Finite Group Databasebeta

← #36355238

Subgroups — #36355238

Order 22144.

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#26011465C212yesmaximal
2#26011286C214yesmaximal, complemented
3#536×C173C211yesmaximal, characteristic
4#19149638—22no—
5#19149594—22no—
6#19149594C2214yes—
7C2×#41×C173C2211yescharacteristic
8C22×Q8×C173C411yescharacteristic, complemented
9C4⋊Q8×C173C412yescomplemented
10#124×C173C2212yes—
11#123×C173—22no—
12C22⋊Q8×C173—22no—
13#117×C173C411yescharacteristic, complemented
14C4⋊Dic173—42no—
15C22⋊Dic173—24no—
16C4×Dic173—42no—
17#41×C173—42no—
18C2×Q8×C173—21no—
19C2×Q8×C173—42no—
20C2×Q8×C173C2×C412yes—
21C2×Q8×C173C2×C411yescharacteristic
22C2×D4×C173D412yes—
23C2×D4×C173C2311yescharacteristic
24C22×C4×C173—21no—
25C22×C4×C173C2×C411yescharacteristic
26C22×C4×C173D412yes—
27C4⋊C4×C173—24no—
28C4⋊C4×C173—21no—
29C22⋊C4×C173—21no—
30C22⋊C4×C173—24no—
31C42×C173C2×C412yes—
32C2×Dic173—44no—
33C23×C173C22⋊C411yescharacteristic
34C23×C173C2×D412yes—
35Q8×C173—41no—
36Q8×C173—42no—
37Q8×C173—42no—
38D4×C173—42no—
39D4×C173—41no—
40C2×C4×C173—22no—
41C2×C4×C173—22no—
42C2×C4×C173—22no—
43C2×C4×C173—22no—
44C2×C4×C173—22no—
45C2×C4×C173—42no—
46C2×C4×C173C22⋊C412yes—
47C2×C4×C173C22⋊C411yescharacteristic
48C2×C4×C173C22×C411yescharacteristic
49C22×C173—21no—
50C22×C173—22no—
51C22×C173—41no—
52C22×C173—42no—
53C22×C173C2×C22⋊C411yescharacteristic
54C4×C173—22no—
55C4×C173—21no—
56C4×C173—42no—
57C4×C173—42no—
58C4×C173#9912yes—
59Dic173—84no—
60C2×C173—21no—
61C2×C173—42no—
62C2×C173—41no—
63C2×C173C2×#9911yescharacteristic
64C173#212911yescharacteristic, complemented, Sylow
65#2129—1731nomaximal, Sylow
66#536D17311yescharacteristic
67#410—1732no—
68#354—1734no—
69C2×#41C2×D17311yescharacteristic
70C22×Q8Dic17311yescharacteristic, complemented
71C4⋊Q8Dic17312yescomplemented
72#124C2×D17312yes—
73#123—22no—
74C22⋊Q8—22no—
75#117—3462no—
76#117Dic17311yescharacteristic, complemented
77#99—3462no—
78#99—1734no—
79#41—42no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8C2×Dic17312yes—
83C2×Q8C2×Dic17311yescharacteristic
84C2×D4C22⋊D17312yes—
85C2×D4C22×D17311yescharacteristic
86C22×C4—21no—
87C22×C4C2×Dic17311yescharacteristic
88C22×C4C22⋊D17312yes—
89C4⋊C4—24no—
90C4⋊C4—21no—
91C4⋊C4—6922no—
92C22⋊C4—24no—
93C22⋊C4—21no—
94C22⋊C4—3464no—
95C42—6922no—
96C42C2×Dic17312yes—
97C23C22⋊Dic17311yescharacteristic
98C23C2×C22⋊D17312yes—
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—6924no—
111C2×C4C22⋊Dic17312yes—
112C2×C4C22⋊Dic17311yescharacteristic
113C2×C4C22×Dic17311yescharacteristic
114C22—21no—
115C22—22no—
116C22—42no—
117C22—41no—
118C22C2×C22⋊Dic17311yescharacteristic
119C4—21no—
120C4—22no—
121C4—42no—
122C4—42no—
123C4—13844no—
124C4#1914959412yes—
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×#1914959411yescharacteristic
subgroupG/H
centerC2C2×#19149594
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#536×C173C2
socleC2×C173C2×#99
prime radicalC2×#41×C173C22
cosocleC2×D4×C173C23
hypercenter#536D173
hypocenterC173#2129
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #26011286C2semidirect
8. C22×Q8×C173C4semidirect
9. C4⋊Q8×C173C4semidirect
13. #117×C173C4semidirect
64. C173#2129semidirect
70. C22×Q8Dic173semidirect
71. C4⋊Q8Dic173semidirect
76. #117Dic173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260114652
#260112862
#536×C1732
#2129173