Finite Group Databasebeta

← #30670892

Subgroups — #30670892

Order 16768.

900 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16862229C212yesmaximal, complemented
2C2×#16862220C211yesmaximal, characteristic, complemented
3C2×#16862215C212yesmaximal, complemented
4#22553248C212yesmaximal, complemented
5#22553221C212yesmaximal, complemented
6#22553212C214yesmaximal, complemented
7C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C131C211yesmaximal, characteristic, complemented
9C22×D524C2211yescharacteristic, complemented
10C22×C22⋊D131C2212yes—
11C22×C22⋊D131C2214yes—
12#16862229—28no—
13#16862220—24no—
14#16862215—28no—
15C2×C4⋊Dic131C2211yescharacteristic, complemented
16C2×Dic131⋊C4C2214yescomplemented
17C2×C22⋊Dic131C2214yes—
18C2×C22⋊Dic131C2212yes—
19C2×(C2×D131)⋊C4C2212yescomplemented
20C2×(C2×D131)⋊C4C2214yescomplemented
21C2×C4×Dic131C2212yescomplemented
22C131⋊#95C2212yescomplemented
23C25×C131C2211yescharacteristic
24C23×C4×C131C2211yescharacteristic
25C23×C4×C131C2211yescharacteristic
26C2×C22⋊C4×C131—24no—
27C2×C22⋊C4×C131C2212yes—
28C2×C22⋊C4×C131C2212yes—
29C23×D131C2312yes—
30C2×D524—22no—
31C2×D524—42no—
32C2×C22⋊D131—28no—
33C2×C22⋊D131—24no—
34C2×C22⋊D131—24no—
35C2×C22⋊D131—24no—
36C2×C22⋊D131—28no—
37C2×C22⋊D131—44no—
38C22×Dic131C2312yes—
39C22×Dic131C2314yes—
40C4⋊Dic131—22no—
41Dic131⋊C4—28no—
42C22⋊Dic131—28no—
43C22⋊Dic131D418yescomplemented
44(C2×D131)⋊C4—28no—
45(C2×D131)⋊C4—24no—
46C4×Dic131—42no—
47C24×C131—22no—
48C24×C131—24no—
49C24×C131—24no—
50C24×C131—42no—
51C24×C131C2312yes—
52C24×C131C2311yescharacteristic
53C22×C4×C131—24no—
54C22×C4×C131—22no—
55C22×C4×C131—42no—
56C22×C4×C131D414yescomplemented
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2311yescharacteristic
60C22⋊C4×C131—24no—
61C22⋊C4×C131—24no—
62C22×D131—22no—
63C22×D131—22no—
64C22×D131—22no—
65C22×D131—24no—
66C22×D131—24no—
67D524—44no—
68C22⋊D131—416no—
69C22⋊D131—48no—
70C2×Dic131—24no—
71C2×Dic131—24no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131C2×D418yes—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131—28no—
79C23×C131—22no—
80C23×C131—24no—
81C23×C131—22no—
82C23×C131—24no—
83C23×C131—416no—
84C23×C131—44no—
85C23×C131—44no—
86C23×C131C2×D414yes—
87C23×C131C2×D412yes—
88C23×C131#4114yes—
89C23×C131C2411yescharacteristic
90C2×C4×C131—22no—
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—24no—
94C2×C4×C131—44no—
95C2×C4×C131—44no—
96C2×C4×C131C2×D412yes—
97C2×C4×C131C2×D412yes—
98C22×C131—22no—
99C22×C131—22no—
100C22×C131—28no—
101C22×C131—24no—
102C22×C131—28no—
103C22×C131—416no—
104C22×C131—42no—
105C22×C131—42no—
106C22×C131—44no—
107C22×C131#12714yes—
108C22×C131C22×D412yes—
109C22×C131C22×D411yescharacteristic
110C22×C131C2×#4112yes—
111C22×C131H(5,𝔽2)12yes—
112C2×D131—42no—
113C2×D131—42no—
114C2×D131—42no—
115C2×D131—44no—
116C2×D131—44no—
117C4×C131—22no—
118C4×C131—22no—
119C4×C131—42no—
120Dic131—28no—
121Dic131—42no—
122C2×C131—24no—
123C2×C131—44no—
124C2×C131#52511yescharacteristic
125C2×C131C2×#12711yescharacteristic
126C2×C131#53411yescharacteristic
127C2×C131#54614yes—
128D131—82no—
129C131#253911yescharacteristic, complemented, Sylow
130#2539—1311nomaximal, Sylow
131C2×#123—1312no—
132C2×#121—1311no—
133C2×#120—1312no—
134C22×C22⋊C4D13111yescharacteristic, complemented
135#399—1312no—
136#392—1312no—
137#390—1314no—
138C4×C22⋊C4—1311no—
139C25C2×D13111yescharacteristic
140C22×D4—1312no—
141C22×D4—1314no—
142C22×D4—1311no—
143C23×C4C2×D13111yescharacteristic
144C23×C4C2×D13111yescharacteristic
145#123—2628no—
146#121—2624no—
147#120—2628no—
148C2×C4⋊C4—1311no—
149C2×C4⋊C4—1314no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—1314no—
152C2×C22⋊C4—1312no—
153C2×C22⋊C4—1314no—
154C2×C22⋊C4—1312no—
155C2×C22⋊C4C2×D13112yes—
156C2×C22⋊C4C2×D13112yes—
157C2×C42—1312no—
158#95—1312no—
159C24—22no—
160C24—24no—
161C24—24no—
162C24—42no—
163C24—1312no—
164C24C22×D13112yes—
165C24C22×D13111yescharacteristic
166C2×D4—2624no—
167C2×D4—2624no—
168C2×D4—2624no—
169C2×D4—2628no—
170C2×D4—2628no—
171C2×D4—2622no—
172C2×D4—5242no—
173C2×D4—5244no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—1312no—
178C22×C4—1314no—
179C22×C4C22⋊D13114yescomplemented
180C22×C4C22×D13112yes—
181C22×C4C22×D13111yescharacteristic
182C22×C4C22×D13111yescharacteristic
183C4⋊C4—2628no—
184C4⋊C4—2622no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—2628no—
188C22⋊C4—2628no—
189C22⋊C4—2624no—
190C22⋊C4—1318no—
191C42—5242no—
192C23—22no—
193C23—22no—
194C23—28no—
195C23—24no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—416no—
200C23—44no—
201C23—44no—
202C23—2622no—
203C23—2622no—
204C23—2622no—
205C23—2624no—
206C23—2624no—
207C23D4×D13114yes—
208C23C2×C22⋊D13112yes—
209C23#1452351314yes—
210C23C23×D13111yescharacteristic
211D4—5244no—
212D4—52416no—
213D4—5248no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—2624no—
221C2×C4—2624no—
222C2×C4—2622no—
223C2×C4—2622no—
224C2×C4—2622no—
225C2×C4—1318no—
226C2×C4C2×C22⋊D13112yes—
227C2×C4C2×C22⋊D13112yes—
228C22—22no—
229C22—28no—
230C22—22no—
231C22—28no—
232C22—24no—
233C22—416no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22—5242no—
238C22—5244no—
239C22—5244no—
240C22—5242no—
241C22—5242no—
242C22#1686224214yes—
243C22C2×D4×D13112yes—
244C22C22×C22⋊D13111yescharacteristic
245C22C2×#1452351312yes—
246C22#1686229112yes—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—2628no—
251C4—5242no—
252C2—24no—
253C2—44no—
254C2—10482no—
255C2#2255382311yescharacteristic
256C2C2×#1686224211yescharacteristic
257C2#2255387111yescharacteristic
258C2#2255395214yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C22⋊C4×C131C2
hypercenterC22×C22⋊C4D131
hypocenterC131#2539
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16862229C2semidirect
2. C2×#16862220C2semidirect
3. C2×#16862215C2semidirect
4. #22553248C2semidirect
5. #22553221C2semidirect
6. #22553212C2semidirect
7. C4×C22⋊Dic131C2semidirect
8. C22×C22⋊C4×C131C2semidirect
9. C22×D524C22semidirect
15. C2×C4⋊Dic131C22semidirect
16. C2×Dic131⋊C4C22semidirect
19. C2×(C2×D131)⋊C4C22semidirect
20. C2×(C2×D131)⋊C4C22semidirect
21. C2×C4×Dic131C22semidirect
22. C131⋊#95C22semidirect
43. C22⋊Dic131D4semidirect
56. C22×C4×C131D4semidirect
129. C131#2539semidirect
134. C22×C22⋊C4D131semidirect
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×#168622292
C2×#168622202
C2×#168622152
#225532482
#225532212
#225532122
C4×C22⋊Dic1312
C22×C22⋊C4×C1312
#2539131