Finite Group Databasebeta

← #30675894

Subgroups — #30675894

Order 16768.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#22553823C211yesmaximal, characteristic
2C2×#16862205C211yesmaximal, characteristic, complemented
3#22553654C214yesmaximal, complemented
4#22553637C214yesmaximal, complemented
5#22553354C212yesmaximal, complemented
6#22553314C212yesmaximal, complemented
7#407×C131C211yesmaximal, characteristic, complemented
8C2×#14523513C2211yescharacteristic
9#16862229—21no—
10#16862225C2212yes—
11#16862220C2212yes—
12#16862215—21no—
13C4×C22⋊D131—22no—
14#16862205—22no—
15#16862205C2211yescharacteristic
16#16862205C2214yes—
17C2×C4⋊Dic131C2212yes—
18C2×C22⋊Dic131—21no—
19C2×C4×Dic131—21no—
20#16862180C2214yescomplemented
21#16862178C2214yescomplemented
22#16862171C2214yescomplemented
23#16862168C2214yescomplemented
24C23×C4×C131C2211yescharacteristic
25C2×M16×C131C2212yes—
26C22⋊C8×C131C2214yes—
27#14523513—24no—
28C2×Dic262C2311yescharacteristic
29C2×D524C2311yescharacteristic
30C2×C22⋊D131—22no—
31C2×C22⋊D131—21no—
32C22×Dic131—22no—
33C2×C4×D131—21no—
34C4⋊Dic131—22no—
35C4⋊Dic131C2312yes—
36C4⋊Dic131C2314yes—
37Dic131⋊C4—22no—
38C22⋊Dic131—22no—
39C22⋊Dic131—21no—
40C22⋊Dic131—24no—
41(C2×D131)⋊C4—22no—
42C4×Dic131—24no—
43C4×Dic131—21no—
44C24×C131D411yescharacteristic
45C22×C4×C131—24no—
46C22×C4×C131D411yescharacteristic
47C22×C4×C131D412yes—
48C22×C4×C131C2312yes—
49C22×C4×C131C2311yescharacteristic
50M16×C131—24no—
51C2×C8×C131C2314yes—
52C22×D131—21no—
53Dic262—22no—
54D524—22no—
55C22⋊D131—44no—
56C22⋊D131—42no—
57C2×Dic131—24no—
58C2×Dic131—22no—
59C2×Dic131—21no—
60C2×Dic131—44no—
61C4×D131—42no—
62C23×C131—22no—
63C23×C131—42no—
64C23×C131C2×D412yes—
65C23×C131C2×D411yescharacteristic
66C2×C4×C131—22no—
67C2×C4×C131—24no—
68C2×C4×C131—24no—
69C2×C4×C131C2×D412yes—
70C2×C4×C131C2×D411yescharacteristic
71C2×C4×C131#4114yes—
72C2×C4×C131C2411yescharacteristic
73C8×C131—24no—
74C22×C131—22no—
75C22×C131—22no—
76C22×C131—22no—
77C22×C131—22no—
78C22×C131—44no—
79C22×C131#12312yes—
80C22×C131C22×D411yescharacteristic
81C2×D131—41no—
82C2×D131—42no—
83C4×C131—22no—
84C4×C131#12312yes—
85C4×C131C2×#4112yes—
86Dic131—41no—
87Dic131—44no—
88Dic131—42no—
89C2×C131—22no—
90C2×C131—42no—
91C2×C131C2×#12311yescharacteristic
92C2×C131#57512yes—
93D131—81no—
94C131#309311yescharacteristic, complemented, Sylow
95#3093—1311nomaximal, Sylow
96#525—1311no—
97C2×#117—1311no—
98#485—1314no—
99#482—1314no—
100#428—1312no—
101#417—1312no—
102#407D13111yescharacteristic, complemented
103C2×#41—1311no—
104C23×C4C2×D13111yescharacteristic
105C2×M16C2×D13112yes—
106#123—2621no—
107C22⋊Q8—1312no—
108#121—1312no—
109#120—2621no—
110C4×D4—2622no—
111#117—2622no—
112#117—1314no—
113#117—1311no—
114C2×C4⋊C4—1312no—
115C2×C22⋊C4—2621no—
116C2×C42—2621no—
117#107—1314no—
118#106—1314no—
119#103—1314no—
120#102—1314no—
121C22⋊C8C2×D13114yes—
122C24D52411yescharacteristic
123#41—2624no—
124C2×Q8—1311no—
125C2×D4—2622no—
126C2×D4—2621no—
127C2×D4—1311no—
128C22×C4—24no—
129C22×C4—2622no—
130C22×C4—2621no—
131C22×C4D52411yescharacteristic
132C22×C4D52412yes—
133C22×C4C22×D13112yes—
134C22×C4C22×D13111yescharacteristic
135M16—24no—
136C2×C8C22×D13114yes—
137C4⋊C4—2622no—
138C4⋊C4—2622no—
139C4⋊C4—1314no—
140C4⋊C4—1312no—
141C22⋊C4—2624no—
142C22⋊C4—2621no—
143C22⋊C4—2622no—
144C22⋊C4—2622no—
145C42—2624no—
146C42—2621no—
147C23—22no—
148C23—42no—
149C23—2621no—
150C23C2×D52412yes—
151C23C2×D52411yescharacteristic
152Q8—2622no—
153D4—2622no—
154D4—5244no—
155D4—5242no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—2624no—
160C2×C4—2622no—
161C2×C4—2621no—
162C2×C4—5244no—
163C2×C4—5242no—
164C2×C4C2×D52412yes—
165C2×C4C2×D52411yescharacteristic
166C2×C4#1452351214yes—
167C2×C4C23×D13111yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—5242no—
175C22—5241no—
176C22#1686223012yes—
177C22C22×D52411yescharacteristic
178C4—22no—
179C4—5244no—
180C4—5242no—
181C4—5241no—
182C4#1686223012yes—
183C4C2×#1452351212yes—
184C2—22no—
185C2—42no—
186C2—10481no—
187C2C2×#1686223011yescharacteristic
188C2#2255411212yes—
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#407×C131C2
socleC22×C131C22×D4
prime radical#22553823C2
hypercenter#407D131
hypocenterC131#3093
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16862205C2semidirect
3. #22553654C2semidirect
4. #22553637C2semidirect
5. #22553354C2semidirect
6. #22553314C2semidirect
7. #407×C131C2semidirect
20. #16862180C22semidirect
21. #16862178C22semidirect
22. #16862171C22semidirect
23. #16862168C22semidirect
94. C131#3093semidirect
102. #407D131semidirect

Maximal subgroups

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

subgroupindex
#225538232
C2×#168622052
#225536542
#225536372
#225533542
#225533142
#407×C1312
#3093131