Finite Group Databasebeta

← C2×#26012100 #36366928

Subgroups — C2×#26012100 #36366928

Order 22144.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012100C2116yesmaximal, direct product factor
2C2×#19149678C211yesmaximal, characteristic, complemented
3C2×#19149674C211yesmaximal, characteristic
4C2×#19149674C211yesmaximal, characteristic
5C2×#19149674C211yesmaximal, characteristic
6C2×#19149673C211yesmaximal, characteristic
7C2×#19149670C211yesmaximal, characteristic, complemented
8C2×#19149669C211yesmaximal, characteristic, complemented
9C2×#19149669C211yesmaximal, characteristic, complemented
10C2×#19149669C211yesmaximal, characteristic, complemented
11C2×#19149657C211yesmaximal, characteristic
12C2×#19149657C211yesmaximal, characteristic
13C2×C4×Dic346C211yesmaximal, characteristic, complemented
14C2×#19149644C211yesmaximal, characteristic, complemented
15C2×#19149637C211yesmaximal, characteristic
16C2×#125×C173C211yesmaximal, characteristic, complemented
17C22×Dic346C2211yescharacteristic
18C22×Dic346C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20#19149678C2218yescomplemented
21#19149674C2218yes—
22#19149674C2218yes—
23#19149674C2218yes—
24#19149673C2218yes—
25#19149670C2218yescomplemented
26#19149669C2218yescomplemented
27#19149669C2218yescomplemented
28#19149669C2218yescomplemented
29#19149657C2218yes—
30#19149657C2218yes—
31C4×Dic346C2218yescomplemented
32#19149644C2218yescomplemented
33#19149637C2218yes—
34C2×C4⋊Dic173C2211yescharacteristic
35C2×C4⋊Dic173C2211yescharacteristic
36C2×C4⋊Dic173C2211yescharacteristic
37C2×C4⋊Dic173C2211yescharacteristic
38C2×#14583805C2211yescharacteristic
39C2×#14583805C2211yescharacteristic
40C2×#14583805C2211yescharacteristic
41C2×#14583805C2211yescharacteristic
42C2×#14583805C2211yescharacteristic
43C2×#14583805C2211yescharacteristic
44C2×#14583805C2211yescharacteristic
45C2×#14583805C2211yescharacteristic
46C2×#14583805C2211yescharacteristic
47C2×#14583805C2211yescharacteristic
48C2×#14583803C2211yescharacteristic
49C2×#14583803C2211yescharacteristic
50C2×#14583803C2211yescharacteristic
51C2×#14583803C2211yescharacteristic
52C2×#14583803C2211yescharacteristic
53C2×#14583803C2211yescharacteristic
54C2×C4×Dic173C2211yescharacteristic
55C2×C4×Dic173C2211yescharacteristic
56C2×C4×Dic173C2211yescharacteristic
57C2×C4×Dic173C2211yescharacteristic
58C2×C4×Dic173C2211yescharacteristic
59#125×C173C2218yescomplemented
60C2×C4⋊C4×C173C2211yescharacteristic
61C2×C4⋊C4×C173C2211yescharacteristic
62C2×C4⋊C4×C173C2211yescharacteristic
63C2×C4⋊C4×C173C2211yescharacteristic
64C2×C4⋊C4×C173C2211yescharacteristic
65C2×C4⋊C4×C173C2211yescharacteristic
66C2×C42×C173C2211yescharacteristic
67C23×D173C2311yescharacteristic
68C2×Dic346—24no—
69C2×Dic346—24no—
70C2×Dic346C2314yes—
71C2×Dic346C2314yes—
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C22×Dic173C2311yescharacteristic
79C2×C4×D173—24no—
80C2×C4×D173C2314yes—
81C4⋊Dic173C2314yes—
82C4⋊Dic173C2314yes—
83C4⋊Dic173C2314yes—
84C4⋊Dic173C2314yes—
85#14583805C2314yes—
86#14583805C2314yes—
87#14583805C2314yes—
88#14583805C2314yes—
89#14583805C2314yes—
90#14583805C2314yes—
91#14583805C2314yes—
92#14583805C2314yes—
93#14583805C2314yes—
94#14583805C2314yes—
95#14583803C2314yes—
96#14583803C2314yes—
97#14583803C2314yes—
98#14583803C2314yes—
99#14583803C2314yes—
100#14583803C2314yes—
101C4×Dic173C2314yes—
102C4×Dic173C2314yes—
103C4×Dic173C2314yes—
104C4×Dic173C2314yes—
105C4×Dic173C2314yes—
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113C4⋊C4×C173C2314yes—
114C4⋊C4×C173C2314yes—
115C4⋊C4×C173C2314yes—
116C4⋊C4×C173C2314yes—
117C4⋊C4×C173C2314yes—
118C4⋊C4×C173C2314yes—
119C42×C173C2314yes—
120C22×D173—22no—
121C22×D173—22no—
122C22×D173—22no—
123C22×D173C2412yes—
124Dic346—28no—
125Dic346—28no—
126C2×Dic173—22no—
127C2×Dic173—22no—
128C2×Dic173—22no—
129C2×Dic173—22no—
130C2×Dic173—22no—
131C2×Dic173—22no—
132C2×Dic173#4114yes—
133C2×Dic173C2412yes—
134C2×Dic173C2412yes—
135C2×Dic173C2412yes—
136C2×Dic173C2412yes—
137C2×Dic173C2412yes—
138C2×Dic173C2412yes—
139C2×Dic173C2412yes—
140C4×D173—28no—
141C23×C173C2411yescharacteristic
142C2×C4×C173—22no—
143C2×C4×C173—22no—
144C2×C4×C173—22no—
145C2×C4×C173—22no—
146C2×C4×C173—22no—
147C2×C4×C173—22no—
148C2×C4×C173—22no—
149C2×C4×C173C2412yes—
150C2×C4×C173C2412yes—
151C2×C4×C173C2412yes—
152C2×C4×C173C2412yes—
153C2×C4×C173C2412yes—
154C2×C4×C173C2412yes—
155C2×C4×C173C2412yes—
156C22×C173C2×#4112yes—
157C22×C1732−1+412yes—
158C22×C1732−1+412yes—
159C22×C173C2511yescharacteristic
160C2×D173—22no—
161C2×D173—22no—
162C2×D173—22no—
163C2×D173—44no—
164C4×C173—22no—
165C4×C173—22no—
166C4×C173—22no—
167C4×C173—22no—
168C4×C173—22no—
169C4×C173—22no—
170C4×C173—22no—
171Dic173—22no—
172Dic173—22no—
173Dic173—22no—
174Dic173—22no—
175Dic173—22no—
176Dic173—22no—
177Dic173C2×#4114yes—
178C2×C173#54114yescomplemented
179C2×C173C22×#4111yescharacteristic
180C2×C173C2×2−1+411yescharacteristic
181C2×C173C2×2−1+411yescharacteristic
182D173—42no—
183C173C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—1731nomaximal, Sylow
185#541—17316no—
186C2×C4⋊Q8—1731no—
187C2×#126—1731no—
188C2×#126—1731no—
189C2×#126—1731no—
190C2×#126—1731no—
191C2×#125—1731no—
192C2×#125—1731no—
193C2×#125—1731no—
194C2×#125—1731no—
195C2×#125D17311yescharacteristic, complemented
196C2×C22⋊Q8—1731no—
197C2×C22⋊Q8—1731no—
198C2×C4×Q8—1731no—
199C2×C4×Q8—1731no—
200C2×#117—1731no—
201C22×Q8—1731no—
202C22×Q8—1731no—
203C23×C4—1731no—
204C4⋊Q8—1738no—
205#126—1738no—
206#126—1738no—
207#126—1738no—
208#126—1738no—
209#125—1738no—
210#125—1738no—
211#125—1738no—
212#125—1738no—
213#125C2×D17318yescomplemented
214C22⋊Q8—1738no—
215C22⋊Q8—1738no—
216C4×Q8—1738no—
217C4×Q8—1738no—
218#117—1738no—
219C2×C4⋊C4—1731no—
220C2×C4⋊C4—1731no—
221C2×C4⋊C4—1731no—
222C2×C4⋊C4—1731no—
223C2×C4⋊C4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4—1731no—
226C2×C4⋊C4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4—1731no—
229C2×C4⋊C4—1731no—
230C2×C4⋊C4—1731no—
231C2×C4⋊C4—1731no—
232C2×C4⋊C4—1731no—
233C2×C4⋊C4C2×D17311yescharacteristic
234C2×C4⋊C4C2×D17311yescharacteristic
235C2×C4⋊C4C2×D17311yescharacteristic
236C2×C4⋊C4C2×D17311yescharacteristic
237C2×C4⋊C4C2×D17311yescharacteristic
238C2×C4⋊C4C2×D17311yescharacteristic
239C2×C22⋊C4—1731no—
240C2×C22⋊C4—1731no—
241C2×C22⋊C4—1731no—
242C2×C22⋊C4—1731no—
243C2×C22⋊C4—1731no—
244C2×C22⋊C4—1731no—
245C2×C42—1731no—
246C2×C42—1731no—
247C2×C42—1731no—
248C2×C42—1731no—
249C2×C42—1731no—
250C2×C42C2×D17311yescharacteristic
251C24—1731no—
252C2×Q8—3464no—
253C2×Q8—3464no—
254C2×Q8—1734no—
255C2×Q8—1734no—
256C22×C4—3464no—
257C22×C4—1731no—
258C22×C4—1731no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4—1734no—
264C22×C4—1731no—
265C22×C4C22×D17311yescharacteristic
266C22×C4C22×D17311yescharacteristic
267C22×C4C22×D17311yescharacteristic
268C22×C4C22×D17311yescharacteristic
269C22×C4C22×D17311yescharacteristic
270C22×C4C22×D17311yescharacteristic
271C22×C4C22×D17311yescharacteristic
272C4⋊C4—1734no—
273C4⋊C4—1734no—
274C4⋊C4—1734no—
275C4⋊C4—1734no—
276C4⋊C4—1734no—
277C4⋊C4—1734no—
278C4⋊C4—1734no—
279C4⋊C4—1734no—
280C4⋊C4—1734no—
281C4⋊C4—1734no—
282C4⋊C4—1734no—
283C4⋊C4—1734no—
284C4⋊C4—1734no—
285C4⋊C4—1734no—
286C4⋊C4C22×D17314yes—
287C4⋊C4C22×D17314yes—
288C4⋊C4C22×D17314yes—
289C4⋊C4C22×D17314yes—
290C4⋊C4C22×D17314yes—
291C4⋊C4C22×D17314yes—
292C22⋊C4—1734no—
293C22⋊C4—1734no—
294C22⋊C4—1734no—
295C22⋊C4—1734no—
296C22⋊C4—1734no—
297C22⋊C4—1734no—
298C42—1734no—
299C42—1734no—
300C42—1734no—
301C42—1734no—
302C42—1734no—
303C42C22×D17314yes—
304C23—3462no—
305C23—3462no—
306C23—3462no—
307C23—1732no—
308C23C23×D17311yescharacteristic
309Q8—3468no—
310Q8—3468no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—3468no—
319C2×C4—3462no—
320C2×C4—3462no—
321C2×C4—3462no—
322C2×C4—3462no—
323C2×C4—3462no—
324C2×C4—3462no—
325C2×C4—1732no—
326C2×C4—1732no—
327C2×C4—1732no—
328C2×C4—1732no—
329C2×C4—1732no—
330C2×C4—1732no—
331C2×C4—1734no—
332C2×C4—1732no—
333C2×C4C23×D17312yes—
334C2×C4C23×D17312yes—
335C2×C4C23×D17312yes—
336C2×C4C23×D17312yes—
337C2×C4C23×D17312yes—
338C2×C4C23×D17312yes—
339C2×C4C23×D17312yes—
340C22—3462no—
341C22—3462no—
342C22—3462no—
343C22—6924no—
344C22#41×D17312yes—
345C22#1914972612yes—
346C22#1914972712yes—
347C22C24×D17311yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—3462no—
356C4—3462no—
357C4—3462no—
358C4—3462no—
359C4—3462no—
360C4—3462no—
361C4—1734no—
362C2—6922no—
363C2#2601210014yesdirect product factor
364C2C2×#41×D17311yescharacteristic
365C2C2×#1914972611yescharacteristic
366C2C2×#1914972711yescharacteristic
subgroupG/H
centerC23C23×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC2×#125×C173C2
socleC23×C173C24
prime radicalC23×D173C23
hypercenterC2×#125D173
hypocenterC173C2×#541
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #26012100C2direct (N × H)
2. C2×#19149678C2semidirect
7. C2×#19149670C2semidirect
8. C2×#19149669C2semidirect
9. C2×#19149669C2semidirect
10. C2×#19149669C2semidirect
13. C2×C4×Dic346C2semidirect
14. C2×#19149644C2semidirect
16. C2×#125×C173C2semidirect
20. #19149678C22semidirect
25. #19149670C22semidirect
26. #19149669C22semidirect
27. #19149669C22semidirect
28. #19149669C22semidirect
31. C4×Dic346C22semidirect
32. #19149644C22semidirect
59. #125×C173C22semidirect
178. C2×C173#541semidirect
183. C173C2×#541semidirect
195. C2×#125D173semidirect
213. #125C2×D173semidirect
363. C2#26012100direct (N × H)

Maximal subgroups

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

subgroupindex
#260121002
C2×#191496782
C2×#191496742
C2×#191496732
C2×#191496702
C2×#191496692
C2×#191496572
C2×C4×Dic3462
C2×#191496442
C2×#191496372
C2×#125×C1732
C2×#541173