Finite Group Databasebeta

← #46835671

Subgroups — #46835671

Order 34688.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 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#31774639C211yesmaximal, characteristic
2#31774292C211yesmaximal, characteristic, complemented
3#31774257C212yesmaximal, complemented
4#31773636C212yesmaximal, complemented
5#329×C271C211yesmaximal, characteristic, complemented
6C2×#16864644—21no—
7C2×#16864642—21no—
8#22576160C2211yescharacteristic, complemented
9#22576149—21no—
10#22576148C2212yes—
11#22576146—22no—
12#22576130—22no—
13#22576128—21no—
14#22576085—22no—
15#22576082—22no—
16#22576071C2211yescharacteristic, complemented
17#124×C271C2211yescharacteristic
18#97×C271C2212yescomplemented
19C23×D271—21no—
20C2×Dic542—21no—
21C2×Dic542D412yes—
22C2×D1084—21no—
23C2×D1084D412yes—
24C2×C22⋊D271—22no—
25#16864644—42no—
26#16864642—42no—
27C2×C271⋊C8—21no—
28C4⋊Dic271—22no—
29C22⋊Dic271—21no—
30(C2×D271)⋊C4—22no—
31(C2×D271)⋊C4—22no—
32(C2×D271)⋊C4—22no—
33C4×Dic271—21no—
34C2×Q8×C271D411yescharacteristic
35C2×D4×C271D411yescharacteristic
36C2×C8×C271—22no—
37C22⋊C4×C271—22no—
38C42×C271C2311yescharacteristic
39C22×D271—22no—
40C22×D271—81no—
41C22×D271—82no—
42Dic542—42no—
43Dic542—41no—
44Dic542—41no—
45D1084—42no—
46D1084—81no—
47C22⋊D271—82no—
48C2×Dic271—22no—
49C2×Dic271—41no—
50C271⋊C8—22no—
51C23×C271—21no—
52Q8×C271—41no—
53D4×C271—41no—
54C2×C4×C271—21no—
55C2×C4×C271C2×D412yes—
56C2×C4×C271C2×D411yescharacteristic
57C8×C271—42no—
58C22×C271—41no—
59C22×C271—42no—
60C22×C271#12011yescharacteristic
61C2×D271—42no—
62C2×D271—42no—
63C2×D271—42no—
64C2×D271—84no—
65C4×C271—21no—
66C4×C271—22no—
67C4×C271—41no—
68Dic271—42no—
69Dic271—81no—
70C2×C271—81no—
71C2×C271#45311yescharacteristic
72C2×C271#45512yes—
73D271—82no—
74C271#165911yescharacteristic, complemented, Sylow
75#1659—2711nomaximal, Sylow
76H(3,𝔽4)—2711no—
77#497—2711no—
78#489—2712no—
79#329—2712no—
80#329D27111yescharacteristic, complemented
81C2×Q16—5421no—
82C2×SD16—5421no—
83C4⋊Q8—2711no—
84#124—2712no—
85#124—5422no—
86#124—5421no—
87#124C2×D27111yescharacteristic
88#120—5422no—
89#120—5421no—
90#103—5422no—
91#102—5422no—
92#97—2711no—
93#97C2×D27112yescomplemented
94C24—5421no—
95C2×Q8—2712no—
96C2×Q8—5421no—
97C2×Q8D108411yescharacteristic
98C2×D4—2712no—
99C2×D4—5421no—
100C2×D4—5422no—
101C2×D4D108411yescharacteristic
102Q16—10842no—
103SD16—10842no—
104C2×C8—22no—
105C2×C8—5421no—
106C4⋊C4—5422no—
107C22⋊C4—22no—
108C22⋊C4—5422no—
109C22⋊C4—5421no—
110C22⋊C4—5422no—
111C22⋊C4—5422no—
112C42—5421no—
113C42C22×D27111yescharacteristic
114C23—21no—
115C23—5422no—
116C23—21681no—
117C23—21682no—
118Q8—41no—
119Q8—10842no—
120Q8—10841no—
121Q8—10841no—
122D4—41no—
123D4—10842no—
124D4—21681no—
125D4—21682no—
126C2×C4—21no—
127C2×C4—5422no—
128C2×C4—10841no—
129C2×C4D4×D27112yes—
130C2×C4C2×D108411yescharacteristic
131C8—42no—
132C8—5422no—
133C22—41no—
134C22—42no—
135C22—10842no—
136C22—10842no—
137C22—10842no—
138C22—21684no—
139C22#2257613011yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—10842no—
144C4—21681no—
145C2—81no—
146C2—21682no—
147C2#3177406711yescharacteristic
148C2#3177407612yes—
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#329×C271C2
socleC22×C271#120
prime radical#31774639C2
hypercenter#329D271
hypocenterC271#1659
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31774292C2semidirect
3. #31774257C2semidirect
4. #31773636C2semidirect
5. #329×C271C2semidirect
8. #22576160C22semidirect
16. #22576071C22semidirect
18. #97×C271C22semidirect
74. C271#1659semidirect
80. #329D271semidirect
93. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746392
#317742922
#317742572
#317736362
#329×C2712
#1659271