Finite Group Databasebeta

← C2×#23757977 #34113664

Subgroups — C2×#23757977 #34113664

Order 20096.

1200 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757977C2116yesmaximal, direct product factor
2C2×#19097769C211yesmaximal, characteristic, complemented
3C2×#19097761C211yesmaximal, characteristic, complemented
4C2×#19097761C211yesmaximal, characteristic, complemented
5C2×#19097749C211yesmaximal, characteristic, complemented
6C2×#19097749C211yesmaximal, characteristic, complemented
7C2×#19097748C211yesmaximal, characteristic, complemented
8C2×#19097748C211yesmaximal, characteristic, complemented
9C2×#19097747C211yesmaximal, characteristic, complemented
10C2×#19097736C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D157C211yesmaximal, characteristic
12C2×#19097731C211yesmaximal, characteristic
13C2×#19097731C211yesmaximal, characteristic
14C22×#14578933C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D157C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C157C211yesmaximal, characteristic, complemented
17C22×Dic314C2211yescharacteristic
18C22×Dic314C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C23×Dic157C2211yescharacteristic
21C23×Dic157C2211yescharacteristic
22C22×C4×D157C2211yescharacteristic
23C22×C4×D157C2211yescharacteristic
24C22×C4×D157C2211yescharacteristic
25#19097769C2218yescomplemented
26#19097761C2218yescomplemented
27#19097761C2218yescomplemented
28#19097749C2218yescomplemented
29#19097749C2218yescomplemented
30#19097748C2218yescomplemented
31#19097748C2218yescomplemented
32#19097747C2218yescomplemented
33#19097736C2218yescomplemented
34C4×C22⋊D157C2218yes—
35#19097731C2218yes—
36#19097731C2218yes—
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×#14578933C2211yescharacteristic
49C2×#14578933C2211yescharacteristic
50C2×#14578933C2218yescomplemented
51C4⋊C4×D157C2218yescomplemented
52C2×C22⋊Dic157C2211yescharacteristic
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—24no—
69C2×Dic314—42no—
70C2×Dic314C2314yes—
71C2×Dic314C2314yes—
72C2×C22⋊D157Q818yescomplemented
73C2×C22⋊D157C2314yes—
74C22×Dic157—24no—
75C22×Dic157—42no—
76C22×Dic157C2311yescharacteristic
77C22×Dic157C2311yescharacteristic
78C22×Dic157C2311yescharacteristic
79C22×Dic157C2311yescharacteristic
80C22×Dic157C2311yescharacteristic
81C22×Dic157C2314yes—
82C22×Dic157C2314yes—
83C22×Dic157C2311yescharacteristic
84C22×Dic157C2311yescharacteristic
85C2×C4×D157—24no—
86C2×C4×D157—42no—
87C2×C4×D157—42no—
88C2×C4×D157C2314yes—
89C2×C4×D157C2314yes—
90C2×C4×D157C2314yes—
91C4⋊Dic157C2314yes—
92C4⋊Dic157C2314yes—
93#14578933C2314yes—
94#14578933C2314yes—
95#14578933C2314yes—
96#14578933C2314yes—
97#14578933C2314yes—
98#14578933C2314yes—
99#14578933C2314yes—
100#14578933C2314yes—
101#14578933C2314yes—
102#14578933C2314yes—
103#14578933C2314yes—
104C22⋊Dic157C2314yes—
105#14578931C2314yes—
106#14578931C2314yes—
107#14578931C2314yes—
108C4×Dic157C2314yes—
109C4×Dic157C2314yes—
110C4×Dic157C2314yes—
111C24×C157C2311yescharacteristic
112C2×Q8×C157—42no—
113C2×Q8×C157C2314yes—
114C22×C4×C157—42no—
115C22×C4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117C22×C4×C157C2311yescharacteristic
118C22×C4×C157C2311yescharacteristic
119C22×C4×C157C2311yescharacteristic
120C22×C4×C157C2311yescharacteristic
121C22×C4×C157C2314yes—
122C4⋊C4×C157C2314yes—
123C4⋊C4×C157C2314yes—
124C4⋊C4×C157C2314yes—
125C22⋊C4×C157C2314yes—
126C22⋊C4×C157C2314yes—
127C22×D157—22no—
128C22×D157—22no—
129C22×D157C2×Q814yes—
130C22×D157C2412yes—
131Dic314—28no—
132Dic314—44no—
133C22⋊D157C2×Q8116yescomplemented
134C2×Dic157—22no—
135C2×Dic157—22no—
136C2×Dic157—22no—
137C2×Dic157—22no—
138C2×Dic157—22no—
139C2×Dic157—28no—
140C2×Dic157—44no—
141C2×Dic157C2×Q814yes—
142C2×Dic157#4114yes—
143C2×Dic157C2412yes—
144C2×Dic157C2412yes—
145C2×Dic157C2412yes—
146C2×Dic157C2412yes—
147C2×Dic157C2412yes—
148C2×Dic157C2412yes—
149C2×Dic157C2412yes—
150C4×D157—28no—
151C4×D157—44no—
152C4×D157—44no—
153C23×C157—22no—
154C23×C157—22no—
155C23×C157C2×Q814yes—
156C23×C157C2412yes—
157C23×C157C2411yescharacteristic
158Q8×C157—44no—
159C2×C4×C157—22no—
160C2×C4×C157—22no—
161C2×C4×C157—22no—
162C2×C4×C157—22no—
163C2×C4×C157—22no—
164C2×C4×C157—22no—
165C2×C4×C157—44no—
166C2×C4×C157C2412yes—
167C2×C4×C157C2412yes—
168C2×C4×C157C2412yes—
169C2×C4×C157C2412yes—
170C2×C4×C157C2412yes—
171C2×C4×C157C2412yes—
172C22×C157—28no—
173C22×C157—22no—
174C22×C157—22no—
175C22×C157C22×Q814yes—
176C22×C157C22×Q812yes—
177C22×C157C2×#4112yes—
178C22×C157H(5,𝔽2)12yes—
179C22×C157C2511yescharacteristic
180C2×D157—28no—
181C2×D157—22no—
182C2×D157—22no—
183C2×D157C22×Q814yes—
184C4×C157—22no—
185C4×C157—22no—
186C4×C157—22no—
187C4×C157—22no—
188C4×C157—22no—
189C4×C157—22no—
190Dic157—22no—
191Dic157—22no—
192Dic157—22no—
193Dic157—22no—
194Dic157—22no—
195Dic157C22×Q814yes—
196Dic157C2×#4114yes—
197C2×C157—24no—
198C2×C157#55414yescomplemented
199C2×C157C23×Q811yescharacteristic
200C2×C157C22×#4111yescharacteristic
201C2×C157C2×H(5,𝔽2)11yescharacteristic
202D157—24no—
203C157C2×#55411yescharacteristic, complemented, Sylow
204C2×#554—1571nomaximal, Sylow
205#554—15716no—
206C2×C4⋊Q8—1571no—
207C2×#125—1571no—
208C2×#125—1571no—
209C2×C22⋊Q8—1571no—
210C2×C22⋊Q8—1571no—
211C2×C22⋊Q8—1571no—
212C2×C22⋊Q8—1571no—
213C2×C22⋊Q8—1571no—
214C2×C22⋊Q8D15711yescharacteristic, complemented
215C2×C4×Q8—1571no—
216C2×C4×D4—1571no—
217C2×C4×D4—1571no—
218C2×C4×D4—1571no—
219C22×C4⋊C4—1571no—
220C22×C4⋊C4—1571no—
221C22×Q8—1571no—
222C22×Q8—1571no—
223C22×Q8C2×D15711yescharacteristic
224C22×D4—1571no—
225C23×C4—1571no—
226C23×C4—1571no—
227C23×C4—1571no—
228C23×C4—1571no—
229C23×C4—1571no—
230C23×C4C2×D15711yescharacteristic
231C4⋊Q8—1578no—
232#125—1578no—
233#125—1578no—
234C22⋊Q8—1578no—
235C22⋊Q8—1578no—
236C22⋊Q8—1578no—
237C22⋊Q8—1578no—
238C22⋊Q8—1578no—
239C22⋊Q8C2×D15718yescomplemented
240C4×Q8—1578no—
241C4×D4—1578no—
242C4×D4—1578no—
243C4×D4—1578no—
244C2×C4⋊C4—1571no—
245C2×C4⋊C4—1571no—
246C2×C4⋊C4—1571no—
247C2×C4⋊C4—1571no—
248C2×C4⋊C4—1571no—
249C2×C4⋊C4—1571no—
250C2×C4⋊C4—1578no—
251C2×C4⋊C4—1571no—
252C2×C4⋊C4—1571no—
253C2×C4⋊C4—1571no—
254C2×C4⋊C4—1571no—
255C2×C4⋊C4—1571no—
256C2×C4⋊C4—1571no—
257C2×C4⋊C4—1571no—
258C2×C4⋊C4—1578no—
259C2×C4⋊C4C2×D15711yescharacteristic
260C2×C4⋊C4C2×D15711yescharacteristic
261C2×C4⋊C4C2×D15711yescharacteristic
262C2×C22⋊C4—1571no—
263C2×C22⋊C4—1571no—
264C2×C22⋊C4—1571no—
265C2×C22⋊C4—1571no—
266C2×C22⋊C4C2×D15711yescharacteristic
267C2×C22⋊C4C2×D15711yescharacteristic
268C2×C42—1571no—
269C2×C42—1571no—
270C2×C42—1571no—
271C24—1571no—
272C24C22×D15711yescharacteristic
273C2×Q8—42no—
274C2×Q8—3144no—
275C2×Q8—6282no—
276C2×Q8—1574no—
277C2×Q8—1574no—
278C2×Q8C22×D15714yes—
279C2×D4—1578no—
280C2×D4—1574no—
281C22×C4—42no—
282C22×C4—3144no—
283C22×C4—3144no—
284C22×C4—6282no—
285C22×C4—6282no—
286C22×C4—6282no—
287C22×C4—1574no—
288C22×C4—1571no—
289C22×C4—1571no—
290C22×C4—1571no—
291C22×C4—1571no—
292C22×C4—1571no—
293C22×C4—1574no—
294C22×C4—1574no—
295C22×C4—1574no—
296C22×C4—1574no—
297C22×C4—1571no—
298C22×C4—1571no—
299C22×C4C22×D15711yescharacteristic
300C22×C4C22×D15711yescharacteristic
301C22×C4C22×D15711yescharacteristic
302C22×C4C22×D15711yescharacteristic
303C22×C4C22×D15711yescharacteristic
304C22×C4C22×D15711yescharacteristic
305C22×C4C22×D15714yes—
306C4⋊C4—1574no—
307C4⋊C4—1574no—
308C4⋊C4—1574no—
309C4⋊C4—1574no—
310C4⋊C4—1574no—
311C4⋊C4—1574no—
312C4⋊C4—1574no—
313C4⋊C4—1574no—
314C4⋊C4—1574no—
315C4⋊C4—1574no—
316C4⋊C4—1574no—
317C4⋊C4—1574no—
318C4⋊C4—1574no—
319C4⋊C4C22×D15714yes—
320C4⋊C4C22×D15714yes—
321C4⋊C4C22×D15714yes—
322C22⋊C4—1574no—
323C22⋊C4—1574no—
324C22⋊C4—1574no—
325C22⋊C4—1574no—
326C22⋊C4C22×D15714yes—
327C22⋊C4C22×D15714yes—
328C42—1574no—
329C42—1574no—
330C42—1574no—
331C23—22no—
332C23—22no—
333C23—3142no—
334C23—3142no—
335C23—1574no—
336C23—1572no—
337C23Q8×D15714yes—
338C23C23×D15712yes—
339C23C23×D15711yescharacteristic
340Q8—44no—
341Q8—3148no—
342Q8—6284no—
343D4—15716no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—3148no—
352C2×C4—3142no—
353C2×C4—3142no—
354C2×C4—3142no—
355C2×C4—3142no—
356C2×C4—3142no—
357C2×C4—3148no—
358C2×C4—6284no—
359C2×C4—6284no—
360C2×C4—6284no—
361C2×C4—1572no—
362C2×C4—1572no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4—1572no—
366C2×C4—1574no—
367C2×C4—1574no—
368C2×C4—1572no—
369C2×C4—1572no—
370C2×C4C23×D15712yes—
371C2×C4C23×D15712yes—
372C2×C4C23×D15712yes—
373C2×C4C23×D15712yes—
374C2×C4C23×D15712yes—
375C2×C4C23×D15712yes—
376C22—22no—
377C22—22no—
378C22—28no—
379C22—3142no—
380C22—3142no—
381C22—3148no—
382C22—1574no—
383C22C2×Q8×D15714yes—
384C22C2×Q8×D15712yes—
385C22#41×D15712yes—
386C22#1909781612yes—
387C22C24×D15711yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—3142no—
395C4—3142no—
396C4—3142no—
397C4—3142no—
398C4—3142no—
399C4—1574no—
400C4—1574no—
401C2—24no—
402C2—3144no—
403C2#2375797714yesdirect product factor
404C2C22×Q8×D15711yescharacteristic
405C2C2×#41×D15711yescharacteristic
406C2C2×#1909781611yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C22⋊Q8×C157C2
socleC23×C157C24
prime radicalC22×C22⋊D157C22
hypercenterC2×C22⋊Q8D157
hypocenterC157C2×#554
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23757977C2direct (N × H)
2. C2×#19097769C2semidirect
3. C2×#19097761C2semidirect
4. C2×#19097761C2semidirect
5. C2×#19097749C2semidirect
6. C2×#19097749C2semidirect
7. C2×#19097748C2semidirect
8. C2×#19097748C2semidirect
9. C2×#19097747C2semidirect
10. C2×#19097736C2semidirect
14. C22×#14578933C2semidirect
15. C2×C4⋊C4×D157C2semidirect
16. C2×C22⋊Q8×C157C2semidirect
25. #19097769C22semidirect
26. #19097761C22semidirect
27. #19097761C22semidirect
28. #19097749C22semidirect
29. #19097749C22semidirect
30. #19097748C22semidirect
31. #19097748C22semidirect
32. #19097747C22semidirect
33. #19097736C22semidirect
50. C2×#14578933C22semidirect
51. C4⋊C4×D157C22semidirect
61. C22⋊Q8×C157C22semidirect
72. C2×C22⋊D157Q8semidirect
133. C22⋊D157C2×Q8semidirect
198. C2×C157#554semidirect
203. C157C2×#554semidirect
214. C2×C22⋊Q8D157semidirect
239. C22⋊Q8C2×D157semidirect
403. C2#23757977direct (N × H)

Maximal subgroups

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

subgroupindex
#237579772
C2×#190977692
C2×#190977612
C2×#190977492
C2×#190977482
C2×#190977472
C2×#190977362
C2×C4×C22⋊D1572
C2×#190977312
C22×#145789332
C2×C4⋊C4×D1572
C2×C22⋊Q8×C1572
C2×#554157