Finite Group Databasebeta

← C2×#22609248 #33067117

Subgroups — C2×#22609248 #33067117

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988997C211yesmaximal, characteristic, complemented
2C2×#17988982C211yesmaximal, characteristic, complemented
3C2×#17988946C211yesmaximal, characteristic, complemented
4#22609248C218yesmaximal, direct product factor
5C2×#17988329C211yesmaximal, characteristic, complemented
6C2×#17988327C211yesmaximal, characteristic, complemented
7C2×#17988327C211yesmaximal, characteristic, complemented
8C2×#562×C73C211yesmaximal, characteristic
9C22×#11924018—21no—
10C22×#11924016—21no—
11C2×#14556591C2211yescharacteristic
12#17988997—22no—
13#17988997C2214yescomplemented
14#17988982—22no—
15#17988982C2214yescomplemented
16#17988946—22no—
17#17988946C2214yescomplemented
18C2×#14556524—21no—
19C2×#14556522—21no—
20C2×#14556515—21no—
21C2×#14556512—21no—
22C2×#14556502C2211yescharacteristic
23C2×#14556502C2211yescharacteristic
24C2×#14556502C2211yescharacteristic
25#17988329C2214yescomplemented
26#17988327C2214yescomplemented
27#17988327C2214yescomplemented
28#562×C73—22no—
29#562×C73—22no—
30#562×C73—22no—
31#562×C73C2214yes—
32C2×#125×C73C2211yescharacteristic
33C2×#124×C73C2211yescharacteristic
34C2×#124×C73C2211yescharacteristic
35C2×#123×C73—21no—
36C2×#123×C73—21no—
37C2×C22⋊Q8×C73—21no—
38C2×C22⋊Q8×C73—21no—
39C2×#121×C73—21no—
40C2×#121×C73—21no—
41C22×Dic146—21no—
42C2×#11924018—24no—
43C2×#11924018—44no—
44C2×#11924016—24no—
45C2×#11924016—44no—
46C22×#10596831—21no—
47C22×#10596831—21no—
48C22×#10596831—21no—
49#14556591—21no—
50#14556591—21no—
51#14556591—21no—
52#14556591C2312yes—
53C2×C4⋊Dic73—21no—
54C2×C4⋊Dic73—21no—
55#14556524—24no—
56#14556522—24no—
57#14556515—24no—
58#14556512—24no—
59#14556502—21no—
60#14556502—21no—
61#14556502—21no—
62#14556502C2312yes—
63#14556502C2312yes—
64#14556502C2312yes—
65C22×Q8×C73D411yescharacteristic
66C22×Q8×C73D411yescharacteristic
67C22×D4×C73—21no—
68C22×D4×C73—21no—
69C22×D4×C73D411yescharacteristic
70C22×D4×C73D411yescharacteristic
71C23×C4×C73—21no—
72C23×C4×C73—21no—
73#125×C73—21no—
74#125×C73—21no—
75#125×C73D412yes—
76#125×C73C2312yes—
77#124×C73—21no—
78#124×C73—21no—
79#124×C73—21no—
80#124×C73—21no—
81#124×C73D412yes—
82#124×C73D412yes—
83#124×C73C2312yes—
84#124×C73C2312yes—
85#123×C73—24no—
86#123×C73—24no—
87#123×C73—24no—
88#123×C73—24no—
89C22⋊Q8×C73—24no—
90C22⋊Q8×C73—24no—
91C22⋊Q8×C73—24no—
92C22⋊Q8×C73—24no—
93#121×C73—24no—
94#121×C73—24no—
95#121×C73—24no—
96#121×C73—24no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73D411yescharacteristic
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×Dic146—22no—
111C2×Dic146—42no—
112C2×Dic146—42no—
113C2×Dic146—41no—
114C22×Dic73—41no—
115#11924018—48no—
116#11924016—48no—
117C2×#10596831—24no—
118C2×#10596831—24no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831—42no—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73—41no—
126C4⋊Dic73—41no—
127C24×C73—21no—
128C24×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—42no—
132C2×Q8×C73—42no—
133C2×Q8×C73C2×D412yes—
134C2×Q8×C73C2×D412yes—
135C2×D4×C73—24no—
136C2×D4×C73—24no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—42no—
140C2×D4×C73—42no—
141C2×D4×C73—82no—
142C2×D4×C73—82no—
143C2×D4×C73C2×D412yes—
144C2×D4×C73C2×D412yes—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—24no—
150C22×C4×C73—24no—
151C22×C4×C73—82no—
152C22×C4×C73—82no—
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C4⋊C4×C73—24no—
157C4⋊C4×C73—24no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73C2×D412yes—
165C4⋊C4×C73C2×D412yes—
166C22⋊C4×C73—24no—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—22no—
170C22⋊C4×C73—22no—
171C22⋊C4×C73—22no—
172C22⋊C4×C73—22no—
173C22⋊C4×C73—22no—
174C22⋊C4×C73—22no—
175C22⋊C4×C73—22no—
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C42×C73C2×D411yescharacteristic
179C42×C73C2411yescharacteristic
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73#12011yescharacteristic
189Q8×C73—42no—
190Q8×C73—42no—
191Q8×C73—81no—
192Q8×C73—81no—
193D4×C73—42no—
194D4×C73—42no—
195D4×C73—81no—
196D4×C73—81no—
197D4×C73—84no—
198D4×C73—84no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—84no—
211C2×C4×C73—84no—
212C2×C4×C73#12011yescharacteristic
213C2×C4×C73#12011yescharacteristic
214C2×C4×C73#12011yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73C22×D411yescharacteristic
218C2×Dic73—42no—
219C2×Dic73—82no—
220Dic146—42no—
221Dic146—42no—
222Dic146—82no—
223#10596831—24no—
224#10596831—24no—
225#10596831—42no—
226C22×C73—42no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—42no—
230C22×C73—42no—
231C22×C73—42no—
232C22×C73—84no—
233C22×C73—84no—
234C22×C73#45312yes—
235C22×C73#45512yes—
236C22×C73#45612yes—
237C22×C73C2×#12011yescharacteristic
238C4×C73—21no—
239C4×C73—21no—
240C4×C73—21no—
241C4×C73—21no—
242C4×C73—21no—
243C4×C73—21no—
244C4×C73—42no—
245C4×C73—42no—
246C4×C73—42no—
247C4×C73—42no—
248Dic73—82no—
249C2×#1661—731nomaximal, Sylow
250C2×C73—82no—
251C2×C73—82no—
252C2×C73#166114yescomplemented
253C2×C73C2×#45311yescharacteristic
254C2×C73C2×#45511yescharacteristic
255C2×C73C2×#45611yescharacteristic
256C2×#562D7311yescharacteristic
257C2×#501—731no—
258C2×#497—731no—
259C2×#489—731no—
260#1661—738no—
261C2×#330—731no—
262C2×#329—731no—
263C2×#329—731no—
264C73C2×#166111yescharacteristic, complemented, Sylow
265C22×Q16—1461no—
266C22×SD16—1461no—
267#562—22no—
268#562—22no—
269#562—22no—
270#562C2×D7314yes—
271C2×C4⋊Q8—731no—
272C2×#125C2×D7311yescharacteristic
273C2×#124C2×D7311yescharacteristic
274C2×#124C2×D7311yescharacteristic
275C2×#123—21no—
276C2×#123—21no—
277C2×C22⋊Q8—21no—
278C2×C22⋊Q8—21no—
279C2×#121—21no—
280C2×#121—21no—
281#501—734no—
282#501—1462no—
283#497—734no—
284#497—1462no—
285#489—734no—
286#489—1462no—
287C2×#107—1461no—
288C2×#106—1461no—
289C2×#103—1461no—
290C2×#102—1461no—
291C2×#97—731no—
292C2×#97—731no—
293C2×#97—731no—
294#330—734no—
295#329—734no—
296#329—734no—
297C22×Q8—1461no—
298C22×Q8C22⋊D7311yescharacteristic
299C22×Q8C22⋊D7311yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4C22⋊D7311yescharacteristic
303C22×D4C22⋊D7311yescharacteristic
304C23×C4—21no—
305C23×C4—21no—
306C2×Q16—2924no—
307C2×Q16—1464no—
308C2×SD16—2924no—
309C2×SD16—1464no—
310C22×C8—1461no—
311C22×C8—1461no—
312C22×C8—1461no—
313C4⋊Q8—732no—
314C4⋊Q8—1461no—
315C4⋊Q8—1461no—
316C4⋊Q8—1461no—
317#125—21no—
318#125—21no—
319#125C22⋊D7312yes—
320#125C22×D7312yes—
321#124—21no—
322#124—21no—
323#124—21no—
324#124—21no—
325#124C22⋊D7312yes—
326#124C22⋊D7312yes—
327#124C22×D7312yes—
328#124C22×D7312yes—
329#123—24no—
330#123—24no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337#121—24no—
338#121—24no—
339#121—24no—
340#121—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—1461no—
346C2×C4⋊C4—1461no—
347C2×C4⋊C4C22⋊D7311yescharacteristic
348C2×C4⋊C4C22⋊D7311yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42C22×D7311yescharacteristic
356#107—1464no—
357#106—1464no—
358#103—1464no—
359#102—1464no—
360#97—732no—
361#97—732no—
362#97—732no—
363#97—1461no—
364#97—1461no—
365#97—1461no—
366C24—21no—
367C24—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—2922no—
373C2×Q8—2922no—
374C2×Q8—2921no—
375C2×Q8—1462no—
376C2×Q8C2×C22⋊D7312yes—
377C2×Q8C2×C22⋊D7312yes—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×C22⋊D7312yes—
387C2×D4C2×C22⋊D7312yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—2921no—
397C22×C4C2×C22⋊D7311yescharacteristic
398C22×C4C2×C22⋊D7311yescharacteristic
399C22×C4C2×C22⋊D7311yescharacteristic
400Q16—2928no—
401SD16—2928no—
402C2×C8—2922no—
403C2×C8—1462no—
404C2×C8—1464no—
405C2×C8—1464no—
406C2×C8—1462no—
407C2×C8—1462no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—2921no—
417C4⋊C4—2921no—
418C4⋊C4—1462no—
419C4⋊C4—1462no—
420C4⋊C4C2×C22⋊D7312yes—
421C4⋊C4C2×C22⋊D7312yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42C2×C22⋊D7311yescharacteristic
433C42C2×C22⋊D7311yescharacteristic
434C42C2×C22⋊D7311yescharacteristic
435C42C23×D7311yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#1455656011yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—81no—
448Q8—81no—
449Q8—2922no—
450Q8—2922no—
451Q8—5842no—
452D4—42no—
453D4—42no—
454D4—81no—
455D4—81no—
456D4—84no—
457D4—84no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—2922no—
472C2×C4—5842no—
473C2×C4#1455656011yescharacteristic
474C2×C4#1455656011yescharacteristic
475C2×C4#1455656011yescharacteristic
476C2×C4C22×C22⋊D7311yescharacteristic
477C2×C4C22×C22⋊D7311yescharacteristic
478C2×C4C22×C22⋊D7311yescharacteristic
479C8—2922no—
480C8—1464no—
481C8—1464no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—84no—
490C22#1798875512yes—
491C22#1798876512yes—
492C22#1798877012yes—
493C22C2×#1455656011yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—5842no—
505C2—82no—
506C2—82no—
507C2#2260924814yesdirect product factor
508C2C2×#1798875511yescharacteristic
509C2C2×#1798876511yescharacteristic
510C2C2×#1798877011yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC2×#562×C73C2
socleC23×C73#120
hypercenterC2×#562D73
hypocenterC73C2×#1661
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. C2×#17988997C2semidirect
2. C2×#17988982C2semidirect
3. C2×#17988946C2semidirect
4. #22609248C2direct (N × H)
5. C2×#17988329C2semidirect
6. C2×#17988327C2semidirect
7. C2×#17988327C2semidirect
13. #17988997C22semidirect
15. #17988982C22semidirect
17. #17988946C22semidirect
25. #17988329C22semidirect
26. #17988327C22semidirect
27. #17988327C22semidirect
252. C2×C73#1661semidirect
264. C73C2×#1661semidirect
507. C2#22609248direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179889972
C2×#179889822
C2×#179889462
#226092482
C2×#179883292
C2×#179883272
C2×#562×C732
C2×#166173