Finite Group Databasebeta

← C2×#11729458 #13355416

Subgroups — C2×#11729458 #13355416

Order 1792.

2102 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#123C211yesmaximal, characteristic
2C2×#90489C211yesmaximal, characteristic
3C22×#6408C211yesmaximal, characteristic
4C2×#89983C212yesmaximal, complemented
5#11729458C218yesmaximal, direct product factor
6C2×#89322C212yesmaximal, complemented
7C23×C7×D4C2211yescharacteristic
8C24×C4×C7C2211yescharacteristic
9C2×C7×#123—24no—
10C2×C7×#123—22no—
11C2×C7×#123C418yescomplemented
12C2×C7×#123C2214yes—
13C22×C7×C4⋊C4—21no—
14C22×C7×C22⋊C4—21no—
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C22⋊Dic7—21no—
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2212yes—
19#90489—22no—
20#90489—24no—
21#90489C2214yes—
22C2×D4×Dic7—24no—
23C2×#6489—21no—
24C2×#6408—24no—
25C2×#6408C2214yes—
26#89983—24no—
27#89983C2218yescomplemented
28C2×C7⋊#95—24no—
29#89322C2218yescomplemented
30C2×#1890—71nomaximal, Sylow
31C25×C7D411yescharacteristic
32C25×C7C2311yescharacteristic
33C22×C7×D4—22no—
34C22×C7×D4—22no—
35C22×C7×D4—24no—
36C22×C7×D4C2×C414yes—
37C22×C7×D4D414yes—
38C22×C7×D4D412yes—
39C22×C7×D4C2312yes—
40C23×C4×C7—22no—
41C23×C4×C7—22no—
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7—22no—
45C23×C4×C7—21no—
46C23×C4×C7—24no—
47C23×C4×C7C2×C414yescomplemented
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2312yes—
50C23×Dic7—22no—
51C23×Dic7—22no—
52C23×Dic7D414yes—
53C7×#123—216no—
54C7×#123—28no—
55C7×#123C2×C4116yescomplemented
56C2×C7×C4⋊C4—28no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C22⋊C4—28no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4C2×C414yescomplemented
63C2×C7×C22⋊C4C2312yes—
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—24no—
71C2×C22⋊Dic7—24no—
72C2×C22⋊Dic7—24no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7D414yes—
75C2×C22⋊Dic7C2314yes—
76C2×C22⋊Dic7C2314yes—
77D4×Dic7—216no—
78D4×Dic7—216no—
79#6489—24no—
80#6489—24no—
81C2×C4⋊Dic7—22no—
82C2×C4×Dic7—22no—
83#6408—28no—
84C7⋊#95—216no—
85C22×#123D711yescharacteristic
86C2×#518—71no—
87C22×#99—71no—
88C2×#381—72no—
89#1890—78no—
90C2×#342—72no—
91C24×C7—21no—
92C24×C7—21no—
93C24×C7—22no—
94C24×C7—44no—
95C24×C7—44no—
96C24×C7C22⋊C414yes—
97C24×C7C22×C412yes—
98C24×C7C2×D411yescharacteristic
99C24×C7C2×D412yes—
100C24×C7C2×D412yes—
101C24×C7C2×D411yescharacteristic
102C24×C7C2×D412yes—
103C24×C7C2×D412yes—
104C24×C7#4114yes—
105C24×C7C2411yescharacteristic
106C2×C7×D4—28no—
107C2×C7×D4—28no—
108C2×C7×D4—24no—
109C2×C7×D4—28no—
110C2×C7×D4—28no—
111C2×C7×D4—24no—
112C2×C7×D4—24no—
113C2×C7×D4—24no—
114C2×C7×D4C22⋊C418yes—
115C2×C7×D4C22×C414yes—
116C2×C7×D4C2×D414yes—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—24no—
121C22×C4×C7—24no—
122C22×C4×C7—24no—
123C22×C4×C7—28no—
124C22×C4×C7—28no—
125C22×C4×C7—24no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—44no—
131C22×C4×C7—41no—
132C22×C4×C7—42no—
133C22×C4×C7—41no—
134C22×C4×C7—44no—
135C22×C4×C7—44no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7C22⋊C414yes—
139C22×C4×C7C22×C414yescomplemented
140C22×C4×C7C2×D412yes—
141C22×Dic7—24no—
142C22×Dic7—28no—
143C22×Dic7—24no—
144C22×Dic7—24no—
145C22×Dic7—48no—
146C22×Dic7—48no—
147C22×Dic7—44no—
148C22×Dic7—84no—
149C22×Dic7C2×D418yes—
150C7×C4⋊C4—216no—
151C7×C22⋊C4—216no—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—22no—
154C7×C22⋊C4C22×C414yescomplemented
155C22⋊Dic7—28no—
156C22⋊Dic7—24no—
157C22⋊Dic7—24no—
158C22⋊Dic7—24no—
159C22⋊Dic7—24no—
160C22⋊Dic7—24no—
161C22⋊Dic7—24no—
162C22⋊Dic7—44no—
163C4⋊Dic7—28no—
164C4×Dic7—28no—
165C23×D4C2×D711yescharacteristic
166C24×C4C2×D711yescharacteristic
167C2×#123—24no—
168C2×#123—22no—
169C2×#123Dic718yescomplemented
170C2×#123C2×D714yes—
171#518—74no—
172#518—144no—
173#518—142no—
174C2×C4×D4—144no—
175C2×#117—141no—
176C22×C4⋊C4—21no—
177C22×C22⋊C4—21no—
178C22×C22⋊C4—72no—
179C22×C22⋊C4—72no—
180C22×C22⋊C4—141no—
181C22×C22⋊C4C2×D711yescharacteristic
182C2×#99—74no—
183C2×#99—144no—
184#381—78no—
185#381—144no—
186C2×#95—144no—
187#342—78no—
188C23×C7—28no—
189C23×C7—24no—
190C23×C7—22no—
191C23×C7—24no—
192C23×C7—22no—
193C23×C7—24no—
194C23×C7—44no—
195C23×C7—44no—
196C23×C7—44no—
197C23×C7—42no—
198C23×C7—48no—
199C23×C7—42no—
200C23×C7—48no—
201C23×C7—44no—
202C23×C7—44no—
203C23×C7—84no—
204C23×C7—82no—
205C23×C7C2×C22⋊C412yes—
206C23×C7C2×C22⋊C414yes—
207C23×C7C2×C22⋊C412yes—
208C23×C7C4×D414yes—
209C23×C7C4×D414yes—
210C23×C7#12011yescharacteristic
211C23×C7#12012yes—
212C23×C7#12011yescharacteristic
213C23×C7#12114yes—
214C23×C7#12114yes—
215C23×C7#12312yes—
216C23×C7#12312yes—
217C23×C7C23×C411yescharacteristic
218C23×C7C22×D412yes—
219C23×C7C22×D411yescharacteristic
220C23×C7C22×D411yescharacteristic
221C23×C7C2×#4112yes—
222C7×D4—28no—
223C7×D4—28no—
224C7×D4—216no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—28no—
232C2×C4×C7—48no—
233C2×C4×C7—44no—
234C2×C4×C7—42no—
235C2×C4×C7—44no—
236C2×C4×C7—42no—
237C2×C4×C7—44no—
238C2×C4×C7—48no—
239C2×C4×C7—44no—
240C2×C4×C7C2×C22⋊C414yes—
241C2×Dic7—28no—
242C2×Dic7—48no—
243C2×Dic7—48no—
244C2×Dic7—48no—
245C2×Dic7—44no—
246C2×Dic7—84no—
247C2×Dic7—88no—
248C2×Dic7—88no—
249C25C22⋊D711yescharacteristic
250C25C22×D711yescharacteristic
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4C2×Dic714yes—
255C22×D4C22⋊D714yes—
256C22×D4C22⋊D712yes—
257C22×D4C22×D712yes—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—24no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—74no—
266C23×C4—142no—
267C23×C4—142no—
268C23×C4C2×Dic714yescomplemented
269C23×C4C22⋊D711yescharacteristic
270C23×C4C22×D712yes—
271#123—216no—
272#123—28no—
273#123C2×Dic7116yescomplemented
274C4×D4—1416no—
275C4×D4—1416no—
276#117—144no—
277#117—144no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—142no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—74no—
286C2×C22⋊C4—74no—
287C2×C22⋊C4—74no—
288C2×C22⋊C4—142no—
289C2×C22⋊C4—144no—
290C2×C22⋊C4—142no—
291C2×C22⋊C4—142no—
292C2×C22⋊C4—142no—
293C2×C22⋊C4—142no—
294C2×C22⋊C4—144no—
295C2×C22⋊C4—144no—
296C2×C22⋊C4—144no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4C2×Dic714yescomplemented
299C2×C22⋊C4C22×D712yes—
300C2×C42—142no—
301#99—148no—
302#95—1416no—
303C22×C7—28no—
304C22×C7—24no—
305C22×C7—24no—
306C22×C7—28no—
307C22×C7—24no—
308C22×C7—24no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7—24no—
312C22×C7—42no—
313C22×C7—44no—
314C22×C7—44no—
315C22×C7—42no—
316C22×C7—41no—
317C22×C7—41no—
318C22×C7—42no—
319C22×C7—44no—
320C22×C7—48no—
321C22×C7—44no—
322C22×C7—82no—
323C22×C7—84no—
324C22×C7#38612yes—
325C22×C7#38614yes—
326C22×C7#38612yes—
327C22×C7#45814yes—
328C22×C7C22×C22⋊C411yescharacteristic
329C22×C7C2×C4×D412yes—
330C22×C7C2×#12011yescharacteristic
331C22×C7C2×#12112yes—
332C22×C7C2×#12311yescharacteristic
333C4×C7—24no—
334C4×C7—44no—
335C4×C7—44no—
336Dic7—48no—
337Dic7—84no—
338C24—21no—
339C24—22no—
340C24—21no—
341C24—44no—
342C24—44no—
343C24C22⋊Dic714yes—
344C24D4×D712yes—
345C24D4×D712yes—
346C24#115914yes—
347C24C22×Dic712yes—
348C24C2×C22⋊D711yescharacteristic
349C24C2×C22⋊D712yes—
350C24C2×C22⋊D711yescharacteristic
351C24C2×C22⋊D712yes—
352C24C23×D711yescharacteristic
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4C22⋊Dic718yes—
362C2×D4C22×Dic714yes—
363C2×D4C2×C22⋊D714yes—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—78no—
386C22×C4—144no—
387C22×C4—148no—
388C22×C4—144no—
389C22×C4—144no—
390C22×C4—288no—
391C22×C4—288no—
392C22×C4—284no—
393C22×C4—564no—
394C22×C4C22⋊Dic714yes—
395C22×C4C22×Dic714yescomplemented
396C22×C4C2×C22⋊D712yes—
397C4⋊C4—216no—
398C4⋊C4—148no—
399C22⋊C4—216no—
400C22⋊C4—24no—
401C22⋊C4—22no—
402C22⋊C4—148no—
403C22⋊C4—144no—
404C22⋊C4—144no—
405C22⋊C4—144no—
406C22⋊C4—144no—
407C22⋊C4—144no—
408C22⋊C4—144no—
409C22⋊C4—284no—
410C22⋊C4C22×Dic714yescomplemented
411C42—148no—
412C2×C7—22no—
413C2×C7—21no—
414C2×C7—21no—
415C2×C7—22no—
416C2×C7—22no—
417C2×C7—24no—
418C2×C7—44no—
419C2×C7#189014yescomplemented
420C2×C7C2×#38611yescharacteristic
421C2×C7C2×#45812yes—
422C23—24no—
423C23—28no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—44no—
429C23—48no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—48no—
435C23—44no—
436C23—44no—
437C23—84no—
438C23—82no—
439C23D4×Dic714yes—
440C23D4×Dic714yes—
441C23#649812yes—
442C23#649812yes—
443C23#650011yescharacteristic
444C23#650012yes—
445C23#650011yescharacteristic
446C23#650214yes—
447C23#650214yes—
448C23C2×C22⋊Dic712yes—
449C23C2×C22⋊Dic714yes—
450C23C2×C22⋊Dic712yes—
451C23C2×D4×D712yes—
452C23C2×#115912yes—
453C23C23×Dic711yescharacteristic
454C23C22×C22⋊D711yescharacteristic
455C23C22×C22⋊D711yescharacteristic
456D4—28no—
457D4—28no—
458D4—216no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—148no—
475C2×C4—288no—
476C2×C4—288no—
477C2×C4—288no—
478C2×C4—284no—
479C2×C4—564no—
480C2×C4—568no—
481C2×C4—568no—
482C2×C4C2×C22⋊Dic714yes—
483C7C2×#189011yescharacteristic, complemented, Sylow
484C22—24no—
485C22—28no—
486C22—24no—
487C22—28no—
488C22—22no—
489C22—24no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—44no—
494C22—42no—
495C22—44no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—48no—
502C22—44no—
503C22—82no—
504C22—84no—
505C22#8982514yes—
506C22#8999212yes—
507C22#8999214yes—
508C22#8999212yes—
509C22C2×D4×Dic712yes—
510C22C2×#649811yescharacteristic
511C22C2×#650011yescharacteristic
512C22C2×#650212yes—
513C22C22×C22⋊Dic711yescharacteristic
514C4—24no—
515C4—44no—
516C4—44no—
517C4—288no—
518C4—564no—
519C2—21no—
520C2—22no—
521C2—21no—
522C2—22no—
523C2—22no—
524C2—24no—
525C2—44no—
526C2#1172945814yesdirect product factor
527C2C2×#8982512yes—
528C2C2×#8999211yescharacteristic
subgroupG/H
centerC23#6500
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC22×C7×#123C2
socleC23×C7#120
prime radicalC23×C7×D4C22
cosocleC24×C7C24
hypercenterC22×#123D7
hypocenterC7C2×#1890
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C25 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
4. C2×#89983C2semidirect
5. #11729458C2direct (N × H)
6. C2×#89322C2semidirect
11. C2×C7×#123C4semidirect
27. #89983C22semidirect
29. #89322C22semidirect
47. C23×C4×C7C2×C4semidirect
55. C7×#123C2×C4semidirect
62. C2×C7×C22⋊C4C2×C4semidirect
139. C22×C4×C7C22×C4semidirect
154. C7×C22⋊C4C22×C4semidirect
169. C2×#123Dic7semidirect
268. C23×C4C2×Dic7semidirect
273. #123C2×Dic7semidirect
298. C2×C22⋊C4C2×Dic7semidirect
395. C22×C4C22×Dic7semidirect
410. C22⋊C4C22×Dic7semidirect
419. C2×C7#1890semidirect
483. C7C2×#1890semidirect
526. C2#11729458direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1232
C2×#904892
C22×#64082
C2×#899832
#117294582
C2×#893222
C2×#18907