Finite Group Databasebeta

← #30670238

Subgroups — #30670238

Order 16768.

700 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 141 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×C22⋊C4×D131C212yesmaximal
2#22553240C212yesmaximal, complemented
3#22553237C211yesmaximal, characteristic, complemented
4#22553186C212yesmaximal, complemented
5#22553183C212yesmaximal, complemented
6#22553174C212yesmaximal, complemented
7#22553174C212yesmaximal, complemented
8#22553167C211yesmaximal, characteristic
9#397×C131C211yesmaximal, characteristic, complemented
10C24×D131C2211yescharacteristic
11C22×C4×D131C2212yes—
12C22×C4×D131C2212yes—
13C2×C4⋊Dic131C2211yescharacteristic, complemented
14C2×Dic131⋊C4C2212yescomplemented
15C2×C22⋊Dic131C2212yes—
16C2×C22⋊Dic131C2211yescharacteristic
17C2×(C2×D131)⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2212yes—
21C22⋊C4×D131—28no—
22C131⋊#95C2211yescharacteristic, complemented
23C131⋊#95C2212yescomplemented
24C131⋊#95C2212yescomplemented
25C131⋊#95C2212yescomplemented
26C131⋊#95C2212yescomplemented
27C2×C4⋊C4×C131C2211yescharacteristic, complemented
28C2×C22⋊C4×C131C2212yes—
29C2×C22⋊C4×C131C2211yescharacteristic
30#95×C131C2211yescharacteristic, complemented
31#95×C131C2212yescomplemented
32C23×D131—24no—
33C23×D131—24no—
34C23×D131—41no—
35C23×D131—41no—
36C23×D131D414yes—
37C23×D131C2312yes—
38C22×Dic131C2312yes—
39C22×Dic131C2312yes—
40C22×Dic131C2312yes—
41C2×C4×D131—24no—
42C2×C4×D131—24no—
43C2×C4×D131—24no—
44C2×C4×D131—24no—
45C2×C4×D131—42no—
46C2×C4×D131—42no—
47C4⋊Dic131—41no—
48Dic131⋊C4—42no—
49C22⋊Dic131—24no—
50C22⋊Dic131—41no—
51(C2×D131)⋊C4—24no—
52(C2×D131)⋊C4—24no—
53(C2×D131)⋊C4—24no—
54(C2×D131)⋊C4—24no—
55C24×C131C2311yescharacteristic
56C22×C4×C131C2312yes—
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2311yescharacteristic
60C4⋊C4×C131—41no—
61C22⋊C4×C131—24no—
62C22⋊C4×C131—41no—
63C22×D131—22no—
64C22×D131—22no—
65C22×D131—22no—
66C22×D131—22no—
67C22×D131—216no—
68C22×D131—44no—
69C22×D131—48no—
70C22×D131—44no—
71C22×D131—44no—
72C22×D131C2×D414yes—
73C22×D131#4114yes—
74C22×D131#4114yes—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C4×D131—48no—
85C4×D131—48no—
86C23×C131—22no—
87C23×C131—22no—
88C23×C131—21no—
89C23×C131—21no—
90C23×C131C2×D412yes—
91C23×C131C2411yescharacteristic
92C2×C4×C131—22no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—22no—
96C2×C4×C131—22no—
97C2×C4×C131—22no—
98C2×C4×C131—21no—
99C2×C4×C131—21no—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C22×C131—24no—
103C22×C131—22no—
104C22×C131—42no—
105C22×C131—41no—
106C22×C131—41no—
107C22×C131C22×D411yescharacteristic
108C22×C131C2×#4112yes—
109C22×C131C2×#4112yes—
110C22×C131H(5,𝔽2)11yescharacteristic
111C22×C131H(5,𝔽2)11yescharacteristic
112C2×D131—24no—
113C2×D131—24no—
114C2×D131—24no—
115C2×D131—24no—
116C2×D131—24no—
117C2×D131—24no—
118C2×D131—416no—
119C2×D131#12318yes—
120C4×C131—42no—
121C4×C131—42no—
122C4×C131—41no—
123C4×C131—41no—
124Dic131—42no—
125Dic131—42no—
126Dic131—42no—
127C2×C131—42no—
128C2×C131C2×#12312yes—
129C2×C131#53411yescharacteristic
130C2×C131#55112yes—
131C2×C131#55111yescharacteristic
132C2×C131#55111yescharacteristic
133D131—28no—
134C131#248311yescharacteristic, complemented, Sylow
135#2483—1311nomaximal, Sylow
136C22×C22⋊C4—1312no—
137#397—1311no—
138#397—1312no—
139#397D13111yescharacteristic, complemented
140#385—1312no—
141#385—1312no—
142#381—1312no—
143#381—1312no—
144#379—1311no—
145C25—1311no—
146C23×C4—1312no—
147C23×C4—1312no—
148C2×C4⋊C4—1312no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4C2×D13111yescharacteristic, complemented
151C2×C22⋊C4—2628no—
152C2×C22⋊C4—1312no—
153C2×C22⋊C4—1312no—
154C2×C22⋊C4—1312no—
155C2×C22⋊C4—1312no—
156C2×C22⋊C4—1312no—
157C2×C22⋊C4—1311no—
158C2×C22⋊C4C2×D13112yes—
159C2×C22⋊C4C2×D13111yescharacteristic
160#95—1311no—
161#95—1312no—
162#95—1312no—
163#95—1312no—
164#95—1312no—
165#95C2×D13111yescharacteristic, complemented
166#95C2×D13112yescomplemented
167C24—2624no—
168C24—2624no—
169C24—5241no—
170C24—5241no—
171C24—1314no—
172C24—1312no—
173C24C22×D13111yescharacteristic
174C22×C4—2624no—
175C22×C4—2624no—
176C22×C4—2624no—
177C22×C4—2624no—
178C22×C4—5242no—
179C22×C4—5242no—
180C22×C4—1312no—
181C22×C4—1312no—
182C22×C4—1312no—
183C22×C4C22×D13112yes—
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13111yescharacteristic
186C22×C4C22×D13112yes—
187C4⋊C4—41no—
188C4⋊C4—5242no—
189C4⋊C4—5241no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—2624no—
193C22⋊C4—2624no—
194C22⋊C4—2624no—
195C22⋊C4—2624no—
196C22⋊C4—2624no—
197C22⋊C4—5241no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—26216no—
203C23—2622no—
204C23—2622no—
205C23—2622no—
206C23—2622no—
207C23—5244no—
208C23—5248no—
209C23—5244no—
210C23—5244no—
211C23—1314no—
212C23—1314no—
213C23—1314no—
214C23D4×D13112yes—
215C23C23×D13111yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—2622no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—2622no—
233C2×C4—2622no—
234C2×C4—2622no—
235C2×C4—5248no—
236C2×C4—5248no—
237C22—22no—
238C22—24no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—2624no—
243C22—2624no—
244C22—2624no—
245C22—2624no—
246C22—2624no—
247C22—2624no—
248C22—52416no—
249C22—1318no—
250C22C2×D4×D13111yescharacteristic
251C22#41×D13112yes—
252C22#41×D13112yes—
253C22#1686229111yescharacteristic
254C22#1686229211yescharacteristic
255C4—42no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—5242no—
260C4—5242no—
261C4—5242no—
262C2—42no—
263C2—2628no—
264C2#123×D13112yes—
265C2#2255387011yescharacteristic
266C2#2255398712yes—
267C2#2255399011yescharacteristic
268C2#2255399211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#397×C131C2
prime radicalC24×D131C22
hypercenter#397D131
hypocenterC131#2483
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #22553240C2semidirect
3. #22553237C2semidirect
4. #22553186C2semidirect
5. #22553183C2semidirect
6. #22553174C2semidirect
7. #22553174C2semidirect
9. #397×C131C2semidirect
13. C2×C4⋊Dic131C22semidirect
14. C2×Dic131⋊C4C22semidirect
22. C131⋊#95C22semidirect
23. C131⋊#95C22semidirect
24. C131⋊#95C22semidirect
25. C131⋊#95C22semidirect
26. C131⋊#95C22semidirect
27. C2×C4⋊C4×C131C22semidirect
30. #95×C131C22semidirect
31. #95×C131C22semidirect
134. C131#2483semidirect
139. #397D131semidirect
150. C2×C4⋊C4C2×D131semidirect
165. #95C2×D131semidirect
166. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1312
#225532402
#225532372
#225531862
#225531832
#225531742
#225531672
#397×C1312
#2483131