Finite Group Databasebeta

← #30674250

Subgroups — #30674250

Order 16768.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D131C211yesmaximal, characteristic, complemented
2C2×#16862227C211yesmaximal, characteristic, complemented
3C2×#16862214C211yesmaximal, characteristic
4#22553188C214yesmaximal, complemented
5#22553183C214yesmaximal, complemented
6#22553175C211yesmaximal, characteristic, complemented
7#22553173C211yesmaximal, characteristic, complemented
8#22553167C211yesmaximal, characteristic, complemented
9C2×#123×C131C211yesmaximal, characteristic, complemented
10C24×D131C2211yescharacteristic
11C22×C22⋊D131C2212yes—
12C23×Dic131C2211yescharacteristic
13C22×C4×D131—24no—
14C22×C4×D131—24no—
15C22×C4×D131—24no—
16C22×C4×D131C2214yes—
17#16862227—24no—
18#16862214—24no—
19C2×Dic131⋊C4C2212yescomplemented
20C2×C22⋊Dic131C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2212yescomplemented
23C2×(C2×D131)⋊C4C2214yes—
24C2×(C2×D131)⋊C4C2212yes—
25C131⋊#95C2214yescomplemented
26C131⋊#95C2214yescomplemented
27C22×D4×C131C2211yescharacteristic
28C23×C4×C131C2211yescharacteristic
29#123×C131—24no—
30C2×C4⋊C4×C131C2212yescomplemented
31C2×C22⋊C4×C131C2211yescharacteristic
32C2×C22⋊C4×C131C2212yescomplemented
33C23×D131—28no—
34C23×D131—22no—
35C23×D131—22no—
36C23×D131D414yes—
37C23×D131C2312yes—
38C2×C22⋊D131—24no—
39C2×C22⋊D131—28no—
40C22×Dic131—22no—
41C22×Dic131—22no—
42C22×Dic131D414yescomplemented
43C22×Dic131C2312yes—
44C22×Dic131C2312yes—
45C22×Dic131C2312yes—
46C2×C4×D131—232no—
47C2×C4×D131—28no—
48C2×C4×D131—28no—
49C2×C4×D131—28no—
50Dic131⋊C4—24no—
51C22⋊Dic131—24no—
52C22⋊Dic131—22no—
53C22⋊Dic131—22no—
54(C2×D131)⋊C4—24no—
55(C2×D131)⋊C4—28no—
56C24×C131C2311yescharacteristic
57C24×C131C2311yescharacteristic
58C2×D4×C131—22no—
59C2×D4×C131—22no—
60C2×D4×C131—22no—
61C22×C4×C131—22no—
62C22×C4×C131—22no—
63C22×C4×C131D414yescomplemented
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C4⋊C4×C131—24no—
68C22⋊C4×C131—22no—
69C22⋊C4×C131—24no—
70C22×D131—216no—
71C22×D131—216no—
72C22×D131—28no—
73C22×D131—22no—
74C22×D131—22no—
75C22×D131—24no—
76C22×D131—216no—
77C22×D131C2×D414yes—
78C22×D131#4118yes—
79C22⋊D131—48no—
80C2×Dic131—28no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—24no—
86C2×Dic131—24no—
87C2×Dic131C2×D414yes—
88C4×D131—232no—
89C23×C131—24no—
90C23×C131—21no—
91C23×C131—21no—
92C23×C131—21no—
93C23×C131—24no—
94C23×C131C2×D412yes—
95C23×C131C2×D412yes—
96C23×C131C2×D412yes—
97C23×C131C2411yescharacteristic
98D4×C131—44no—
99C2×C4×C131—28no—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—21no—
104C2×C4×C131—21no—
105C2×C4×C131—21no—
106C2×C4×C131—24no—
107C2×C4×C131C2×D414yes—
108C22×C131—24no—
109C22×C131—24no—
110C22×C131—24no—
111C22×C131—41no—
112C22×C131—42no—
113C22×C131—42no—
114C22×C131—42no—
115C22×C131#12014yes—
116C22×C131C22×D411yescharacteristic
117C22×C131C22×D411yescharacteristic
118C22×C131C22×D411yescharacteristic
119C22×C131C2×#4114yes—
120C2×D131—232no—
121C2×D131—24no—
122C2×D131—24no—
123C2×D131—28no—
124C2×D131—28no—
125C2×D131#12318yescomplemented
126C4×C131—24no—
127C4×C131—41no—
128C4×C131—42no—
129Dic131—24no—
130Dic131—42no—
131Dic131—42no—
132C2×C131—24no—
133C2×C131—81no—
134C2×C131C2×#12011yescharacteristic
135C2×C131C2×#12312yes—
136C2×C131#52512yes—
137C2×C131#52512yes—
138D131—28no—
139C131#284911yescharacteristic, complemented, Sylow
140#2849—1311nomaximal, Sylow
141C24×C4—1311no—
142C2×#123—1311no—
143C2×#123D13111yescharacteristic, complemented
144C2×#120—1311no—
145#386—1314no—
146#385—1314no—
147#381—1311no—
148#381—1311no—
149#379—1311no—
150C25—1311no—
151C22×D4—1312no—
152C22×D4C2×D13111yescharacteristic
153C23×C4—2624no—
154C23×C4—2624no—
155C23×C4—2624no—
156C23×C4—1314no—
157C23×C4—1311no—
158C23×C4C2×D13111yescharacteristic
159#123—24no—
160#123—2624no—
161#120—2624no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4C2×D13112yescomplemented
164C2×C22⋊C4—1311no—
165C2×C22⋊C4—1311no—
166C2×C22⋊C4—1312no—
167C2×C22⋊C4—1312no—
168C2×C22⋊C4—1314no—
169C2×C22⋊C4C2×D13111yescharacteristic
170C2×C22⋊C4C2×D13112yescomplemented
171#95—1314no—
172#95—1314no—
173C24—2628no—
174C24—2622no—
175C24—2622no—
176C24—1314no—
177C24—1312no—
178C24C22×D13111yescharacteristic
179C24C22×D13111yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—2624no—
184C2×D4—2628no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—26232no—
188C22×C4—2628no—
189C22×C4—2628no—
190C22×C4—2622no—
191C22×C4—2622no—
192C22×C4—2628no—
193C22×C4—1312no—
194C22×C4—1312no—
195C22×C4—1314no—
196C22×C4—1312no—
197C22×C4C22⋊D13114yescomplemented
198C22×C4C22×D13112yes—
199C22×C4C22×D13111yescharacteristic
200C22×C4C22×D13112yes—
201C4⋊C4—24no—
202C4⋊C4—2624no—
203C22⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—2628no—
206C22⋊C4—2624no—
207C22⋊C4—2622no—
208C22⋊C4—2622no—
209C22⋊C4—2624no—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—24no—
215C23—26216no—
216C23—26216no—
217C23—26216no—
218C23—2628no—
219C23—2622no—
220C23—2622no—
221C23—2624no—
222C23—1314no—
223C23—1318no—
224C23D4×D13112yes—
225C23D4×D13112yes—
226C23C2×C22⋊D13112yes—
227C23C23×D13111yescharacteristic
228D4—44no—
229D4—5248no—
230C2×C4—28no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—26232no—
239C2×C4—2628no—
240C2×C4—2622no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2622no—
244C2×C4—2624no—
245C2×C4—2624no—
246C2×C4—1314no—
247C2×C4C2×C22⋊D13114yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—26232no—
256C22—2624no—
257C22—2624no—
258C22—2628no—
259C22—2628no—
260C22—1318no—
261C22#1686221414yes—
262C22C2×D4×D13111yescharacteristic
263C22C2×D4×D13111yescharacteristic
264C22C22×C22⋊D13111yescharacteristic
265C22#41×D13114yes—
266C4—24no—
267C4—41no—
268C4—42no—
269C4—2624no—
270C4—5242no—
271C4—5242no—
272C2—24no—
273C2—2628no—
274C2—81no—
275C2C2×#1686221411yescharacteristic
276C2#123×D13112yes—
277C2#2255381912yes—
278C2#2255382012yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
prime radicalC2×#16862214C2
hypercenterC2×#123D131
hypocenterC131#2849
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×D131C2semidirect
2. C2×#16862227C2semidirect
4. #22553188C2semidirect
5. #22553183C2semidirect
6. #22553175C2semidirect
7. #22553173C2semidirect
8. #22553167C2semidirect
9. C2×#123×C131C2semidirect
19. C2×Dic131⋊C4C22semidirect
22. C2×C22⋊Dic131C22semidirect
25. C131⋊#95C22semidirect
26. C131⋊#95C22semidirect
30. C2×C4⋊C4×C131C22semidirect
32. C2×C22⋊C4×C131C22semidirect
42. C22×Dic131D4semidirect
63. C22×C4×C131D4semidirect
125. C2×D131#123semidirect
139. C131#2849semidirect
143. C2×#123D131semidirect
163. C2×C4⋊C4C2×D131semidirect
170. C2×C22⋊C4C2×D131semidirect
197. C22×C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1312
C2×#168622272
C2×#168622142
#225531882
#225531832
#225531752
#225531732
#225531672
C2×#123×C1312
#2849131