Finite Group Databasebeta

← #46836889

Subgroups — #46836889

Order 34688.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#31774664C211yesmaximal, characteristic, complemented
2#31774461C211yesmaximal, characteristic
3#31773901C212yesmaximal, complemented
4#31773653C212yesmaximal, complemented
5#410×C271C211yesmaximal, characteristic, complemented
6C22×D1084C2211yescharacteristic
7#22576167—24no—
8C2×#16864638—22no—
9C2×#16864638C2212yes—
10#22576156C412yescomplemented
11#22576149C412yescomplemented
12#22576134—22no—
13#22576128—22no—
14#22576119C2211yescharacteristic
15#22576080—24no—
16C2×#41×C271C2211yescharacteristic
17#117×C271C2212yes—
18#99×C271—22no—
19C23×D271—21no—
20C2×D1084—22no—
21C2×D1084—21no—
22C2×D1084—22no—
23C2×D1084C2×C412yes—
24C2×C22⋊D271—24no—
25#16864638—24no—
26#16864638—22no—
27#16864638—22no—
28C2×C271⋊C8—24no—
29C2×C271⋊C8D414yescomplemented
30C4⋊Dic271—21no—
31C22⋊Dic271—21no—
32(C2×D271)⋊C4—24no—
33C4×Dic271C2×C412yes—
34#41×C271—22no—
35#41×C271—22no—
36C2×Q8×C271C2×C411yescharacteristic
37C2×D4×C271C2×C411yescharacteristic
38C2×D4×C271D412yes—
39C22×C4×C271D412yes—
40C22×C4×C271C2311yescharacteristic
41C4⋊C4×C271—22no—
42C22⋊C4×C271—22no—
43C42×C271—22no—
44C22×D271—22no—
45C22×D271—42no—
46C22×D271—82no—
47D1084—24no—
48D1084—41no—
49D1084—41no—
50C22⋊D271—84no—
51C2×Dic271—22no—
52C271⋊C8—24no—
53C23×C271C22⋊C412yes—
54C23×C271C2×D411yescharacteristic
55Q8×C271—22no—
56D4×C271—24no—
57D4×C271—22no—
58C2×C4×C271—22no—
59C2×C4×C271—22no—
60C2×C4×C271—42no—
61C2×C4×C271C22⋊C412yes—
62C2×C4×C271C22×C411yescharacteristic
63C2×C4×C271C2×D412yes—
64C2×C4×C271C2×D411yescharacteristic
65C2×C4×C271#4112yes—
66C22×C271—22no—
67C22×C271—41no—
68C22×C271—42no—
69C22×C271C2×C22⋊C411yescharacteristic
70C22×C271#12112yes—
71C2×D271—44no—
72C2×D271—42no—
73C2×D271—84no—
74C4×C271—22no—
75C4×C271—82no—
76C4×C271C4×D412yes—
77C4×C271#12011yescharacteristic
78C4×C271#12311yescharacteristic
79Dic271—42no—
80C2×C271—21no—
81C2×C271—22no—
82C2×C271—42no—
83C2×C271#38611yescharacteristic
84D271—82no—
85C271#191011yescharacteristic, complemented, Sylow
86#1910—2711nomaximal, Sylow
87#568—2711no—
88#535—2711no—
89#418—2712no—
90#410D27111yescharacteristic, complemented
91H(3,ℤ/4)—2712no—
92C2×#41C2×D27111yescharacteristic
93C22×D4—2711no—
94#131—5424no—
95C2×M16—2712no—
96C2×M16—5422no—
97#127—2712no—
98#124—2712no—
99#121—5422no—
100#120—5422no—
101#117—2711no—
102#117C2×D27112yes—
103#102—5424no—
104#99—22no—
105C24—5421no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D27111yescharacteristic
109C2×D4—2712no—
110C2×D4—5424no—
111C2×D4—5422no—
112C2×D4—5421no—
113C2×D4—5422no—
114C2×D4C4×D27111yescharacteristic
115C2×D4C22⋊D27112yes—
116C22×C4D108412yes—
117C22×C4C22×D27111yescharacteristic
118M16—5424no—
119M16—5422no—
120M16—5422no—
121C2×C8—2714no—
122C2×C8—5424no—
123C4⋊C4—22no—
124C4⋊C4—5421no—
125C22⋊C4—22no—
126C22⋊C4—5421no—
127C22⋊C4—5424no—
128C42—22no—
129C42—2712no—
130C23—5422no—
131C23—10842no—
132C23—21682no—
133C23(C2×D271)⋊C412yes—
134C23C2×C22⋊D27111yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—5424no—
139D4—10841no—
140D4—10841no—
141D4—21684no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—5422no—
146C2×C4(C2×D271)⋊C412yes—
147C2×C4C2×C4×D27111yescharacteristic
148C2×C4D4×D27112yes—
149C2×C4C2×D108411yescharacteristic
150C2×C4#1686465312yes—
151C8—5424no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—10844no—
156C22—10842no—
157C22—21684no—
158C22C2×(C2×D271)⋊C411yescharacteristic
159C22#2257613212yes—
160C4—22no—
161C4—82no—
162C4—10842no—
163C4(C22⋊D271)⋊C412yes—
164C4#2257613011yescharacteristic
165C4#2257614411yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—21682no—
170C2#3177377411yescharacteristic
subgroupG/H
centerC2#31773774
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#410×C271C2
socleC2×C271#386
prime radical#31774461C2
cosocleC22×C4×C271C23
hypercenter#410D271
hypocenterC271#1910
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #31774664C2semidirect
3. #31773901C2semidirect
4. #31773653C2semidirect
5. #410×C271C2semidirect
10. #22576156C4semidirect
11. #22576149C4semidirect
29. C2×C271⋊C8D4semidirect
85. C271#1910semidirect
90. #410D271semidirect

Maximal subgroups

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

subgroupindex
#317746642
#317744612
#317739012
#317736532
#410×C2712
#1910271