Finite Group Databasebeta

← #46836896

Subgroups — #46836896

Order 34688.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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#31774464C211yesmaximal, characteristic
2#31773915C212yesmaximal, complemented
3C2×#22576077C211yesmaximal, characteristic, complemented
4#31773653C212yesmaximal, complemented
5#518×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2211yescharacteristic
7C2×#16864638C2212yes—
8#22576156C412yescomplemented
9#22576147C412yescomplemented
10#22576131—22no—
11#22576129—22no—
12#22576119C2211yescharacteristic
13#22576088—24no—
14#22576077—22no—
15C22×D4×C271C2211yescharacteristic
16C4×D4×C271—24no—
17#117×C271C2212yes—
18C2×C22⋊C4×C271—22no—
19#16864654—22no—
20#16864654—22no—
21C2×Dic542C2×C411yescharacteristic, complemented
22C2×D1084C2×C411yescharacteristic, complemented
23C2×C22⋊D271—24no—
24C2×C22⋊D271—21no—
25C2×C4×D271—21no—
26#16864638—22no—
27#16864638—22no—
28C2×C271⋊C8—22no—
29C4⋊Dic271—21no—
30C22⋊Dic271—24no—
31C22⋊Dic271—21no—
32C4×Dic271C2×C412yescomplemented
33C24×C271D412yes—
34C2×D4×C271—22no—
35C2×D4×C271—22no—
36C2×D4×C271C2×C412yes—
37C2×D4×C271D412yes—
38C22×C4×C271—24no—
39C22×C4×C271C2311yescharacteristic
40C4⋊C4×C271—22no—
41C22⋊C4×C271—24no—
42C22⋊C4×C271—22no—
43C42×C271D414yescomplemented
44C22×D271—21no—
45Dic542—22no—
46D1084—22no—
47C22⋊D271—41no—
48C22⋊D271—41no—
49C22⋊D271—84no—
50C2×Dic271—22no—
51C2×Dic271—21no—
52C4×D271—42no—
53C271⋊C8—42no—
54C23×C271—22no—
55C23×C271—22no—
56C23×C271—82no—
57C23×C271C22⋊C414yes—
58C23×C271C2×D411yescharacteristic
59D4×C271—24no—
60D4×C271—24no—
61C2×C4×C271—24no—
62C2×C4×C271—84no—
63C2×C4×C271C22×C411yescharacteristic
64C2×C4×C271C2×D412yes—
65C2×C4×C271C2×D411yescharacteristic
66C2×C4×C271#4112yes—
67C22×C271—24no—
68C22×C271—41no—
69C22×C271—44no—
70C22×C271—44no—
71C22×C271—82no—
72C22×C271C2×C22⋊C411yescharacteristic
73C22×C271#12011yescharacteristic
74C22×C271#12311yescharacteristic
75C2×D271—42no—
76C2×D271—41no—
77C4×C271—44no—
78C4×C271C4×D412yes—
79C4×C271#12112yes—
80Dic271—42no—
81Dic271—41no—
82C2×C271—21no—
83C2×C271—21no—
84C2×C271—21no—
85C2×C271—44no—
86C2×C271#38611yescharacteristic
87D271—81no—
88C271#191111yescharacteristic, complemented, Sylow
89#1911—2711nomaximal, Sylow
90#535—2711no—
91#518D27111yescharacteristic, complemented
92#421—2712no—
93C2×#100—2711no—
94H(3,ℤ/4)—2712no—
95C2×#41—2711no—
96C22×D4C2×D27111yescharacteristic
97C2×M16—2712no—
98#127—2712no—
99#124—2712no—
100#121—5422no—
101#120—5422no—
102C4×D4—24no—
103#117—2711no—
104#117C2×D27112yes—
105C2×C22⋊C4—22no—
106#104—5424no—
107#100—5422no—
108C24C22⋊D27112yes—
109#41—5422no—
110#41—5422no—
111C2×Q8—2711no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—2711no—
115C2×D4—5421no—
116C2×D4—5424no—
117C2×D4C4×D27112yes—
118C2×D4D108412yes—
119C22×C4—24no—
120C22×C4—5421no—
121C22×C4C22×D27111yescharacteristic
122M16—5422no—
123M16—5422no—
124C2×C8—5422no—
125C4⋊C4—22no—
126C4⋊C4—5421no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—5424no—
130C22⋊C4—5421no—
131C42—2712no—
132C42C22⋊D27114yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—5421no—
137C23(C2×D271)⋊C414yes—
138C23C2×C22⋊D27111yescharacteristic
139Q8—5422no—
140D4—24no—
141D4—24no—
142D4—5422no—
143D4—10841no—
144D4—10841no—
145D4—21684no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—5422no—
149C2×C4—5421no—
150C2×C4—10842no—
151C2×C4C2×C4×D27111yescharacteristic
152C2×C4C2×C22⋊D27112yes—
153C2×C4C2×D108411yescharacteristic
154C2×C4#1686465412yes—
155C8—10842no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—10842no—
162C22—10841no—
163C22C2×(C2×D271)⋊C411yescharacteristic
164C22#2257612911yescharacteristic
165C22#2257614311yescharacteristic
166C4—44no—
167C4—10842no—
168C4—10841no—
169C4C4×C22⋊D27112yes—
170C4#2257613412yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—21681no—
176C2#3177377611yescharacteristic
subgroupG/H
centerC2#31773776
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#518×C271C2
socleC2×C271#386
prime radical#31774464C2
cosocleC22×C4×C271C23
hypercenter#518D271
hypocenterC271#1911
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #31773915C2semidirect
3. C2×#22576077C2semidirect
4. #31773653C2semidirect
5. #518×C271C2semidirect
8. #22576156C4semidirect
9. #22576147C4semidirect
21. C2×Dic542C2×C4semidirect
22. C2×D1084C2×C4semidirect
32. C4×Dic271C2×C4semidirect
43. C42×C271D4semidirect
88. C271#1911semidirect
91. #518D271semidirect
132. C42C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317744642
#317739152
C2×#225760772
#317736532
#518×C2712
#1911271