Finite Group Databasebeta

← #46835685

Subgroups — #46835685

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#31774259C212yesmaximal, complemented
3#31773639C211yesmaximal, characteristic, complemented
4#31773636C212yesmaximal, complemented
5#491×C271C211yesmaximal, characteristic, complemented
6#22576148C2212yes—
7#22576145—22no—
8#22576138—22no—
9#22576135—22no—
10#22576083—22no—
11#22576080—22no—
12#22576071C2212yes—
13C4⋊Q8×C271C2211yescharacteristic
14#125×C271C2211yescharacteristic
15#103×C271—22no—
16#97×C271C2211yescharacteristic
17C2×Dic542D412yes—
18C2×D1084—22no—
19C2×D1084D412yes—
20C2×C4×D271—22no—
21C2×C271⋊C8—22no—
22Dic271⋊C4—22no—
23(C2×D271)⋊C4—22no—
24(C2×D271)⋊C4—22no—
25(C2×D271)⋊C4—22no—
26C2×Q8×C271—21no—
27C2×C8×C271—21no—
28C4⋊C4×C271—22no—
29C4⋊C4×C271—22no—
30C4⋊C4×C271D412yes—
31C42×C271C2311yescharacteristic
32C22×D271—22no—
33Dic542—42no—
34D1084—42no—
35D1084—82no—
36C2×Dic271—22no—
37C4×D271—82no—
38C271⋊C8—42no—
39Q8×C271—42no—
40C2×C4×C271—22no—
41C2×C4×C271—41no—
42C2×C4×C271C2×D412yes—
43C2×C4×C271C2×D411yescharacteristic
44C8×C271—41no—
45C22×C271#12011yescharacteristic
46C2×D271—42no—
47C2×D271—42no—
48C2×D271—42no—
49C4×C271—21no—
50C4×C271—22no—
51C4×C271—42no—
52C4×C271—81no—
53Dic271—42no—
54C2×C271#45411yescharacteristic
55C2×C271#45512yes—
56D271—82no—
57C271#166211yescharacteristic, complemented, Sylow
58#1662—2711nomaximal, Sylow
59#562—2711no—
60#491D27111yescharacteristic, complemented
61#489—2712no—
62#330—2711no—
63#329—2712no—
64C4⋊Q8C2×D27111yescharacteristic
65#125C2×D27111yescharacteristic
66#124—2712no—
67#123—5422no—
68C22⋊Q8—5422no—
69#121—5422no—
70#103—22no—
71#103—5422no—
72#102—5422no—
73#97—2712no—
74#97C2×D27111yescharacteristic
75C2×Q8—21no—
76C2×Q8—2712no—
77C2×D4—2712no—
78C2×D4—5422no—
79C22×C4—5422no—
80C2×C8—21no—
81C2×C8—5422no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—5422no—
85C4⋊C4C22⋊D27112yes—
86C22⋊C4—5422no—
87C22⋊C4—5422no—
88C22⋊C4—5422no—
89C42C22×D27111yescharacteristic
90C23—5422no—
91Q8—42no—
92Q8—10842no—
93D4—10842no—
94D4—21682no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—5422no—
98C2×C4—21682no—
99C2×C4D4×D27112yes—
100C2×C4C2×C22⋊D27111yescharacteristic
101C8—41no—
102C8—10842no—
103C22—10842no—
104C22—10842no—
105C22—10842no—
106C22#2257612811yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—10842no—
112C2—21682no—
113C2#3177406911yescharacteristic
114C2#3177407312yes—
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#491×C271C2
socleC22×C271#120
prime radical#31774645C2
hypercenter#491D271
hypocenterC271#1662
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #31774259C2semidirect
3. #31773639C2semidirect
4. #31773636C2semidirect
5. #491×C271C2semidirect
57. C271#1662semidirect
60. #491D271semidirect

Maximal subgroups

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

subgroupindex
#317746452
#317742592
#317736392
#317736362
#491×C2712
#1662271