Finite Group Databasebeta

← #46835340

Subgroups — #46835340

Order 34688.

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

normal subgroups only

This group has 238 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#31774055C212yesmaximal
3#31773817C211yesmaximal, characteristic
4#31773635C212yesmaximal
5#324×C271C211yesmaximal, characteristic
6C22×Dic542C2211yescharacteristic
7C2×Dic1084—22no—
8#22576159—22no—
9#22576151—22no—
10#22576139—21no—
11#22576139C2212yes—
12#22576119—21no—
13C2×Dic271⋊C4—21no—
14#22576085—22no—
15C271⋊#95—21no—
16C271⋊#95C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19C2×Dic542—21no—
20C2×Dic542—22no—
21C2×Dic542—42no—
22C2×Dic542D412yes—
23C22×Dic271—21no—
24C22×Dic271—21no—
25Dic1084—44no—
26C4⋊Dic271—21no—
27C4⋊Dic271—21no—
28C4⋊Dic271—21no—
29C4⋊Dic271D412yes—
30Dic271⋊C4—22no—
31Dic271⋊C4—22no—
32Dic271⋊C4—42no—
33C22⋊Dic271—22no—
34C4×Dic271—22no—
35C22×C4×C271D411yescharacteristic
36C22×C4×C271D411yescharacteristic
37C22×C4×C271C2311yescharacteristic
38C2×C8×C271—22no—
39C4⋊C4×C271—22no—
40Dic542—24no—
41Dic542—42no—
42Dic542—42no—
43Dic542—81no—
44C2×Dic271—22no—
45C2×Dic271—22no—
46C2×Dic271—22no—
47C2×Dic271—22no—
48C2×Dic271—41no—
49C2×Dic271—41no—
50C2×Dic271—41no—
51C2×Dic271—41no—
52C23×C271C2×D411yescharacteristic
53C2×C4×C271—21no—
54C2×C4×C271—21no—
55C2×C4×C271—21no—
56C2×C4×C271—41no—
57C2×C4×C271—41no—
58C2×C4×C271Q1612yescomplemented
59C2×C4×C271C2×D412yes—
60C8×C271—42no—
61C22×C271—21no—
62C22×C271—21no—
63C22×C271#12011yescharacteristic
64C22×C271C2×Q1611yescharacteristic
65C22×C271#13711yescharacteristic
66C4×C271—22no—
67C4×C271—22no—
68C4×C271—41no—
69Dic271—42no—
70Dic271—42no—
71Dic271—42no—
72Dic271—42no—
73C2×C271—21no—
74C2×C271—21no—
75C2×C271#45111yescharacteristic
76C2×C271#45611yescharacteristic
77C2×C271B(4,𝔽2)11yescharacteristic
78C271#160511yescharacteristic, complemented, Sylow
79#1605—2711nomaximal, Sylow
80#544—2711no—
81#451—2712no—
82#395—2711no—
83#328—2712no—
84#324D27111yescharacteristic
85C22×Q8—2711no—
86C2×Q16—5422no—
87C4⋊Q8—5422no—
88#125—5422no—
89C22⋊Q8—2712no—
90C22⋊Q8—5421no—
91#117—5421no—
92C2×C4⋊C4—5421no—
93C2×C4⋊C4C2×D27111yescharacteristic
94#103—5422no—
95C22⋊C8C2×D27112yes—
96#95—2711no—
97#95—5421no—
98C2×Q8—2712no—
99C2×Q8—5421no—
100C2×Q8—5422no—
101C2×Q8—10842no—
102C22×C4—5421no—
103C22×C4—5421no—
104C22×C4D108411yescharacteristic
105C22×C4D108411yescharacteristic
106C22×C4C22×D27111yescharacteristic
107Q16—10844no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—2712no—
111C4⋊C4—5421no—
112C4⋊C4—5421no—
113C4⋊C4—5422no—
114C4⋊C4—5421no—
115C4⋊C4—5422no—
116C4⋊C4—10842no—
117C22⋊C4—5422no—
118C42—5422no—
119C23C2×D108411yescharacteristic
120Q8—5424no—
121Q8—10842no—
122Q8—10842no—
123Q8—21681no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—5422no—
130C2×C4—5422no—
131C2×C4—5422no—
132C2×C4—5422no—
133C2×C4—10841no—
134C2×C4—10841no—
135C2×C4—10841no—
136C2×C4—10841no—
137C2×C4Dic108412yescomplemented
138C2×C4D4×D27112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2257613011yescharacteristic
143C22C2×Dic108411yescharacteristic
144C22#2257619311yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—10842no—
149C4—10842no—
150C4—10842no—
151C4—10842no—
152C2—21no—
153C2—21no—
154C2#3177405511yescharacteristic
155C2#3177408211yescharacteristic
156C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radicalC23×C271C2×D4
hypercenter#324D271
hypocenterC271#1605
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
58. C2×C4×C271Q16semidirect
78. C271#1605semidirect
137. C2×C4Dic1084semidirect

Maximal subgroups

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

subgroupindex
#317745282
#317740552
#317738172
#317736352
#324×C2712
#1605271