Finite Group Databasebeta

← #30666402

Subgroups — #30666402

Order 16768.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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
2C22×C4×Dic131C211yesmaximal, characteristic, complemented
3C2×#16862172C212yesmaximal, complemented
4#22553073C212yesmaximal, complemented
5#407×C131C211yesmaximal, characteristic, complemented
6C2×#14523513C2211yescharacteristic
7C23×Dic131—21no—
8#16862229—21no—
9#16862225C412yescomplemented
10#16862220C412yescomplemented
11#16862215—21no—
12C4×C22⋊D131—22no—
13#16862205C2211yescharacteristic
14C2×C4×Dic131—24no—
15C2×C4×Dic131—21no—
16C2×C4×Dic131C2212yes—
17#16862172—24no—
18C23×C4×C131C2211yescharacteristic
19C2×M16×C131C2212yes—
20C22⋊C8×C131—22no—
21#14523513—22no—
22#14523513—22no—
23C2×Dic262C2×C411yescharacteristic, complemented
24C2×D524C2×C411yescharacteristic, complemented
25C2×C22⋊D131—22no—
26C2×C22⋊D131—21no—
27C22×Dic131—28no—
28C22×Dic131—24no—
29C22×Dic131—22no—
30C2×C4×D131—21no—
31C4⋊Dic131C2×C412yescomplemented
32Dic131⋊C4—22no—
33C22⋊Dic131—22no—
34C22⋊Dic131—21no—
35(C2×D131)⋊C4—22no—
36C4×Dic131—24no—
37C4×Dic131—22no—
38C4×Dic131—21no—
39C4×Dic131D414yescomplemented
40C24×C131D411yescharacteristic
41C22×C4×C131—22no—
42C22×C4×C131—22no—
43C22×C4×C131C2×C412yes—
44C22×C4×C131D412yes—
45C22×C4×C131D411yescharacteristic
46C22×C4×C131C2311yescharacteristic
47M16×C131—22no—
48M16×C131—22no—
49C2×C8×C131—22no—
50C22×D131—21no—
51Dic262—22no—
52D524—22no—
53C22⋊D131—44no—
54C22⋊D131—41no—
55C22⋊D131—41no—
56C2×Dic131—216no—
57C2×Dic131—28no—
58C2×Dic131—24no—
59C2×Dic131—22no—
60C2×Dic131—21no—
61C4×D131—42no—
62C23×C131—24no—
63C23×C131—21no—
64C23×C131—21no—
65C23×C131C22⋊C412yes—
66C23×C131C2×D411yescharacteristic
67C2×C4×C131—21no—
68C2×C4×C131—21no—
69C2×C4×C131—24no—
70C2×C4×C131—24no—
71C2×C4×C131C22⋊C412yes—
72C2×C4×C131C22×C411yescharacteristic
73C2×C4×C131C2×D412yes—
74C2×C4×C131C2×D411yescharacteristic
75C2×C4×C131#4112yes—
76C8×C131—42no—
77C22×C131—24no—
78C22×C131—24no—
79C22×C131—22no—
80C22×C131—22no—
81C22×C131—22no—
82C22×C131#10414yescomplemented
83C22×C131C2×C22⋊C411yescharacteristic
84C22×C131#12011yescharacteristic
85C22×C131#12311yescharacteristic
86C2×D131—41no—
87C2×D131—42no—
88C4×C131—22no—
89C4×C131C4×D412yes—
90C4×C131#12112yes—
91Dic131—28no—
92Dic131—41no—
93Dic131—42no—
94C2×C131—21no—
95C2×C131—24no—
96C2×C131—21no—
97C2×C131#38611yescharacteristic
98C2×C131C2×#10412yes—
99D131—81no—
100C131#189611yescharacteristic, complemented, Sylow
101#1896—1311nomaximal, Sylow
102#525—1311no—
103C22×C42—1311no—
104C2×#104—1312no—
105#407D13111yescharacteristic, complemented
106#339—1312no—
107C2×#41—1311no—
108C23×C4—2621no—
109C23×C4C2×D13111yescharacteristic
110C2×M16C2×D13112yes—
111#123—2621no—
112C22⋊Q8—1312no—
113#121—1312no—
114#120—2621no—
115C4×D4—2622no—
116#117—1311no—
117C2×C42—2621no—
118C2×C42—2624no—
119C2×C42—1312no—
120#104—2624no—
121C22⋊C8—22no—
122C24D52411yescharacteristic
123#41—2622no—
124#41—2622no—
125C2×Q8—1311no—
126C2×D4—2622no—
127C2×D4—2621no—
128C2×D4—1311no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—2624no—
132C22×C4—2628no—
133C22×C4—2622no—
134C22×C4—2621no—
135C22×C4C4×D13112yes—
136C22×C4C22⋊D13112yes—
137C22×C4D52411yescharacteristic
138C22×C4C22×D13111yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—2622no—
143C4⋊C4—1312no—
144C22⋊C4—2622no—
145C22⋊C4—2621no—
146C22⋊C4—2622no—
147C42—2624no—
148C42—2622no—
149C42—2621no—
150C42—1314no—
151C23—21no—
152C23—21no—
153C23—24no—
154C23—2621no—
155C23(C2×D131)⋊C412yes—
156C23C2×D52411yescharacteristic
157Q8—2622no—
158D4—2622no—
159D4—5241no—
160D4—5241no—
161D4—5244no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—26216no—
167C2×C4—2628no—
168C2×C4—2624no—
169C2×C4—2622no—
170C2×C4—2621no—
171C2×C4—5242no—
172C2×C4(C2×D131)⋊C412yes—
173C2×C4C2×C4×D13111yescharacteristic
174C2×C4D4×D13112yes—
175C2×C4C2×C22⋊D13111yescharacteristic
176C2×C4#1452351212yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—5241no—
184C22—5242no—
185C22#1686217214yescomplemented
186C22C2×(C2×D131)⋊C411yescharacteristic
187C22#1686221611yescharacteristic
188C22#1686223011yescharacteristic
189C4—22no—
190C4—2628no—
191C4—5241no—
192C4—5242no—
193C4(C22⋊D131)⋊C412yes—
194C4#1686221812yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—10481no—
199C2#2255318911yescharacteristic
200C2C2×#1686217212yes—
subgroupG/H
centerC2×C4C2×C22⋊D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#407×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553823C2
cosocleC22×C4×C131C23
hypercenter#407D131
hypocenterC131#1896
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic131C2semidirect
3. C2×#16862172C2semidirect
4. #22553073C2semidirect
5. #407×C131C2semidirect
9. #16862225C4semidirect
10. #16862220C4semidirect
23. C2×Dic262C2×C4semidirect
24. C2×D524C2×C4semidirect
31. C4⋊Dic131C2×C4semidirect
39. C4×Dic131D4semidirect
82. C22×C131#104semidirect
100. C131#1896semidirect
105. #407D131semidirect
185. C22#16862172semidirect

Maximal subgroups

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

subgroupindex
#225538232
C22×C4×Dic1312
C2×#168621722
#225530732
#407×C1312
#1896131