Finite Group Databasebeta

← #30675539

Subgroups — #30675539

Order 16768.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16862266C211yesmaximal, characteristic, complemented
2C23×C131⋊C8C211yesmaximal, characteristic, complemented
3#22553823C211yesmaximal, characteristic, complemented
4#22553551C212yesmaximal, complemented
5#22553546C212yesmaximal, complemented
6#22553543C212yesmaximal, complemented
7#22553539C212yesmaximal, complemented
8#22553348C211yesmaximal, characteristic, complemented
9#22553308C211yesmaximal, characteristic, complemented
10#22553306C211yesmaximal, characteristic, complemented
11#525×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13#16862266—24no—
14C2×#14523503C2211yescharacteristic, complemented
15C2×#14523501C2211yescharacteristic, complemented
16C2×#14523500C2211yescharacteristic, complemented
17C2×#14523498C2211yescharacteristic, complemented
18C22×C131⋊C8—24no—
19C22×C131⋊C8C2212yes—
20C22×C131⋊C8C2212yes—
21C22×C131⋊C8C2211yescharacteristic
22C22×C131⋊C8C2211yescharacteristic
23#16862229—21no—
24#16862225C2212yes—
25#16862220C2212yes—
26#16862215—21no—
27C4×C22⋊D131—22no—
28#16862205C2211yescharacteristic
29#16862179C2212yescomplemented
30#16862177C2212yescomplemented
31#16862170C2212yescomplemented
32#16862169C2212yescomplemented
33#16862167C2212yescomplemented
34#16862166C2212yescomplemented
35C2×#41×C131C2211yescharacteristic
36C23×C4×C131C2211yescharacteristic
37#123×C131—21no—
38C22⋊Q8×C131C2212yes—
39#121×C131C2212yes—
40#120×C131—21no—
41C4×D4×C131—22no—
42#117×C131C2211yescharacteristic
43#14523513—22no—
44#14523513—22no—
45C2×Dic262C2311yescharacteristic
46C2×D524C2311yescharacteristic
47C2×C22⋊D131—22no—
48C2×C22⋊D131—21no—
49C2×C4×D131—21no—
50#14523503—22no—
51#14523501—22no—
52#14523500—22no—
53#14523498—22no—
54C2×C131⋊C8—24no—
55C2×C131⋊C8—24no—
56C2×C131⋊C8—22no—
57C2×C131⋊C8D414yescomplemented
58C2×C131⋊C8C2312yes—
59C2×C131⋊C8C2311yescharacteristic
60C2×C131⋊C8C2311yescharacteristic
61C4⋊Dic131C2312yes—
62Dic131⋊C4—22no—
63C22⋊Dic131—22no—
64C22⋊Dic131—21no—
65(C2×D131)⋊C4—22no—
66C4×Dic131—21no—
67C24×C131D411yescharacteristic
68#41×C131—22no—
69#41×C131—22no—
70C2×Q8×C131C2311yescharacteristic
71C2×D4×C131—22no—
72C2×D4×C131—21no—
73C2×D4×C131C2311yescharacteristic
74C22×C4×C131—22no—
75C22×C4×C131—22no—
76C22×C4×C131—21no—
77C22×C4×C131D411yescharacteristic
78C22×C4×C131D412yes—
79C22×C4×C131C2312yes—
80C22×C4×C131C2311yescharacteristic
81C4⋊C4×C131—22no—
82C4⋊C4×C131C2312yes—
83C22⋊C4×C131—21no—
84C22⋊C4×C131—22no—
85C22⋊C4×C131—22no—
86C42×C131—21no—
87C22×D131—21no—
88Dic262—21no—
89Dic262—21no—
90D524—21no—
91D524—21no—
92C22⋊D131—42no—
93C22⋊D131—44no—
94C2×Dic131—22no—
95C2×Dic131—21no—
96C4×D131—42no—
97C131⋊C8—22no—
98C131⋊C8C2×D412yes—
99C131⋊C8C2×D412yes—
100C23×C131—22no—
101C23×C131—24no—
102C23×C131—21no—
103C23×C131C2×D411yescharacteristic
104C23×C131C2×D412yes—
105Q8×C131—21no—
106Q8×C131—21no—
107D4×C131—21no—
108D4×C131—21no—
109D4×C131—42no—
110D4×C131—44no—
111C2×C4×C131—22no—
112C2×C4×C131—24no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—21no—
117C2×C4×C131—42no—
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2×D411yescharacteristic
121C2×C4×C131#4112yes—
122C2×C4×C131C2411yescharacteristic
123C22×C131—28no—
124C22×C131—22no—
125C22×C131—22no—
126C22×C131—22no—
127C22×C131—41no—
128C22×C131—42no—
129C22×C131#12112yes—
130C22×C131#13514yescomplemented
131C22×C131C22×D411yescharacteristic
132C2×D131—41no—
133C2×D131—42no—
134C4×C131—22no—
135C4×C131—41no—
136C4×C131—42no—
137C4×C131#12112yes—
138C4×C131C22×D411yescharacteristic
139C4×C131C2×#4111yescharacteristic
140Dic131—41no—
141Dic131—42no—
142C2×C131—22no—
143C2×C131—24no—
144C2×C131—81no—
145C2×C131C2×#12111yescharacteristic
146C2×C131C2×#13512yes—
147D131—81no—
148C131#305011yescharacteristic, complemented, Sylow
149#3050—1311nomaximal, Sylow
150C2×#135—1311no—
151C23×C8—1311no—
152#525—1311no—
153#525D13111yescharacteristic, complemented
154#467—1312no—
155#466—1312no—
156#465—1312no—
157#465—1312no—
158#427—1311no—
159#416—1311no—
160#416—1311no—
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C23×C4C2×D13111yescharacteristic
164#135—2624no—
165C2×Q16—1311no—
166C2×SD16—1311no—
167C2×SD16—1311no—
168C2×D8—1311no—
169C22×C8—2624no—
170C22×C8—1311no—
171C22×C8—1312no—
172C22×C8—1312no—
173C22×C8—1311no—
174#123—21no—
175#123—2621no—
176C22⋊Q8—1312no—
177C22⋊Q8C2×D13112yes—
178#121—1312no—
179#121C2×D13112yes—
180#120—21no—
181#120—2621no—
182C4×D4—22no—
183C4×D4—2622no—
184#117—1311no—
185#117C2×D13111yescharacteristic
186#107—1312no—
187#106—1312no—
188#103—1312no—
189#103—1312no—
190#102—1312no—
191#102—1312no—
192C24C22⋊D13111yescharacteristic
193#41—22no—
194#41—22no—
195#41—2622no—
196#41—2622no—
197C2×Q8—1311no—
198C2×Q8C22×D13111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—2622no—
202C2×D4—2621no—
203C2×D4—1311no—
204C2×D4C22×D13111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2621no—
209C22×C4C22⋊D13111yescharacteristic
210C22×C4C22⋊D13112yes—
211C22×C4C22×D13112yes—
212C22×C4C22×D13111yescharacteristic
213Q16—2622no—
214SD16—2622no—
215SD16—2622no—
216D8—2622no—
217C2×C8—2622no—
218C2×C8—2624no—
219C2×C8—2624no—
220C2×C8—1312no—
221C2×C8—1314no—
222C2×C8—1311no—
223C2×C8—1311no—
224C4⋊C4—22no—
225C4⋊C4—2622no—
226C4⋊C4—1312no—
227C4⋊C4C22×D13112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2621no—
232C22⋊C4—2622no—
233C22⋊C4—2622no—
234C42—21no—
235C42—2621no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—2621no—
240C23C2×C22⋊D13111yescharacteristic
241C23C2×C22⋊D13112yes—
242Q8—21no—
243Q8—21no—
244Q8—2621no—
245Q8—2621no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—2621no—
251D4—2621no—
252D4—5242no—
253D4—5244no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—2622no—
262C2×C4—2621no—
263C2×C4—5242no—
264C2×C4D4×D13112yes—
265C2×C4C2×C22⋊D13112yes—
266C2×C4C2×C22⋊D13111yescharacteristic
267C2×C4#1452351212yes—
268C2×C4C23×D13111yescharacteristic
269C8—2622no—
270C8—1312no—
271C8—1312no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—5241no—
279C22—5242no—
280C22#1686221812yes—
281C22#1686226614yescomplemented
282C22C22×C22⋊D13111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—5241no—
287C4—5242no—
288C4#1686221812yes—
289C4C2×D4×D13111yescharacteristic
290C4C2×#1452351211yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—10481no—
295C2C2×#1686221811yescharacteristic
296C2C2×#1686226612yes—
subgroupG/H
centerC2×C4C2×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#525×C131C2
socleC22×C131C22×D4
hypercenter#525D131
hypocenterC131#3050
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16862266C2semidirect
2. C23×C131⋊C8C2semidirect
3. #22553823C2semidirect
4. #22553551C2semidirect
5. #22553546C2semidirect
6. #22553543C2semidirect
7. #22553539C2semidirect
8. #22553348C2semidirect
9. #22553308C2semidirect
10. #22553306C2semidirect
11. #525×C131C2semidirect
14. C2×#14523503C22semidirect
15. C2×#14523501C22semidirect
16. C2×#14523500C22semidirect
17. C2×#14523498C22semidirect
29. #16862179C22semidirect
30. #16862177C22semidirect
31. #16862170C22semidirect
32. #16862169C22semidirect
33. #16862167C22semidirect
34. #16862166C22semidirect
57. C2×C131⋊C8D4semidirect
130. C22×C131#135semidirect
148. C131#3050semidirect
153. #525D131semidirect
281. C22#16862266semidirect

Maximal subgroups

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

subgroupindex
C2×#168622662
C23×C131⋊C82
#225538232
#225535512
#225535462
#225535432
#225535392
#225533482
#225533082
#225533062
#525×C1312
#3050131