Finite Group Databasebeta

← #15784536

Subgroups — #15784536

Order 3328.

2878 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 427 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×#11718616C213yesmaximal
2#13113074C212yesmaximal, complemented
3#13113073C213yesmaximal, complemented
4#13113072C216yesmaximal, complemented
5C13×#1437C211yesmaximal, characteristic, complemented
6C25×D13C2211yescharacteristic
7C22×C13×C22⋊C4C2213yes—
8C22×C42×C13C2211yescharacteristic, complemented
9C22×C22⋊Dic13C2213yes—
10C22×#6103C2216yes—
11C22×#6103C2216yes—
12C22×C4×Dic13C2213yescomplemented
13C2×C13×#95C2213yescomplemented
14C2×#88071C2213yescomplemented
15C2×#88071C2216yescomplemented
16#11718616C4124yescomplemented
17C25×C13C2311yescharacteristic
18C24×D13—46no—
19C24×D13—43no—
20C24×D13—43no—
21C24×D13C2×C4112yes—
22C24×D13C2312yes—
23C23×C4×C13C2313yes—
24C23×C4×C13C2313yes—
25C23×Dic13C2316yes—
26C2×C13×C22⋊C4—43no—
27C2×C13×C22⋊C4—43no—
28C2×C13×C22⋊C4C2×C4112yes—
29C2×C42×C13—43no—
30C2×C22⋊Dic13—46no—
31C2×C22⋊Dic13C2×C4112yes—
32C2×#6103—46no—
33C2×#6103—46no—
34C2×#6103—46no—
35C2×#6103—46no—
36C2×#6103C2×C4124yes—
37C2×#6103C2×C4124yes—
38C2×C4×Dic13—43no—
39C2×C4×Dic13—46no—
40C13×#95—43no—
41#88071—43no—
42#88071—46no—
43#13827—131nomaximal, Sylow
44C24×C13—26no—
45C24×C13—23no—
46C24×C13—23no—
47C24×C13C22×C416yes—
48C24×C13C2411yescharacteristic
49C23×D13—26no—
50C23×D13—26no—
51C23×D13—26no—
52C23×D13—26no—
53C23×D13—424no—
54C23×D13—424no—
55C23×D13—412no—
56C23×D13—412no—
57C23×D13—44no—
58C23×D13—424no—
59C23×D13—412no—
60C23×D13—48no—
61C23×D13—44no—
62C23×D13—412no—
63C23×D13C42116yescomplemented
64C23×D13C22×C4112yes—
65C22×C4×C13—26no—
66C22×C4×C13—23no—
67C22×C4×C13—26no—
68C22×C4×C13—23no—
69C22×C4×C13—26no—
70C22×C4×C13—23no—
71C22×C4×C13—26no—
72C22×C4×C13—23no—
73C22×C4×C13C22×C416yes—
74C22×C4×C13C22×C416yes—
75C22×Dic13—26no—
76C22×Dic13—26no—
77C22×Dic13—26no—
78C22×Dic13—26no—
79C22×Dic13—26no—
80C22×Dic13—26no—
81C22×Dic13C22×C4112yes—
82C13×C22⋊C4—412no—
83C42×C13—44no—
84C22⋊Dic13—412no—
85#6103—424no—
86#6103—424no—
87C4×Dic13—412no—
88C2×#379—133no—
89#1437—132no—
90#1437—136no—
91#1437—133no—
92#1437D1311yescharacteristic, complemented
93C23×C13—26no—
94C23×C13—26no—
95C23×C13—212no—
96C23×C13—212no—
97C23×C13—41no—
98C23×C13—46no—
99C23×C13—43no—
100C23×C13—42no—
101C23×C13—41no—
102C23×C13—43no—
103C23×C13C2×C4214yes—
104C23×C13C23×C413yes—
105C23×C13H(5,𝔽2)13yes—
106C23×C13H(5,𝔽2)13yes—
107C23×C13H(5,𝔽2)16yes—
108C22×D13—212no—
109C22×D13—212no—
110C22×D13—212no—
111C22×D13—212no—
112C22×D13—212no—
113C22×D13—212no—
114C22×D13—448no—
115C22×D13—448no—
116C22×D13—448no—
117C22×D13—496no—
118C22×D13—42no—
119C22×D13—46no—
120C22×D13—46no—
121C22×D13—46no—
122C22×D13—42no—
123C22×D13—42no—
124C22×D13—42no—
125C22×D13—46no—
126C22×D13C2×C4218yes—
127C2×C4×C13—212no—
128C2×C4×C13—212no—
129C2×C4×C13—26no—
130C2×C4×C13—26no—
131C2×C4×C13—43no—
132C2×C4×C13—46no—
133C2×C4×C13—46no—
134C2×C4×C13—43no—
135C2×C4×C13—43no—
136C2×C4×C13—43no—
137C2×Dic13—212no—
138C2×Dic13—212no—
139C2×Dic13—212no—
140C2×Dic13—46no—
141C2×Dic13—46no—
142C2×Dic13—46no—
143C2×Dic13—46no—
144C26—131no—
145C22×C22⋊C4—133no—
146C22×C22⋊C4—136no—
147C22×C22⋊C4—136no—
148C22×C22⋊C4C2×D1313yes—
149C22×C42—133no—
150C22×C42C2×D1311yescharacteristic, complemented
151#379—1324no—
152C2×#95—136no—
153C2×#95—133no—
154C2×#95C2×D1313yescomplemented
155C22×C13—212no—
156C22×C13—46no—
157C22×C13—46no—
158C22×C13—412no—
159C22×C13C22×C4211yescharacteristic
160C22×C13#51813yes—
161C22×C13#51813yes—
162C22×C13#51816yes—
163C22×C13#51816yes—
164C22×C13#56011yescharacteristic
165C22×C13#56016yes—
166C22×C13#56013yes—
167C22×C13#56012yes—
168C22×C13H(3,𝔽4)11yescharacteristic
169C22×C13H(3,𝔽4)13yes—
170C2×D13—224no—
171C2×D13—464no—
172C2×D13—412no—
173C2×D13—412no—
174C2×D13—412no—
175C2×D13—412no—
176C4×C13—46no—
177C4×C13—46no—
178Dic13—412no—
179C25—132no—
180C25—1312no—
181C25—523no—
182C25—523no—
183C25—526no—
184C25C22×D1311yescharacteristic
185C23×C4—136no—
186C23×C4C22×D1313yes—
187C23×C4C22×D1313yes—
188C2×C22⋊C4—43no—
189C2×C22⋊C4—43no—
190C2×C22⋊C4—1312no—
191C2×C22⋊C4—1324no—
192C2×C22⋊C4—1324no—
193C2×C22⋊C4—526no—
194C2×C22⋊C4—526no—
195C2×C22⋊C4—526no—
196C2×C22⋊C4—526no—
197C2×C22⋊C4—526no—
198C2×C22⋊C4C4×D13112yes—
199C2×C42—43no—
200C2×C42—523no—
201C2×C42—526no—
202#95—43no—
203#95—523no—
204#95—526no—
205C2×C13—44no—
206C2×C13#227313yes—
207C2×C13#237513yes—
208C2×C13#237513yes—
209C2×C13#237516yes—
210D13—48no—
211C24—23no—
212C24—23no—
213C24—26no—
214C24—1312no—
215C24—1316no—
216C24—266no—
217C24—266no—
218C24—266no—
219C24—266no—
220C24—5212no—
221C24—5212no—
222C24—5224no—
223C24—5224no—
224C24—524no—
225C24—524no—
226C24—5224no—
227C24—5212no—
228C24—5212no—
229C24—528no—
230C24C2×C4×D1316yes—
231C24C23×D1311yescharacteristic
232C22×C4—23no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—26no—
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—1312no—
241C22×C4—266no—
242C22×C4—266no—
243C22×C4—266no—
244C22×C4—266no—
245C22×C4—266no—
246C22×C4—266no—
247C22×C4C2×C4×D1316yes—
248C22×C4C2×C4×D1316yes—
249C22⋊C4—412no—
250C22⋊C4—5224no—
251C22⋊C4—5224no—
252C22⋊C4—5212no—
253C42—44no—
254C42—5212no—
255C13#1382711yescharacteristic, complemented, Sylow
256C23—26no—
257C23—26no—
258C23—212no—
259C23—212no—
260C23—41no—
261C23—41no—
262C23—46no—
263C23—43no—
264C23—43no—
265C23—42no—
266C23—138no—
267C23—2612no—
268C23—2612no—
269C23—2612no—
270C23—2612no—
271C23—2612no—
272C23—2612no—
273C23—5248no—
274C23—5248no—
275C23—5248no—
276C23—5296no—
277C23—522no—
278C23—522no—
279C23—526no—
280C23—526no—
281C23—526no—
282C23—526no—
283C23—522no—
284C23—522no—
285C23C42×D1314yes—
286C23C22×C4×D1313yes—
287C23#8825113yes—
288C23#8825113yes—
289C23#8825616yes—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—212no—
293C2×C4—212no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—2612no—
301C2×C4—2612no—
302C2×C4—2612no—
303C2×C4—526no—
304C2×C4—526no—
305C2×C4—526no—
306C2×C4—526no—
307C22—212no—
308C22—46no—
309C22—46no—
310C22—412no—
311C22—2624no—
312C22—5264no—
313C22—5212no—
314C22—5212no—
315C22—5212no—
316C22—5212no—
317C22C2×C42×D1311yescharacteristic
318C22#1171919113yes—
319C22#1171922613yes—
320C22#1171925016yes—
321C22#1171925016yes—
322C22#1171929911yescharacteristic
323C22#1171932516yes—
324C22#1171937113yes—
325C22#1171938813yes—
326C22#1171941612yes—
327C22#1171942511yescharacteristic
328C4—46no—
329C4—46no—
330C4—5212no—
331C2—44no—
332C2—528no—
333C2#1311718813yes—
334C2#1311797313yes—
335C2#1311797413yes—
336C2#1311797616yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC22×C13C22×C42
FittingC13×#1437C2
socle, cosocleC24×C13C24
prime radicalC25×D13C22
hypercenter#1437D13
hypocenterC13#13827
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #13113074C2semidirect
3. #13113073C2semidirect
4. #13113072C2semidirect
5. C13×#1437C2semidirect
8. C22×C42×C13C22semidirect
12. C22×C4×Dic13C22semidirect
13. C2×C13×#95C22semidirect
14. C2×#88071C22semidirect
15. C2×#88071C22semidirect
16. #11718616C4semidirect
63. C23×D13C42semidirect
92. #1437D13semidirect
150. C22×C42C2×D13semidirect
154. C2×#95C2×D13semidirect
255. C13#13827semidirect

Maximal subgroups

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

subgroupindex
C2×#117186162
#131130742
#131130732
#131130722
C13×#14372
#1382713