Finite Group Databasebeta

← C2×#22601322 #32878354

Subgroups — C2×#22601322 #32878354

Order 18176.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986728C211yesmaximal, characteristic, complemented
2C2×#17986671C211yesmaximal, characteristic
3#22601322C2116yesmaximal, direct product factor
4C2×#17986372C213yesmaximal, complemented
5C2×#17986200C213yesmaximal, complemented
6C2×#17986205C213yesmaximal, complemented
7C2×C4×#11765854C213yesmaximal, complemented
8C2×C8×Q8×C71C211yesmaximal, characteristic, complemented
9C22×#11765866—21no—
10C22×#11765854C2213yes—
11C22×#11765854C2216yes—
12#17986728—24no—
13#17986728C2218yescomplemented
14#17986671—24no—
15#17986671C2218yes—
16C2×#14556128C2213yes—
17C2×#14556125C2213yes—
18C2×#14556119—23no—
19C2×#14556103—23no—
20C2×Q8×Dic71—21no—
21C2×C4×Dic142C2213yes—
22C2×C4×D284C2213yes—
23#17986372C22124yescomplemented
24#17986205C22124yescomplemented
25#17986200C22124yescomplemented
26C4×#11765854C22124yescomplemented
27C2×#14556060C2211yescharacteristic
28C2×#14556053C2213yes—
29C2×#14556050C2213yes—
30C2×C4×Q8×C71C2211yescharacteristic
31C8×Q8×C71C2218yescomplemented
32C2×C4⋊C8×C71C2213yes—
33C2×C4×C8×C71C2213yes—
34C2×#11765866—28no—
35C2×#11765866—48no—
36C22×Dic142C2313yes—
37C22×D284—23no—
38C22×D284—23no—
39C22×D284C2313yes—
40C22×C4×D71—23no—
41C2×#11765854—212no—
42C2×#11765854—412no—
43C2×#11765854C23112yes—
44C2×#11765854C23124yes—
45#14556128—26no—
46#14556128C23112yes—
47#14556125—26no—
48#14556125C23112yes—
49#14556119—224no—
50#14556103—224no—
51Q8×Dic71—28no—
52C4×Dic142—26no—
53C4×Dic142C23112yes—
54C4×D284—26no—
55C4×D284C23112yes—
56C2×C4⋊Dic71—23no—
57C2×C4⋊Dic71—23no—
58C2×C4⋊Dic71C2311yescharacteristic
59C2×C4⋊Dic71C2311yescharacteristic
60C2×Dic71⋊C4—23no—
61C2×(C2×D71)⋊C4—23no—
62C2×C4×Dic71—23no—
63#14556060C2314yes—
64#14556053C23112yes—
65#14556050C23112yes—
66C22×Q8×C71D411yescharacteristic
67C22×C8×C71C2313yes—
68C22×C8×C71C2311yescharacteristic
69C4×Q8×C71D414yes—
70C4×Q8×C71C2314yes—
71C2×C4⋊C4×C71D413yes—
72C2×C42×C71C2313yes—
73C4⋊C8×C71C23112yes—
74C4×C8×C71C23112yes—
75C23×D71—23no—
76#11765866—432no—
77C2×Dic142—26no—
78C2×Dic142—26no—
79C2×Dic142—23no—
80C2×Dic142C2416yes—
81C2×D284—26no—
82C2×D284—26no—
83C2×D284—212no—
84C2×D284—23no—
85C2×D284—212no—
86C2×D284—412no—
87C2×D284—86no—
88C2×D284C2416yes—
89C22×Dic71—21no—
90C22×Dic71—23no—
91C22×Dic71—21no—
92C2×C4×D71—212no—
93C2×C4×D71—412no—
94#11765854—224no—
95#11765854—424no—
96C4⋊Dic71—212no—
97C4⋊Dic71—212no—
98C4⋊Dic71—21no—
99C4⋊Dic71—21no—
100C4⋊Dic71C2412yes—
101C4⋊Dic71C2412yes—
102Dic71⋊C4—212no—
103(C2×D71)⋊C4—212no—
104C4×Dic71—212no—
105C2×Q8×C71SD1618yescomplemented
106C2×Q8×C71C2×D414yes—
107C22×C4×C71C2×D413yes—
108C22×C4×C71C2×D413yes—
109C22×C4×C71C2411yescharacteristic
110C2×C8×C71—26no—
111C2×C8×C71#4114yescomplemented
112C2×C8×C71C2416yes—
113C2×C8×C71C2412yes—
114C4⋊C4×C71C2×D4112yes—
115C42×C71C2×D416yes—
116C42×C71C2416yes—
117C23×C71C22×D411yescharacteristic
118C22×D71—26no—
119C22×D71—46no—
120C22×D71—46no—
121C22×D71—46no—
122Q8×C71C2×SD16116yescomplemented
123C2×C4×C71—26no—
124C2×C4×C71C2×SD16112yes—
125C2×C4×C71C22×D416yes—
126C2×C4×C71C22×D411yescharacteristic
127C2×C4×C71C22×D416yes—
128C2×C4×C71C2×#4112yes—
129C2×C4×C71H(5,𝔽2)12yes—
130C2×C4×C71C2511yescharacteristic
131C2×Dic71—22no—
132C2×Dic71—26no—
133C2×Dic71—22no—
134C2×Dic71—46no—
135C2×Dic71—42no—
136C2×Dic71—42no—
137D284—26no—
138D284—26no—
139D284—46no—
140D284—424no—
141D284—812no—
142C4×D71—424no—
143Dic142—26no—
144Dic142—26no—
145Dic142—46no—
146C8×C71—26no—
147C8×C71C2×#4114yescomplemented
148C22×C71#55012yes—
149C22×C71C22×SD1612yes—
150C22×C71#57712yes—
151C22×C71C23×D411yescharacteristic
152C2×D71—46no—
153C2×D71—46no—
154C2×D71—46no—
155C2×D71—812no—
156C4×C71—26no—
157C4×C71#55012yes—
158C4×C71C22×SD16112yes—
159C4×C71C22×#4111yescharacteristic
160C4×C71C2×H(5,𝔽2)11yescharacteristic
161Dic71—46no—
162Dic71—42no—
163Dic71—42no—
164C2×#3392—711nomaximal, Sylow
165C2×C71#339214yescomplemented
166C2×C71C2×#55011yescharacteristic
167C2×C71C23×SD1611yescharacteristic
168C2×C71C2×#57711yescharacteristic
169D71—86no—
170C2×#557—711no—
171C2×#550—711no—
172#3392—7116no—
173C2×#492—713no—
174C2×#460—713no—
175C2×#461—713no—
176C2×C8×Q8D7111yescharacteristic, complemented
177C2×C4×SD16—713no—
178C71C2×#339211yescharacteristic, complemented, Sylow
179C22×#41—1421no—
180C22×SD16—716no—
181C22×SD16—713no—
182#557—718no—
183#557—1424no—
184#550—718no—
185#550—1424no—
186C2×C4⋊Q8—713no—
187C2×#127—713no—
188C2×#125—1423no—
189C2×#121—1423no—
190C2×C4×Q8—713no—
191C2×C4×Q8—1421no—
192C2×C4×Q8C2×D7111yescharacteristic
193C2×C4×D4—713no—
194#492—7124no—
195#461—7124no—
196#460—7124no—
197C8×Q8C2×D7118yescomplemented
198C4×SD16—7124no—
199C2×#106—711no—
200C2×C4⋊C8C2×D7113yes—
201C2×#103—713no—
202C2×#102—713no—
203C2×C4×C8C2×D7113yes—
204C2×#41—2848no—
205C2×#41—1428no—
206C22×Q8—713no—
207C22×Q8D28411yescharacteristic
208C22×D4—713no—
209C22×D4—1423no—
210C22×D4—1423no—
211C23×C4—1423no—
212C2×SD16—28412no—
213C2×SD16—7124no—
214C2×SD16—7112no—
215C2×SD16—14212no—
216C22×C8C22×D7113yes—
217C22×C8C22×D7111yescharacteristic
218C4⋊Q8—7112no—
219C4⋊Q8—1426no—
220#127—7112no—
221#127—1426no—
222#125—14224no—
223#121—14224no—
224C4×Q8—7112no—
225C4×Q8—1428no—
226C4×Q8—1426no—
227C4×Q8D28414yes—
228C4×Q8C22×D7114yes—
229C4×D4—7112no—
230C4×D4—1426no—
231C2×C4⋊C4—711no—
232C2×C4⋊C4—711no—
233C2×C4⋊C4—1423no—
234C2×C4⋊C4—1423no—
235C2×C4⋊C4—1423no—
236C2×C4⋊C4D28413yes—
237C2×C22⋊C4—1423no—
238C2×C42—1423no—
239C2×C42C22×D7113yes—
240#106—714no—
241C4⋊C8C22×D71112yes—
242#103—7112no—
243#102—7112no—
244C4×C8C22×D71112yes—
245C24—1423no—
246#41—28432no—
247C2×Q8—716no—
248C2×Q8—1426no—
249C2×Q8—1426no—
250C2×Q8—1423no—
251C2×Q8#1176585418yescomplemented
252C2×Q8C2×D28414yes—
253C2×D4—28412no—
254C2×D4—5686no—
255C2×D4—716no—
256C2×D4—1423no—
257C2×D4—14212no—
258C2×D4—14212no—
259C2×D4—1426no—
260C2×D4—1426no—
261C22×C4—28412no—
262C22×C4—1421no—
263C22×C4—1423no—
264C22×C4—1421no—
265C22×C4—14212no—
266C22×C4C2×D28413yes—
267C22×C4C2×D28413yes—
268C22×C4C23×D7111yescharacteristic
269SD16—28424no—
270SD16—14224no—
271C2×C8—26no—
272C2×C8#1176586614yescomplemented
273C2×C8C23×D7116yes—
274C2×C8C23×D7112yes—
275C4⋊C4—712no—
276C4⋊C4—712no—
277C4⋊C4—14212no—
278C4⋊C4—1421no—
279C4⋊C4—1421no—
280C4⋊C4—14212no—
281C4⋊C4—14212no—
282C4⋊C4C2×D284112yes—
283C22⋊C4—14212no—
284C42—14212no—
285C42C2×D28416yes—
286C42C23×D7116yes—
287C23—2846no—
288C23—2846no—
289C23—2846no—
290C23—1426no—
291C23C22×D28411yescharacteristic
292Q8—2846no—
293Q8—1426no—
294Q8—1426no—
295Q8C2×#11765854116yescomplemented
296D4—2846no—
297D4—28424no—
298D4—56812no—
299D4—1426no—
300D4—1426no—
301C2×C4—26no—
302C2×C4—2846no—
303C2×C4—2842no—
304C2×C4—2842no—
305C2×C4—28424no—
306C2×C4—1422no—
307C2×C4—1426no—
308C2×C4—1422no—
309C2×C4C2×#11765854112yes—
310C2×C4C22×D28416yes—
311C2×C4C22×D28411yescharacteristic
312C2×C4C22×D28416yes—
313C2×C4C2×#1176586612yes—
314C2×C4#1455617412yes—
315C2×C4C24×D7111yescharacteristic
316C8—26no—
317C8C2×#1176586614yescomplemented
318C22—2846no—
319C22—2846no—
320C22—2846no—
321C22—56812no—
322C22#1798667112yes—
323C22C22×#1176585412yes—
324C22#1798681112yes—
325C22C23×D28411yescharacteristic
326C4—26no—
327C4—2842no—
328C4—2846no—
329C4—2842no—
330C4#1798667112yes—
331C4C22×#11765854112yes—
332C4C22×#1176586611yescharacteristic
333C4C2×#1455617411yescharacteristic
334C2—5686no—
335C2#2260132214yesdirect product factor
336C2C2×#1798667111yescharacteristic
337C2C23×#1176585411yescharacteristic
338C2C2×#1798681111yescharacteristic
subgroupG/H
centerC23C22×D284
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×C8×Q8×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986671C2
hypercenterC2×C8×Q8D71
hypocenterC71C2×#3392
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17986728C2semidirect
3. #22601322C2direct (N × H)
4. C2×#17986372C2semidirect
5. C2×#17986200C2semidirect
6. C2×#17986205C2semidirect
7. C2×C4×#11765854C2semidirect
8. C2×C8×Q8×C71C2semidirect
13. #17986728C22semidirect
23. #17986372C22semidirect
24. #17986205C22semidirect
25. #17986200C22semidirect
26. C4×#11765854C22semidirect
31. C8×Q8×C71C22semidirect
105. C2×Q8×C71SD16semidirect
111. C2×C8×C71#41semidirect
122. Q8×C71C2×SD16semidirect
147. C8×C71C2×#41semidirect
165. C2×C71#3392semidirect
176. C2×C8×Q8D71semidirect
178. C71C2×#3392semidirect
197. C8×Q8C2×D71semidirect
251. C2×Q8#11765854semidirect
272. C2×C8#11765866semidirect
295. Q8C2×#11765854semidirect
317. C8C2×#11765866semidirect
335. C2#22601322direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179867282
C2×#179866712
#226013222
C2×#179863722
C2×#179862002
C2×#179862052
C2×C4×#117658542
C2×C8×Q8×C712
C2×#339271