Finite Group Databasebeta

← #30674244

Subgroups — #30674244

Order 16768.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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×#16862221C212yesmaximal, complemented
3C2×#16862220C211yesmaximal, characteristic, complemented
4C2×#16862216C212yesmaximal, complemented
5#22553190C214yesmaximal, complemented
6#22553189C212yesmaximal, complemented
7#22553169C212yesmaximal, complemented
8#380×C131C211yesmaximal, characteristic, complemented
9C24×D131C2211yescharacteristic
10C22×D524C2214yes—
11C22×D524C2211yescharacteristic, complemented
12C22×C22⋊D131C2212yes—
13C23×Dic131C2211yescharacteristic
14C22×C4×D131—28no—
15C22×C4×D131—24no—
16C22×C4×D131C2212yes—
17C22×C4×D131C2212yes—
18#16862221—28no—
19#16862220—24no—
20#16862216—28no—
21C2×C4⋊Dic131C2211yescharacteristic, complemented
22C2×(C2×D131)⋊C4C2214yes—
23C2×(C2×D131)⋊C4C2214yescomplemented
24C2×(C2×D131)⋊C4C2212yescomplemented
25C131⋊#95C2214yescomplemented
26C23×C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2212yescomplemented
28C2×C22⋊C4×C131C2212yes—
29#95×C131C2212yescomplemented
30C23×D131—24no—
31C23×D131—24no—
32C23×D131—22no—
33C23×D131—22no—
34C23×D131D414yescomplemented
35C23×D131C2312yes—
36C23×D131C2312yes—
37C2×D524—28no—
38C2×D524—28no—
39C2×D524—28no—
40C2×D524—22no—
41C2×D524—24no—
42C2×C22⋊D131—24no—
43C2×C22⋊D131—24no—
44C2×C22⋊D131—24no—
45C22×Dic131—24no—
46C22×Dic131—22no—
47C22×Dic131C2312yes—
48C22×Dic131C2312yes—
49C2×C4×D131—232no—
50C2×C4×D131—24no—
51C2×C4×D131—24no—
52C2×C4×D131—24no—
53C2×C4×D131—24no—
54C2×C4×D131—24no—
55C2×C4×D131D418yescomplemented
56C4⋊Dic131—22no—
57(C2×D131)⋊C4—28no—
58(C2×D131)⋊C4—28no—
59(C2×D131)⋊C4—24no—
60C24×C131C2311yescharacteristic
61C22×C4×C131—24no—
62C22×C4×C131D414yescomplemented
63C22×C4×C131C2314yes—
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2311yescharacteristic
66C4⋊C4×C131—24no—
67C22⋊C4×C131—24no—
68C22×D131—28no—
69C22×D131—216no—
70C22×D131—28no—
71C22×D131—28no—
72C22×D131—22no—
73C22×D131—22no—
74C22×D131—22no—
75C22×D131—22no—
76C22×D131—216no—
77C22×D131—22no—
78C22×D131—24no—
79C22×D131—22no—
80C22×D131—22no—
81C22×D131—24no—
82C22×D131C2×D414yes—
83C22×D131C2×D414yes—
84C22×D131#4114yes—
85D524—416no—
86D524—44no—
87C22⋊D131—48no—
88C2×Dic131—28no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131C2×D414yes—
95C4×D131—216no—
96C4×D131—216no—
97C23×C131—22no—
98C23×C131—22no—
99C23×C131C2×D412yes—
100C23×C131C2×D412yes—
101C23×C131#4112yes—
102C23×C131C2411yescharacteristic
103C2×C4×C131—22no—
104C2×C4×C131—28no—
105C2×C4×C131—24no—
106C2×C4×C131—24no—
107C2×C4×C131—24no—
108C2×C4×C131C2×D414yes—
109C2×C4×C131C2×D412yes—
110C2×C4×C131C2×D412yes—
111C22×C131—22no—
112C22×C131—24no—
113C22×C131—22no—
114C22×C131—24no—
115C22×C131#12114yescomplemented
116C22×C131C22×D412yes—
117C22×C131C22×D411yescharacteristic
118C22×C131C22×D411yescharacteristic
119C22×C131C2×#4112yes—
120C22×C131C2×#4111yescharacteristic
121C2×D131—232no—
122C2×D131—24no—
123C2×D131—24no—
124C2×D131—24no—
125C2×D131—24no—
126C2×D131—24no—
127C2×D131—24no—
128C2×D131—42no—
129C2×D131—42no—
130C2×D131—44no—
131C2×D131—42no—
132C2×D131—44no—
133C2×D131#12118yescomplemented
134C4×C131—22no—
135C4×C131—44no—
136C4×C131#12014yescomplemented
137Dic131—24no—
138Dic131—42no—
139C2×C131—24no—
140C2×C131C2×#12011yescharacteristic
141C2×C131C2×#12112yes—
142C2×C131C2×#12111yescharacteristic
143C2×C131#52511yescharacteristic
144C2×C131#52512yes—
145D131—28no—
146D131—82no—
147C131#284711yescharacteristic, complemented, Sylow
148#2847—1311nomaximal, Sylow
149C24×C4—1311no—
150C2×#121—1311no—
151C2×#121—1312no—
152C2×#120—1312no—
153#386—1312no—
154#386—1314no—
155#380—1312no—
156#380D13111yescharacteristic, complemented
157C25—1311no—
158C22×D4—1312no—
159C22×D4—1314no—
160C22×D4—1311no—
161C23×C4—2628no—
162C23×C4—2624no—
163C23×C4—1311no—
164C23×C4—1312no—
165C23×C4—1312no—
166C23×C4C2×D13111yescharacteristic
167#121—2628no—
168#121—2624no—
169#120—2628no—
170C2×C4⋊C4—1311no—
171C2×C4⋊C4C2×D13112yescomplemented
172C2×C22⋊C4—1314no—
173C2×C22⋊C4—1314no—
174C2×C22⋊C4—1312no—
175C2×C22⋊C4C2×D13112yes—
176#95—1314no—
177#95C2×D13112yescomplemented
178C24—2624no—
179C24—2624no—
180C24—2622no—
181C24—2622no—
182C24—1312no—
183C24—1314no—
184C24—1312no—
185C24C22×D13111yescharacteristic
186C2×D4—2628no—
187C2×D4—2628no—
188C2×D4—2628no—
189C2×D4—2624no—
190C2×D4—2624no—
191C2×D4—2624no—
192C2×D4—2624no—
193C2×D4—2622no—
194C22×C4—24no—
195C22×C4—26232no—
196C22×C4—2624no—
197C22×C4—2624no—
198C22×C4—2622no—
199C22×C4—2624no—
200C22×C4—2624no—
201C22×C4—2624no—
202C22×C4—2624no—
203C22×C4—1312no—
204C22×C4—1312no—
205C22×C4—1318no—
206C22×C4D52414yescomplemented
207C22×C4C22×D13114yes—
208C22×C4C22×D13111yescharacteristic
209C22×C4C22×D13111yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—2622no—
212C22⋊C4—24no—
213C22⋊C4—2628no—
214C22⋊C4—2628no—
215C22⋊C4—2624no—
216C23—22no—
217C23—22no—
218C23—2628no—
219C23—2628no—
220C23—26216no—
221C23—26216no—
222C23—2628no—
223C23—2622no—
224C23—2622no—
225C23—2622no—
226C23—2622no—
227C23—2624no—
228C23—2622no—
229C23—2622no—
230C23—2624no—
231C23—2622no—
232C23—1314no—
233C23—1314no—
234C23—1314no—
235C23D4×D13112yes—
236C23C2×D52412yes—
237C23#1452351412yes—
238C23C23×D13111yescharacteristic
239D4—52416no—
240D4—5248no—
241D4—5244no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—26216no—
248C2×C4—2628no—
249C2×C4—2622no—
250C2×C4—2622no—
251C2×C4—26216no—
252C2×C4—2622no—
253C2×C4—2622no—
254C2×C4—2622no—
255C2×C4—1314no—
256C2×C4D4×D13114yes—
257C2×C4C2×D52412yes—
258C2×C4C2×D52412yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—26232no—
264C22—2624no—
265C22—2624no—
266C22—2624no—
267C22—2624no—
268C22—2624no—
269C22—2624no—
270C22—5242no—
271C22—5244no—
272C22—5242no—
273C22—5242no—
274C22—5244no—
275C22—1318no—
276C22#1686222114yescomplemented
277C22C2×D4×D13112yes—
278C22C2×D4×D13111yescharacteristic
279C22C22×D52411yescharacteristic
280C22#41×D13112yes—
281C22C2×#1452351411yescharacteristic
282C4—22no—
283C4—44no—
284C4—2624no—
285C4—5242no—
286C4#1686221614yescomplemented
287C2—24no—
288C2—2628no—
289C2—10482no—
290C2C2×#1686221611yescharacteristic
291C2#121×D13112yes—
292C2C2×#1686222111yescharacteristic
293C2#2255382411yescharacteristic
294C2#2255382512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#380×C131C2
hypercenter#380D131
hypocenterC131#2847
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C4×D131C2semidirect
2. C2×#16862221C2semidirect
3. C2×#16862220C2semidirect
4. C2×#16862216C2semidirect
5. #22553190C2semidirect
6. #22553189C2semidirect
7. #22553169C2semidirect
8. #380×C131C2semidirect
11. C22×D524C22semidirect
21. C2×C4⋊Dic131C22semidirect
23. C2×(C2×D131)⋊C4C22semidirect
24. C2×(C2×D131)⋊C4C22semidirect
25. C131⋊#95C22semidirect
27. C2×C4⋊C4×C131C22semidirect
29. #95×C131C22semidirect
34. C23×D131D4semidirect
55. C2×C4×D131D4semidirect
62. C22×C4×C131D4semidirect
115. C22×C131#121semidirect
133. C2×D131#121semidirect
136. C4×C131#120semidirect
147. C131#2847semidirect
156. #380D131semidirect
171. C2×C4⋊C4C2×D131semidirect
177. #95C2×D131semidirect
206. C22×C4D524semidirect
276. C22#16862221semidirect
286. C4#16862216semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1312
C2×#168622212
C2×#168622202
C2×#168622162
#225531902
#225531892
#225531692
#380×C1312
#2847131