Finite Group Databasebeta

← #46837413

Subgroups — #46837413

Order 34688.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#22576193C212yesmaximal, complemented
2#31774478C211yesmaximal, characteristic
3C2×#22576086C212yesmaximal, complemented
4#31773802C211yesmaximal, characteristic, complemented
5#343×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2212yes—
7C22×Dic542C2211yescharacteristic
8#22576193—24no—
9#22576193—24no—
10C2×Dic1084—22no—
11C2×#16864643—22no—
12#22576140—22no—
13#22576135—22no—
14#22576134—22no—
15C2×C4×Dic271C2211yescharacteristic
16#22576086—24no—
17C271⋊#95—22no—
18C2×M16×C271C2212yes—
19C2×C4⋊C4×C271C2211yescharacteristic
20#16864654—22no—
21#16864654—22no—
22#16864654—22no—
23#16864654—22no—
24C2×Dic542—22no—
25C2×Dic542—22no—
26C2×Dic542—21no—
27C2×Dic542D412yes—
28C2×Dic542D412yescomplemented
29C2×D1084—22no—
30C2×D1084D412yes—
31C2×C22⋊D271—22no—
32C22×Dic271—21no—
33C22×Dic271—21no—
34C2×C4×D271—22no—
35Dic1084—28no—
36#16864643—28no—
37C4⋊Dic271—22no—
38(C2×D271)⋊C4—24no—
39C4×Dic271—22no—
40C22×C4×C271D412yes—
41C22×C4×C271C2311yescharacteristic
42M16×C271—22no—
43M16×C271D414yescomplemented
44C2×C8×C271—22no—
45C4⋊C4×C271—22no—
46C22×D271—22no—
47Dic542—24no—
48Dic542—22no—
49Dic542—22no—
50Dic542—42no—
51D1084—22no—
52D1084—22no—
53D1084—82no—
54C22⋊D271—42no—
55C22⋊D271—42no—
56C2×Dic271—22no—
57C2×Dic271—22no—
58C2×Dic271—22no—
59C2×Dic271—41no—
60C2×Dic271—41no—
61C2×Dic271—41no—
62C2×Dic271—41no—
63C4×D271—42no—
64C4×D271—42no—
65C23×C271C2×D411yescharacteristic
66C2×C4×C271—22no—
67C2×C4×C271—42no—
68C2×C4×C271C2×D411yescharacteristic
69C2×C4×C271C2×D412yes—
70C2×C4×C271C2×D412yes—
71C2×C4×C271#4111yescharacteristic
72C8×C271—24no—
73C22×C271—21no—
74C22×C271—21no—
75C22×C271#12011yescharacteristic
76C22×C271#12111yescharacteristic
77C22×C271#12711yescharacteristic
78C2×D271—42no—
79C2×D271—42no—
80C2×D271—42no—
81C4×C271—42no—
82C4×C271#12012yes—
83C4×C271#12112yes—
84Dic271—42no—
85Dic271—42no—
86Dic271—42no—
87C2×C271—21no—
88C2×C271—21no—
89C2×C271#39211yescharacteristic
90C2×C271#45511yescharacteristic
91C2×C271#45611yescharacteristic
92D271—82no—
93C271#200711yescharacteristic, complemented, Sylow
94#2007—2711nomaximal, Sylow
95C2×#137—2712no—
96#537—2711no—
97C2×#104—2712no—
98#391—2711no—
99#343D27111yescharacteristic, complemented
100C2×#41—2712no—
101C22×Q8—2711no—
102#137—5424no—
103#137—5424no—
104C2×Q16—5422no—
105C2×SD16—5422no—
106C2×M16C2×D27112yes—
107C22⋊Q8—5422no—
108#121—5422no—
109#121—5422no—
110C2×C4⋊C4C2×D27111yescharacteristic
111C2×C42—2711no—
112#104—5424no—
113#95—5422no—
114#41—5422no—
115#41—5422no—
116#41—5422no—
117#41—5422no—
118C2×Q8—2712no—
119C2×Q8—2712no—
120C2×Q8—5421no—
121C2×Q8—5422no—
122C2×Q8—5422no—
123C2×D4—2712no—
124C2×D4—5422no—
125C2×D4—5422no—
126C22×C4—5421no—
127C22×C4—5421no—
128C22×C4—5422no—
129C22×C4D108412yes—
130C22×C4C22×D27111yescharacteristic
131Q16—5428no—
132SD16—5428no—
133M16—22no—
134M16D108414yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—5422no—
138C22⋊C4—5424no—
139C42—5422no—
140C23—5422no—
141C23C2×D108411yescharacteristic
142Q8—5424no—
143Q8—5422no—
144Q8—5422no—
145Q8—10842no—
146D4—5422no—
147D4—5422no—
148D4—10842no—
149D4—10842no—
150D4—21682no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—5422no—
154C2×C4—5422no—
155C2×C4—5422no—
156C2×C4—10841no—
157C2×C4—10841no—
158C2×C4—10841no—
159C2×C4—10841no—
160C2×C4—10842no—
161C2×C4—10842no—
162C2×C4D4×D27111yescharacteristic
163C2×C4D4×D27112yes—
164C2×C4C2×D108412yes—
165C2×C4#1686465511yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—10842no—
170C22—10842no—
171C22—10842no—
172C22#2257613011yescharacteristic
173C22#2257613511yescharacteristic
174C22#2257615711yescharacteristic
175C4—42no—
176C4—10842no—
177C4—10842no—
178C4—10842no—
179C4#2257613012yes—
180C4#2257613512yes—
181C2—21no—
182C2—21no—
183C2—21682no—
184C2#3177380711yescharacteristic
185C2#3177407711yescharacteristic
186C2#3177408211yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#343×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#343D271
hypocenterC271#2007
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
3. C2×#22576086C2semidirect
4. #31773802C2semidirect
5. #343×C271C2semidirect
28. C2×Dic542D4semidirect
43. M16×C271D4semidirect
93. C271#2007semidirect
99. #343D271semidirect
134. M16D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
#317744782
C2×#225760862
#317738022
#343×C2712
#2007271