Finite Group Databasebeta

← #46834957

Subgroups — #46834957

Order 34688.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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#31774453C211yesmaximal, characteristic
2#31773849C212yesmaximal
3#31773639C211yesmaximal, characteristic, complemented
4#31773638C212yesmaximal, complemented
5#330×C271C211yesmaximal, characteristic, complemented
6C2×#16864637—22no—
7#22576159C412yes—
8#22576152C2211yescharacteristic
9#22576138—22no—
10#22576125—22no—
11#22576118C412yes—
12C42×D271C2211yescharacteristic
13#22576071C2212yes—
14C8×Dic271—22no—
15#125×C271C2211yescharacteristic
16#97×C271C2212yes—
17C2×Dic542—22no—
18C2×C4×D271C2×C411yescharacteristic
19C2×C4×D271D412yes—
20#16864637—28no—
21C2×C271⋊C8—22no—
22C4⋊Dic271C2×C412yes—
23Dic271⋊C4—22no—
24Dic271⋊C4—22no—
25(C2×D271)⋊C4—22no—
26C4×Dic271—22no—
27C4×Dic271C2×C411yescharacteristic
28C4×Dic271D412yes—
29C2×C8×C271—22no—
30C4⋊C4×C271—22no—
31C4⋊C4×C271C2×C412yes—
32C42×C271C2311yescharacteristic
33C22×D271C22⋊C411yescharacteristic
34Dic542—44no—
35C2×Dic271—22no—
36C2×Dic271C22⋊C411yescharacteristic
37C2×Dic271C22⋊C412yes—
38C4×D271—22no—
39C4×D271—24no—
40C271⋊C8—24no—
41C2×C4×C271—22no—
42C2×C4×C271C22×C411yescharacteristic
43C2×C4×C271C2×D412yes—
44C8×C271—24no—
45C22×C271C2×C22⋊C411yescharacteristic
46C2×D271—21no—
47C2×D271—22no—
48C4×C271—21no—
49C4×C271—22no—
50C4×C271—42no—
51Dic271—42no—
52Dic271#10112yes—
53Dic271#10414yes—
54C2×C271C2×#10111yescharacteristic
55C2×C271C2×#10412yes—
56D271—41no—
57C271#152611yescharacteristic, complemented, Sylow
58#1526—2711nomaximal, Sylow
59#533—2711no—
60C4×M16—2712no—
61#330—2711no—
62#330—2712no—
63#330D27111yescharacteristic, complemented
64C2×M16—5422no—
65C4⋊Q8—2712no—
66#125—2711no—
67#125C2×D27111yescharacteristic
68C22⋊Q8—5422no—
69C4×Q8—5422no—
70#117—2712no—
71C2×C42—2711no—
72#97—2712no—
73#97C2×D27112yes—
74C4×C8—5422no—
75C2×Q8—5422no—
76C22×C4—2711no—
77C22×C4—2712no—
78M16—5428no—
79C2×C8—22no—
80C2×C8—5422no—
81C4⋊C4—22no—
82C4⋊C4—2712no—
83C4⋊C4—5422no—
84C4⋊C4—5422no—
85C4⋊C4C4×D27112yes—
86C22⋊C4—5422no—
87C42—2711no—
88C42—2712no—
89C42—5422no—
90C42C22×D27111yescharacteristic
91C23—2711no—
92Q8—10844no—
93C2×C4—22no—
94C2×C4—2711no—
95C2×C4—2712no—
96C2×C4—5424no—
97C2×C4—5422no—
98C2×C4—5422no—
99C2×C4C2×C4×D27111yescharacteristic
100C2×C4D4×D27112yes—
101C8—24no—
102C8—5424no—
103C22—5421no—
104C22—5422no—
105C22C22⋊C4×D27111yescharacteristic
106C4—21no—
107C4—22no—
108C4—42no—
109C4—2712no—
110C4—2714no—
111C4—10842no—
112C2—10841no—
113C2#101×D27111yescharacteristic
114C2#104×D27112yes—
subgroupG/H
centerC22C22⋊C4×D271
commutatorC2×C4×C271C22×C4
FrattiniC42C22×D271
Fitting#330×C271C2
socleC22×C271C2×C22⋊C4
prime radicalC22×D271C22⋊C4
cosocleC42×C271C23
hypercenter#330D271
hypocenterC271#1526
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #31773639C2semidirect
4. #31773638C2semidirect
5. #330×C271C2semidirect
57. C271#1526semidirect
63. #330D271semidirect

Maximal subgroups

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

subgroupindex
#317744532
#317738492
#317736392
#317736382
#330×C2712
#1526271