Finite Group Databasebeta

← C2×#23763040 #34276010

Subgroups — C2×#23763040 #34276010

Order 20224.

966 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098939C211yesmaximal, characteristic
2C2×#19098937C211yesmaximal, characteristic, complemented
3C2×#19098915C211yesmaximal, characteristic, complemented
4C2×#19098910C211yesmaximal, characteristic, complemented
5C2×#19098909C211yesmaximal, characteristic, complemented
6#23763040C218yesmaximal, direct product factor
7C2×#19098284C211yesmaximal, characteristic, complemented
8C2×#486×C79C211yesmaximal, characteristic, complemented
9C22×#11929611—22no—
10C22×#11929609—22no—
11C2×#14579161C2211yescharacteristic
12C2×#14579158C2211yescharacteristic
13#19098939—22no—
14#19098939C2214yes—
15#19098937—22no—
16#19098937C2214yescomplemented
17#19098915—22no—
18#19098915C2214yescomplemented
19#19098910—22no—
20#19098910C2214yescomplemented
21#19098909—22no—
22#19098909C2214yescomplemented
23C2×#14579084—22no—
24C2×#14579083—22no—
25C2×#14579082—22no—
26C2×C4×C79⋊C8C2211yescharacteristic
27C2×C4×C79⋊C8C2211yescharacteristic
28#19098284C2214yescomplemented
29C2×C4⋊Q8×C79C2211yescharacteristic
30C2×#127×C79C2211yescharacteristic
31#486×C79—22no—
32#486×C79C2214yescomplemented
33C2×#102×C79—22no—
34C2×C4×C8×C79C2211yescharacteristic
35C22×Dic158—21no—
36C22×D316—21no—
37C22×D316—21no—
38C22×D316—21no—
39C2×#11929611—28no—
40C2×#11929611—48no—
41C2×#11929609—28no—
42C2×#11929609—48no—
43C22×C79⋊C8D412yes—
44C22×C79⋊C8D412yes—
45#14579161—21no—
46#14579161—21no—
47#14579161—21no—
48#14579161C2312yes—
49#14579158—21no—
50#14579158—21no—
51#14579158—21no—
52#14579158C2312yes—
53C2×C4⋊Dic79—21no—
54C2×C4⋊Dic79—21no—
55#14579084—28no—
56#14579083—28no—
57#14579082—28no—
58C4×C79⋊C8D412yes—
59C4×C79⋊C8D412yes—
60C4×C79⋊C8C2312yes—
61C4×C79⋊C8C2312yes—
62C22×Q8×C79—21no—
63C22×D4×C79—21no—
64C22×D4×C79—21no—
65C22×D4×C79—21no—
66C22×C8×C79D412yes—
67C4⋊Q8×C79—21no—
68C4⋊Q8×C79—21no—
69C4⋊Q8×C79—21no—
70C4⋊Q8×C79C2312yes—
71#127×C79—21no—
72#127×C79—21no—
73#127×C79—21no—
74#127×C79C2312yes—
75C2×C4⋊C4×C79—21no—
76C2×C4⋊C4×C79—21no—
77C2×C42×C79C2311yescharacteristic
78#102×C79—28no—
79C4×C8×C79D412yes—
80C4×C8×C79C2312yes—
81C23×D79—41no—
82C2×Dic158—22no—
83C2×Dic158—41no—
84C2×Dic158—44no—
85C2×D316—22no—
86C2×D316—22no—
87C2×D316—22no—
88C2×D316—41no—
89C2×D316—41no—
90C2×D316—41no—
91C2×D316—44no—
92C2×D316—44no—
93C2×D316—44no—
94C22×Dic79—41no—
95#11929611—416no—
96#11929609—416no—
97C2×C79⋊C8—24no—
98C2×C79⋊C8—24no—
99C2×C79⋊C8C2×D414yes—
100C2×C79⋊C8C2×D414yes—
101C4⋊Dic79—22no—
102C4⋊Dic79—22no—
103C4⋊Dic79—41no—
104C4⋊Dic79—41no—
105C24×C79—41no—
106C2×Q8×C79—22no—
107C2×Q8×C79—41no—
108C2×Q8×C79—44no—
109C2×D4×C79—22no—
110C2×D4×C79—22no—
111C2×D4×C79—22no—
112C2×D4×C79—41no—
113C2×D4×C79—41no—
114C2×D4×C79—41no—
115C2×D4×C79—44no—
116C2×D4×C79—44no—
117C2×D4×C79—44no—
118C22×C4×C79—41no—
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C2×C8×C79—24no—
123C2×C8×C79C2×D414yes—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—22no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C42×C79C2×D411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79C2411yescharacteristic
132C23×C79—42no—
133C23×C79—82no—
134C23×C79—82no—
135C23×C79—82no—
136C23×C79#12711yescharacteristic
137C22×D79—42no—
138C22×D79—82no—
139C22×D79—82no—
140C22×D79—82no—
141Q8×C79—44no—
142Q8×C79—82no—
143D4×C79—44no—
144D4×C79—44no—
145D4×C79—44no—
146D4×C79—82no—
147D4×C79—82no—
148D4×C79—82no—
149C2×C4×C79—42no—
150C2×C4×C79—82no—
151C2×C4×C79#12711yescharacteristic
152C2×C4×C79#12711yescharacteristic
153C2×C4×C79#12711yescharacteristic
154C2×C4×C79#13514yes—
155C2×C4×C79AGL(1,ℤ/8)14yes—
156C2×C4×C79AGL(1,ℤ/8)14yes—
157C2×C4×C79C22×D411yescharacteristic
158C2×C4×C79C22×D411yescharacteristic
159C2×C4×C79C22×D411yescharacteristic
160C2×Dic79—42no—
161C2×Dic79—82no—
162D316—44no—
163D316—44no—
164D316—44no—
165D316—82no—
166D316—82no—
167D316—82no—
168Dic158—44no—
169Dic158—82no—
170C8×C79—24no—
171C79⋊C8—24no—
172C79⋊C8—24no—
173C22×C79—82no—
174C22×C79—82no—
175C22×C79—82no—
176C22×C79—164no—
177C22×C79#49512yes—
178C22×C79#49612yes—
179C22×C79#49612yes—
180C22×C79C2×#12711yescharacteristic
181C2×D79—82no—
182C2×D79—82no—
183C2×D79—82no—
184C2×D79—164no—
185C4×C79—82no—
186C4×C79#49512yes—
187C4×C79#49612yes—
188C4×C79#49612yes—
189C4×C79C2×#13512yes—
190C4×C79C2×AGL(1,ℤ/8)12yes—
191C4×C79C2×AGL(1,ℤ/8)12yes—
192Dic79—82no—
193C2×#1717—791nomaximal, Sylow
194C2×C79—162no—
195C2×C79#171714yescomplemented
196C2×C79C2×#49511yescharacteristic
197C2×C79C2×#49611yescharacteristic
198C2×C79C2×#49611yescharacteristic
199D79—162no—
200C2×#493—791no—
201C2×#492—791no—
202C2×#487—791no—
203C2×#486—791no—
204C2×#486—791no—
205C2×#486D7911yescharacteristic, complemented
206#1717—798no—
207C2×#322—791no—
208C79C2×#171711yescharacteristic, complemented, Sylow
209C22×SD16—1582no—
210C22×D8—1582no—
211C2×C4⋊Q8—791no—
212C2×C4⋊Q8C2×D7911yescharacteristic
213C2×#127—791no—
214C2×#127C2×D7911yescharacteristic
215#493—794no—
216#493—1582no—
217#492—794no—
218#492—1582no—
219#487—794no—
220#487—1582no—
221#486—22no—
222#486—794no—
223#486—794no—
224#486—1582no—
225#486—1582no—
226#486C2×D7914yescomplemented
227C2×#103—1582no—
228C2×#102—22no—
229C2×#102—1582no—
230C2×#102—1582no—
231C2×C4×C8—791no—
232C2×C4×C8—791no—
233C2×C4×C8C2×D7911yescharacteristic
234#322—794no—
235C22×Q8—21no—
236C22×Q8—1581no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—1581no—
241C22×D4—1581no—
242C22×D4—1581no—
243C2×SD16—3168no—
244C2×SD16—1588no—
245C2×D8—3168no—
246C2×D8—1588no—
247C22×C8—792no—
248C22×C8—792no—
249C22×C8C22⋊D7912yes—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—792no—
254C4⋊Q8—1581no—
255C4⋊Q8—1581no—
256C4⋊Q8—1581no—
257C4⋊Q8C22×D7912yes—
258#127—21no—
259#127—21no—
260#127—21no—
261#127—792no—
262#127—1581no—
263#127—1581no—
264#127—1581no—
265#127C22×D7912yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1581no—
269C2×C4⋊C4—1581no—
270C2×C42C22×D7911yescharacteristic
271#103—1588no—
272#102—28no—
273#102—1588no—
274#102—1588no—
275C4×C8—792no—
276C4×C8—792no—
277C4×C8—792no—
278C4×C8—792no—
279C4×C8C22⋊D7912yes—
280C4×C8C22×D7912yes—
281C24—41no—
282C24—3161no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—44no—
286C2×Q8—3161no—
287C2×Q8—3164no—
288C2×Q8—1582no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—3161no—
299C2×D4—3161no—
300C2×D4—3161no—
301C2×D4—3164no—
302C2×D4—3164no—
303C2×D4—3164no—
304C2×D4—1582no—
305C2×D4—1582no—
306C2×D4—1582no—
307C22×C4—41no—
308C22×C4—3161no—
309C22×C4D4×D7911yescharacteristic
310C22×C4D4×D7911yescharacteristic
311C22×C4C2×C22⋊D7911yescharacteristic
312SD16—31616no—
313D8—31616no—
314C2×C8—24no—
315C2×C8—794no—
316C2×C8—794no—
317C2×C8—1584no—
318C2×C8—1584no—
319C2×C8C2×C22⋊D7914yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—3161no—
325C4⋊C4—3161no—
326C4⋊C4—1582no—
327C4⋊C4—1582no—
328C42D4×D7911yescharacteristic
329C42D4×D7911yescharacteristic
330C42C2×C22⋊D7911yescharacteristic
331C42C23×D7911yescharacteristic
332C23—42no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—3162no—
337C23—6322no—
338C23—6322no—
339C23—6322no—
340C23#1457915711yescharacteristic
341Q8—44no—
342Q8—82no—
343Q8—3164no—
344Q8—6322no—
345D4—44no—
346D4—44no—
347D4—44no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—3164no—
352D4—3164no—
353D4—3164no—
354D4—6322no—
355D4—6322no—
356D4—6322no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—3162no—
360C2×C4—6322no—
361C2×C4#1457915711yescharacteristic
362C2×C4#1457915711yescharacteristic
363C2×C4#1457915711yescharacteristic
364C2×C4#1457918114yes—
365C2×C4#1457918514yes—
366C2×C4#1457918814yes—
367C2×C4C2×D4×D7911yescharacteristic
368C2×C4C2×D4×D7911yescharacteristic
369C2×C4C22×C22⋊D7911yescharacteristic
370C8—24no—
371C8—1584no—
372C8—1584no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—164no—
377C22—6322no—
378C22—6322no—
379C22—6322no—
380C22—12644no—
381C22#1909894512yes—
382C22#1909894912yes—
383C22#1909895112yes—
384C22C2×#1457915711yescharacteristic
385C4—82no—
386C4—6322no—
387C4#1909894512yes—
388C4#1909894912yes—
389C4#1909895112yes—
390C4C2×#1457918112yes—
391C4C2×#1457918512yes—
392C4C2×#1457918812yes—
393C2—162no—
394C2—12642no—
395C2#2376304014yesdirect product factor
396C2C2×#1909894511yescharacteristic
397C2C2×#1909894911yescharacteristic
398C2C2×#1909895111yescharacteristic
subgroupG/H
centerC23#14579157
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#486×C79C2
socleC23×C79#127
prime radicalC2×#19098939C2
hypercenterC2×#486D79
hypocenterC79C2×#1717
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. C2×#19098937C2semidirect
3. C2×#19098915C2semidirect
4. C2×#19098910C2semidirect
5. C2×#19098909C2semidirect
6. #23763040C2direct (N × H)
7. C2×#19098284C2semidirect
8. C2×#486×C79C2semidirect
16. #19098937C22semidirect
18. #19098915C22semidirect
20. #19098910C22semidirect
22. #19098909C22semidirect
28. #19098284C22semidirect
32. #486×C79C22semidirect
195. C2×C79#1717semidirect
205. C2×#486D79semidirect
208. C79C2×#1717semidirect
226. #486C2×D79semidirect
395. C2#23763040direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190989392
C2×#190989372
C2×#190989152
C2×#190989102
C2×#190989092
#237630402
C2×#190982842
C2×#486×C792
C2×#171779