Finite Group Databasebeta

← C2×#26041947 #37547198

Subgroups — C2×#26041947 #37547198

Order 23168.

1736 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668449C211yesmaximal, characteristic, complemented
2C22×#15668448C211yesmaximal, characteristic, complemented
3#26041947C2116yesmaximal, direct product factor
4C2×#19153812C211yesmaximal, characteristic, complemented
5C2×#19153812C212yesmaximal, complemented
6C2×#19153791C211yesmaximal, characteristic, complemented
7C2×#19153789C212yesmaximal, complemented
8C2×#19153788C212yesmaximal, complemented
9C2×#19153781C211yesmaximal, characteristic, complemented
10C2×C4×C22⋊D181C211yesmaximal, characteristic, complemented
11C2×#19153776C212yesmaximal, complemented
12C2×#121×C181C211yesmaximal, characteristic, complemented
13C2×#15668449—28no—
14C2×#15668449C2218yescomplemented
15C2×#15668448—28no—
16C2×#15668448C2218yescomplemented
17C22×Dic362C2211yescharacteristic
18C22×D724C2211yescharacteristic
19C22×D724C2212yes—
20C22×C22⋊D181C2212yes—
21C22×C22⋊D181C2211yescharacteristic
22C22×C22⋊D181C2212yes—
23C22×C22⋊D181C2212yes—
24C22×C22⋊D181C2212yes—
25C23×Dic181C2212yes—
26C22×C4×D181C2212yes—
27C22×C4×D181C2211yescharacteristic
28#19153812C2218yescomplemented
29#19153812C22116yescomplemented
30#19153791C2218yescomplemented
31#19153789C22116yescomplemented
32#19153788C22116yescomplemented
33#19153781C2218yescomplemented
34C4×C22⋊D181C2218yescomplemented
35#19153776C22116yescomplemented
36C2×#15668428C2211yescharacteristic
37C2×#15668428C2212yes—
38C2×C22⋊Dic181C2211yescharacteristic
39C2×#15668426C2212yes—
40C2×#15668426C2211yescharacteristic
41C2×C4×Dic181C2212yes—
42C2×C4×Dic181C2211yescharacteristic
43C22×D4×C181C2211yescharacteristic
44C22×D4×C181C2212yes—
45C23×C4×C181C2211yescharacteristic
46#121×C181C2218yescomplemented
47C2×C4⋊C4×C181C2211yescharacteristic
48C2×C22⋊C4×C181C2212yes—
49C23×D181C2311yescharacteristic
50C23×D181C2312yes—
51#15668449—232no—
52#15668448—232no—
53C2×Dic362D418yescomplemented
54C2×Dic362C2314yes—
55C2×D724—24no—
56C2×D724—44no—
57C2×D724C2314yes—
58C2×D724C2318yes—
59C2×C22⋊D181—28no—
60C2×C22⋊D181—28no—
61C2×C22⋊D181—28no—
62C2×C22⋊D181—24no—
63C2×C22⋊D181—44no—
64C2×C22⋊D181C2318yes—
65C2×C22⋊D181C2314yes—
66C2×C22⋊D181C2318yes—
67C2×C22⋊D181C2318yes—
68C2×C22⋊D181C2318yes—
69C22×Dic181—28no—
70C22×Dic181C2311yescharacteristic
71C22×Dic181C2311yescharacteristic
72C22×Dic181C2318yes—
73C22×Dic181C2312yes—
74C22×Dic181C2311yescharacteristic
75C2×C4×D181—28no—
76C2×C4×D181—24no—
77C2×C4×D181C2318yes—
78C2×C4×D181C2314yes—
79#15668428C2314yes—
80#15668428C2318yes—
81C22⋊Dic181C2314yes—
82#15668426C2318yes—
83#15668426C2314yes—
84C4×Dic181C2318yes—
85C4×Dic181C2314yes—
86C24×C181C2312yes—
87C24×C181C2311yescharacteristic
88C2×D4×C181—24no—
89C2×D4×C181—44no—
90C2×D4×C181C2314yes—
91C2×D4×C181C2318yes—
92C22×C4×C181—24no—
93C22×C4×C181C2311yescharacteristic
94C22×C4×C181C2312yes—
95C22×C4×C181C2314yes—
96C22×C4×C181C2311yescharacteristic
97C4⋊C4×C181C2314yes—
98C22⋊C4×C181C2318yes—
99C22×D181—22no—
100C22×D181—24no—
101C22×D181—22no—
102C22×D181—24no—
103C22×D181—24no—
104C22×D181—22no—
105C22×D181C2412yes—
106C22×D181C2414yes—
107Dic362C2×D4116yescomplemented
108D724—28no—
109D724—48no—
110C22⋊D181—216no—
111C22⋊D181—216no—
112C22⋊D181—216no—
113C22⋊D181—28no—
114C22⋊D181—48no—
115C2×Dic181—22no—
116C2×Dic181—22no—
117C2×Dic181—216no—
118C2×Dic181C2×D418yes—
119C2×Dic181#4114yescomplemented
120C2×Dic181C2412yes—
121C2×Dic181C2412yes—
122C2×Dic181C2414yes—
123C2×Dic181C2412yes—
124C4×D181—28no—
125C4×D181—216no—
126C23×C181—22no—
127C23×C181—24no—
128C23×C181—24no—
129C23×C181—22no—
130C23×C181—22no—
131C23×C181—24no—
132C23×C181C2414yes—
133C23×C181C2412yes—
134C23×C181C2411yescharacteristic
135D4×C181—28no—
136D4×C181—48no—
137C2×C4×C181—22no—
138C2×C4×C181—24no—
139C2×C4×C181—28no—
140C2×C4×C181C2×D414yes—
141C2×C4×C181C2412yes—
142C2×C4×C181C2414yes—
143C2×C4×C181C2412yes—
144C22×C181—22no—
145C22×C181—24no—
146C22×C181—24no—
147C22×C181—22no—
148C22×C181—22no—
149C22×C181—24no—
150C22×C181—44no—
151C22×C181—48no—
152C22×C181C22×D412yes—
153C22×C181C2×#4112yes—
154C22×C181H(5,𝔽2)12yes—
155C22×C181C2511yescharacteristic
156C2×D181—22no—
157C2×D181—24no—
158C2×D181—22no—
159C2×D181—24no—
160C2×D181—24no—
161C2×D181—22no—
162C2×D181—44no—
163C2×D181—48no—
164C4×C181—22no—
165C4×C181—24no—
166C4×C181C22×D414yes—
167Dic181—22no—
168Dic181—22no—
169Dic181C22×D418yes—
170Dic181C2×#4114yescomplemented
171C2×C181—42no—
172C2×C181—44no—
173C2×C181#55014yescomplemented
174C2×C181C23×D411yescharacteristic
175C2×C181C22×#4111yescharacteristic
176C2×C181C2×H(5,𝔽2)11yescharacteristic
177D181—42no—
178D181—44no—
179C181C2×#55011yescharacteristic, complemented, Sylow
180C2×#550—1811nomaximal, Sylow
181C22×#41—1811no—
182C22×#41—1811no—
183#550—18116no—
184C2×#127—1812no—
185C2×#127—1811no—
186C2×#121—1812no—
187C2×#121—1811no—
188C2×#121—1812no—
189C2×#121D18111yescharacteristic, complemented
190C2×C4×Q8—1811no—
191C2×C4×D4—1811no—
192C2×C4×D4—1812no—
193C2×#41—3628no—
194C2×#41—3628no—
195C2×#41—1818no—
196C2×#41—1818no—
197C22×Q8—1811no—
198C22×D4—1811no—
199C22×D4—1812no—
200C22×D4—1812no—
201C22×D4—1812no—
202C22×D4—1812no—
203C22×D4—1811no—
204C22×D4—1812no—
205C22×D4C2×D18111yescharacteristic
206C22×D4C2×D18112yes—
207C23×C4—1812no—
208C23×C4—1812no—
209C23×C4—1811no—
210C23×C4C2×D18111yescharacteristic
211#127—18116no—
212#127—1818no—
213#121—18116no—
214#121—1818no—
215#121—18116no—
216#121C2×D18118yescomplemented
217C4×Q8—1818no—
218C4×D4—1818no—
219C4×D4—18116no—
220C2×C4⋊C4—1811no—
221C2×C4⋊C4—1812no—
222C2×C4⋊C4C2×D18111yescharacteristic
223C2×C22⋊C4—1811no—
224C2×C22⋊C4—1811no—
225C2×C22⋊C4—1812no—
226C2×C22⋊C4C2×D18112yes—
227C2×C42—1811no—
228C2×C42—1812no—
229C24—1811no—
230C24—1812no—
231C24C22×D18111yescharacteristic
232C24C22×D18112yes—
233#41—36232no—
234#41—36232no—
235C2×Q8—1818no—
236C2×Q8—1814no—
237C2×D4—24no—
238C2×D4—44no—
239C2×D4—3628no—
240C2×D4—3628no—
241C2×D4—3624no—
242C2×D4—3628no—
243C2×D4—3624no—
244C2×D4—1814no—
245C2×D4—1818no—
246C2×D4—1818no—
247C2×D4—1818no—
248C2×D4—1818no—
249C2×D4—1814no—
250C2×D4—1818no—
251C2×D4—7244no—
252C2×D4—7244no—
253C2×D4C22×D18114yes—
254C2×D4C22×D18118yes—
255C22×C4—24no—
256C22×C4—3628no—
257C22×C4—3628no—
258C22×C4—3624no—
259C22×C4—1818no—
260C22×C4—1811no—
261C22×C4—1811no—
262C22×C4—1818no—
263C22×C4—1814no—
264C22×C4—1811no—
265C22×C4—1812no—
266C22×C4C22×D18114yes—
267C22×C4C22×D18112yes—
268C22×C4C22×D18111yescharacteristic
269C22×C4C22×D18111yescharacteristic
270C4⋊C4—1814no—
271C4⋊C4—1818no—
272C4⋊C4C22×D18114yes—
273C22⋊C4—1814no—
274C22⋊C4—1814no—
275C22⋊C4—1818no—
276C22⋊C4C22×D18118yes—
277C42—1814no—
278C42—1818no—
279C23—22no—
280C23—24no—
281C23—22no—
282C23—24no—
283C23—22no—
284C23—24no—
285C23—3624no—
286C23—3622no—
287C23—3622no—
288C23—3624no—
289C23—3622no—
290C23—3624no—
291C23—1812no—
292C23—1814no—
293C23C23×D18112yes—
294C23C23×D18114yes—
295C23C23×D18111yescharacteristic
296Q8—18116no—
297D4—28no—
298D4—48no—
299D4—36216no—
300D4—36216no—
301D4—3628no—
302D4—36216no—
303D4—3628no—
304D4—7248no—
305D4—7248no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—28no—
309C2×C4—36216no—
310C2×C4—3622no—
311C2×C4—3622no—
312C2×C4—36216no—
313C2×C4—3628no—
314C2×C4—1812no—
315C2×C4—1812no—
316C2×C4—1818no—
317C2×C4—1814no—
318C2×C4—1812no—
319C2×C4—1814no—
320C2×C4D4×D18114yes—
321C2×C4C23×D18114yes—
322C2×C4C23×D18112yes—
323C2×C4C23×D18112yes—
324C22—22no—
325C22—24no—
326C22—22no—
327C22—24no—
328C22—22no—
329C22—24no—
330C22—44no—
331C22—48no—
332C22—3624no—
333C22—3622no—
334C22—3622no—
335C22—3624no—
336C22—3622no—
337C22—3624no—
338C22—7248no—
339C22—7244no—
340C22C2×D4×D18112yes—
341C22#41×D18112yes—
342C22#1915386112yes—
343C22C24×D18111yescharacteristic
344C4—22no—
345C4—24no—
346C4—3622no—
347C4—3622no—
348C4—1818no—
349C4—1814no—
350C4C2×D4×D18114yes—
351C2—42no—
352C2—44no—
353C2—7244no—
354C2—7242no—
355C2#2604194714yesdirect product factor
356C2C22×D4×D18111yescharacteristic
357C2C2×#41×D18111yescharacteristic
358C2C2×#1915386111yescharacteristic
subgroupG/H
centerC23C23×D181
commutator, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC2×#121×C181C2
socleC23×C181C24
hypercenterC2×#121D181
hypocenterC181C2×#550
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#15668449C2semidirect
2. C22×#15668448C2semidirect
3. #26041947C2direct (N × H)
4. C2×#19153812C2semidirect
5. C2×#19153812C2semidirect
6. C2×#19153791C2semidirect
7. C2×#19153789C2semidirect
8. C2×#19153788C2semidirect
9. C2×#19153781C2semidirect
10. C2×C4×C22⋊D181C2semidirect
11. C2×#19153776C2semidirect
12. C2×#121×C181C2semidirect
14. C2×#15668449C22semidirect
16. C2×#15668448C22semidirect
28. #19153812C22semidirect
29. #19153812C22semidirect
30. #19153791C22semidirect
31. #19153789C22semidirect
32. #19153788C22semidirect
33. #19153781C22semidirect
34. C4×C22⋊D181C22semidirect
35. #19153776C22semidirect
46. #121×C181C22semidirect
53. C2×Dic362D4semidirect
107. Dic362C2×D4semidirect
119. C2×Dic181#41semidirect
170. Dic181C2×#41semidirect
173. C2×C181#550semidirect
179. C181C2×#550semidirect
189. C2×#121D181semidirect
216. #121C2×D181semidirect
355. C2#26041947direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684492
C22×#156684482
#260419472
C2×#191538122
C2×#191537912
C2×#191537892
C2×#191537882
C2×#191537812
C2×C4×C22⋊D1812
C2×#191537762
C2×#121×C1812
C2×#550181