Finite Group Databasebeta

← C2×#20299350 #28404706

Subgroups — C2×#20299350 #28404706

Order 14464.

1024 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20299350C2116yesmaximal, direct product factor
2C2×#16813622C211yesmaximal, characteristic, complemented
3C2×#16813621C211yesmaximal, characteristic, complemented
4C2×#16813617C211yesmaximal, characteristic, complemented
5C2×#16813616C211yesmaximal, characteristic, complemented
6C2×#16813615C211yesmaximal, characteristic, complemented
7C2×#16813613C211yesmaximal, characteristic, complemented
8C2×#16813612C211yesmaximal, characteristic, complemented
9C2×#16813608C211yesmaximal, characteristic
10C2×#16813606C211yesmaximal, characteristic
11C2×#16813604C211yesmaximal, characteristic
12C2×#16813602C211yesmaximal, characteristic, complemented
13C2×#16813600C211yesmaximal, characteristic, complemented
14C2×#16813599C211yesmaximal, characteristic, complemented
15C2×#16813599C211yesmaximal, characteristic, complemented
16C2×#126×C113C211yesmaximal, characteristic, complemented
17C22×Dic226C2211yescharacteristic
18C22×Dic226C2211yescharacteristic
19C22×Dic226C2211yescharacteristic
20C22×C22⋊D113C2211yescharacteristic
21C23×Dic113C2211yescharacteristic
22C22×C4×D113C2211yescharacteristic
23#16813622C2218yescomplemented
24#16813621C2218yescomplemented
25#16813617C2218yescomplemented
26#16813616C2218yescomplemented
27#16813615C2218yescomplemented
28#16813613C2218yescomplemented
29#16813612C2218yescomplemented
30#16813608C2218yes—
31#16813606C2218yes—
32#16813604C2218yes—
33#16813602C2218yescomplemented
34#16813600C2218yescomplemented
35#16813599C2218yescomplemented
36#16813599C2218yescomplemented
37C2×C4⋊Dic113C2211yescharacteristic
38C2×C4⋊Dic113C2211yescharacteristic
39C2×C4⋊Dic113C2211yescharacteristic
40C2×C4⋊Dic113C2211yescharacteristic
41C2×C4⋊Dic113C2211yescharacteristic
42C2×C4⋊Dic113C2211yescharacteristic
43C2×#14268626C2211yescharacteristic
44C2×#14268626C2211yescharacteristic
45C2×#14268626C2211yescharacteristic
46C2×#14268626C2211yescharacteristic
47C2×#14268626C2211yescharacteristic
48C2×#14268626C2211yescharacteristic
49C2×#14268626C2211yescharacteristic
50C2×C22⋊Dic113C2211yescharacteristic
51C2×C22⋊Dic113C2211yescharacteristic
52C2×#14268624C2211yescharacteristic
53C2×#14268624C2211yescharacteristic
54C2×#14268624C2211yescharacteristic
55C2×#14268624C2211yescharacteristic
56C2×#14268624C2211yescharacteristic
57C2×C4×Dic113C2211yescharacteristic
58C2×C4×Dic113C2211yescharacteristic
59#126×C113C2218yescomplemented
60C2×C4⋊C4×C113C2211yescharacteristic
61C2×C4⋊C4×C113C2211yescharacteristic
62C2×C4⋊C4×C113C2211yescharacteristic
63C2×C22⋊C4×C113C2211yescharacteristic
64C2×C22⋊C4×C113C2211yescharacteristic
65C2×C22⋊C4×C113C2211yescharacteristic
66C2×C42×C113C2211yescharacteristic
67C23×D113C2311yescharacteristic
68C2×Dic226—42no—
69C2×Dic226—42no—
70C2×Dic226—42no—
71C2×Dic226C2314yes—
72C2×Dic226C2314yes—
73C2×Dic226C2314yes—
74C2×C22⋊D113—42no—
75C2×C22⋊D113C2314yes—
76C22×Dic113—42no—
77C22×Dic113C2311yescharacteristic
78C22×Dic113C2311yescharacteristic
79C22×Dic113C2311yescharacteristic
80C22×Dic113C2311yescharacteristic
81C22×Dic113C2311yescharacteristic
82C22×Dic113C2311yescharacteristic
83C22×Dic113C2311yescharacteristic
84C22×Dic113C2314yes—
85C2×C4×D113—42no—
86C2×C4×D113C2314yes—
87C4⋊Dic113C2314yes—
88C4⋊Dic113C2314yes—
89C4⋊Dic113C2314yes—
90C4⋊Dic113C2314yes—
91C4⋊Dic113C2314yes—
92C4⋊Dic113C2314yes—
93#14268626C2314yes—
94#14268626C2314yes—
95#14268626C2314yes—
96#14268626C2314yes—
97#14268626C2314yes—
98#14268626C2314yes—
99#14268626C2314yes—
100C22⋊Dic113C2314yes—
101C22⋊Dic113C2314yes—
102#14268624C2314yes—
103#14268624C2314yes—
104#14268624C2314yes—
105#14268624C2314yes—
106#14268624C2314yes—
107C4×Dic113C2314yes—
108C4×Dic113C2314yes—
109C24×C113C2311yescharacteristic
110C22×C4×C113C2311yescharacteristic
111C22×C4×C113C2311yescharacteristic
112C22×C4×C113C2311yescharacteristic
113C22×C4×C113C2311yescharacteristic
114C22×C4×C113C2311yescharacteristic
115C22×C4×C113C2311yescharacteristic
116C4⋊C4×C113C2314yes—
117C4⋊C4×C113C2314yes—
118C4⋊C4×C113C2314yes—
119C22⋊C4×C113C2314yes—
120C22⋊C4×C113C2314yes—
121C22⋊C4×C113C2314yes—
122C42×C113C2314yes—
123C23×C113—22no—
124C23×C113—22no—
125C23×C113—22no—
126C23×C113C2412yes—
127C23×C113C2411yescharacteristic
128C22×D113—22no—
129C22×D113—22no—
130C22×D113—22no—
131C22×D113C2412yes—
132C2×C4×C113—22no—
133C2×C4×C113—22no—
134C2×C4×C113—22no—
135C2×C4×C113—22no—
136C2×C4×C113—22no—
137C2×C4×C113—22no—
138C2×C4×C113C2412yes—
139C2×C4×C113C2412yes—
140C2×C4×C113C2412yes—
141C2×C4×C113C2412yes—
142C2×C4×C113C2412yes—
143C2×C4×C113C2412yes—
144C22⋊D113—44no—
145C2×Dic113—22no—
146C2×Dic113—22no—
147C2×Dic113—22no—
148C2×Dic113—22no—
149C2×Dic113—22no—
150C2×Dic113—22no—
151C2×Dic113—22no—
152C2×Dic113—44no—
153C2×Dic113C2412yes—
154C2×Dic113C2412yes—
155C2×Dic113C2412yes—
156C2×Dic113C2412yes—
157C2×Dic113C2412yes—
158C2×Dic113C2412yes—
159C2×Dic113C2412yes—
160C4×D113—44no—
161Dic226—44no—
162Dic226—44no—
163Dic226—44no—
164C22×C113—22no—
165C22×C113—22no—
166C22×C113—22no—
167C22×C113—44no—
168C22×C113H(5,𝔽2)12yes—
169C22×C1132−1+412yes—
170C22×C1132−1+412yes—
171C22×C113C2511yescharacteristic
172C2×D113—22no—
173C2×D113—22no—
174C2×D113—22no—
175C2×D113—44no—
176C4×C113—22no—
177C4×C113—22no—
178C4×C113—22no—
179C4×C113—22no—
180C4×C113—22no—
181C4×C113—22no—
182Dic113—22no—
183Dic113—22no—
184Dic113—22no—
185Dic113—22no—
186Dic113—22no—
187Dic113—22no—
188Dic113—22no—
189C2×C113—42no—
190C2×C113#56314yescomplemented
191C2×C113C2×H(5,𝔽2)11yescharacteristic
192C2×C113C2×2−1+411yescharacteristic
193C2×C113C2×2−1+411yescharacteristic
194D113—42no—
195C2×#563—1131nomaximal, Sylow
196C113C2×#56311yescharacteristic, complemented, Sylow
197#563—11316no—
198C2×C4⋊Q8—1131no—
199C2×C4⋊Q8—1131no—
200C2×#126—1131no—
201C2×#126—1131no—
202C2×#126—1131no—
203C2×#126D11311yescharacteristic, complemented
204C2×#125—1131no—
205C2×#125—1131no—
206C2×#124—1131no—
207C2×#123—1131no—
208C2×#123—1131no—
209C2×C22⋊Q8—1131no—
210C2×C22⋊Q8—1131no—
211C2×C22⋊Q8—1131no—
212C2×C22⋊Q8—1131no—
213C22×Q8—1131no—
214C22×Q8—1131no—
215C22×Q8—1131no—
216C22×D4—1131no—
217C23×C4—1131no—
218C23×C4—1131no—
219C4⋊Q8—1138no—
220C4⋊Q8—1138no—
221#126—1138no—
222#126—1138no—
223#126—1138no—
224#126C2×D11318yescomplemented
225#125—1138no—
226#125—1138no—
227#124—1138no—
228#123—1138no—
229#123—1138no—
230C22⋊Q8—1138no—
231C22⋊Q8—1138no—
232C22⋊Q8—1138no—
233C22⋊Q8—1138no—
234C2×C4⋊C4—1131no—
235C2×C4⋊C4—1131no—
236C2×C4⋊C4—1131no—
237C2×C4⋊C4—1131no—
238C2×C4⋊C4—1131no—
239C2×C4⋊C4—1131no—
240C2×C4⋊C4—1131no—
241C2×C4⋊C4—1131no—
242C2×C4⋊C4—1131no—
243C2×C4⋊C4—1131no—
244C2×C4⋊C4—1131no—
245C2×C4⋊C4—1131no—
246C2×C4⋊C4—1131no—
247C2×C4⋊C4C2×D11311yescharacteristic
248C2×C4⋊C4C2×D11311yescharacteristic
249C2×C4⋊C4C2×D11311yescharacteristic
250C2×C22⋊C4—1131no—
251C2×C22⋊C4—1131no—
252C2×C22⋊C4—1131no—
253C2×C22⋊C4—1131no—
254C2×C22⋊C4—1131no—
255C2×C22⋊C4—1131no—
256C2×C22⋊C4—1131no—
257C2×C22⋊C4C2×D11311yescharacteristic
258C2×C22⋊C4C2×D11311yescharacteristic
259C2×C22⋊C4C2×D11311yescharacteristic
260C2×C42—1131no—
261C2×C42—1131no—
262C2×C42C2×D11311yescharacteristic
263C24—1131no—
264C24C22×D11311yescharacteristic
265C2×Q8—1134no—
266C2×Q8—1134no—
267C2×Q8—1134no—
268C2×Q8—4522no—
269C2×Q8—4522no—
270C2×Q8—4522no—
271C2×D4—1134no—
272C2×D4—4522no—
273C22×C4—1134no—
274C22×C4—1131no—
275C22×C4—1131no—
276C22×C4—1131no—
277C22×C4—1131no—
278C22×C4—1131no—
279C22×C4—1131no—
280C22×C4—1131no—
281C22×C4—1134no—
282C22×C4—4522no—
283C22×C4—4522no—
284C22×C4C22×D11311yescharacteristic
285C22×C4C22×D11311yescharacteristic
286C22×C4C22×D11311yescharacteristic
287C22×C4C22×D11311yescharacteristic
288C22×C4C22×D11311yescharacteristic
289C22×C4C22×D11311yescharacteristic
290C4⋊C4—1134no—
291C4⋊C4—1134no—
292C4⋊C4—1134no—
293C4⋊C4—1134no—
294C4⋊C4—1134no—
295C4⋊C4—1134no—
296C4⋊C4—1134no—
297C4⋊C4—1134no—
298C4⋊C4—1134no—
299C4⋊C4—1134no—
300C4⋊C4—1134no—
301C4⋊C4—1134no—
302C4⋊C4—1134no—
303C4⋊C4C22×D11314yes—
304C4⋊C4C22×D11314yes—
305C4⋊C4C22×D11314yes—
306C22⋊C4—1134no—
307C22⋊C4—1134no—
308C22⋊C4—1134no—
309C22⋊C4—1134no—
310C22⋊C4—1134no—
311C22⋊C4—1134no—
312C22⋊C4—1134no—
313C22⋊C4C22×D11314yes—
314C22⋊C4C22×D11314yes—
315C22⋊C4C22×D11314yes—
316C42—1134no—
317C42—1134no—
318C42C22×D11314yes—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—1132no—
323C23—2262no—
324C23—2262no—
325C23—2262no—
326C23C23×D11312yes—
327C23C23×D11311yescharacteristic
328Q8—4524no—
329Q8—4524no—
330Q8—4524no—
331D4—4524no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—1132no—
339C2×C4—1132no—
340C2×C4—1132no—
341C2×C4—1132no—
342C2×C4—1132no—
343C2×C4—1132no—
344C2×C4—1132no—
345C2×C4—4524no—
346C2×C4—4524no—
347C2×C4—2262no—
348C2×C4—2262no—
349C2×C4—2262no—
350C2×C4—2262no—
351C2×C4—2262no—
352C2×C4—2262no—
353C2×C4—2262no—
354C2×C4C23×D11312yes—
355C2×C4C23×D11312yes—
356C2×C4C23×D11312yes—
357C2×C4C23×D11312yes—
358C2×C4C23×D11312yes—
359C2×C4C23×D11312yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—4524no—
365C22—2262no—
366C22—2262no—
367C22—2262no—
368C22#1681366812yes—
369C22#1681366912yes—
370C22#1681366912yes—
371C22C24×D11311yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—2262no—
379C4—2262no—
380C4—2262no—
381C4—2262no—
382C4—2262no—
383C4—2262no—
384C4—2262no—
385C2—42no—
386C2—4522no—
387C2#2029935014yesdirect product factor
388C2C2×#1681366811yescharacteristic
389C2C2×#1681366911yescharacteristic
390C2C2×#1681366911yescharacteristic
subgroupG/H
centerC23C23×D113
commutator, cosocleC22×C113C25
FrattiniC22C24×D113
FittingC2×#126×C113C2
socleC23×C113C24
prime radicalC22×C22⋊D113C22
hypercenterC2×#126D113
hypocenterC113C2×#563
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20299350C2direct (N × H)
2. C2×#16813622C2semidirect
3. C2×#16813621C2semidirect
4. C2×#16813617C2semidirect
5. C2×#16813616C2semidirect
6. C2×#16813615C2semidirect
7. C2×#16813613C2semidirect
8. C2×#16813612C2semidirect
12. C2×#16813602C2semidirect
13. C2×#16813600C2semidirect
14. C2×#16813599C2semidirect
15. C2×#16813599C2semidirect
16. C2×#126×C113C2semidirect
23. #16813622C22semidirect
24. #16813621C22semidirect
25. #16813617C22semidirect
26. #16813616C22semidirect
27. #16813615C22semidirect
28. #16813613C22semidirect
29. #16813612C22semidirect
33. #16813602C22semidirect
34. #16813600C22semidirect
35. #16813599C22semidirect
36. #16813599C22semidirect
59. #126×C113C22semidirect
190. C2×C113#563semidirect
196. C113C2×#563semidirect
203. C2×#126D113semidirect
224. #126C2×D113semidirect
387. C2#20299350direct (N × H)

Maximal subgroups

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

subgroupindex
#202993502
C2×#168136222
C2×#168136212
C2×#168136172
C2×#168136162
C2×#168136152
C2×#168136132
C2×#168136122
C2×#168136082
C2×#168136062
C2×#168136042
C2×#168136022
C2×#168136002
C2×#168135992
C2×#126×C1132
C2×#563113