Finite Group Databasebeta

← #46835673

Subgroups — #46835673

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#31774647C211yesmaximal, characteristic
2#31774303C212yesmaximal, complemented
3#31773638C212yesmaximal, complemented
4#31773637C211yesmaximal, characteristic
5#497×C271C211yesmaximal, characteristic
6#22576158—21no—
7#22576153—22no—
8#22576152C2212yes—
9#22576141—21no—
10#22576137—22no—
11#22576093—22no—
12#22576091—22no—
13#22576071C2212yes—
14C2×Q16×C271—21no—
15C2×SD16×C271—21no—
16C4⋊Q8×C271C2211yescharacteristic
17#124×C271C2211yescharacteristic
18#97×C271C2211yescharacteristic
19C2×Dic542—21no—
20C22×Dic271—21no—
21C2×C271⋊C8—22no—
22C4⋊Dic271D412yes—
23C4⋊Dic271D412yes—
24Dic271⋊C4—22no—
25Dic271⋊C4—22no—
26Dic271⋊C4—22no—
27C22⋊Dic271—22no—
28C22⋊Dic271—21no—
29C4×Dic271—21no—
30C2×Q8×C271—21no—
31C2×Q8×C271D411yescharacteristic
32C2×D4×C271D411yescharacteristic
33Q16×C271—42no—
34SD16×C271—42no—
35C2×C8×C271—21no—
36C4⋊C4×C271—22no—
37C22⋊C4×C271—22no—
38C42×C271C2311yescharacteristic
39Dic542—81no—
40C2×Dic271—22no—
41C2×Dic271—22no—
42C2×Dic271—81no—
43C271⋊C8—24no—
44C23×C271—21no—
45Q8×C271—41no—
46Q8×C271—41no—
47Q8×C271—41no—
48D4×C271—41no—
49C2×C4×C271—21no—
50C2×C4×C271—41no—
51C2×C4×C271C2×D412yes—
52C2×C4×C271C2×D411yescharacteristic
53C8×C271—22no—
54C22×C271—41no—
55C22×C271—42no—
56C22×C271#12011yescharacteristic
57C4×C271—21no—
58C4×C271—22no—
59C4×C271—41no—
60C4×C271—81no—
61Dic271—42no—
62Dic271—42no—
63C2×C271—81no—
64C2×C271#45311yescharacteristic
65C2×C271#45612yes—
66C271#166011yescharacteristic, complemented, Sylow
67#1660—2711nomaximal, Sylow
68#563—2711no—
69#501—2712no—
70#497D27111yescharacteristic
71#330—2712no—
72#329—2711no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—5421no—
76C4⋊Q8C2×D27111yescharacteristic
77#126—5422no—
78#125—2712no—
79#124C2×D27111yescharacteristic
80#123—5421no—
81C22⋊Q8—5422no—
82#107—5422no—
83#106—5422no—
84#97—2712no—
85#97C2×D27111yescharacteristic
86C2×Q8—21no—
87C2×Q8—5421no—
88C2×Q8C22⋊D27111yescharacteristic
89C2×D4C22⋊D27111yescharacteristic
90C22×C4—5421no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—5422no—
95C4⋊C4—22no—
96C4⋊C4—2712no—
97C4⋊C4—2712no—
98C4⋊C4—5422no—
99C4⋊C4—5422no—
100C4⋊C4—5422no—
101C22⋊C4—22no—
102C22⋊C4—5422no—
103C22⋊C4—5421no—
104C42—5421no—
105C42C22×D27111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—21681no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—5422no—
115C2×C4—5422no—
116C2×C4—21681no—
117C2×C4D4×D27112yes—
118C2×C4C2×C22⋊D27111yescharacteristic
119C8—22no—
120C8—5424no—
121C22—41no—
122C22—42no—
123C22#2257612811yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—10842no—
129C4—10842no—
130C2—81no—
131C2#3177406411yescharacteristic
132C2#3177407812yes—
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#497×C271C2
socleC22×C271#120
prime radicalC2×D4×C271D4
hypercenter#497D271
hypocenterC271#1660
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #31774303C2semidirect
3. #31773638C2semidirect
66. C271#1660semidirect

Maximal subgroups

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

subgroupindex
#317746472
#317743032
#317736382
#317736372
#497×C2712
#1660271