Finite Group Databasebeta

← #46835374

Subgroups — #46835374

Order 34688.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774528C211yesmaximal, characteristic
2#31774228C212yesmaximal
3#31773827C211yesmaximal, characteristic
4#31773635C212yesmaximal
5#324×C271C211yesmaximal, characteristic
6#22576159—22no—
7#22576151—22no—
8#22576139—21no—
9#22576139C2212yes—
10#22576119—21no—
11C2×C4⋊Dic271C2211yescharacteristic
12C2×Dic271⋊C4—21no—
13#22576092—22no—
14#22576085—22no—
15C271⋊#95—21no—
16C271⋊#95C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19C2×Dic542—21no—
20C2×Dic542D412yes—
21C22×Dic271—21no—
22C22×Dic271—21no—
23C4⋊Dic271—21no—
24C4⋊Dic271—21no—
25C4⋊Dic271—21no—
26C4⋊Dic271—22no—
27C4⋊Dic271D412yes—
28Dic271⋊C4—22no—
29Dic271⋊C4—22no—
30Dic271⋊C4—42no—
31C22⋊Dic271—22no—
32C4×Dic271—22no—
33C22×C4×C271D411yescharacteristic
34C22×C4×C271D411yescharacteristic
35C22×C4×C271C2311yescharacteristic
36C2×C8×C271—22no—
37C4⋊C4×C271—22no—
38Dic542—42no—
39Dic542—81no—
40C2×Dic271—22no—
41C2×Dic271—22no—
42C2×Dic271—22no—
43C2×Dic271—41no—
44C2×Dic271—41no—
45C2×Dic271—41no—
46C2×Dic271—41no—
47C2×Dic271—41no—
48C23×C271C2×D411yescharacteristic
49C2×C4×C271—21no—
50C2×C4×C271—21no—
51C2×C4×C271—21no—
52C2×C4×C271—41no—
53C2×C4×C271—41no—
54C2×C4×C271SD1612yes—
55C2×C4×C271C2×D412yes—
56C8×C271—42no—
57C22×C271—21no—
58C22×C271—21no—
59C22×C271#12011yescharacteristic
60C22×C271C2×SD1611yescharacteristic
61C22×C271#13711yescharacteristic
62C4×C271—22no—
63C4×C271—22no—
64C4×C271—41no—
65Dic271—42no—
66Dic271—42no—
67Dic271—42no—
68Dic271—81no—
69C2×C271—21no—
70C2×C271—21no—
71C2×C271#44811yescharacteristic
72C2×C271#45611yescharacteristic
73C2×C271#45811yescharacteristic
74C271#161111yescharacteristic, complemented, Sylow
75#1611—2711nomaximal, Sylow
76#544—2711no—
77#483—2712no—
78#398—2711no—
79#328—2712no—
80#324D27111yescharacteristic
81C4⋊Q8—5422no—
82#125—5422no—
83C22⋊Q8—2712no—
84C22⋊Q8—5421no—
85#117—5421no—
86C2×C4⋊C4—2711no—
87C2×C4⋊C4—5421no—
88C2×C4⋊C4C2×D27111yescharacteristic
89#106—5422no—
90#103—5422no—
91C22⋊C8C2×D27112yes—
92#95—2711no—
93#95—5421no—
94C2×Q8—2712no—
95C2×Q8—5421no—
96C22×C4—5421no—
97C22×C4—5421no—
98C22×C4D108411yescharacteristic
99C22×C4D108411yescharacteristic
100C22×C4C22×D27111yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—2712no—
104C4⋊C4—5421no—
105C4⋊C4—5421no—
106C4⋊C4—5422no—
107C4⋊C4—5421no—
108C4⋊C4—5422no—
109C4⋊C4—5422no—
110C4⋊C4—10842no—
111C22⋊C4—5422no—
112C42—5422no—
113C23C2×D108411yescharacteristic
114Q8—10842no—
115Q8—21681no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—5422no—
122C2×C4—5422no—
123C2×C4—5422no—
124C2×C4—10841no—
125C2×C4—10841no—
126C2×C4—10841no—
127C2×C4—10841no—
128C2×C4—10841no—
129C2×C4#1686464312yes—
130C2×C4D4×D27112yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#2257613011yescharacteristic
135C22C2×#1686464311yescharacteristic
136C22#2257619311yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—10842no—
141C4—10842no—
142C4—10842no—
143C4—21681no—
144C2—21no—
145C2—21no—
146C2#3177403811yescharacteristic
147C2#3177408211yescharacteristic
148C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radicalC23×C271C2×D4
hypercenter#324D271
hypocenterC271#1611
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
74. C271#1611semidirect

Maximal subgroups

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

subgroupindex
#317745282
#317742282
#317738272
#317736352
#324×C2712
#1611271