Finite Group Databasebeta

← C2×#22621146 #33958137

Subgroups — C2×#22621146 #33958137

Order 18688.

1758 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989143C211yesmaximal, characteristic
2C2×#17989142C211yesmaximal, characteristic
3#22621146C2116yesmaximal, direct product factor
4C2×#17988993C211yesmaximal, characteristic, complemented
5C2×#17988957C211yesmaximal, characteristic, complemented
6C2×#17988956C211yesmaximal, characteristic, complemented
7C2×#17988931C212yesmaximal
8C2×#17988930C212yesmaximal
9C2×#17988881C212yesmaximal, complemented
10C2×#17988871C212yesmaximal, complemented
11C2×#17988642C211yesmaximal, characteristic
12C2×#501×C73C211yesmaximal, characteristic, complemented
13#17989143—24no—
14#17989143C2218yes—
15#17989142—24no—
16#17989142C2218yes—
17C2×#14556591C2211yescharacteristic
18C2×#14556590C2211yescharacteristic
19C2×#14556590C2211yescharacteristic
20C2×#14556583C2211yescharacteristic
21C2×#14556582—21no—
22C2×#14556571C2212yes—
23C2×#14556569—21no—
24C2×#14556569—21no—
25C2×#14556567—21no—
26C2×Q8×Dic73—21no—
27C2×#14556556—21no—
28C2×#14556556—21no—
29C2×#14556556C2212yes—
30C2×#14556549—21no—
31C2×#14556549C2211yescharacteristic
32C2×#14556549C2211yescharacteristic
33C2×C42×D73C2211yescharacteristic
34#17988993C2218yescomplemented
35#17988957C2218yescomplemented
36#17988956C2218yescomplemented
37#17988931C22116yes—
38#17988930C22116yes—
39#17988881C22116yescomplemented
40#17988871C22116yescomplemented
41#17988642C2218yes—
42C2×#14556524C2212yes—
43C2×#14556522C2212yes—
44C2×#14556520C2212yes—
45C2×#14556516C2212yes—
46C2×#14556516C2212yes—
47C2×#14556514C2212yes—
48C2×#14556514C2212yes—
49C2×#14556503C2212yes—
50C2×#14556502C2211yescharacteristic
51C2×C4⋊Q8×C73C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53#501×C73C2218yescomplemented
54C2×#107×C73C2212yes—
55C2×#106×C73C2212yes—
56C2×#97×C73C2211yescharacteristic
57C22×Dic146—21no—
58C22×Dic146—21no—
59C22×Dic146C2312yes—
60C22×C4×D73D411yescharacteristic
61C22×C4×D73D411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2312yes—
64#14556591—22no—
65#14556591C2314yes—
66#14556590—22no—
67#14556590—22no—
68#14556590C2314yes—
69#14556590C2314yes—
70#14556583—22no—
71#14556583C2314yes—
72#14556582—28no—
73#14556571—24no—
74#14556571C2318yes—
75#14556569—28no—
76#14556569—28no—
77#14556567—28no—
78Q8×Dic73—28no—
79#14556556—24no—
80#14556556—28no—
81#14556556—28no—
82#14556556C2318yes—
83#14556549—28no—
84#14556549—22no—
85#14556549—22no—
86#14556549C2314yes—
87#14556549C2314yes—
88C2×C4⋊Dic73—22no—
89C2×C4⋊Dic73—21no—
90C2×C4⋊Dic73—21no—
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—22no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C2×#11924005—22no—
99C2×#11924005—21no—
100C2×#11924005—21no—
101C2×C4×Dic73—22no—
102C2×C4×Dic73—21no—
103C2×C4×Dic73—21no—
104C2×C4×Dic73D411yescharacteristic
105C2×C4×Dic73D411yescharacteristic
106C2×C4×Dic73C2311yescharacteristic
107C42×D73D414yes—
108C42×D73C2314yes—
109#14556524C2318yes—
110#14556522C2318yes—
111#14556520C2318yes—
112#14556516C2318yes—
113#14556516C2318yes—
114#14556514C2318yes—
115#14556514C2318yes—
116#14556503C2318yes—
117#14556502C2314yes—
118C22×Q8×C73—21no—
119C22×C8×C73C2312yes—
120C4⋊Q8×C73—22no—
121C4⋊Q8×C73C2314yes—
122#125×C73—22no—
123#125×C73C2314yes—
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73C2312yes—
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C42×C73C2311yescharacteristic
131#107×C73C2318yes—
132#106×C73C2318yes—
133#97×C73C2314yes—
134C23×D73C2×D411yescharacteristic
135C2×Dic146—28no—
136C2×Dic146—22no—
137C2×Dic146—24no—
138C2×Dic146—24no—
139C2×Dic146—44no—
140C2×Dic146—44no—
141C2×Dic146C2414yes—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—22no—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C2×C4×D73—24no—
149C2×C4×D73—24no—
150C2×C4×D73—42no—
151C2×C4×D73C2×D412yes—
152C2×C4×D73C2×D414yes—
153C2×C4×D73C2×D414yes—
154C2×C4×D73C2412yes—
155C2×#10596831—24no—
156C2×#10596831C2414yes—
157C4⋊Dic73—24no—
158C4⋊Dic73—24no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—28no—
162C4⋊Dic73C2412yes—
163C4⋊Dic73C2412yes—
164#11924007—24no—
165#11924007—24no—
166#11924007—24no—
167#11924007—24no—
168#11924007—28no—
169#11924005—28no—
170#11924005—24no—
171#11924005—24no—
172C4×Dic73—28no—
173C4×Dic73—24no—
174C4×Dic73—24no—
175C4×Dic73C2×D412yes—
176C4×Dic73C2×D414yes—
177C4×Dic73C2×D414yes—
178C4×Dic73C2412yes—
179C2×Q8×C73—24no—
180C2×Q8×C73—44no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—22no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C2×C8×C73—24no—
188C2×C8×C73C2414yes—
189C4⋊C4×C73—24no—
190C4⋊C4×C73—24no—
191C4⋊C4×C73—24no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73C2412yes—
196C4⋊C4×C73C2414yes—
197C4⋊C4×C73C2412yes—
198C42×C73C2×D412yes—
199C42×C73C2412yes—
200C23×C73C22×D411yescharacteristic
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—22no—
204C22×D73C22×D412yes—
205Q8×C73—48no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—24no—
211C2×C4×C73—42no—
212C2×C4×C73—44no—
213C2×C4×C73—42no—
214C2×C4×C73C22×D412yes—
215C2×C4×C73C22×D412yes—
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C732−1+414yes—
218C2×C4×C73C2511yescharacteristic
219C2×Dic73—22no—
220C2×Dic73—22no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—24no—
224C2×Dic73—44no—
225C2×Dic73—42no—
226C2×Dic73—42no—
227C2×Dic73#13714yes—
228C2×Dic73C22×D412yes—
229C2×Dic73C22×D412yes—
230C2×Dic73C22×D412yes—
231C4×D73—24no—
232C4×D73—24no—
233C4×D73—28no—
234C4×D73—44no—
235Dic146—28no—
236Dic146—48no—
237Dic146—44no—
238Dic146—48no—
239C8×C73—24no—
240#10596831—24no—
241C22×C73#53612yes—
242C22×C73C2×#13712yes—
243C22×C73#57512yes—
244C22×C73C23×D411yescharacteristic
245C2×D73—22no—
246C2×D73—22no—
247C2×D73—22no—
248C2×D73—44no—
249C4×C73—22no—
250C4×C73—22no—
251C4×C73—42no—
252C4×C73—42no—
253C4×C73—44no—
254C4×C73#53612yes—
255C4×C73C2×2−1+412yes—
256Dic73—22no—
257Dic73—22no—
258Dic73—44no—
259Dic73—42no—
260Dic73—42no—
261Dic73C2×#13714yes—
262C2×#3237—731nomaximal, Sylow
263C2×C73#323714yescomplemented
264C2×C73C2×#53611yescharacteristic
265C2×C73C22×#13711yescharacteristic
266C2×C73C2×#57511yescharacteristic
267D73—42no—
268C2×#533—731no—
269C2×#533—731no—
270#3237—7316no—
271C2×#501—731no—
272C2×#501D7311yescharacteristic, complemented
273C2×#491—731no—
274C2×#491—731no—
275C2×#485—732no—
276C2×#485—732no—
277C2×#477—732no—
278C2×#475—732no—
279C2×#432—731no—
280C73C2×#323711yescharacteristic, complemented, Sylow
281#533—738no—
282#533—738no—
283#533—1464no—
284#533—1464no—
285C2×C4⋊Q8—731no—
286C2×C4⋊Q8—731no—
287C2×C4⋊Q8—731no—
288C2×C4⋊Q8C2×D7311yescharacteristic
289C2×#125—731no—
290C2×#125—1461no—
291C2×#125C2×D7311yescharacteristic
292C2×C22⋊Q8—732no—
293C2×C22⋊Q8—1461no—
294C2×C22⋊Q8—1461no—
295C2×C22⋊Q8—1461no—
296C2×C4×Q8—732no—
297C2×C4×Q8—1461no—
298C2×C4×Q8—1461no—
299C2×C4×Q8—1461no—
300C2×#117—731no—
301C2×#117—731no—
302C2×#117—1461no—
303C22×C42—731no—
304#501—738no—
305#501C2×D7318yescomplemented
306#491—738no—
307#491—738no—
308#485—7316no—
309#485—7316no—
310#477—7316no—
311#475—7316no—
312#432—738no—
313C2×#107—732no—
314C2×#107C2×D7312yes—
315C2×#106—732no—
316C2×#106C2×D7312yes—
317C2×C4⋊C8—732no—
318C2×#103—732no—
319C2×#103—732no—
320C2×#103—732no—
321C2×#103—732no—
322C2×C22⋊C8—732no—
323C2×#97—731no—
324C2×#97C2×D7311yescharacteristic
325C22×Q8—21no—
326C22×Q8—732no—
327C22×Q8—1461no—
328C22×Q8—1461no—
329C23×C4—731no—
330C23×C4—731no—
331C23×C4—731no—
332C22×C8—732no—
333C22×C8C22×D7312yes—
334C4⋊Q8—22no—
335C4⋊Q8—734no—
336C4⋊Q8—734no—
337C4⋊Q8—734no—
338C4⋊Q8—1462no—
339C4⋊Q8—1462no—
340C4⋊Q8—1462no—
341C4⋊Q8C22×D7314yes—
342#125—22no—
343#125—734no—
344#125—1468no—
345#125—1462no—
346#125C22×D7314yes—
347C22⋊Q8—738no—
348C22⋊Q8—1464no—
349C22⋊Q8—1468no—
350C22⋊Q8—1468no—
351C22⋊Q8—1468no—
352C4×Q8—738no—
353C4×Q8—1464no—
354C4×Q8—1468no—
355C4×Q8—1468no—
356C4×Q8—1468no—
357#117—734no—
358#117—734no—
359#117—1462no—
360#117—1462no—
361#117—1468no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7312yes—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1462no—
381C2×C42—731no—
382C2×C42—731no—
383C2×C42—731no—
384C2×C42—734no—
385C2×C42—734no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42—1462no—
389C2×C42C22×D7311yescharacteristic
390#107—738no—
391#107C22×D7318yes—
392#106—738no—
393#106C22×D7318yes—
394C4⋊C8—738no—
395#103—738no—
396#103—738no—
397#103—738no—
398#103—738no—
399C22⋊C8—738no—
400#97—734no—
401#97C22×D7314yes—
402C24—731no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—2924no—
406C2×Q8—2924no—
407C2×Q8—734no—
408C2×Q8—1464no—
409C2×Q8—1464no—
410C2×Q8—1468no—
411C2×Q8—1462no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—2922no—
416C22×C4—734no—
417C22×C4—731no—
418C22×C4—731no—
419C22×C4—731no—
420C22×C4—734no—
421C22×C4—732no—
422C22×C4—732no—
423C22×C4—1464no—
424C22×C4—1461no—
425C22×C4—1461no—
426C22×C4—1462no—
427C22×C4—1464no—
428C22×C4D4×D7311yescharacteristic
429C22×C4D4×D7311yescharacteristic
430C22×C4C23×D7311yescharacteristic
431C2×C8—24no—
432C2×C8—734no—
433C2×C8—1464no—
434C2×C8C23×D7314yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—732no—
442C4⋊C4—732no—
443C4⋊C4—1468no—
444C4⋊C4—1464no—
445C4⋊C4—1464no—
446C4⋊C4—1468no—
447C4⋊C4—1464no—
448C4⋊C4—1464no—
449C4⋊C4—1461no—
450C4⋊C4—1461no—
451C4⋊C4—1464no—
452C4⋊C4—1464no—
453C4⋊C4C23×D7312yes—
454C4⋊C4C23×D7312yes—
455C4⋊C4C23×D7314yes—
456C22⋊C4—1464no—
457C22⋊C4—1464no—
458C22⋊C4—1468no—
459C42—734no—
460C42—734no—
461C42—732no—
462C42—732no—
463C42—1464no—
464C42—1464no—
465C42—1468no—
466C42D4×D7312yes—
467C42C23×D7312yes—
468C23—732no—
469C23—1462no—
470C23—1462no—
471C23—1462no—
472C23C2×D4×D7311yescharacteristic
473Q8—48no—
474Q8—2928no—
475Q8—2928no—
476Q8—2924no—
477Q8—1468no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—2924no—
487C2×C4—2922no—
488C2×C4—2922no—
489C2×C4—2924no—
490C2×C4—734no—
491C2×C4—732no—
492C2×C4—732no—
493C2×C4—732no—
494C2×C4—1468no—
495C2×C4—1462no—
496C2×C4—1462no—
497C2×C4—1464no—
498C2×C4—1462no—
499C2×C4—1462no—
500C2×C4—1464no—
501C2×C4—1464no—
502C2×C4C2×D4×D7312yes—
503C2×C4C2×D4×D7312yes—
504C2×C4C2×D4×D7311yescharacteristic
505C2×C4#1455663914yes—
506C2×C4C24×D7311yescharacteristic
507C8—24no—
508C8—1464no—
509C22—2924no—
510C22—1462no—
511C22—1462no—
512C22—1462no—
513C22#1798916212yes—
514C22#137×D7312yes—
515C22#1798938312yes—
516C22C22×D4×D7311yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—2924no—
523C4—2922no—
524C4—2922no—
525C4—734no—
526C4—1462no—
527C4—1462no—
528C4#1798916212yes—
529C4C2×#1455663912yes—
530C2—2922no—
531C2#2262114614yesdirect product factor
532C2C2×#1798916211yescharacteristic
533C2C2×#137×D7311yescharacteristic
534C2C2×#1798938311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#501×C73C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
hypercenterC2×#501D73
hypocenterC73C2×#3237
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #22621146C2direct (N × H)
4. C2×#17988993C2semidirect
5. C2×#17988957C2semidirect
6. C2×#17988956C2semidirect
9. C2×#17988881C2semidirect
10. C2×#17988871C2semidirect
12. C2×#501×C73C2semidirect
34. #17988993C22semidirect
35. #17988957C22semidirect
36. #17988956C22semidirect
39. #17988881C22semidirect
40. #17988871C22semidirect
53. #501×C73C22semidirect
263. C2×C73#3237semidirect
272. C2×#501D73semidirect
280. C73C2×#3237semidirect
305. #501C2×D73semidirect
531. C2#22621146direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179891432
C2×#179891422
#226211462
C2×#179889932
C2×#179889572
C2×#179889562
C2×#179889312
C2×#179889302
C2×#179888812
C2×#179888712
C2×#179886422
C2×#501×C732
C2×#323773