Finite Group Databasebeta

← C22×#11718905 #16729632

Subgroups — C22×#11718905 #16729632

Order 3328.

7014 conjugacy classes of subgroups; only the 827 normal subgroups are shown.

normal subgroups only

This group has 7014 conjugacy classes of subgroups — too many to draw. Only its 827 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
1C22×#88191C213yesmaximal, complemented
2C2×#11718905C2124yesmaximal, direct product factor
3C22×#88074C213yesmaximal, complemented
4C22×C13×H(5,𝔽2)C211yesmaximal, characteristic, complemented
5C25×D13C2211yescharacteristic, complemented
6C2×C13×H(5,𝔽2)—218no—
7C2×C13×H(5,𝔽2)C22112yescomplemented
8C22×C13×#41—23no—
9C23×D4×C13—23no—
10C23×D4×C13C2213yes—
11C23×C22⋊D13—23no—
12C22×C22⋊Dic13C2213yescomplemented
13C2×#88191—218no—
14C2×#88191C22136yescomplemented
15C22×#6103—23no—
16#11718905C22164yesdirect product factor
17C2×#88074C22136yescomplemented
18C25×C13—21no—
19C25×C13D413yescomplemented
20C25×C13C2311yescharacteristic
21C24×D13—23no—
22C24×D13—29no—
23C24×D13—48no—
24C24×D13C2316yescomplemented
25C13×H(5,𝔽2)—248no—
26C13×H(5,𝔽2)—272no—
27C13×H(5,𝔽2)C23116yescomplemented
28C2×C13×#41—236no—
29C2×C13×#41—236no—
30C22×Q8×C13—21no—
31C22×D4×C13—23no—
32C22×D4×C13—236no—
33C22×D4×C13—26no—
34C22×D4×C13—218no—
35C22×D4×C13—236no—
36C22×D4×C13D4118yescomplemented
37C22×D4×C13C23118yes—
38C23×C4×C13—23no—
39C23×C4×C13D413yescomplemented
40C22×C22⋊D13—236no—
41C22×C22⋊D13—424no—
42C23×Dic13—23no—
43C2×C22⋊Dic13—29no—
44C2×C22⋊Dic13C23118yescomplemented
45#88191—272no—
46#88191C23148yescomplemented
47C2×#6103—236no—
48#88074C23148yescomplemented
49C22×B(4,𝔽2)—131nomaximal, Sylow
50C24×C13—212no—
51C24×C13—23no—
52C24×C13—29no—
53C24×C13—44no—
54C24×C13—412no—
55C24×C13—84no—
56C24×C13C2×D413yes—
57C24×C13C2×D419yes—
58C24×C13C2×D4118yescomplemented
59C24×C13C2413yes—
60C23×D13—224no—
61C23×D13—218no—
62C23×D13—218no—
63C23×D13—424no—
64C23×D13—472no—
65C23×D13—412no—
66C23×D13—49no—
67C23×D13—46no—
68C23×D13—41no—
69C23×D13C2414yescomplemented
70C13×#41—2144no—
71C13×#41—248no—
72C2×Q8×C13—212no—
73C2×D4×C13—236no—
74C2×D4×C13—2144no—
75C2×D4×C13—236no—
76C2×D4×C13—236no—
77C2×D4×C13—236no—
78C2×D4×C13—236no—
79C2×D4×C13—248no—
80C2×D4×C13C2×D4136yescomplemented
81C2×D4×C13C24112yes—
82C22×C4×C13—236no—
83C22×C4×C13—23no—
84C22×C4×C13—29no—
85C22×C4×C13C2×D4118yescomplemented
86C2×C22⋊D13—248no—
87C2×C22⋊D13—4144no—
88C22×Dic13—218no—
89C22×Dic13—412no—
90C22⋊Dic13—218no—
91C22⋊Dic13C24112yescomplemented
92#6103—248no—
93C22×H(5,𝔽2)D1311yescharacteristic, complemented
94C22×#120—133no—
95C2×B(4,𝔽2)—1324no—
96C22×#99—133no—
97C23×C13—212no—
98C23×C13—216no—
99C23×C13—218no—
100C23×C13—218no—
101C23×C13—418no—
102C23×C13—46no—
103C23×C13—436no—
104C23×C13—412no—
105C23×C13—818no—
106C23×C13—824no—
107C23×C13#12019yes—
108C23×C13#12016yes—
109C23×C13#12011yescharacteristic
110C23×C13C22×D419yes—
111C23×C13C22×D419yes—
112C23×C13C22×D4112yescomplemented
113C23×C13C2511yescharacteristic
114C22×D13—272no—
115C22×D13—212no—
116C22×D13—48no—
117C22×D13—472no—
118C22×D13—436no—
119C22×D13—472no—
120C22×D13—448no—
121C22×D13—418no—
122C22×D13—46no—
123C22×D13—48no—
124Q8×C13—216no—
125D4×C13—248no—
126D4×C13—272no—
127D4×C13—224no—
128C2×C4×C13—248no—
129C2×C4×C13—218no—
130C2×C4×C13—218no—
131C2×C4×C13C22×D4112yescomplemented
132C22⋊D13—496no—
133C2×Dic13—212no—
134C2×Dic13—436no—
135C26—131no—
136C2×H(5,𝔽2)—218no—
137C2×H(5,𝔽2)C2×D13112yescomplemented
138C22×#41—23no—
139C23×D4—23no—
140C23×D4—263no—
141C23×D4C2×D1313yes—
142C2×#120—1336no—
143C2×#120—2618no—
144C22×C22⋊C4—133no—
145C22×C22⋊C4—263no—
146B(4,𝔽2)—1364no—
147C2×#99—1336no—
148C22×C13—218no—
149C22×C13—218no—
150C22×C13—212no—
151C22×C13—46no—
152C22×C13—49no—
153C22×C13—41no—
154C22×C13—412no—
155C22×C13—436no—
156C22×C13—818no—
157C22×C13—816no—
158C22×C13B(4,𝔽2)14yescomplemented
159C22×C13C2×#12019yes—
160C22×C13C2×#12013yes—
161C22×C13C23×D413yes—
162C2×D13—224no—
163C2×D13—412no—
164C2×D13—472no—
165C2×D13—44no—
166C2×D13—424no—
167C4×C13—212no—
168Dic13—412no—
169C25—21no—
170C25—136no—
171C25—263no—
172C25—269no—
173C25—528no—
174C25C22⋊D1313yescomplemented
175C25C22×D1311yescharacteristic
176H(5,𝔽2)—248no—
177H(5,𝔽2)—272no—
178H(5,𝔽2)C22×D13116yescomplemented
179C2×#41—236no—
180C2×#41—236no—
181C22×Q8—21no—
182C22×D4—23no—
183C22×D4—236no—
184C22×D4—26no—
185C22×D4—218no—
186C22×D4—236no—
187C22×D4—2636no—
188C22×D4—5224no—
189C22×D4C22⋊D13118yescomplemented
190C22×D4C22×D13118yes—
191C23×C4—23no—
192C23×C4—263no—
193C23×C4C22⋊D1313yescomplemented
194#120—1348no—
195#120—2672no—
196C2×C22⋊C4—1318no—
197C2×C22⋊C4—269no—
198C2×C22⋊C4—2636no—
199#99—1348no—
200C2×C13—23no—
201C2×C13—29no—
202C2×C13—412no—
203C2×C13C2×B(4,𝔽2)16yescomplemented
204C2×C13C22×#12011yescharacteristic
205D13—48no—
206C24—212no—
207C24—23no—
208C24—29no—
209C24—44no—
210C24—412no—
211C24—84no—
212C24—134no—
213C24—2624no—
214C24—2618no—
215C24—2618no—
216C24—5212no—
217C24—5224no—
218C24—5272no—
219C24—529no—
220C24—526no—
221C24—521no—
222C24C2×C22⋊D1313yes—
223C24C2×C22⋊D1319yes—
224C24C2×C22⋊D13118yescomplemented
225C24C23×D1313yes—
226#41—2144no—
227#41—248no—
228C2×Q8—212no—
229C2×D4—236no—
230C2×D4—2144no—
231C2×D4—236no—
232C2×D4—236no—
233C2×D4—236no—
234C2×D4—236no—
235C2×D4—248no—
236C2×D4—2648no—
237C2×D4—52144no—
238C2×D4C2×C22⋊D13136yescomplemented
239C2×D4C23×D13112yes—
240C22×C4—236no—
241C22×C4—29no—
242C22×C4—23no—
243C22×C4—2618no—
244C22×C4—5212no—
245C22×C4C2×C22⋊D13118yescomplemented
246C22⋊C4—1312no—
247C22⋊C4—2618no—
248C22⋊C4—2648no—
249C13C22×B(4,𝔽2)11yescharacteristic, complemented, Sylow
250C23—212no—
251C23—216no—
252C23—218no—
253C23—218no—
254C23—418no—
255C23—46no—
256C23—436no—
257C23—412no—
258C23—818no—
259C23—824no—
260C23—2672no—
261C23—2612no—
262C23—5236no—
263C23—5272no—
264C23—528no—
265C23—5272no—
266C23—5248no—
267C23—5218no—
268C23—526no—
269C23—528no—
270C23#8820719yes—
271C23#8820716yes—
272C23#8820711yescharacteristic
273C23C22×C22⋊D1319yes—
274C23C22×C22⋊D1319yes—
275C23C22×C22⋊D13112yescomplemented
276C23C24×D1311yescharacteristic
277Q8—216no—
278D4—248no—
279D4—272no—
280D4—224no—
281D4—5296no—
282C2×C4—248no—
283C2×C4—218no—
284C2×C4—218no—
285C2×C4—2612no—
286C2×C4—5236no—
287C2×C4C22×C22⋊D13112yescomplemented
288C22—218no—
289C22—218no—
290C22—212no—
291C22—46no—
292C22—49no—
293C22—41no—
294C22—412no—
295C22—436no—
296C22—818no—
297C22—816no—
298C22—2624no—
299C22—5212no—
300C22—5272no—
301C22—524no—
302C22—5224no—
303C22#1171890514yesdirect product factor
304C22C2×#8820719yes—
305C22C2×#8820713yes—
306C22C23×C22⋊D1313yes—
307C4—212no—
308C4—5212no—
309C2—23no—
310C2—29no—
311C2—412no—
312C2—528no—
313C2C2×#1171890516yesdirect product factor
314C2C22×#8820711yescharacteristic
subgroupG/H
centerC23#88207
commutator, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC22×C13×H(5,𝔽2)C2
socleC23×C13#120
hypercenterC22×H(5,𝔽2)D13
hypocenterC13C22×B(4,𝔽2)
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#88191C2semidirect
2. C2×#11718905C2direct (N × H)
3. C22×#88074C2semidirect
4. C22×C13×H(5,𝔽2)C2semidirect
5. C25×D13C22semidirect
7. C2×C13×H(5,𝔽2)C22semidirect
12. C22×C22⋊Dic13C22semidirect
14. C2×#88191C22semidirect
16. #11718905C22direct (N × H)
17. C2×#88074C22semidirect
19. C25×C13D4semidirect
24. C24×D13C23semidirect
27. C13×H(5,𝔽2)C23semidirect
36. C22×D4×C13D4semidirect
39. C23×C4×C13D4semidirect
44. C2×C22⋊Dic13C23semidirect
46. #88191C23semidirect
48. #88074C23semidirect
58. C24×C13C2×D4semidirect
69. C23×D13C24semidirect
80. C2×D4×C13C2×D4semidirect
85. C22×C4×C13C2×D4semidirect
91. C22⋊Dic13C24semidirect
93. C22×H(5,𝔽2)D13semidirect
112. C23×C13C22×D4semidirect
131. C2×C4×C13C22×D4semidirect
137. C2×H(5,𝔽2)C2×D13semidirect
158. C22×C13B(4,𝔽2)semidirect
174. C25C22⋊D13semidirect
178. H(5,𝔽2)C22×D13semidirect
189. C22×D4C22⋊D13semidirect
193. C23×C4C22⋊D13semidirect
203. C2×C13C2×B(4,𝔽2)semidirect
224. C24C2×C22⋊D13semidirect
238. C2×D4C2×C22⋊D13semidirect
245. C22×C4C2×C22⋊D13semidirect
249. C13C22×B(4,𝔽2)semidirect
275. C23C22×C22⋊D13semidirect
287. C2×C4C22×C22⋊D13semidirect
303. C22#11718905direct (N × H)
313. C2C2×#11718905direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#881912
C2×#117189052
C22×#880742
C22×C13×H(5,𝔽2)2
C22×B(4,𝔽2)13