Finite Group Databasebeta

← #30675352

Subgroups — #30675352

Order 16768.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#22554148C211yesmaximal, characteristic, complemented
2#22553823C212yesmaximal, complemented
3#22553501C214yesmaximal, complemented
4#22553498C214yesmaximal, complemented
5#22553286C211yesmaximal, characteristic, complemented
6#22553285C212yesmaximal, complemented
7#525×C131C211yesmaximal, characteristic, complemented
8#16862294—21no—
9#16862293—22no—
10#16862292—22no—
11#16862291—21no—
12#16862291C2214yes—
13C2×#14523513—22no—
14C2×#14523513C2212yes—
15#41×D131—24no—
16#16862229C2214yescomplemented
17#16862225—22no—
18#16862220—22no—
19#16862215C2214yescomplemented
20C4×C22⋊D131—24no—
21#16862205C2212yescomplemented
22#16862161C2214yescomplemented
23#16862160C2218yescomplemented
24C2×#41×C131C2211yescharacteristic
25C23×C4×C131C2211yescharacteristic, complemented
26#123×C131C2212yescomplemented
27C22⋊Q8×C131—21no—
28#121×C131—21no—
29#120×C131C2212yescomplemented
30C4×D4×C131—22no—
31#117×C131C2211yescharacteristic, complemented
32#14523514—24no—
33#14523513—24no—
34#14523513—24no—
35#14523513—22no—
36#14523513—22no—
37#14523513—24no—
38#14523513—22no—
39#14523513—22no—
40#14523512—24no—
41#14523512—28no—
42C2×Dic262—22no—
43C2×Dic262D412yescomplemented
44Q8×D131—24no—
45C2×D524—22no—
46C2×D524D412yescomplemented
47C2×C22⋊D131—24no—
48C2×C22⋊D131—24no—
49C2×C22⋊D131C2314yes—
50D4×D131—24no—
51D4×D131—28no—
52C2×C4×D131—24no—
53C2×C4×D131D414yescomplemented
54C4⋊Dic131—22no—
55Dic131⋊C4—24no—
56C22⋊Dic131—24no—
57C22⋊Dic131C2314yes—
58(C2×D131)⋊C4—24no—
59C4×Dic131—22no—
60C24×C131C2311yescharacteristic
61#41×C131—22no—
62#41×C131—22no—
63C2×Q8×C131D411yescharacteristic, complemented
64C2×D4×C131—22no—
65C2×D4×C131D411yescharacteristic, complemented
66C2×D4×C131C2312yes—
67C22×C4×C131—22no—
68C22×C4×C131—21no—
69C22×C4×C131—21no—
70C22×C4×C131—22no—
71C22×C4×C131D412yescomplemented
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2311yescharacteristic
74C4⋊C4×C131—21no—
75C4⋊C4×C131—22no—
76C22⋊C4×C131—22no—
77C22⋊C4×C131—22no—
78C22⋊C4×C131C2312yes—
79C42×C131—21no—
80C22×D131—24no—
81C22×D131C2×D414yes—
82Dic262—24no—
83Dic262—22no—
84Dic262—22no—
85D524—24no—
86D524—22no—
87D524—22no—
88C22⋊D131—28no—
89C22⋊D131—24no—
90C22⋊D131—24no—
91C22⋊D131—48no—
92C2×Dic131—24no—
93C2×Dic131—24no—
94C2×Dic131C2×D414yes—
95C4×D131—28no—
96C4×D131—24no—
97C4×D131—24no—
98C23×C131—21no—
99C23×C131—22no—
100C23×C131—42no—
101C23×C131C2×D412yes—
102C23×C131C2411yescharacteristic
103Q8×C131—22no—
104D4×C131—24no—
105D4×C131—22no—
106D4×C131—44no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—41no—
113C2×C4×C131—42no—
114C2×C4×C131—41no—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131C2×D411yescharacteristic
120C22×C131—22no—
121C22×C131—24no—
122C22×C131—22no—
123C22×C131—41no—
124C22×C131—42no—
125C22×C131—42no—
126C22×C131—41no—
127C22×C131—41no—
128C22×C131C22×D412yes—
129C22×C131C22×D411yescharacteristic
130C2×D131—24no—
131C2×D131—44no—
132C2×D131—84no—
133C4×C131—22no—
134C4×C131—42no—
135C4×C131—41no—
136C4×C131#12011yescharacteristic
137C4×C131#12012yes—
138C4×C131#12011yescharacteristic
139Dic131—24no—
140Dic131—44no—
141C2×C131—22no—
142C2×C131—21no—
143C2×C131—42no—
144C2×C131—42no—
145C2×C131C2×#12011yescharacteristic
146D131—44no—
147C131#302511yescharacteristic, complemented, Sylow
148#3025—1311nomaximal, Sylow
149#585—1311no—
150#525—1312no—
151#525D13111yescharacteristic, complemented
152#458—1314no—
153B(4,𝔽2)—1314no—
154#410—1311no—
155#410—1312no—
1562−1+4—2622no—
1572−1+4—2621no—
158H(5,𝔽2)—2622no—
159H(5,𝔽2)—2621no—
160H(5,𝔽2)—1314no—
161C2×#41—2624no—
162C2×#41—2622no—
163C2×#41—1312no—
164C2×#41C2×D13111yescharacteristic
165C23×C4C2×D13111yescharacteristic, complemented
166#123—1314no—
167#123C2×D13112yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—2622no—
170#121—21no—
171#121—2622no—
172#120—1314no—
173#120C2×D13112yescomplemented
174C4×D4—22no—
175C4×D4—2624no—
176#117—1312no—
177#117C2×D13111yescharacteristic, complemented
178#99—1314no—
179#99—1318no—
180C24C22×D13111yescharacteristic
181#41—22no—
182#41—22no—
183#41—2624no—
184#41—2624no—
185#41—2624no—
186#41—2624no—
187#41—2628no—
188#41—2622no—
189#41—2622no—
190#41—2622no—
191#41—2624no—
192#41—2622no—
193C2×Q8—2622no—
194C2×Q8—2624no—
195C2×Q8—1312no—
196C2×Q8C22⋊D13111yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—2622no—
199C2×D4—2624no—
200C2×D4—2628no—
201C2×D4—2624no—
202C2×D4—2624no—
203C2×D4—1314no—
204C2×D4—1312no—
205C2×D4C22⋊D13111yescharacteristic, complemented
206C2×D4C22×D13112yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2624no—
212C22×C4—1314no—
213C22×C4C22⋊D13112yescomplemented
214C22×C4C22×D13111yescharacteristic
215C22×C4C22×D13111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—2624no—
219C4⋊C4—2622no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—2624no—
223C22⋊C4—2624no—
224C22⋊C4—1314no—
225C22⋊C4C22×D13112yes—
226C42—21no—
227C42—2622no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—2624no—
232C23—1314no—
233C23C2×C22⋊D13112yes—
234C23C23×D13111yescharacteristic
235Q8—22no—
236Q8—2624no—
237Q8—2622no—
238Q8—2622no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—2624no—
243D4—2628no—
244D4—2624no—
245D4—2624no—
246D4—2622no—
247D4—2622no—
248D4—5248no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—2628no—
258C2×C4—2624no—
259C2×C4—2624no—
260C2×C4—2624no—
261C2×C4—2624no—
262C2×C4—1314no—
263C2×C4D4×D13112yes—
264C2×C4D4×D13112yes—
265C2×C4C2×C22⋊D13112yes—
266C2×C4C2×C22⋊D13111yescharacteristic
267C2×C4C2×C22⋊D13111yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—2624no—
277C22—5244no—
278C22—10484no—
279C22C2×D4×D13112yes—
280C22C22×C22⋊D13111yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—2624no—
285C4—5244no—
286C4#1686221411yescharacteristic
287C4#1686221412yes—
288C4#1686221411yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—5244no—
294C2C2×#1686221411yescharacteristic
subgroupG/H
centerC4#16862214
commutator, cosocleC23×C131C24
FrattiniC23C23×D131
Fitting#525×C131C2
socleC2×C131C2×#120
hypercenter#525D131
hypocenterC131#3025
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #22554148C2semidirect
2. #22553823C2semidirect
3. #22553501C2semidirect
4. #22553498C2semidirect
5. #22553286C2semidirect
6. #22553285C2semidirect
7. #525×C131C2semidirect
16. #16862229C22semidirect
19. #16862215C22semidirect
21. #16862205C22semidirect
22. #16862161C22semidirect
23. #16862160C22semidirect
25. C23×C4×C131C22semidirect
26. #123×C131C22semidirect
29. #120×C131C22semidirect
31. #117×C131C22semidirect
43. C2×Dic262D4semidirect
46. C2×D524D4semidirect
53. C2×C4×D131D4semidirect
63. C2×Q8×C131D4semidirect
65. C2×D4×C131D4semidirect
71. C22×C4×C131D4semidirect
147. C131#3025semidirect
151. #525D131semidirect
165. C23×C4C2×D131semidirect
167. #123C2×D131semidirect
173. #120C2×D131semidirect
177. #117C2×D131semidirect
196. C2×Q8C22⋊D131semidirect
205. C2×D4C22⋊D131semidirect
213. C22×C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225541482
#225538232
#225535012
#225534982
#225532862
#225532852
#525×C1312
#3025131