Finite Group Databasebeta

← #30664945

Subgroups — #30664945

Order 16768.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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#22553870C211yesmaximal, characteristic
2#22553625C211yesmaximal, characteristic, complemented
3#22553238C211yesmaximal, characteristic, complemented
4#22553047C211yesmaximal, characteristic, complemented
5#22553046C211yesmaximal, characteristic, complemented
6#22553035C211yesmaximal, characteristic, complemented
7#480×C131C211yesmaximal, characteristic, complemented
8C2×D4×D131—21no—
9#16862231—21no—
10#16862228—21no—
11#16862220C2211yescharacteristic
12#16862218—21no—
13#16862216—21no—
14#16862214—21no—
15C2×(C2×D131)⋊C4—21no—
16C2×(C2×D131)⋊C4C2211yescharacteristic
17#16862179—21no—
18#16862166—21no—
19C22⋊(C131⋊C8)C2211yescharacteristic, complemented
20C131⋊#95—21no—
21C131⋊#95C2211yescharacteristic, complemented
22#121×C131C2211yescharacteristic
23C2×C4⋊C4×C131C2211yescharacteristic, complemented
24#107×C131—21no—
25#102×C131—21no—
26C22⋊C8×C131C2211yescharacteristic, complemented
27C23×D131—21no—
28C23×D131D411yescharacteristic
29C2×D524—21no—
30C2×D524D411yescharacteristic
31C2×C22⋊D131—21no—
32C2×C22⋊D131—21no—
33C2×C22⋊D131—21no—
34D4×D131—42no—
35D4×D131—42no—
36C22×Dic131—21no—
37C22×Dic131D411yescharacteristic
38C2×C4×D131—21no—
39C2×C131⋊C8—21no—
40C4⋊Dic131D411yescharacteristic
41Dic131⋊C4—21no—
42C22⋊Dic131—21no—
43(C2×D131)⋊C4—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—21no—
46(C2×D131)⋊C4—21no—
47(C2×D131)⋊C4—42no—
48C2×D4×C131—21no—
49C2×D4×C131D411yescharacteristic
50C22×C4×C131—21no—
51C22×C4×C131C2311yescharacteristic
52C2×C8×C131—21no—
53C4⋊C4×C131—21no—
54C4⋊C4×C131—21no—
55C4⋊C4×C131D411yescharacteristic
56C22⋊C4×C131—21no—
57C22×D131—21no—
58C22×D131—21no—
59C22×D131—21no—
60C22×D131—42no—
61C22×D131—41no—
62C22×D131—41no—
63C22×D131—81no—
64C22×D131—81no—
65C22×D131D812yes—
66D524—41no—
67D524—42no—
68C22⋊D131—42no—
69C22⋊D131—41no—
70C22⋊D131—41no—
71C22⋊D131—81no—
72C2×Dic131—21no—
73C2×Dic131—21no—
74C2×Dic131—41no—
75C2×Dic131—41no—
76C2×Dic131—41no—
77C2×Dic131—41no—
78C4×D131—42no—
79C131⋊C8—41no—
80C23×C131—21no—
81C23×C131C2×D411yescharacteristic
82D4×C131—41no—
83D4×C131—41no—
84D4×C131—41no—
85C2×C4×C131—21no—
86C2×C4×C131—41no—
87C2×C4×C131—41no—
88C2×C4×C131—41no—
89C2×C4×C131—41no—
90C2×C4×C131C2×D411yescharacteristic
91C2×C4×C131C2×D411yescharacteristic
92C8×C131—41no—
93C22×C131—21no—
94C22×C131—21no—
95C22×C131—41no—
96C22×C131—41no—
97C22×C131—41no—
98C22×C131#12011yescharacteristic
99C22×C131C2×D811yescharacteristic
100C22×C131AGL(1,ℤ/8)11yescharacteristic
101C2×D131—22no—
102C2×D131—41no—
103C2×D131—41no—
104C2×D131—42no—
105C2×D131—41no—
106C2×D131—41no—
107C2×D131—41no—
108C2×D131—81no—
109C2×D131—84no—
110C4×C131—21no—
111C4×C131—21no—
112C4×C131—41no—
113C4×C131—81no—
114Dic131—41no—
115Dic131—41no—
116Dic131—81no—
117C2×C131—21no—
118C2×C131—21no—
119C2×C131—81no—
120C2×C131#44711yescharacteristic
121C2×C131#45511yescharacteristic
122C2×C131#45811yescharacteristic
123D131—42no—
124D131—81no—
125C131#160711yescharacteristic, complemented, Sylow
126#1607—1311nomaximal, Sylow
127#534—1311no—
128#480—1311no—
129#480D13111yescharacteristic, complemented
130#397—1311no—
131#327—1311no—
132#327—1311no—
133#323—1311no—
134C22×D4—2621no—
135#123—2621no—
136#123—2621no—
137#121—2621no—
138#121—1311no—
139#121C2×D13111yescharacteristic
140#120—2621no—
141#120—2621no—
142C2×C4⋊C4C2×D13111yescharacteristic, complemented
143C2×C22⋊C4—2621no—
144C2×C22⋊C4—1311no—
145#107—21no—
146#107—2621no—
147#102—21no—
148#102—2621no—
149C22⋊C8—1311no—
150C22⋊C8C2×D13111yescharacteristic, complemented
151#95—2621no—
152#95—1311no—
153C24—2621no—
154C24—1311no—
155C2×D4—21no—
156C2×D4—2621no—
157C2×D4—2621no—
158C2×D4—2621no—
159C2×D4—2621no—
160C2×D4—5242no—
161C2×D4—5242no—
162C2×D4—1311no—
163C2×D4C22⋊D13111yescharacteristic
164C22×C4—21no—
165C22×C4—2621no—
166C22×C4—2621no—
167C22×C4—1311no—
168C22×C4C22×D13111yescharacteristic
169C2×C8—21no—
170C2×C8—2621no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—2621no—
174C4⋊C4—1311no—
175C4⋊C4C22⋊D13111yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—2621no—
178C22⋊C4—2621no—
179C22⋊C4—2621no—
180C22⋊C4—2621no—
181C22⋊C4—2621no—
182C22⋊C4—5242no—
183C23—21no—
184C23—2621no—
185C23—2621no—
186C23—2621no—
187C23—5241no—
188C23—5241no—
189C23—5242no—
190C23—10481no—
191C23—10481no—
192C23—1312no—
193C23D4×D13111yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—5241no—
198D4—5242no—
199D4—5242no—
200D4—5241no—
201D4—5241no—
202D4—10481no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2621no—
209C2×C4—2621no—
210C2×C4—5242no—
211C2×C4—5241no—
212C2×C4—5241no—
213C2×C4—5241no—
214C2×C4—5241no—
215C2×C4D4×D13111yescharacteristic
216C2×C4C2×C22⋊D13111yescharacteristic
217C8—41no—
218C8—5241no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2622no—
225C22—5242no—
226C22—5241no—
227C22—5241no—
228C22—5241no—
229C22—5241no—
230C22—5241no—
231C22—10481no—
232C22—10484no—
233C22#1686221411yescharacteristic
234C22D8×D13111yescharacteristic
235C22#1686227011yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—5241no—
241C4—5241no—
242C4—10481no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—5242no—
247C2—10481no—
248C2#2255344411yescharacteristic
249C2#2255348811yescharacteristic
250C2#2255350111yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#480×C131C2
socleC22×C131#120
prime radical#22553870C2
hypercenter#480D131
hypocenterC131#1607
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #22553625C2semidirect
3. #22553238C2semidirect
4. #22553047C2semidirect
5. #22553046C2semidirect
6. #22553035C2semidirect
7. #480×C131C2semidirect
19. C22⋊(C131⋊C8)C22semidirect
21. C131⋊#95C22semidirect
23. C2×C4⋊C4×C131C22semidirect
26. C22⋊C8×C131C22semidirect
125. C131#1607semidirect
129. #480D131semidirect
142. C2×C4⋊C4C2×D131semidirect
150. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538702
#225536252
#225532382
#225530472
#225530462
#225530352
#480×C1312
#1607131