Finite Group Databasebeta

← #30670893

Subgroups — #30670893

Order 16768.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16862227C212yesmaximal, complemented
2C2×#16862214C212yesmaximal, complemented
3C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
4#22553247C212yesmaximal, complemented
5#22553220C212yesmaximal, complemented
6#22553212C214yesmaximal, complemented
7C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
8C2×#121×C131C211yesmaximal, characteristic, complemented
9C24×D131C2211yescharacteristic
10C22×C22⋊D131C2214yes—
11C23×Dic131C2211yescharacteristic
12#16862227—28no—
13#16862214—28no—
14C2×Dic131⋊C4C2214yescomplemented
15C2×C22⋊Dic131C2212yes—
16C2×C22⋊Dic131C2214yescomplemented
17C2×(C2×D131)⋊C4—24no—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2214yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×C4×Dic131C2212yescomplemented
22C131⋊#95C2212yescomplemented
23C22×D4×C131C2212yes—
24C22×D4×C131C2211yescharacteristic, complemented
25C23×C4×C131C2211yescharacteristic
26#121×C131—24no—
27C2×C4⋊C4×C131C2211yescharacteristic, complemented
28C2×C22⋊C4×C131C2212yescomplemented
29C23×D131—24no—
30C23×D131—22no—
31C23×D131—24no—
32C23×D131—42no—
33C23×D131C2312yes—
34C2×C22⋊D131—28no—
35C2×C22⋊D131—28no—
36C2×C22⋊D131—44no—
37C22×Dic131—24no—
38C22×Dic131—22no—
39C22×Dic131C2312yes—
40C22×Dic131C2314yes—
41Dic131⋊C4—28no—
42C22⋊Dic131—28no—
43C22⋊Dic131D418yescomplemented
44(C2×D131)⋊C4—28no—
45(C2×D131)⋊C4—24no—
46(C2×D131)⋊C4—24no—
47C4×Dic131—42no—
48C24×C131C2312yes—
49C24×C131C2311yescharacteristic
50C2×D4×C131—24no—
51C2×D4×C131—24no—
52C2×D4×C131—24no—
53C2×D4×C131—22no—
54C2×D4×C131—42no—
55C22×C4×C131—42no—
56C22×C4×C131D414yescomplemented
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2311yescharacteristic
60C4⋊C4×C131—22no—
61C22⋊C4×C131—24no—
62C22×D131—28no—
63C22×D131—22no—
64C22×D131—24no—
65C22×D131—22no—
66C22×D131—24no—
67C22×D131—416no—
68C22×D131—44no—
69C22×D131—44no—
70C22×D131#4114yes—
71C22⋊D131—416no—
72C2×Dic131—24no—
73C2×Dic131—24no—
74C2×Dic131—22no—
75C2×Dic131—24no—
76C2×Dic131—44no—
77C2×Dic131C2×D418yes—
78C23×C131—22no—
79C23×C131—22no—
80C23×C131—22no—
81C23×C131—22no—
82C23×C131—22no—
83C23×C131—24no—
84C23×C131—24no—
85C23×C131C2×D414yes—
86C23×C131C2×D412yes—
87C23×C131C2411yescharacteristic
88D4×C131—44no—
89D4×C131—48no—
90C2×C4×C131—22no—
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—44no—
96C2×C4×C131C2×D412yes—
97C2×C4×C131C2×D412yes—
98C22×C131—22no—
99C22×C131—22no—
100C22×C131—28no—
101C22×C131—42no—
102C22×C131—42no—
103C22×C131—42no—
104C22×C131—44no—
105C22×C131—44no—
106C22×C131#12714yes—
107C22×C131C22×D412yes—
108C22×C131C22×D411yescharacteristic
109C22×C131C2×#4112yes—
110C22×C131H(5,𝔽2)12yes—
111C2×D131—24no—
112C2×D131—28no—
113C2×D131—416no—
114C2×D131—42no—
115C2×D131—42no—
116C2×D131—44no—
117C4×C131—22no—
118C4×C131—22no—
119C4×C131—42no—
120Dic131—28no—
121Dic131—42no—
122C2×C131—24no—
123C2×C131—82no—
124C2×C131#52511yescharacteristic
125C2×C131C2×#12711yescharacteristic
126C2×C131#53411yescharacteristic
127C2×C131#54614yes—
128D131—44no—
129C131#253911yescharacteristic, complemented, Sylow
130#2539—1311nomaximal, Sylow
131C2×#123—1312no—
132C2×#121D13111yescharacteristic, complemented
133C2×#120—1312no—
134C22×C22⋊C4—1311no—
135#399—1312no—
136#392—1312no—
137#390—1314no—
138C4×C22⋊C4—1311no—
139C25—1311no—
140C22×D4—1314no—
141C22×D4C2×D13112yes—
142C22×D4C2×D13111yescharacteristic, complemented
143C23×C4—1311no—
144C23×C4C2×D13111yescharacteristic
145#123—2628no—
146#121—24no—
147#120—2628no—
148C2×C4⋊C4—1314no—
149C2×C4⋊C4C2×D13111yescharacteristic, complemented
150C2×C22⋊C4—2624no—
151C2×C22⋊C4—1314no—
152C2×C22⋊C4—1312no—
153C2×C22⋊C4—1312no—
154C2×C22⋊C4—1314no—
155C2×C22⋊C4—1312no—
156C2×C22⋊C4C2×D13112yescomplemented
157C2×C42—1312no—
158#95—1312no—
159C24—2624no—
160C24—2622no—
161C24—2624no—
162C24—5242no—
163C24—1312no—
164C24C22×D13112yes—
165C24C22×D13111yescharacteristic
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—2628no—
172C2×D4—2628no—
173C2×D4—5244no—
174C22×C4—42no—
175C22×C4—2624no—
176C22×C4—2622no—
177C22×C4—1312no—
178C22×C4—1314no—
179C22×C4C22⋊D13114yescomplemented
180C22×C4C22×D13112yes—
181C22×C4C22×D13111yescharacteristic
182C22×C4C22×D13111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—2628no—
185C22⋊C4—24no—
186C22⋊C4—2628no—
187C22⋊C4—2628no—
188C22⋊C4—2624no—
189C22⋊C4—2624no—
190C22⋊C4—1318no—
191C42—5242no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—2628no—
200C23—2624no—
201C23—2622no—
202C23—2622no—
203C23—2624no—
204C23—52416no—
205C23—5244no—
206C23—5244no—
207C23—1314no—
208C23D4×D13114yes—
209C23C2×C22⋊D13112yes—
210C23C23×D13111yescharacteristic
211D4—44no—
212D4—48no—
213D4—52416no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—2624no—
221C2×C4—2624no—
222C2×C4—2624no—
223C2×C4—2622no—
224C2×C4—5244no—
225C2×C4—1318no—
226C2×C4C2×C22⋊D13112yes—
227C2×C4C2×C22⋊D13112yes—
228C22—22no—
229C22—28no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—44no—
234C22—42no—
235C22—42no—
236C22—2628no—
237C22—2624no—
238C22—52416no—
239C22—5242no—
240C22—5244no—
241C22—5242no—
242C22#1686224214yes—
243C22C2×D4×D13112yes—
244C22C22×C22⋊D13111yescharacteristic
245C22#41×D13112yes—
246C22#1686229112yes—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—2628no—
251C4—5242no—
252C2—24no—
253C2—82no—
254C2—5244no—
255C2#2255381911yescharacteristic
256C2C2×#1686224211yescharacteristic
257C2#2255387111yescharacteristic
258C2#2255395114yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#121×C131C2
hypercenterC2×#121D131
hypocenterC131#2539
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16862227C2semidirect
2. C2×#16862214C2semidirect
3. C22×(C2×D131)⋊C4C2semidirect
4. #22553247C2semidirect
5. #22553220C2semidirect
6. #22553212C2semidirect
7. C4×C22⋊Dic131C2semidirect
8. C2×#121×C131C2semidirect
14. C2×Dic131⋊C4C22semidirect
16. C2×C22⋊Dic131C22semidirect
21. C2×C4×Dic131C22semidirect
22. C131⋊#95C22semidirect
24. C22×D4×C131C22semidirect
27. C2×C4⋊C4×C131C22semidirect
28. C2×C22⋊C4×C131C22semidirect
43. C22⋊Dic131D4semidirect
56. C22×C4×C131D4semidirect
129. C131#2539semidirect
132. C2×#121D131semidirect
142. C22×D4C2×D131semidirect
149. C2×C4⋊C4C2×D131semidirect
156. C2×C22⋊C4C2×D131semidirect
179. C22×C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622272
C2×#168622142
C22×(C2×D131)⋊C42
#225532472
#225532202
#225532122
C4×C22⋊Dic1312
C2×#121×C1312
#2539131