Finite Group Databasebeta

← #16792511

Subgroups — #16792511

Order 3328.

1530 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 272 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×#11719059C211yesmaximal, characteristic, complemented
2C2×#11719061C211yesmaximal, characteristic, complemented
3#13130021C216yesmaximal, complemented
4#13130019C216yesmaximal
5C2×D13×#126C211yesmaximal, characteristic
6C22×C22⋊#261C2214yes—
7C22×C4⋊#261C2216yes—
8C2×C13×#126C411yescharacteristic, complemented
9D13×#126—24no—
10D13×#126—24no—
11C2×#88152C413yescomplemented
12C2×C4⋊C4×D13C2213yes—
13C2×C22⋊C4×D13C2213yes—
14C2×#88133C413yescomplemented
15C2×#88130C411yescharacteristic, complemented
16C2×C42×D13C2211yescharacteristic
17C2×(C2×#261)⋊C4C2216yes—
18C2×(C2×#261)⋊C4C22112yes—
19#11719061—24no—
20#11719059—24no—
21C24×D13C2311yescharacteristic
22C23×#261C2314yes—
23C23×#261C2314yes—
24C22×C4×D13C2313yes—
25C22×C4×D13C2313yes—
26C2×C22⋊#261—412no—
27C2×C4⋊#261—212no—
28C2×C4⋊#261—412no—
29C13×#126—24no—
30C2×C13×C4⋊C4C2×C413yes—
31C2×C13×C22⋊C4C2×C413yes—
32C2×C42×C13C2×C411yescharacteristic
33C2×C22⋊Dic13C2×C413yes—
34C2×#6103C2×C413yes—
35C2×#6103C2×C413yes—
36C2×C4⋊Dic13C2×C413yes—
37C2×#6101C2×C413yes—
38C2×#6101C2×C413yes—
39C2×C4×Dic13C2×C413yes—
40#88152—212no—
41C4⋊C4×D13—26no—
42C4⋊C4×D13—26no—
43C22⋊C4×D13—26no—
44C22⋊C4×D13—26no—
45#88133—212no—
46#88130—24no—
47C42×D13D414yes—
48C42×D13D414yes—
49(C2×#261)⋊C4—212no—
50(C2×#261)⋊C4—412no—
51#14665—131nomaximal, Sylow
52C24×C13C22×C411yescharacteristic
53C23×D13—21no—
54C23×D13—21no—
55C23×D13—26no—
56C23×D13—26no—
57C23×D13C22×C411yescharacteristic
58C23×D13C2411yescharacteristic
59C22×#261—24no—
60C22×#261—24no—
61C22×#261—212no—
62C22×#261—212no—
63C22×#261—212no—
64C22×#261—212no—
65C22×C4×C13C22×C413yes—
66C22×C4×C13C22×C413yes—
67C22×Dic13C22×C413yes—
68C22×Dic13C22×C413yes—
69C2×C4×D13—26no—
70C2×C4×D13—26no—
71C2×C4×D13—23no—
72C2×C4×D13—23no—
73C2×C4×D13—26no—
74C2×C4×D13—26no—
75C2×C4×D13C2×D416yes—
76C2×C4×D13C2×D416yes—
77C22⋊#261—416no—
78C4⋊#261—424no—
79C13×C4⋊C4—26no—
80C13×C22⋊C4—26no—
81C42×C13C22⋊C414yescomplemented
82C22⋊Dic13—26no—
83#6103—26no—
84#6103—26no—
85C4⋊Dic13—26no—
86#6101—26no—
87#6101—26no—
88C4×Dic13C22⋊C4112yescomplemented
89C22×#126—131no—
90C2×#384—131no—
91C2×#383—131no—
92#1441—136no—
93#1439—136no—
94C23×C13—21no—
95C23×C13—26no—
96C23×C13C23×C411yescharacteristic
97C22×D13—21no—
98C22×D13—26no—
99C22×D13—43no—
100C22×D13—43no—
101C22×D13—46no—
102C22×D13—44no—
103C22×D13—412no—
104C22×D13C22×D411yescharacteristic
105C22×D13C22×D411yescharacteristic
106C22×D13H(5,𝔽2)16yes—
107C22×D132−1+416yes—
108C2×#261—412no—
109C2×#261—412no—
110C2×#261—44no—
111C2×#261—44no—
112C2×#261—412no—
113C2×#261—412no—
114C2×C4×C13—26no—
115C2×C4×C13—23no—
116C2×C4×C13—26no—
117C2×C4×C13C2×C22⋊C416yes—
118C2×Dic13—26no—
119C2×Dic13—23no—
120C2×Dic13—26no—
121C2×Dic13C2×C22⋊C416yes—
122C4×D13—26no—
123C4×D13—26no—
124C4×D13—212no—
125C4×D13—43no—
126C4×D13—43no—
127C4×D13—46no—
128C2×#126—133no—
129C2×#126—133no—
130C2×#126—131no—
131C2×#126—264no—
132C2×#126—264no—
133C2×#126#26111yescharacteristic, complemented
134C22×C4⋊C4—133no—
135C22×C4⋊C4—136no—
136C22×C22⋊C4—133no—
137C22×C22⋊C4—134no—
138C22×C42—131no—
139#384—264no—
140#383—264no—
141C2×#95—136no—
142C2×#95—1312no—
143C22×C13—43no—
144C22×C13—44no—
145C22×C13C22×C22⋊C411yescharacteristic
146C22×C13#52016yes—
147C2×D13—43no—
148C2×D13—44no—
149C2×D13—81no—
150C2×D13—83no—
151C2×D13—84no—
152C2×D13#53513yes—
153C2×D13#53613yes—
154C2×D13#53716yes—
155C2×D13#56014yes—
156C2×D13#563112yes—
157#261—84no—
158#261—84no—
159C4×C13—26no—
160C4×C13—43no—
161Dic13—26no—
162Dic13—43no—
163C25—131no—
164C23×C4—133no—
165C23×C4—133no—
166C23×C4—134no—
167C23×C4—134no—
168#126—24no—
169#126—2612no—
170#126—2612no—
171#126—264no—
172C2×C4⋊C4—133no—
173C2×C4⋊C4—133no—
174C2×C4⋊C4—133no—
175C2×C4⋊C4—266no—
176C2×C4⋊C4—266no—
177C2×C4⋊C4—2612no—
178C2×C4⋊C4—5212no—
179C2×C4⋊C4C2×#26113yes—
180C2×C22⋊C4—133no—
181C2×C22⋊C4—133no—
182C2×C22⋊C4—133no—
183C2×C22⋊C4—266no—
184C2×C22⋊C4—266no—
185C2×C22⋊C4—5212no—
186C2×C22⋊C4C2×#26113yes—
187C2×C42—133no—
188C2×C42—134no—
189C2×C42—134no—
190C2×C42C2×#26111yescharacteristic
191#95—2612no—
192#95—5212no—
193C2×C13—81no—
194C2×C13#231113yes—
195C2×C13#238314yes—
196D13—81no—
197D13#281811yescharacteristic
198D13#282113yes—
199D13#282314yes—
200C24—131no—
201C24—131no—
202C24—261no—
203C24—261no—
204C24—266no—
205C24—266no—
206C24C22×#26111yescharacteristic
207C22×C4—133no—
208C22×C4—136no—
209C22×C4—136no—
210C22×C4—133no—
211C22×C4—266no—
212C22×C4—266no—
213C22×C4—266no—
214C22×C4—263no—
215C22×C4—263no—
216C22×C4—266no—
217C22×C4—264no—
218C22×C4—2612no—
219C22×C4—2612no—
220C22×C4—2612no—
221C22×C4—264no—
222C22×C4—2612no—
223C22×C4C22×#26113yes—
224C22×C4C22×#26113yes—
225C4⋊C4—26no—
226C4⋊C4—266no—
227C4⋊C4—266no—
228C4⋊C4—266no—
229C4⋊C4—5224no—
230C22⋊C4—26no—
231C22⋊C4—266no—
232C22⋊C4—266no—
233C22⋊C4—266no—
234C22⋊C4—5216no—
235C42—1312no—
236C42C22⋊#26114yescomplemented
237C13#1466511yescharacteristic, complemented, Sylow
238C23—26no—
239C23—21no—
240C23—136no—
241C23—136no—
242C23—131no—
243C23—131no—
244C23—266no—
245C23—261no—
246C23—524no—
247C23—523no—
248C23—523no—
249C23—526no—
250C23—5212no—
251C23C23×#26111yescharacteristic
252C2×C4—26no—
253C2×C4—26no—
254C2×C4—23no—
255C2×C4—136no—
256C2×C4—266no—
257C2×C4—266no—
258C2×C4—2612no—
259C2×C4—266no—
260C2×C4—266no—
261C2×C4—263no—
262C2×C4—523no—
263C2×C4—523no—
264C2×C4—526no—
265C2×C4—524no—
266C2×C4—524no—
267C2×C4—5212no—
268C2×C4—5212no—
269C2×C4—5212no—
270C2×C4—5212no—
271C2×C4C2×C22⋊#26116yes—
272C22—43no—
273C22—44no—
274C22—133no—
275C22—133no—
276C22—136no—
277C22—134no—
278C22—1312no—
279C22—523no—
280C22—524no—
281C22—1041no—
282C22—1044no—
283C22—1043no—
284C22#1171963116yes—
285C22C22×C22⋊#26111yescharacteristic
286C4—26no—
287C4—43no—
288C4—266no—
289C4—523no—
290C4—1044no—
291C4—1044no—
292C2—81no—
293C2—131no—
294C2—134no—
295C2—133no—
296C2—1041no—
297C2#1313131113yes—
298C2#1313138314yes—
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC2×C13×#126C4
prime radicalC24×D13C23
cosocleC23×D13C24
hypercenterC2×#126#261
hypocenterC13#14665
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#11719059C2semidirect
2. C2×#11719061C2semidirect
3. #13130021C2semidirect
8. C2×C13×#126C4semidirect
11. C2×#88152C4semidirect
14. C2×#88133C4semidirect
15. C2×#88130C4semidirect
81. C42×C13C22⋊C4semidirect
88. C4×Dic13C22⋊C4semidirect
133. C2×#126#261semidirect
236. C42C22⋊#261semidirect
237. C13#14665semidirect

Maximal subgroups

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

subgroupindex
C2×#117190592
C2×#117190612
#131300212
#131300192
C2×D13×#1262
#1466513