Finite Group Databasebeta

← #22590015

Subgroups — #22590015

Order 9088.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#17985973C212yesmaximal
2#17985794C214yesmaximal, complemented
3#536×C71C211yesmaximal, characteristic
4#14556087—22no—
5#14556043—22no—
6#14556043C2214yes—
7C2×#41×C71C2211yescharacteristic
8C22×Q8×C71C411yescharacteristic, complemented
9C4⋊Q8×C71C412yescomplemented
10#124×C71C2212yes—
11#123×C71—22no—
12C22⋊Q8×C71—22no—
13#117×C71C411yescharacteristic, complemented
14C4⋊Dic71—42no—
15C22⋊Dic71—24no—
16C4×Dic71—42no—
17#41×C71—42no—
18C2×Q8×C71—21no—
19C2×Q8×C71—42no—
20C2×Q8×C71C2×C412yes—
21C2×Q8×C71C2×C411yescharacteristic
22C2×D4×C71D412yes—
23C2×D4×C71C2311yescharacteristic
24C22×C4×C71—21no—
25C22×C4×C71C2×C411yescharacteristic
26C22×C4×C71D412yes—
27C4⋊C4×C71—21no—
28C4⋊C4×C71—24no—
29C22⋊C4×C71—21no—
30C22⋊C4×C71—24no—
31C42×C71C2×C412yes—
32C23×C71C22⋊C411yescharacteristic
33C23×C71C2×D412yes—
34Q8×C71—41no—
35Q8×C71—42no—
36Q8×C71—42no—
37D4×C71—42no—
38D4×C71—41no—
39C2×C4×C71—22no—
40C2×C4×C71—22no—
41C2×C4×C71—22no—
42C2×C4×C71—22no—
43C2×C4×C71—22no—
44C2×C4×C71—42no—
45C2×C4×C71C22⋊C412yes—
46C2×C4×C71C22⋊C411yescharacteristic
47C2×C4×C71C22×C411yescharacteristic
48C2×Dic71—44no—
49C22×C71—22no—
50C22×C71—21no—
51C22×C71—41no—
52C22×C71—42no—
53C22×C71C2×C22⋊C411yescharacteristic
54C4×C71—22no—
55C4×C71—21no—
56C4×C71—42no—
57C4×C71—42no—
58C4×C71#9912yes—
59Dic71—84no—
60C2×C71—21no—
61C2×C71—42no—
62C2×C71—41no—
63C2×C71C2×#9911yescharacteristic
64#2129—711nomaximal, Sylow
65C71#212911yescharacteristic, complemented, Sylow
66#536D7111yescharacteristic
67#410—712no—
68#354—714no—
69C2×#41C2×D7111yescharacteristic
70C22×Q8Dic7111yescharacteristic, complemented
71C4⋊Q8Dic7112yescomplemented
72#124C2×D7112yes—
73#123—22no—
74C22⋊Q8—22no—
75#117—1422no—
76#117Dic7111yescharacteristic, complemented
77#99—714no—
78#99—1422no—
79#41—42no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8C2×Dic7112yes—
83C2×Q8C2×Dic7111yescharacteristic
84C2×D4C22⋊D7112yes—
85C2×D4C22×D7111yescharacteristic
86C22×C4—21no—
87C22×C4C2×Dic7111yescharacteristic
88C22×C4C22⋊D7112yes—
89C4⋊C4—24no—
90C4⋊C4—21no—
91C4⋊C4—2842no—
92C22⋊C4—21no—
93C22⋊C4—24no—
94C22⋊C4—1424no—
95C42—2842no—
96C42C2×Dic7112yes—
97C23C22⋊Dic7111yescharacteristic
98C23C2×C22⋊D7112yes—
99Q8—41no—
100Q8—42no—
101Q8—42no—
102D4—42no—
103D4—41no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—2844no—
111C2×C4C22⋊Dic7112yes—
112C2×C4C22⋊Dic7111yescharacteristic
113C2×C4C22×Dic7111yescharacteristic
114C22—21no—
115C22—22no—
116C22—41no—
117C22—42no—
118C22C2×C22⋊Dic7111yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C4—5684no—
124C4#1455604312yes—
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×#1455604311yescharacteristic
subgroupG/H
centerC2C2×#14556043
commutatorC2×C4×C71C22×C4
FrattiniC2×D4C22×D71
Fitting#536×C71C2
socleC2×C71C2×#99
hypercenter#536D71
hypocenterC71#2129
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #17985794C2semidirect
8. C22×Q8×C71C4semidirect
9. C4⋊Q8×C71C4semidirect
13. #117×C71C4semidirect
65. C71#2129semidirect
70. C22×Q8Dic71semidirect
71. C4⋊Q8Dic71semidirect
76. #117Dic71semidirect