Finite Group Databasebeta

← #46835688

Subgroups — #46835688

Order 34688.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#31774645C211yesmaximal, characteristic
2#31774267C211yesmaximal, characteristic, complemented
3#31774257C212yesmaximal, complemented
4#31773636C212yesmaximal, complemented
5#330×C271C211yesmaximal, characteristic, complemented
6#22576160C2211yescharacteristic
7#22576148C2212yes—
8#22576145—22no—
9#22576138—22no—
10#22576135—22no—
11#22576085—22no—
12#22576083—22no—
13#22576082—22no—
14#22576071C2211yescharacteristic
15#125×C271C2211yescharacteristic
16#97×C271C2212yes—
17C2×Dic542—21no—
18C2×Dic542D412yes—
19C2×D1084—22no—
20C2×D1084D412yes—
21C2×C4×D271—22no—
22C2×C271⋊C8—21no—
23C4⋊Dic271—22no—
24Dic271⋊C4—22no—
25(C2×D271)⋊C4—22no—
26(C2×D271)⋊C4—22no—
27(C2×D271)⋊C4—22no—
28C2×C8×C271—22no—
29C4⋊C4×C271—22no—
30C4⋊C4×C271D412yes—
31C42×C271C2311yescharacteristic
32C22×D271—22no—
33Dic542—42no—
34Dic542—42no—
35D1084—42no—
36D1084—82no—
37C2×Dic271—22no—
38C2×Dic271—41no—
39C4×D271—82no—
40C271⋊C8—41no—
41C2×C4×C271—22no—
42C2×C4×C271C2×D412yes—
43C2×C4×C271C2×D411yescharacteristic
44C8×C271—42no—
45C22×C271#12011yescharacteristic
46C2×D271—42no—
47C2×D271—42no—
48C2×D271—42no—
49C4×C271—21no—
50C4×C271—22no—
51C4×C271—42no—
52Dic271—42no—
53Dic271—81no—
54C2×C271#45411yescharacteristic
55C2×C271#45512yes—
56D271—82no—
57C271#166211yescharacteristic, complemented, Sylow
58#1662—2711nomaximal, Sylow
59#562—2711no—
60#491—2711no—
61#489—2712no—
62#330D27111yescharacteristic, complemented
63#329—2712no—
64C4⋊Q8—2711no—
65#125C2×D27111yescharacteristic
66#124—2712no—
67#123—5422no—
68C22⋊Q8—5422no—
69#121—5422no—
70#103—5422no—
71#103—5422no—
72#102—5422no—
73#97—2711no—
74#97C2×D27112yes—
75C2×Q8—2712no—
76C2×Q8—5421no—
77C2×D4—2712no—
78C2×D4—5422no—
79C22×C4—5422no—
80C2×C8—22no—
81C2×C8—5421no—
82C4⋊C4—22no—
83C4⋊C4—5422no—
84C4⋊C4—5422no—
85C4⋊C4D108412yes—
86C22⋊C4—5422no—
87C22⋊C4—5422no—
88C22⋊C4—5422no—
89C42C22×D27111yescharacteristic
90C23—5422no—
91Q8—10842no—
92Q8—10842no—
93D4—10842no—
94D4—21682no—
95C2×C4—22no—
96C2×C4—5422no—
97C2×C4—10841no—
98C2×C4—21682no—
99C2×C4D4×D27112yes—
100C2×C4C2×D108411yescharacteristic
101C8—42no—
102C8—10841no—
103C22—10842no—
104C22—10842no—
105C22—10842no—
106C22#2257613011yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—10842no—
111C4—21681no—
112C2—21682no—
113C2#3177407211yescharacteristic
114C2#3177407612yes—
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#330×C271C2
socleC22×C271#120
prime radical#31774645C2
hypercenter#330D271
hypocenterC271#1662
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #31774267C2semidirect
3. #31774257C2semidirect
4. #31773636C2semidirect
5. #330×C271C2semidirect
57. C271#1662semidirect
62. #330D271semidirect

Maximal subgroups

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

subgroupindex
#317746452
#317742672
#317742572
#317736362
#330×C2712
#1662271