Finite Group Databasebeta

← C22×#17986186 #32864165

Subgroups — C22×#17986186 #32864165

Order 18176.

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×#14556096C213yesmaximal, complemented
2C2×#17986186C2124yesmaximal, direct product factor
3C22×#14556043C213yesmaximal, complemented
4C22×H(5,𝔽2)×C71C211yesmaximal, characteristic, complemented
5C25×D71C2211yescharacteristic, complemented
6C23×C22⋊D71—23no—
7C2×#14556096—218no—
8C2×#14556096C22136yescomplemented
9C22×C22⋊Dic71C2213yescomplemented
10C22×(C2×D71)⋊C4—23no—
11#17986186C22164yesdirect product factor
12C2×#14556043C22136yescomplemented
13C2×H(5,𝔽2)×C71—218no—
14C2×H(5,𝔽2)×C71C22112yescomplemented
15C22×#41×C71—23no—
16C23×D4×C71—23no—
17C23×D4×C71C2213yes—
18C24×D71—23no—
19C24×D71—29no—
20C24×D71—48no—
21C24×D71C2316yescomplemented
22C22×C22⋊D71—236no—
23C22×C22⋊D71—424no—
24C23×Dic71—23no—
25#14556096—272no—
26#14556096C23148yescomplemented
27C2×C22⋊Dic71—29no—
28C2×C22⋊Dic71C23118yescomplemented
29C2×(C2×D71)⋊C4—236no—
30#14556043C23148yescomplemented
31C25×C71—21no—
32C25×C71D413yescomplemented
33C25×C71C2311yescharacteristic
34H(5,𝔽2)×C71—248no—
35H(5,𝔽2)×C71—272no—
36H(5,𝔽2)×C71C23116yescomplemented
37C2×#41×C71—236no—
38C2×#41×C71—236no—
39C22×Q8×C71—21no—
40C22×D4×C71—23no—
41C22×D4×C71—236no—
42C22×D4×C71—26no—
43C22×D4×C71—218no—
44C22×D4×C71—236no—
45C22×D4×C71D4118yescomplemented
46C22×D4×C71C23118yes—
47C23×C4×C71—23no—
48C23×C4×C71D413yescomplemented
49C23×D71—224no—
50C23×D71—218no—
51C23×D71—218no—
52C23×D71—424no—
53C23×D71—472no—
54C23×D71—412no—
55C23×D71—49no—
56C23×D71—46no—
57C23×D71—41no—
58C23×D71C2414yescomplemented
59C2×C22⋊D71—248no—
60C2×C22⋊D71—4144no—
61C22×Dic71—218no—
62C22×Dic71—412no—
63C22⋊Dic71—218no—
64C22⋊Dic71C24112yescomplemented
65(C2×D71)⋊C4—248no—
66C24×C71—212no—
67C24×C71—23no—
68C24×C71—29no—
69C24×C71—44no—
70C24×C71—412no—
71C24×C71—84no—
72C24×C71C2×D413yes—
73C24×C71C2×D419yes—
74C24×C71C2×D4118yescomplemented
75C24×C71C2413yes—
76#41×C71—2144no—
77#41×C71—248no—
78C2×Q8×C71—212no—
79C2×D4×C71—236no—
80C2×D4×C71—2144no—
81C2×D4×C71—236no—
82C2×D4×C71—236no—
83C2×D4×C71—236no—
84C2×D4×C71—236no—
85C2×D4×C71—248no—
86C2×D4×C71C2×D4136yescomplemented
87C2×D4×C71C24112yes—
88C22×C4×C71—236no—
89C22×C4×C71—23no—
90C22×C4×C71—29no—
91C22×C4×C71C2×D4118yescomplemented
92C23×C71—212no—
93C23×C71—216no—
94C23×C71—218no—
95C23×C71—218no—
96C23×C71—418no—
97C23×C71—46no—
98C23×C71—436no—
99C23×C71—412no—
100C23×C71—818no—
101C23×C71—824no—
102C23×C71#12019yes—
103C23×C71#12016yes—
104C23×C71#12011yescharacteristic
105C23×C71C22×D419yes—
106C23×C71C22×D419yes—
107C23×C71C22×D4112yescomplemented
108C23×C71C2511yescharacteristic
109C22×D71—272no—
110C22×D71—212no—
111C22×D71—48no—
112C22×D71—472no—
113C22×D71—436no—
114C22×D71—472no—
115C22×D71—448no—
116C22×D71—418no—
117C22×D71—46no—
118C22×D71—48no—
119Q8×C71—216no—
120D4×C71—248no—
121D4×C71—272no—
122D4×C71—224no—
123C2×C4×C71—248no—
124C2×C4×C71—218no—
125C2×C4×C71—218no—
126C2×C4×C71C22×D4112yescomplemented
127C22⋊D71—496no—
128C2×Dic71—212no—
129C2×Dic71—436no—
130C22×C71—218no—
131C22×C71—218no—
132C22×C71—212no—
133C22×C71—46no—
134C22×C71—49no—
135C22×C71—41no—
136C22×C71—412no—
137C22×C71—436no—
138C22×C71—818no—
139C22×C71—816no—
140C22×C71B(4,𝔽2)14yescomplemented
141C22×C71C2×#12019yes—
142C22×C71C2×#12013yes—
143C22×C71C23×D413yes—
144C2×D71—224no—
145C2×D71—412no—
146C2×D71—472no—
147C2×D71—44no—
148C2×D71—424no—
149C4×C71—212no—
150Dic71—412no—
151C22×B(4,𝔽2)—711nomaximal, Sylow
152C2×C71—23no—
153C2×C71—29no—
154C2×C71—412no—
155C2×C71C2×B(4,𝔽2)16yescomplemented
156C2×C71C22×#12011yescharacteristic
157D71—48no—
158C22×H(5,𝔽2)D7111yescharacteristic, complemented
159C22×#120—713no—
160C2×B(4,𝔽2)—7124no—
161C22×#99—713no—
162C71C22×B(4,𝔽2)11yescharacteristic, complemented, Sylow
163C26—711no—
164C2×H(5,𝔽2)—218no—
165C2×H(5,𝔽2)C2×D71112yescomplemented
166C22×#41—23no—
167C23×D4—23no—
168C23×D4—1423no—
169C23×D4C2×D7113yes—
170C2×#120—7136no—
171C2×#120—14218no—
172C22×C22⋊C4—713no—
173C22×C22⋊C4—1423no—
174B(4,𝔽2)—7164no—
175C2×#99—7136no—
176C25—21no—
177C25—2848no—
178C25—716no—
179C25—1429no—
180C25—1423no—
181C25C22⋊D7113yescomplemented
182C25C22×D7111yescharacteristic
183H(5,𝔽2)—248no—
184H(5,𝔽2)—272no—
185H(5,𝔽2)C22×D71116yescomplemented
186C2×#41—236no—
187C2×#41—236no—
188C22×Q8—21no—
189C22×D4—23no—
190C22×D4—236no—
191C22×D4—26no—
192C22×D4—218no—
193C22×D4—236no—
194C22×D4—28424no—
195C22×D4—14236no—
196C22×D4C22⋊D71118yescomplemented
197C22×D4C22×D71118yes—
198C23×C4—23no—
199C23×C4—1423no—
200C23×C4C22⋊D7113yescomplemented
201#120—7148no—
202#120—14272no—
203C2×C22⋊C4—7118no—
204C2×C22⋊C4—1429no—
205C2×C22⋊C4—14236no—
206#99—7148no—
207C24—212no—
208C24—23no—
209C24—29no—
210C24—44no—
211C24—412no—
212C24—84no—
213C24—28412no—
214C24—28472no—
215C24—28424no—
216C24—2849no—
217C24—2846no—
218C24—2841no—
219C24—714no—
220C24—14224no—
221C24—14218no—
222C24—14218no—
223C24C2×C22⋊D7113yes—
224C24C2×C22⋊D7119yes—
225C24C2×C22⋊D71118yescomplemented
226C24C23×D7113yes—
227#41—2144no—
228#41—248no—
229C2×Q8—212no—
230C2×D4—236no—
231C2×D4—2144no—
232C2×D4—236no—
233C2×D4—236no—
234C2×D4—248no—
235C2×D4—236no—
236C2×D4—236no—
237C2×D4—284144no—
238C2×D4—14248no—
239C2×D4C2×C22⋊D71136yescomplemented
240C2×D4C23×D71112yes—
241C22×C4—236no—
242C22×C4—29no—
243C22×C4—23no—
244C22×C4—28412no—
245C22×C4—14218no—
246C22×C4C2×C22⋊D71118yescomplemented
247C22⋊C4—7112no—
248C22⋊C4—14218no—
249C22⋊C4—14248no—
250C23—212no—
251C23—216no—
252C23—218no—
253C23—218no—
254C23—418no—
255C23—46no—
256C23—436no—
257C23—412no—
258C23—818no—
259C23—824no—
260C23—28436no—
261C23—28472no—
262C23—28448no—
263C23—28472no—
264C23—2848no—
265C23—2848no—
266C23—28418no—
267C23—2846no—
268C23—14272no—
269C23—14212no—
270C23#1455609719yes—
271C23#1455609716yes—
272C23#1455609711yescharacteristic
273C23C22×C22⋊D7119yes—
274C23C22×C22⋊D7119yes—
275C23C22×C22⋊D71112yescomplemented
276C23C24×D7111yescharacteristic
277Q8—216no—
278D4—248no—
279D4—224no—
280D4—272no—
281D4—28496no—
282C2×C4—248no—
283C2×C4—218no—
284C2×C4—218no—
285C2×C4—28436no—
286C2×C4—14212no—
287C2×C4C22×C22⋊D71112yescomplemented
288C22—218no—
289C22—218no—
290C22—212no—
291C22—49no—
292C22—41no—
293C22—46no—
294C22—412no—
295C22—436no—
296C22—818no—
297C22—816no—
298C22—28412no—
299C22—28472no—
300C22—2844no—
301C22—28424no—
302C22—14224no—
303C22#1798618614yesdirect product factor
304C22C2×#1455609719yes—
305C22C2×#1455609713yes—
306C22C23×C22⋊D7113yes—
307C4—212no—
308C4—28412no—
309C2—23no—
310C2—29no—
311C2—412no—
312C2—2848no—
313C2C2×#1798618616yesdirect product factor
314C2C22×#1455609711yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC22×H(5,𝔽2)×C71C2
socleC23×C71#120
hypercenterC22×H(5,𝔽2)D71
hypocenterC71C22×B(4,𝔽2)
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#14556096C2semidirect
2. C2×#17986186C2direct (N × H)
3. C22×#14556043C2semidirect
4. C22×H(5,𝔽2)×C71C2semidirect
5. C25×D71C22semidirect
8. C2×#14556096C22semidirect
9. C22×C22⋊Dic71C22semidirect
11. #17986186C22direct (N × H)
12. C2×#14556043C22semidirect
14. C2×H(5,𝔽2)×C71C22semidirect
21. C24×D71C23semidirect
26. #14556096C23semidirect
28. C2×C22⋊Dic71C23semidirect
30. #14556043C23semidirect
32. C25×C71D4semidirect
36. H(5,𝔽2)×C71C23semidirect
45. C22×D4×C71D4semidirect
48. C23×C4×C71D4semidirect
58. C23×D71C24semidirect
64. C22⋊Dic71C24semidirect
74. C24×C71C2×D4semidirect
86. C2×D4×C71C2×D4semidirect
91. C22×C4×C71C2×D4semidirect
107. C23×C71C22×D4semidirect
126. C2×C4×C71C22×D4semidirect
140. C22×C71B(4,𝔽2)semidirect
155. C2×C71C2×B(4,𝔽2)semidirect
158. C22×H(5,𝔽2)D71semidirect
162. C71C22×B(4,𝔽2)semidirect
165. C2×H(5,𝔽2)C2×D71semidirect
181. C25C22⋊D71semidirect
185. H(5,𝔽2)C22×D71semidirect
196. C22×D4C22⋊D71semidirect
200. C23×C4C22⋊D71semidirect
225. C24C2×C22⋊D71semidirect
239. C2×D4C2×C22⋊D71semidirect
246. C22×C4C2×C22⋊D71semidirect
275. C23C22×C22⋊D71semidirect
287. C2×C4C22×C22⋊D71semidirect
303. C22#17986186direct (N × H)
313. C2C2×#17986186direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560962
C2×#179861862
C22×#145560432
C22×H(5,𝔽2)×C712
C22×B(4,𝔽2)71