Finite Group Databasebeta

← #36325813

Subgroups — #36325813

Order 21376.

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#24923243C212yesmaximal
2#24923064C214yesmaximal, complemented
3#536×C167C211yesmaximal, characteristic
4#19128791—22no—
5#19128747—22no—
6#19128747C2214yes—
7C2×#41×C167C2211yescharacteristic
8C22×Q8×C167C411yescharacteristic, complemented
9C4⋊Q8×C167C412yescomplemented
10#124×C167C2212yes—
11#123×C167—22no—
12C22⋊Q8×C167—22no—
13#117×C167C411yescharacteristic, complemented
14C4⋊Dic167—42no—
15C22⋊Dic167—24no—
16C4×Dic167—42no—
17#41×C167—42no—
18C2×Q8×C167—21no—
19C2×Q8×C167—42no—
20C2×Q8×C167C2×C412yes—
21C2×Q8×C167C2×C411yescharacteristic
22C2×D4×C167D412yes—
23C2×D4×C167C2311yescharacteristic
24C22×C4×C167—21no—
25C22×C4×C167C2×C411yescharacteristic
26C22×C4×C167D412yes—
27C4⋊C4×C167—21no—
28C4⋊C4×C167—24no—
29C22⋊C4×C167—21no—
30C22⋊C4×C167—24no—
31C42×C167C2×C412yes—
32C2×Dic167—44no—
33C23×C167C22⋊C411yescharacteristic
34C23×C167C2×D412yes—
35Q8×C167—41no—
36Q8×C167—42no—
37Q8×C167—42no—
38D4×C167—42no—
39D4×C167—41no—
40C2×C4×C167—22no—
41C2×C4×C167—22no—
42C2×C4×C167—22no—
43C2×C4×C167—22no—
44C2×C4×C167—22no—
45C2×C4×C167—42no—
46C2×C4×C167C22⋊C412yes—
47C2×C4×C167C22⋊C411yescharacteristic
48C2×C4×C167C22×C411yescharacteristic
49C22×C167—21no—
50C22×C167—22no—
51C22×C167—41no—
52C22×C167—42no—
53C22×C167C2×C22⋊C411yescharacteristic
54C4×C167—22no—
55C4×C167—21no—
56C4×C167—42no—
57C4×C167—42no—
58C4×C167#9912yes—
59Dic167—84no—
60C2×C167—21no—
61C2×C167—42no—
62C2×C167—41no—
63C2×C167C2×#9911yescharacteristic
64C167#212911yescharacteristic, complemented, Sylow
65#2129—1671nomaximal, Sylow
66#536D16711yescharacteristic
67#410—1672no—
68#354—1674no—
69C2×#41C2×D16711yescharacteristic
70C22×Q8Dic16711yescharacteristic, complemented
71C4⋊Q8Dic16712yescomplemented
72#124C2×D16712yes—
73#123—22no—
74C22⋊Q8—22no—
75#117—3342no—
76#117Dic16711yescharacteristic, complemented
77#99—3342no—
78#99—1674no—
79#41—42no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8C2×Dic16712yes—
83C2×Q8C2×Dic16711yescharacteristic
84C2×D4C22⋊D16712yes—
85C2×D4C22×D16711yescharacteristic
86C22×C4—21no—
87C22×C4C2×Dic16711yescharacteristic
88C22×C4C22⋊D16712yes—
89C4⋊C4—24no—
90C4⋊C4—21no—
91C4⋊C4—6682no—
92C22⋊C4—21no—
93C22⋊C4—24no—
94C22⋊C4—3344no—
95C42—6682no—
96C42C2×Dic16712yes—
97C23C22⋊Dic16711yescharacteristic
98C23C2×C22⋊D16712yes—
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—6684no—
111C2×C4C22⋊Dic16712yes—
112C2×C4C22⋊Dic16711yescharacteristic
113C2×C4C22×Dic16711yescharacteristic
114C22—21no—
115C22—22no—
116C22—41no—
117C22—42no—
118C22C2×C22⋊Dic16711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C4—13364no—
124C4#1912874712yes—
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×#1912874711yescharacteristic
subgroupG/H
centerC2C2×#19128747
commutatorC2×C4×C167C22×C4
FrattiniC2×D4C22×D167
Fitting#536×C167C2
socleC2×C167C2×#99
prime radicalC2×#41×C167C22
cosocleC2×D4×C167C23
hypercenter#536D167
hypocenterC167#2129
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #24923064C2semidirect
8. C22×Q8×C167C4semidirect
9. C4⋊Q8×C167C4semidirect
13. #117×C167C4semidirect
64. C167#2129semidirect
70. C22×Q8Dic167semidirect
71. C4⋊Q8Dic167semidirect
76. #117Dic167semidirect

Maximal subgroups

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

subgroupindex
#249232432
#249230642
#536×C1672
#2129167