Finite Group Databasebeta

← C22×#17986741 #32904324

Subgroups — C22×#17986741 #32904324

Order 18176.

10246 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986741C2148yesmaximal, direct product factor
2C22×#14556117C212yesmaximal, complemented
3C22×#14556116C212yesmaximal, complemented
4C22×#14556114C214yesmaximal, complemented
5C22×#14556098C214yesmaximal, complemented
6C22×#14556096C212yesmaximal, complemented
7C22×#124×C71C211yesmaximal, characteristic, complemented
8C25×D71C2212yescomplemented
9C23×Dic142C2212yescomplemented
10C23×D284C2212yes—
11C23×D284C2212yes—
12C23×C22⋊D71C2214yes—
13#17986741C221256yesdirect product factor
14C2×#14556117C22148yescomplemented
15C2×#14556116C22148yescomplemented
16C2×#14556114C22196yescomplemented
17C2×#14556098C22196yescomplemented
18C2×#14556096C22148yescomplemented
19C22×C22⋊Dic71C2212yescomplemented
20C22×(C2×D71)⋊C4C2214yescomplemented
21C22×(C2×D71)⋊C4C2214yescomplemented
22C22×(C2×D71)⋊C4C2214yescomplemented
23C22×C4×Dic71C2212yescomplemented
24C23×Q8×C71C2211yescharacteristic, complemented
25C23×D4×C71C2211yescharacteristic
26C2×#124×C71C22124yescomplemented
27C22×C22⋊C4×C71C2214yescomplemented
28C22×C42×C71C2211yescharacteristic, complemented
29C24×D71—416no—
30C24×D71—48no—
31C24×D71C2314yes—
32C24×D71C23124yescomplemented
33C22×Dic142—48no—
34C22×Dic142C23124yescomplemented
35C22×D284—48no—
36C22×D284—48no—
37C22×D284C23124yes—
38C22×D284C23124yes—
39C22×C22⋊D71—416no—
40C22×C22⋊D71C23148yes—
41C23×Dic71C2314yes—
42#14556117C231128yescomplemented
43#14556116C231128yescomplemented
44#14556114C231256yescomplemented
45#14556098C231256yescomplemented
46#14556096C231128yescomplemented
47C2×C22⋊Dic71C23124yescomplemented
48C2×(C2×D71)⋊C4C23148yescomplemented
49C2×(C2×D71)⋊C4C23148yescomplemented
50C2×(C2×D71)⋊C4C23148yescomplemented
51C2×C4×Dic71C23124yescomplemented
52C25×C71C2312yes—
53C22×Q8×C71—44no—
54C22×Q8×C71C23112yescomplemented
55C22×D4×C71—44no—
56C22×D4×C71C23112yes—
57C23×C4×C71C2312yes—
58C23×C4×C71C2312yes—
59C23×C4×C71C2311yescharacteristic
60#124×C71C23164yescomplemented
61C2×C22⋊C4×C71C23148yescomplemented
62C2×C42×C71C23112yescomplemented
63C23×D71—216no—
64C23×D71—216no—
65C23×D71—216no—
66C23×D71—496no—
67C23×D71—448no—
68C23×D71—4128no—
69C23×D71C24124yes—
70C23×D71C24132yescomplemented
71C2×Dic142—448no—
72C2×Dic142C24132yescomplemented
73C2×D284—448no—
74C2×D284—448no—
75C2×D284C24132yes—
76C2×D284C24132yes—
77C2×C22⋊D71—496no—
78C2×C22⋊D71C24164yes—
79C22×Dic71—216no—
80C22×Dic71C24124yes—
81C22⋊Dic71C24132yescomplemented
82(C2×D71)⋊C4C24164yescomplemented
83(C2×D71)⋊C4C24164yescomplemented
84(C2×D71)⋊C4C24164yescomplemented
85C4×Dic71C24132yescomplemented
86C24×C71—216no—
87C24×C71—28no—
88C24×C71C24112yes—
89C24×C71C2411yescharacteristic
90C2×Q8×C71—424no—
91C2×Q8×C71C24116yescomplemented
92C2×D4×C71—424no—
93C2×D4×C71C24116yes—
94C22×C4×C71—28no—
95C22×C4×C71—28no—
96C22×C4×C71—24no—
97C22×C4×C71C24112yes—
98C22×C4×C71C24112yes—
99C22×C4×C71C2416yes—
100C22⋊C4×C71C24164yescomplemented
101C42×C71C24116yescomplemented
102C23×C71—224no—
103C23×C71—248no—
104C23×C71—432no—
105C23×C71H(5,𝔽2)14yes—
106C23×C71H(5,𝔽2)18yes—
107C23×C71C2518yes—
108C23×C71C2513yes—
109C22×D71—248no—
110C22×D71—248no—
111C22×D71—248no—
112C22×D71—432no—
113C22×D71—464no—
114C22×D71—4384no—
115C22×D71—464no—
116C22×D71C25116yes—
117Q8×C71—416no—
118D4×C71—416no—
119C2×C4×C71—212no—
120C2×C4×C71—224no—
121C2×C4×C71—224no—
122C2×C4×C71C2518yes—
123C2×C4×C71C2514yes—
124C2×C4×C71C2518yes—
125C22⋊D71—464no—
126C2×Dic71—248no—
127C2×Dic71C25116yes—
128D284—432no—
129D284—432no—
130Dic142—432no—
131C22×C71—216no—
132C22×C71—28no—
133C22×C71—448no—
134C22×C71H(3,𝔽4)116yescomplemented
135C22×C71C2×H(5,𝔽2)16yes—
136C22×C71C2×H(5,𝔽2)112yes—
137C22×C71C2611yescharacteristic
138C2×D71—216no—
139C2×D71—216no—
140C2×D71—216no—
141C2×D71—4128no—
142C2×D71—496no—
143C4×C71—28no—
144C4×C71—28no—
145C4×C71—24no—
146Dic71—216no—
147C22×H(3,𝔽4)—711nomaximal, Sylow
148C2×C71—48no—
149C2×C71C2×H(3,𝔽4)112yescomplemented
150C2×C71C22×H(5,𝔽2)11yescharacteristic
151C2×C71C22×H(5,𝔽2)12yes—
152D71—416no—
153C2×H(3,𝔽4)—7148no—
154C22×#124—712no—
155C22×#124—712no—
156C22×#124—714no—
157C22×#124D7111yescharacteristic, complemented
158C22×#120—712no—
159C22×#120—714no—
160C71C22×H(3,𝔽4)11yescharacteristic, complemented, Sylow
161C26—712no—
162C23×Q8—712no—
163C23×Q8C2×D7111yescharacteristic, complemented
164C23×D4—714no—
165C23×D4—712no—
166C23×D4—712no—
167C23×D4C2×D7111yescharacteristic
168H(3,𝔽4)—71256no—
169C2×#124—7148no—
170C2×#124—7196no—
171C2×#124—7148no—
172C2×#124C2×D71124yescomplemented
173C2×#120—7148no—
174C2×#120—7196no—
175C22×C22⋊C4—712no—
176C22×C22⋊C4—714no—
177C22×C22⋊C4—714no—
178C22×C22⋊C4—714no—
179C22×C22⋊C4C2×D7114yescomplemented
180C22×C42—712no—
181C22×C42C2×D7111yescharacteristic, complemented
182C25—2848no—
183C25—28416no—
184C25—7124no—
185C25—714no—
186C25C22×D7112yes—
187C22×Q8—44no—
188C22×Q8—2848no—
189C22×Q8—7124no—
190C22×Q8C22×D71112yescomplemented
191C22×D4—44no—
192C22×D4—2848no—
193C22×D4—2848no—
194C22×D4—28416no—
195C22×D4—7148no—
196C22×D4—7124no—
197C22×D4—7124no—
198C22×D4C22×D71112yes—
199C23×C4—714no—
200C23×C4C22×D7111yescharacteristic
201C23×C4C22×D7112yes—
202C23×C4C22×D7112yes—
203#124—71128no—
204#124—71256no—
205#124—71128no—
206#124C22×D71164yescomplemented
207#120—71128no—
208#120—71256no—
209C2×C22⋊C4—7124no—
210C2×C22⋊C4—7148no—
211C2×C22⋊C4—7148no—
212C2×C22⋊C4—7148no—
213C2×C22⋊C4C22×D71148yescomplemented
214C2×C42—7124no—
215C2×C42C22×D71112yescomplemented
216C24—28no—
217C24—216no—
218C24—28448no—
219C24—28496no—
220C24—284128no—
221C24—7132no—
222C24—7124no—
223C24—14216no—
224C24—14216no—
225C24—14216no—
226C24C23×D71112yes—
227C24C23×D7111yescharacteristic
228C2×Q8—424no—
229C2×Q8—28448no—
230C2×Q8—7132no—
231C2×Q8C23×D71116yescomplemented
232C2×D4—424no—
233C2×D4—28448no—
234C2×D4—28448no—
235C2×D4—28496no—
236C2×D4—7164no—
237C2×D4—7132no—
238C2×D4—7132no—
239C2×D4C23×D71116yes—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—7124no—
244C22×C4—14216no—
245C22×C4C23×D71112yes—
246C22×C4C23×D7116yes—
247C22×C4C23×D71112yes—
248C22⋊C4—7132no—
249C22⋊C4—7164no—
250C22⋊C4—7164no—
251C22⋊C4—7164no—
252C22⋊C4C23×D71164yescomplemented
253C42—7132no—
254C42C23×D71116yescomplemented
255C23—224no—
256C23—248no—
257C23—432no—
258C23—28432no—
259C23—28464no—
260C23—284384no—
261C23—28464no—
262C23—7116no—
263C23—14248no—
264C23—14248no—
265C23—14248no—
266C23#1455617314yes—
267C23#1455617418yes—
268C23C24×D7118yes—
269C23C24×D7113yes—
270Q8—416no—
271Q8—28432no—
272D4—416no—
273D4—28432no—
274D4—28432no—
275D4—28464no—
276C2×C4—212no—
277C2×C4—224no—
278C2×C4—224no—
279C2×C4—7116no—
280C2×C4—14248no—
281C2×C4C24×D7118yes—
282C2×C4C24×D7114yes—
283C2×C4C24×D7118yes—
284C22—28no—
285C22—216no—
286C22—448no—
287C22—284128no—
288C22—28496no—
289C22—14216no—
290C22—14216no—
291C22—14216no—
292C22#17986741116yesdirect product factor
293C22C2×#1455617316yes—
294C22C2×#14556174112yes—
295C22C25×D7111yescharacteristic
296C4—24no—
297C4—28no—
298C4—28no—
299C4—14216no—
300C2—48no—
301C2—28416no—
302C2C2×#17986741112yesdirect product factor
303C2C22×#1455617311yescharacteristic
304C2C22×#1455617412yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC22×#124×C71C2
socleC24×C71C24
hypercenterC22×#124D71
hypocenterC71C22×H(3,𝔽4)
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C2×#17986741C2direct (N × H)
2. C22×#14556117C2semidirect
3. C22×#14556116C2semidirect
4. C22×#14556114C2semidirect
5. C22×#14556098C2semidirect
6. C22×#14556096C2semidirect
7. C22×#124×C71C2semidirect
8. C25×D71C22semidirect
9. C23×Dic142C22semidirect
13. #17986741C22direct (N × H)
14. C2×#14556117C22semidirect
15. C2×#14556116C22semidirect
16. C2×#14556114C22semidirect
17. C2×#14556098C22semidirect
18. C2×#14556096C22semidirect
19. C22×C22⋊Dic71C22semidirect
20. C22×(C2×D71)⋊C4C22semidirect
21. C22×(C2×D71)⋊C4C22semidirect
22. C22×(C2×D71)⋊C4C22semidirect
23. C22×C4×Dic71C22semidirect
24. C23×Q8×C71C22semidirect
26. C2×#124×C71C22semidirect
27. C22×C22⋊C4×C71C22semidirect
28. C22×C42×C71C22semidirect
32. C24×D71C23semidirect
34. C22×Dic142C23semidirect
42. #14556117C23semidirect
43. #14556116C23semidirect
44. #14556114C23semidirect
45. #14556098C23semidirect
46. #14556096C23semidirect
47. C2×C22⋊Dic71C23semidirect
48. C2×(C2×D71)⋊C4C23semidirect
49. C2×(C2×D71)⋊C4C23semidirect
50. C2×(C2×D71)⋊C4C23semidirect
51. C2×C4×Dic71C23semidirect
54. C22×Q8×C71C23semidirect
60. #124×C71C23semidirect
61. C2×C22⋊C4×C71C23semidirect
62. C2×C42×C71C23semidirect
70. C23×D71C24semidirect
72. C2×Dic142C24semidirect
81. C22⋊Dic71C24semidirect
82. (C2×D71)⋊C4C24semidirect
83. (C2×D71)⋊C4C24semidirect
84. (C2×D71)⋊C4C24semidirect
85. C4×Dic71C24semidirect
91. C2×Q8×C71C24semidirect
100. C22⋊C4×C71C24semidirect
101. C42×C71C24semidirect
134. C22×C71H(3,𝔽4)semidirect
149. C2×C71C2×H(3,𝔽4)semidirect
157. C22×#124D71semidirect
160. C71C22×H(3,𝔽4)semidirect
163. C23×Q8C2×D71semidirect
172. C2×#124C2×D71semidirect
179. C22×C22⋊C4C2×D71semidirect
181. C22×C42C2×D71semidirect
190. C22×Q8C22×D71semidirect
206. #124C22×D71semidirect
213. C2×C22⋊C4C22×D71semidirect
215. C2×C42C22×D71semidirect
231. C2×Q8C23×D71semidirect
252. C22⋊C4C23×D71semidirect
254. C42C23×D71semidirect
292. C22#17986741direct (N × H)
302. C2C2×#17986741direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179867412
C22×#145561172
C22×#145561162
C22×#145561142
C22×#145560982
C22×#145560962
C22×#124×C712
C22×H(3,𝔽4)71