Finite Group Databasebeta

← C2×#23758024 #34114267

Subgroups — C2×#23758024 #34114267

Order 20096.

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#23758024C2116yesmaximal, direct product factor
2C2×#19097770C211yesmaximal, characteristic, complemented
3C2×#19097769C211yesmaximal, characteristic, complemented
4C2×#19097766C211yesmaximal, characteristic, complemented
5C2×#19097766C211yesmaximal, characteristic, complemented
6C2×#19097764C211yesmaximal, characteristic, complemented
7C2×#19097764C211yesmaximal, characteristic, complemented
8C2×#19097762C211yesmaximal, characteristic, complemented
9C2×#19097761C211yesmaximal, characteristic, complemented
10C2×#19097757C211yesmaximal, characteristic
11C2×#19097754C211yesmaximal, characteristic
12C2×#19097753C211yesmaximal, characteristic
13C2×#19097750C211yesmaximal, characteristic, complemented
14C2×#19097749C211yesmaximal, characteristic, complemented
15C2×#19097747C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C157C211yesmaximal, characteristic, complemented
17C22×Dic314C2211yescharacteristic
18C22×Dic314C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21#19097770C2218yescomplemented
22#19097769C2218yescomplemented
23#19097766C2218yescomplemented
24#19097766C2218yescomplemented
25#19097764C2218yescomplemented
26#19097764C2218yescomplemented
27#19097762C2218yescomplemented
28#19097761C2218yescomplemented
29#19097757C2218yes—
30#19097754C2218yes—
31#19097753C2218yes—
32#19097750C2218yescomplemented
33#19097749C2218yescomplemented
34#19097747C2218yescomplemented
35C2×C4⋊Dic157C2211yescharacteristic
36C2×C4⋊Dic157C2211yescharacteristic
37C2×C4⋊Dic157C2211yescharacteristic
38C2×C4⋊Dic157C2211yescharacteristic
39C2×#14578933C2211yescharacteristic
40C2×#14578933C2211yescharacteristic
41C2×#14578933C2211yescharacteristic
42C2×#14578933C2211yescharacteristic
43C2×#14578933C2211yescharacteristic
44C2×#14578933C2211yescharacteristic
45C2×#14578933C2211yescharacteristic
46C2×#14578933C2211yescharacteristic
47C2×#14578933C2211yescharacteristic
48C2×C22⋊Dic157C2211yescharacteristic
49C2×C22⋊Dic157C2211yescharacteristic
50C2×C22⋊Dic157C2211yescharacteristic
51C2×#14578931C2211yescharacteristic
52C2×#14578931C2211yescharacteristic
53C2×#14578931C2211yescharacteristic
54C2×#14578931C2211yescharacteristic
55C2×#14578931C2211yescharacteristic
56C2×C4×Dic157C2211yescharacteristic
57C2×C4×Dic157C2211yescharacteristic
58C2×C4×Dic157C2211yescharacteristic
59C22×Q8×C157C2211yescharacteristic
60C23×C4×C157C2211yescharacteristic
61C22⋊Q8×C157C2218yescomplemented
62C2×C4⋊C4×C157C2211yescharacteristic
63C2×C4⋊C4×C157C2211yescharacteristic
64C2×C4⋊C4×C157C2211yescharacteristic
65C2×C22⋊C4×C157C2211yescharacteristic
66C2×C22⋊C4×C157C2211yescharacteristic
67C23×D157C2311yescharacteristic
68C2×Dic314—42no—
69C2×Dic314—42no—
70C2×Dic314C2314yes—
71C2×Dic314C2314yes—
72C2×C22⋊D157—42no—
73C2×C22⋊D157C2314yes—
74C22×Dic157C2311yescharacteristic
75C22×Dic157C2311yescharacteristic
76C22×Dic157C2311yescharacteristic
77C22×Dic157C2311yescharacteristic
78C22×Dic157C2311yescharacteristic
79C22×Dic157C2311yescharacteristic
80C22×Dic157C2311yescharacteristic
81C2×C4×D157—42no—
82C2×C4×D157C2314yes—
83C4⋊Dic157C2314yes—
84C4⋊Dic157C2314yes—
85C4⋊Dic157C2314yes—
86C4⋊Dic157C2314yes—
87#14578933C2314yes—
88#14578933C2314yes—
89#14578933C2314yes—
90#14578933C2314yes—
91#14578933C2314yes—
92#14578933C2314yes—
93#14578933C2314yes—
94#14578933C2314yes—
95#14578933C2314yes—
96C22⋊Dic157C2314yes—
97C22⋊Dic157C2314yes—
98C22⋊Dic157C2314yes—
99#14578931C2314yes—
100#14578931C2314yes—
101#14578931C2314yes—
102#14578931C2314yes—
103#14578931C2314yes—
104C4×Dic157C2314yes—
105C4×Dic157C2314yes—
106C4×Dic157C2314yes—
107C24×C157C2311yescharacteristic
108C2×Q8×C157—42no—
109C2×Q8×C157C2314yes—
110C22×C4×C157—42no—
111C22×C4×C157C2311yescharacteristic
112C22×C4×C157C2311yescharacteristic
113C22×C4×C157C2311yescharacteristic
114C22×C4×C157C2311yescharacteristic
115C22×C4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117C22×C4×C157C2314yes—
118C4⋊C4×C157C2314yes—
119C4⋊C4×C157C2314yes—
120C4⋊C4×C157C2314yes—
121C22⋊C4×C157C2314yes—
122C22⋊C4×C157C2314yes—
123C22×D157—22no—
124C22×D157—22no—
125C22×D157—22no—
126C22×D157C2412yes—
127Dic314—44no—
128Dic314—44no—
129C22⋊D157—44no—
130C2×Dic157—22no—
131C2×Dic157—22no—
132C2×Dic157—22no—
133C2×Dic157—22no—
134C2×Dic157—22no—
135C2×Dic157—22no—
136C2×Dic157—22no—
137C2×Dic157C2412yes—
138C2×Dic157C2412yes—
139C2×Dic157C2412yes—
140C2×Dic157C2412yes—
141C2×Dic157C2412yes—
142C2×Dic157C2412yes—
143C2×Dic157C2412yes—
144C4×D157—44no—
145C23×C157—22no—
146C23×C157—22no—
147C23×C157—22no—
148C23×C157C2412yes—
149C23×C157C2411yescharacteristic
150Q8×C157—44no—
151C2×C4×C157—22no—
152C2×C4×C157—22no—
153C2×C4×C157—22no—
154C2×C4×C157—22no—
155C2×C4×C157—22no—
156C2×C4×C157—22no—
157C2×C4×C157—44no—
158C2×C4×C157C2412yes—
159C2×C4×C157C2412yes—
160C2×C4×C157C2412yes—
161C2×C4×C157C2412yes—
162C2×C4×C157C2412yes—
163C2×C4×C157C2412yes—
164C22×C157—22no—
165C22×C157—22no—
166C22×C157—22no—
167C22×C157—44no—
168C22×C157H(5,𝔽2)12yes—
169C22×C1572−1+412yes—
170C22×C1572−1+412yes—
171C22×C157C2511yescharacteristic
172C2×D157—22no—
173C2×D157—22no—
174C2×D157—22no—
175C2×D157—44no—
176C4×C157—22no—
177C4×C157—22no—
178C4×C157—22no—
179C4×C157—22no—
180C4×C157—22no—
181C4×C157—22no—
182Dic157—22no—
183Dic157—22no—
184Dic157—22no—
185Dic157—22no—
186Dic157—22no—
187Dic157—22no—
188Dic157—22no—
189C2×C157—42no—
190C2×C157#56314yescomplemented
191C2×C157C2×H(5,𝔽2)11yescharacteristic
192C2×C157C2×2−1+411yescharacteristic
193C2×C157C2×2−1+411yescharacteristic
194D157—42no—
195C157C2×#56311yescharacteristic, complemented, Sylow
196C2×#563—1571nomaximal, Sylow
197#563—15716no—
198C2×C4⋊Q8—1571no—
199C2×C4⋊Q8—1571no—
200C2×#126—1571no—
201C2×#126—1571no—
202C2×#126—1571no—
203C2×#126—1571no—
204C2×#125—1571no—
205C2×#125—1571no—
206C2×#124—1571no—
207C2×#123—1571no—
208C2×#123—1571no—
209C2×C22⋊Q8—1571no—
210C2×C22⋊Q8—1571no—
211C2×C22⋊Q8—1571no—
212C2×C22⋊Q8D15711yescharacteristic, complemented
213C22×Q8—1571no—
214C22×Q8—1571no—
215C22×Q8C2×D15711yescharacteristic
216C22×D4—1571no—
217C23×C4—1571no—
218C23×C4C2×D15711yescharacteristic
219C4⋊Q8—1578no—
220C4⋊Q8—1578no—
221#126—1578no—
222#126—1578no—
223#126—1578no—
224#126—1578no—
225#125—1578no—
226#125—1578no—
227#124—1578no—
228#123—1578no—
229#123—1578no—
230C22⋊Q8—1578no—
231C22⋊Q8—1578no—
232C22⋊Q8—1578no—
233C22⋊Q8C2×D15718yescomplemented
234C2×C4⋊C4—1571no—
235C2×C4⋊C4—1571no—
236C2×C4⋊C4—1571no—
237C2×C4⋊C4—1571no—
238C2×C4⋊C4—1571no—
239C2×C4⋊C4—1571no—
240C2×C4⋊C4—1571no—
241C2×C4⋊C4—1571no—
242C2×C4⋊C4—1571no—
243C2×C4⋊C4—1571no—
244C2×C4⋊C4—1571no—
245C2×C4⋊C4—1571no—
246C2×C4⋊C4—1571no—
247C2×C4⋊C4C2×D15711yescharacteristic
248C2×C4⋊C4C2×D15711yescharacteristic
249C2×C4⋊C4C2×D15711yescharacteristic
250C2×C22⋊C4—1571no—
251C2×C22⋊C4—1571no—
252C2×C22⋊C4—1571no—
253C2×C22⋊C4—1571no—
254C2×C22⋊C4—1571no—
255C2×C22⋊C4—1571no—
256C2×C22⋊C4—1571no—
257C2×C22⋊C4—1571no—
258C2×C22⋊C4C2×D15711yescharacteristic
259C2×C22⋊C4C2×D15711yescharacteristic
260C2×C42—1571no—
261C2×C42—1571no—
262C2×C42—1571no—
263C24—1571no—
264C24C22×D15711yescharacteristic
265C2×Q8—42no—
266C2×Q8—6282no—
267C2×Q8—6282no—
268C2×Q8—1574no—
269C2×Q8—1574no—
270C2×Q8C22×D15714yes—
271C2×D4—6282no—
272C2×D4—1574no—
273C22×C4—42no—
274C22×C4—6282no—
275C22×C4—1571no—
276C22×C4—1571no—
277C22×C4—1571no—
278C22×C4—1571no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1574no—
283C22×C4C22×D15711yescharacteristic
284C22×C4C22×D15711yescharacteristic
285C22×C4C22×D15711yescharacteristic
286C22×C4C22×D15711yescharacteristic
287C22×C4C22×D15711yescharacteristic
288C22×C4C22×D15711yescharacteristic
289C22×C4C22×D15714yes—
290C4⋊C4—1574no—
291C4⋊C4—1574no—
292C4⋊C4—1574no—
293C4⋊C4—1574no—
294C4⋊C4—1574no—
295C4⋊C4—1574no—
296C4⋊C4—1574no—
297C4⋊C4—1574no—
298C4⋊C4—1574no—
299C4⋊C4—1574no—
300C4⋊C4—1574no—
301C4⋊C4—1574no—
302C4⋊C4—1574no—
303C4⋊C4C22×D15714yes—
304C4⋊C4C22×D15714yes—
305C4⋊C4C22×D15714yes—
306C22⋊C4—1574no—
307C22⋊C4—1574no—
308C22⋊C4—1574no—
309C22⋊C4—1574no—
310C22⋊C4—1574no—
311C22⋊C4—1574no—
312C22⋊C4—1574no—
313C22⋊C4—1574no—
314C22⋊C4C22×D15714yes—
315C22⋊C4C22×D15714yes—
316C42—1574no—
317C42—1574no—
318C42—1574no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—3142no—
323C23—3142no—
324C23—3142no—
325C23—1572no—
326C23C23×D15712yes—
327C23C23×D15711yescharacteristic
328Q8—44no—
329Q8—6284no—
330Q8—6284no—
331D4—6284no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—3142no—
340C2×C4—3142no—
341C2×C4—3142no—
342C2×C4—3142no—
343C2×C4—3142no—
344C2×C4—3142no—
345C2×C4—3142no—
346C2×C4—6284no—
347C2×C4—1572no—
348C2×C4—1572no—
349C2×C4—1572no—
350C2×C4—1572no—
351C2×C4—1572no—
352C2×C4—1572no—
353C2×C4—1572no—
354C2×C4C23×D15712yes—
355C2×C4C23×D15712yes—
356C2×C4C23×D15712yes—
357C2×C4C23×D15712yes—
358C2×C4C23×D15712yes—
359C2×C4C23×D15712yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3142no—
365C22—3142no—
366C22—3142no—
367C22—6284no—
368C22#1909781612yes—
369C22#1909781812yes—
370C22#1909781912yes—
371C22C24×D15711yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—3142no—
379C4—3142no—
380C4—3142no—
381C4—3142no—
382C4—3142no—
383C4—3142no—
384C4—3142no—
385C2—42no—
386C2—6282no—
387C2#2375802414yesdirect product factor
388C2C2×#1909781611yescharacteristic
389C2C2×#1909781811yescharacteristic
390C2C2×#1909781911yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C22⋊Q8×C157C2
socleC23×C157C24
prime radicalC22×C22⋊D157C22
hypercenterC2×C22⋊Q8D157
hypocenterC157C2×#563
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23758024C2direct (N × H)
2. C2×#19097770C2semidirect
3. C2×#19097769C2semidirect
4. C2×#19097766C2semidirect
5. C2×#19097766C2semidirect
6. C2×#19097764C2semidirect
7. C2×#19097764C2semidirect
8. C2×#19097762C2semidirect
9. C2×#19097761C2semidirect
13. C2×#19097750C2semidirect
14. C2×#19097749C2semidirect
15. C2×#19097747C2semidirect
16. C2×C22⋊Q8×C157C2semidirect
21. #19097770C22semidirect
22. #19097769C22semidirect
23. #19097766C22semidirect
24. #19097766C22semidirect
25. #19097764C22semidirect
26. #19097764C22semidirect
27. #19097762C22semidirect
28. #19097761C22semidirect
32. #19097750C22semidirect
33. #19097749C22semidirect
34. #19097747C22semidirect
61. C22⋊Q8×C157C22semidirect
190. C2×C157#563semidirect
195. C157C2×#563semidirect
212. C2×C22⋊Q8D157semidirect
233. C22⋊Q8C2×D157semidirect
387. C2#23758024direct (N × H)

Maximal subgroups

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

subgroupindex
#237580242
C2×#190977702
C2×#190977692
C2×#190977662
C2×#190977642
C2×#190977622
C2×#190977612
C2×#190977572
C2×#190977542
C2×#190977532
C2×#190977502
C2×#190977492
C2×#190977472
C2×C22⋊Q8×C1572
C2×#563157