Finite Group Databasebeta

← #30669319

Subgroups — #30669319

Order 16768.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D131C211yesmaximal, characteristic, complemented
2C2×#16862223C212yesmaximal, complemented
3C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
4#22553233C211yesmaximal, characteristic, complemented
5#22553231C211yesmaximal, characteristic, complemented
6#22553224C211yesmaximal, characteristic, complemented
7#22553216C212yesmaximal, complemented
8#22553188C212yesmaximal
9#22553186C212yesmaximal, complemented
10#22553171C211yesmaximal, characteristic, complemented
11#393×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×Dic262C2212yes—
14C2×Q8×D131—28no—
15C2×Q8×D131—24no—
16C22×C22⋊D131C2211yescharacteristic
17C23×Dic131C2211yescharacteristic
18C22×C4×D131C2212yes—
19C22×C4×D131C2211yescharacteristic
20#16862223—28no—
21(C22⋊D131)⋊C4—24no—
22C2×C4⋊Dic131C2212yes—
23C2×Dic131⋊C4C2212yes—
24C2×Dic131⋊C4C2212yes—
25C2×Dic131⋊C4C2211yescharacteristic
26C2×C22⋊Dic131C2212yes—
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×C4×Dic131C2211yescharacteristic
30C131⋊#95C2211yescharacteristic
31C131⋊#95C2212yes—
32C131⋊#95C2212yes—
33C131⋊#95C2212yes—
34C22×Q8×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36C2×C22⋊C4×C131C2211yescharacteristic
37#95×C131C2211yescharacteristic
38#95×C131C2212yes—
39C23×D131C2311yescharacteristic
40C2×Dic262—24no—
41C2×Dic262—24no—
42C2×Dic262—24no—
43C2×Dic262—22no—
44C2×Dic262D418yescomplemented
45Q8×D131—232no—
46C2×C22⋊D131—22no—
47C2×C22⋊D131—22no—
48C2×C22⋊D131Q814yescomplemented
49C22×Dic131—24no—
50C22×Dic131—22no—
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2312yes—
53C22×Dic131C2312yes—
54C22×Dic131C2311yescharacteristic
55C22×Dic131C2311yescharacteristic
56C2×C4×D131—24no—
57C2×C4×D131—24no—
58C2×C4×D131—24no—
59C2×C4×D131—24no—
60C2×C4×D131D414yes—
61C4⋊Dic131—24no—
62Dic131⋊C4—24no—
63Dic131⋊C4—24no—
64Dic131⋊C4—22no—
65C22⋊Dic131—24no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—24no—
68C4×Dic131—22no—
69C24×C131C2311yescharacteristic
70C2×Q8×C131—24no—
71C2×Q8×C131—22no—
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2312yes—
74C22×C4×C131C2312yes—
75C22×C4×C131C2311yescharacteristic
76C22⋊C4×C131—24no—
77C22⋊C4×C131—22no—
78C22×D131—22no—
79C22×D131—22no—
80C22×D131C2×D412yes—
81C22×D131C2×Q812yes—
82C22×D131#4112yes—
83Dic262—28no—
84Dic262—216no—
85C22⋊D131—28no—
86C2×Dic131—21no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—21no—
90C2×Dic131—21no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—44no—
96C2×Dic131C2×D414yes—
97C2×Dic131C2×D412yes—
98C2×Dic131C2×D414yes—
99C2×Dic131C2×Q812yescomplemented
100C4×D131—216no—
101C4×D131—28no—
102C23×C131—22no—
103C23×C131—21no—
104C23×C131—21no—
105C23×C131—22no—
106C23×C131C2×Q812yescomplemented
107C23×C131C2411yescharacteristic
108Q8×C131—28no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131C2×D414yes—
118C2×C4×C131C2×D412yes—
119C22×C131—22no—
120C22×C131—24no—
121C22×C131—41no—
122C22×C131—41no—
123C22×C131—42no—
124C22×C131C22×D411yescharacteristic
125C22×C131C22×D412yes—
126C22×C131C22×Q811yescharacteristic
127C22×C131C2×#4111yescharacteristic
128C22×C1312−1+412yes—
129C2×D131—22no—
130C2×D131—22no—
131C2×D131—24no—
132C2×D131—24no—
133C2×D131C22⋊Q814yescomplemented
134C4×C131—24no—
135C4×C131—22no—
136C4×C131—41no—
137C4×C131—42no—
138Dic131—24no—
139Dic131—22no—
140Dic131—41no—
141Dic131—42no—
142Dic131#12014yes—
143C2×C131—42no—
144C2×C131C2×#12011yescharacteristic
145C2×C131C2×C22⋊Q811yescharacteristic
146C2×C131#53611yescharacteristic
147C2×C131#54812yes—
148C2×C131D4×Q812yes—
149D131—24no—
150C131#240911yescharacteristic, complemented, Sylow
151#2409—1311nomaximal, Sylow
152C23×Q8—1311no—
153C2×C22⋊Q8—1312no—
154C2×C4×D4—1311no—
155#395—1311no—
156#395—1311no—
157#393—1311no—
158#393D13111yescharacteristic, complemented
159#391—1312no—
160#386—1312no—
161#385—1312no—
162#380—1311no—
163C22×Q8—2628no—
164C22×Q8—2624no—
165C22×Q8—1311no—
166C22×Q8—1312no—
167C22×Q8C2×D13111yescharacteristic
168C22×D4—1311no—
169C23×C4—1311no—
170C23×C4—1312no—
171C23×C4—1311no—
172C22⋊Q8—2628no—
173C4×D4—2624no—
174C2×C4⋊C4—1311no—
175C2×C4⋊C4—1312no—
176C2×C4⋊C4—1312no—
177C2×C4⋊C4—1312no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1311no—
180C2×C22⋊C4—1312no—
181C2×C22⋊C4C2×D13111yescharacteristic
182C2×C22⋊C4C2×D13112yes—
183C2×C42—1311no—
184#95—1311no—
185#95—1312no—
186#95—1312no—
187#95—1312no—
188#95C2×D13111yescharacteristic
189#95C2×D13112yes—
190C24—1311no—
191C24C22×D13111yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—2624no—
195C2×Q8—2624no—
196C2×Q8—2624no—
197C2×Q8—26232no—
198C2×Q8—2622no—
199C2×Q8—1318no—
200C2×D4—2622no—
201C2×D4—2622no—
202C2×D4—1314no—
203C22×C4—2624no—
204C22×C4—2624no—
205C22×C4—2624no—
206C22×C4—2624no—
207C22×C4—2622no—
208C22×C4—2624no—
209C22×C4—1312no—
210C22×C4—1311no—
211C22×C4—1311no—
212C22×C4—1312no—
213C22×C4—1314no—
214C22×C4—1311no—
215C22×C4C22×D13111yescharacteristic
216C22×C4C22×D13112yes—
217C22×C4C22×D13112yes—
218C22×C4C22×D13111yescharacteristic
219C4⋊C4—2624no—
220C4⋊C4—2624no—
221C4⋊C4—2622no—
222C4⋊C4—2624no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—2624no—
226C22⋊C4—2622no—
227C22⋊C4—2624no—
228C42—2622no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—2622no—
234C23—2622no—
235C23—1312no—
236C23—1312no—
237C23—1312no—
238C23Q8×D13112yescomplemented
239C23C23×D13111yescharacteristic
240Q8—28no—
241Q8—2628no—
242Q8—26216no—
243D4—2628no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—26216no—
253C2×C4—2628no—
254C2×C4—2621no—
255C2×C4—2622no—
256C2×C4—2622no—
257C2×C4—2621no—
258C2×C4—2622no—
259C2×C4—2621no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—5244no—
264C2×C4—1314no—
265C2×C4—1312no—
266C2×C4—1312no—
267C2×C4—1314no—
268C2×C4D4×D13114yes—
269C2×C4D4×D13112yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—2622no—
276C22—2624no—
277C22—2624no—
278C22—2622no—
279C22—1314no—
280C22C2×D4×D13111yescharacteristic
281C22C2×D4×D13112yes—
282C22C2×Q8×D13111yescharacteristic
283C22#41×D13111yescharacteristic
284C22#1686229312yes—
285C4—24no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—2624no—
290C4—2622no—
291C4—5242no—
292C4—5241no—
293C4—1314no—
294C2—42no—
295C2—2624no—
296C2#120×D13111yescharacteristic
297C2C22⋊Q8×D13111yescharacteristic
298C2#2255388211yescharacteristic
299C2#2255397012yes—
300C2#2255397512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#393×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#393D131
hypocenterC131#2409
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D131C2semidirect
2. C2×#16862223C2semidirect
4. #22553233C2semidirect
5. #22553231C2semidirect
6. #22553224C2semidirect
7. #22553216C2semidirect
9. #22553186C2semidirect
10. #22553171C2semidirect
11. #393×C131C2semidirect
44. C2×Dic262D4semidirect
48. C2×C22⋊D131Q8semidirect
99. C2×Dic131C2×Q8semidirect
106. C23×C131C2×Q8semidirect
133. C2×D131C22⋊Q8semidirect
150. C131#2409semidirect
158. #393D131semidirect
238. C23Q8×D131semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1312
C2×#168622232
C2×(C22⋊D131)⋊C42
#225532332
#225532312
#225532242
#225532162
#225531882
#225531862
#225531712
#393×C1312
#2409131