—
95C22×C4D28412yes—
96C22×C4C22×D7111yescharacteristic
97M16—24no—
98C2×C8C22×D7114yes—
99C4⋊C4—714no—
100C4⋊C4—1424no—
101C4⋊C4—1424no—
102C4⋊C4—1424no—
103C4⋊C4—1422no—
104C22⋊C4—1422no—
105C42—1422no—
106C42D28412yes—
107C42C22×D7111yescharacteristic
108C42C22×D7111yescharacteristic
109C23C2×D28411yescharacteristic
110Q8—2844no—
111Q8—2844no—
112Q8—1428no—
113C2×C4—24no—
114C2×C4—2844no—
115C2×C4—1424no—
116C2×C4—1424no—
117C2×C4C2×D28412yes—
118C2×C4C2×D28412yes—
119C2×C4C2×D28411yescharacteristic
120C2×C4#1176586514yes—
121C2×C4C23×D7111yescharacteristic
122C8—24no—
123C22—22no—
124C22#1455611612yes—
125C22C22×D28411yescharacteristic
126C4—22no—
127C4—2844no—
128C4—2844no—
129C4#1455611612yes—
130C4#1455616114yes—
131C4C2×#1176586512yes—
132C2—22no—
133C2C2×#1455611611yescharacteristic
134C2C2×#1455616112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC4×M16×C71C2
socleC22×C71C22×D4
hypercenterC4×M16D71
hypocenterC71#3139
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
3. #17986368C2semidirect
4. #17986350C2semidirect
68. C71#3139semidirect