Finite Group Databasebeta

← #46835371

Subgroups — #46835371

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#31774227C212yesmaximal
3#31773633C212yesmaximal
4#31773623C211yesmaximal, characteristic
5#398×C271C211yesmaximal, characteristic
6#22576159—22no—
7#22576151—22no—
8#22576139—21no—
9#22576139C2212yes—
10#22576119—21no—
11#22576091—22no—
12#22576083—22no—
13C22⋊(C271⋊C8)C2212yes—
14C2×C4⋊C4×C271—21no—
15C2×C4⋊C4×C271C2211yescharacteristic
16C2×C4⋊C4×C271C2211yescharacteristic
17#95×C271—21no—
18#95×C271C2211yescharacteristic
19C2×Dic542—21no—
20C2×Dic542D412yes—
21C2×C271⋊C8—22no—
22C4⋊Dic271—21no—
23C4⋊Dic271—21no—
24C4⋊Dic271—21no—
25C4⋊Dic271D412yes—
26Dic271⋊C4—22no—
27Dic271⋊C4—22no—
28C22⋊Dic271—22no—
29C4×Dic271—22no—
30C22×C4×C271—21no—
31C22×C4×C271—21no—
32C22×C4×C271D411yescharacteristic
33C22×C4×C271D411yescharacteristic
34C22×C4×C271C2311yescharacteristic
35C4⋊C4×C271—22no—
36C4⋊C4×C271—22no—
37C4⋊C4×C271—42no—
38Dic542—42no—
39Dic542—81no—
40C2×Dic271—22no—
41C2×Dic271—22no—
42C271⋊C8—42no—
43C23×C271C2×D411yescharacteristic
44C2×C4×C271—21no—
45C2×C4×C271—21no—
46C2×C4×C271—22no—
47C2×C4×C271—21no—
48C2×C4×C271—41no—
49C2×C4×C271—41no—
50C2×C4×C271—41no—
51C2×C4×C271—41no—
52C2×C4×C271—41no—
53C2×C4×C271—41no—
54C2×C4×C271—41no—
55C2×C4×C271SD1612yes—
56C2×C4×C271C2×D412yes—
57C22×C271—21no—
58C22×C271—21no—
59C22×C271#12011yescharacteristic
60C22×C271C2×SD1611yescharacteristic
61C22×C271#13711yescharacteristic
62C4×C271—22no—
63C4×C271—22no—
64C4×C271—42no—
65C4×C271—41no—
66C4×C271—81no—
67Dic271—42no—
68Dic271—42no—
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#398D27111yescharacteristic
79#328—2712no—
80#324—2711no—
81C4⋊Q8—5422no—
82#125—5422no—
83C22⋊Q8—2712no—
84C22⋊Q8—5421no—
85#117—5421no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D27111yescharacteristic
88C2×C4⋊C4C2×D27111yescharacteristic
89#106—5422no—
90#103—5422no—
91C22⋊C8—2712no—
92#95—21no—
93#95C2×D27111yescharacteristic
94C2×Q8—2712no—
95C2×Q8—5421no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D27111yescharacteristic
99C22×C4C22⋊D27111yescharacteristic
100C22×C4C22×D27111yescharacteristic
101C2×C8—5422no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—2712no—
106C4⋊C4—5421no—
107C4⋊C4—5421no—
108C4⋊C4—5422no—
109C4⋊C4—5421no—
110C4⋊C4—5422no—
111C22⋊C4—5422no—
112C42—5422no—
113C23C2×C22⋊D27111yescharacteristic
114Q8—10842no—
115Q8—21681no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—5422no—
128C2×C4—5422no—
129C2×C4#1686464212yes—
130C2×C4D4×D27112yes—
131C8—10842no—
132C22—21no—
133C22—21no—
134C22#2257612811yescharacteristic
135C22C2×#1686464211yescharacteristic
136C22#2257619011yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—10842no—
143C4—10842no—
144C2—21no—
145C2—21no—
146C2#3177403511yescharacteristic
147C2#3177407911yescharacteristic
148C2#3177408611yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#398×C271C2
socleC22×C271#120
prime radicalC23×C271C2×D4
hypercenter#398D271
hypocenterC271#1611
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

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
#317742272
#317736332
#317736232
#398×C2712
#1611271