Finite Group Databasebeta

← C2×#22609988 #33085656

Subgroups — C2×#22609988 #33085656

Order 18688.

1374 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556598C211yesmaximal, characteristic
2C22×#14556526C211yesmaximal, characteristic, complemented
3C22×#14556522C211yesmaximal, characteristic, complemented
4C2×#17988588C211yesmaximal, characteristic
5#22609988C218yesmaximal, direct product factor
6C2×#17988356C211yesmaximal, characteristic, complemented
7C2×#17988355C211yesmaximal, characteristic, complemented
8C2×#417×C73C211yesmaximal, characteristic
9C2×#14556598—24no—
10C2×#14556598—22no—
11C2×#14556598C418yescomplemented
12C2×#14556598C2214yes—
13C22×#11924012—21no—
14C22×#11924012C2211yescharacteristic
15C22×#11924012C2211yescharacteristic
16C23×#10596831—21no—
17C23×#10596831C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C2×#14556526—24no—
20C2×#14556526C2214yescomplemented
21C2×#14556522—21no—
22C2×#14556522—21no—
23C2×#14556522—22no—
24C2×#14556522C2214yescomplemented
25#17988588—22no—
26#17988588C2214yes—
27C2×#14556515—21no—
28C2×#14556512—21no—
29#17988356C2214yescomplemented
30#17988355C2214yescomplemented
31C22×#41×C73C2211yescharacteristic
32C22×M16×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34#417×C73—22no—
35#417×C73C2214yes—
36C2×#103×C73—21no—
37C2×#102×C73—21no—
38C23×Dic73—21no—
39#14556598—216no—
40#14556598—28no—
41#14556598C2×C4116yescomplemented
42C2×#11924012—28no—
43C2×#11924012—24no—
44C2×#11924012—22no—
45C2×#11924012—21no—
46C2×#11924012—21no—
47C2×#11924012—22no—
48C2×#11924012—22no—
49C2×#11924012C2×C414yes—
50C2×#11924012C2312yes—
51C2×#11924012C2312yes—
52C22×#10596831—28no—
53C22×#10596831—24no—
54C22×#10596831—21no—
55C22×#10596831—21no—
56C22×#10596831C2×C414yes—
57C22×#10596831D411yescharacteristic
58C22×#10596831D411yescharacteristic
59C22×#10596831D412yes—
60C22×#10596831D414yes—
61C22×#10596831C2312yes—
62C2×C4⋊Dic73—21no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2312yes—
68#14556526—28no—
69#14556522—24no—
70#14556522—24no—
71#14556515—24no—
72#14556512—24no—
73C2×#41×C73—22no—
74C2×#41×C73—21no—
75C2×#41×C73—21no—
76C2×#41×C73—22no—
77C2×#41×C73C2×C414yes—
78C2×#41×C73D412yes—
79C2×#41×C73Q814yes—
80C2×#41×C73C2312yes—
81C22×Q8×C73D411yescharacteristic
82C22×D4×C73—21no—
83C22×D4×C73D411yescharacteristic
84C23×C4×C73—21no—
85C23×C4×C73—21no—
86C23×C4×C73C2311yescharacteristic
87C2×M16×C73—21no—
88C2×M16×C73—22no—
89C2×M16×C73—22no—
90C2×M16×C73C2312yes—
91C22×C8×C73—21no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73C2312yes—
98#103×C73—24no—
99#102×C73—24no—
100C22×Dic73—22no—
101C22×Dic73—41no—
102C22×Dic73—41no—
103C22×Dic73—82no—
104#11924012—24no—
105#11924012—22no—
106#11924012—216no—
107#11924012—22no—
108#11924012—22no—
109#11924012—42no—
110#11924012C22×C414yes—
111C2×#10596831—24no—
112C2×#10596831—22no—
113C2×#10596831—216no—
114C2×#10596831—22no—
115C2×#10596831—42no—
116C2×#10596831C22⋊C414yes—
117C2×#10596831C22⋊C414yes—
118C2×#10596831C22⋊C418yes—
119C2×#10596831C22×C414yes—
120C2×#10596831C2×D412yes—
121C2×#10596831C2×D412yes—
122C2×#10596831C2×D414yes—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73—41no—
126C4⋊Dic73—41no—
127C24×C73—21no—
128C24×C73#4111yescharacteristic
129#41×C73—24no—
130#41×C73—24no—
131#41×C73—24no—
132#41×C73—24no—
133#41×C73—24no—
134#41×C73—22no—
135#41×C73—22no—
136#41×C73C4⋊C418yes—
137#41×C73C22×C414yes—
138#41×C73C2×Q814yes—
139C2×Q8×C73—21no—
140C2×Q8×C73—22no—
141C2×Q8×C73C4⋊C414yes—
142C2×Q8×C73C2×D412yes—
143C2×D4×C73—24no—
144C2×D4×C73—22no—
145C2×D4×C73—24no—
146C2×D4×C73—22no—
147C2×D4×C73—22no—
148C2×D4×C73—21no—
149C2×D4×C73C4⋊C414yes—
150C2×D4×C73C2×D412yes—
151C22×C4×C73—21no—
152C22×C4×C73—24no—
153C22×C4×C73—22no—
154C22×C4×C73—24no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—41no—
158C22×C4×C73—41no—
159C22×C4×C73—82no—
160C22×C4×C73C22×C412yes—
161C22×C4×C73C2×D412yes—
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×Q812yes—
167C22×C4×C73#4111yescharacteristic
168C22×C4×C73#4112yes—
169C22×C4×C73C2411yescharacteristic
170M16×C73—22no—
171M16×C73—22no—
172M16×C73—42no—
173C2×C8×C73—22no—
174C2×C8×C73—42no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—41no—
178C4⋊C4×C73—41no—
179C23×C73—22no—
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—24no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73#11712yes—
186C23×C73#12111yescharacteristic
187C23×C73#12111yescharacteristic
188C23×C73C22⋊Q812yes—
189C23×C73C2×#4111yescharacteristic
190Q8×C73—22no—
191Q8×C73—24no—
192Q8×C73C2×C4⋊C414yes—
193D4×C73—24no—
194D4×C73—24no—
195D4×C73—28no—
196D4×C73—22no—
197D4×C73—24no—
198D4×C73C2×C4⋊C414yes—
199C2×C4×C73—22no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—28no—
204C2×C4×C73—42no—
205C2×C4×C73—82no—
206C2×C4×C73—82no—
207C2×C4×C73—82no—
208C2×C4×C73C2×C22⋊C412yes—
209C2×C4×C73C2×C22⋊C412yes—
210C2×C4×C73C2×C4⋊C412yes—
211C2×C4×C73C2×C4⋊C412yes—
212C2×C4×C73#11712yes—
213C2×C4×C73#12112yes—
214C2×C4×C73#12112yes—
215C2×C4×C73#12111yescharacteristic
216C2×C4×C73#12111yescharacteristic
217C2×C4×C73C22⋊Q812yes—
218C2×C4×C73C22⋊Q812yes—
219C2×C4×C73C22⋊Q812yes—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C22×D411yescharacteristic
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C22×Q811yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×Dic73—42no—
227C2×Dic73—82no—
228C2×Dic73—82no—
229C2×Dic73—82no—
230C8×C73—42no—
231#10596831—24no—
232#10596831—42no—
233#10596831C2×C22⋊C414yes—
234#10596831C2×C22⋊C414yes—
235C22×C73—24no—
236C22×C73—24no—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—24no—
242C22×C73—48no—
243C22×C73—42no—
244C22×C73—42no—
245C22×C73#38012yes—
246C22×C73#38012yes—
247C22×C73#47112yes—
248C22×C73#47112yes—
249C22×C73C2×#11711yescharacteristic
250C22×C73C2×#12111yescharacteristic
251C22×C73C2×C22⋊Q811yescharacteristic
252C4×C73—24no—
253C4×C73—82no—
254C4×C73#38012yes—
255C4×C73#38012yes—
256C4×C73C22×C22⋊C411yescharacteristic
257C4×C73C22×C4⋊C411yescharacteristic
258C4×C73C2×#12111yescharacteristic
259C4×C73C2×C22⋊Q811yescharacteristic
260Dic73—82no—
261C2×#1815—731nomaximal, Sylow
262C2×C73—21no—
263C2×C73—21no—
264C2×C73—22no—
265C2×C73—44no—
266C2×C73#181514yescomplemented
267C2×C73C2×#38011yescharacteristic
268C2×C73C2×#47111yescharacteristic
269C2×C73C2×#47111yescharacteristic
270C22×#131—731no—
271C22×#108—731no—
272C22×#106—731no—
273C2×#417—731no—
274C2×#417D7311yescharacteristic
275#1815—738no—
276C2×#343—731no—
277C2×#343—731no—
278C73C2×#181511yescharacteristic, complemented, Sylow
279C22×#41C2×D7311yescharacteristic
280C2×#131—738no—
281C2×#131—734no—
282C2×#131—1462no—
283C2×#131—1464no—
284C22×M16—731no—
285C22×M16—731no—
286C22×M16—1461no—
287C22×M16C2×D7311yescharacteristic
288C23×C8—731no—
289C23×C8—1461no—
290C22×C4⋊C4—731no—
291C22×C4⋊C4C2×D7311yescharacteristic
292C2×#108—734no—
293C2×#108—1464no—
294C2×#106—734no—
295C2×#106—1461no—
296C2×#106—1461no—
297C2×#106—1462no—
298#417—22no—
299#417—734no—
300#417—1462no—
301#417C2×D7314yes—
302C2×#103—21no—
303C2×#103—1461no—
304C2×#102—21no—
305C2×#102—1461no—
306#343—734no—
307#343—734no—
308C2×#41—22no—
309C2×#41—21no—
310C2×#41—21no—
311C2×#41—22no—
312C2×#41Dic14614yes—
313C2×#41C4×D7314yes—
314C2×#41C22⋊D7312yes—
315C2×#41C22×D7312yes—
316C22×Q8C22⋊D7311yescharacteristic
317C22×D4—21no—
318C22×D4C22⋊D7311yescharacteristic
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—1461no—
322C23×C4C22×D7311yescharacteristic
323#131—7316no—
324#131—14616no—
325#131—1468no—
326C2×M16—21no—
327C2×M16—22no—
328C2×M16—22no—
329C2×M16—734no—
330C2×M16—732no—
331C2×M16—732no—
332C2×M16—1462no—
333C2×M16—1462no—
334C2×M16—1468no—
335C2×M16—1464no—
336C2×M16—1461no—
337C2×M16—1462no—
338C2×M16—1461no—
339C2×M16C22×D7312yes—
340C22×C8—21no—
341C22×C8—734no—
342C22×C8—731no—
343C22×C8—731no—
344C22×C8—732no—
345C22×C8—734no—
346C22×C8—732no—
347C22×C8—1461no—
348C22×C8—1468no—
349C22×C8—1464no—
350C22×C8—1461no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—732no—
357C2×C4⋊C4—1461no—
358C2×C4⋊C4—1461no—
359C2×C4⋊C4—1461no—
360C2×C4⋊C4—1461no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4C22×D7312yes—
363#108—1468no—
364#106—1464no—
365#106—1464no—
366#103—24no—
367#103—1464no—
368#102—24no—
369#102—1464no—
370C24—21no—
371C24#1192402711yescharacteristic
372#41—24no—
373#41—24no—
374#41—24no—
375#41—24no—
376#41—24no—
377#41—22no—
378#41—22no—
379#41#1192400718yes—
380#41C2×C4×D7314yes—
381#41C2×Dic14614yes—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8#1192400714yes—
385C2×Q8C2×C22⋊D7312yes—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4#1192400714yes—
393C2×D4C2×C22⋊D7312yes—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4—2921no—
404C22×C4—2921no—
405C22×C4—5842no—
406C22×C4—1462no—
407C22×C4C2×C4×D7312yes—
408C22×C4D4×D7312yes—
409C22×C4D4×D7311yescharacteristic
410C22×C4D4×D7311yescharacteristic
411C22×C4C2×C22⋊D7311yescharacteristic
412C22×C4C2×C22⋊D7311yescharacteristic
413C22×C4C2×Dic14612yes—
414C22×C4#1192402711yescharacteristic
415C22×C4#1192402712yes—
416C22×C4C23×D7311yescharacteristic
417M16—22no—
418M16—22no—
419M16—42no—
420M16—2922no—
421M16—734no—
422M16—1462no—
423M16—1462no—
424M16—14616no—
425M16—1464no—
426M16—1462no—
427C2×C8—22no—
428C2×C8—42no—
429C2×C8—2922no—
430C2×C8—734no—
431C2×C8—734no—
432C2×C8—738no—
433C2×C8—734no—
434C2×C8—732no—
435C2×C8—732no—
436C2×C8—734no—
437C2×C8—1462no—
438C2×C8—14616no—
439C2×C8—1464no—
440C2×C8—1462no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—2921no—
446C4⋊C4—2921no—
447C4⋊C4—1462no—
448C4⋊C4—1462no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23#1455654712yes—
456C23#1455656311yescharacteristic
457C23#1455656311yescharacteristic
458C23#1455656812yes—
459C23C2×#1192402711yescharacteristic
460Q8—24no—
461Q8—22no—
462Q8C2×#1192400714yes—
463D4—28no—
464D4—24no—
465D4—24no—
466D4—24no—
467D4—22no—
468D4C2×#1192400714yes—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—2922no—
479C2×C4—5842no—
480C2×C4—5842no—
481C2×C4—5842no—
482C2×C4C22⋊C4×D7312yes—
483C2×C4C22⋊C4×D7312yes—
484C2×C4C2×#1192400712yes—
485C2×C4C2×#1192400712yes—
486C2×C4#1455654712yes—
487C2×C4#1455656311yescharacteristic
488C2×C4#1455656311yescharacteristic
489C2×C4#1455656312yes—
490C2×C4#1455656312yes—
491C2×C4#1455656812yes—
492C2×C4#1455656812yes—
493C2×C4#1455656812yes—
494C2×C4C22×C4×D7311yescharacteristic
495C2×C4C2×D4×D7311yescharacteristic
496C2×C4C2×D4×D7311yescharacteristic
497C2×C4C22×C22⋊D7311yescharacteristic
498C2×C4C22×Dic14611yescharacteristic
499C2×C4C2×#1192402711yescharacteristic
500C8—42no—
501C8—2922no—
502C8—734no—
503C8—734no—
504C8—1464no—
505C22—24no—
506C22—24no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—24no—
512C22—48no—
513C22—42no—
514C22—42no—
515C22#1798844512yes—
516C22#1798844512yes—
517C22#1798884812yes—
518C22#1798885212yes—
519C22C2×#1455654711yescharacteristic
520C22C2×#1455656311yescharacteristic
521C22C2×#1455656811yescharacteristic
522C4—24no—
523C4—82no—
524C4—5842no—
525C4#1798844512yes—
526C4#1798844512yes—
527C4C2×C22⋊C4×D7311yescharacteristic
528C4C22×#1192400711yescharacteristic
529C4C2×#1455656311yescharacteristic
530C4C2×#1455656811yescharacteristic
531C2—22no—
532C2—21no—
533C2—21no—
534C2—44no—
535C2#2260998814yesdirect product factor
536C2C2×#1798844511yescharacteristic
537C2C2×#1798884811yescharacteristic
538C2C2×#1798885211yescharacteristic
subgroupG/H
centerC23#14556563
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#417×C73C2
socleC23×C73#121
prime radicalC22×#41×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#417D73
hypocenterC73C2×#1815
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14556526C2semidirect
3. C22×#14556522C2semidirect
5. #22609988C2direct (N × H)
6. C2×#17988356C2semidirect
7. C2×#17988355C2semidirect
11. C2×#14556598C4semidirect
20. C2×#14556526C22semidirect
24. C2×#14556522C22semidirect
29. #17988356C22semidirect
30. #17988355C22semidirect
41. #14556598C2×C4semidirect
266. C2×C73#1815semidirect
278. C73C2×#1815semidirect
535. C2#22609988direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565982
C22×#145565262
C22×#145565222
C2×#179885882
#226099882
C2×#179883562
C2×#179883552
C2×#417×C732
C2×#181573