Finite Group Databasebeta

← C2×#11727777 #13355413

Subgroups — C2×#11727777 #13355413

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×#6498C211yesmaximal, characteristic
2C2×#90489C211yesmaximal, characteristic
3C22×C7×#99C211yesmaximal, characteristic
4C2×#89715C212yesmaximal, complemented
5C2×#89322C212yesmaximal, complemented
6#11727777C218yesmaximal, direct product factor
7C23×C7×D4C2211yescharacteristic
8C24×Dic7C2211yescharacteristic
9C22×C7×C22⋊C4C2212yes—
10C22×C22⋊Dic7—21no—
11C22×C22⋊Dic7—21no—
12C22×C22⋊Dic7C2212yes—
13C22×C22⋊Dic7C2211yescharacteristic
14#90489—22no—
15#90489—24no—
16#90489C2214yes—
17C2×#6498—22no—
18C2×#6498—24no—
19C2×#6498C418yescomplemented
20C2×#6498C2214yes—
21C2×D4×Dic7—24no—
22C2×#6489—21no—
23C22×Dic7⋊C4—21no—
24C2×C7×#99—24no—
25C2×C7×#99C2214yes—
26C2×C7⋊#95—24no—
27#89715—24no—
28#89715C2218yescomplemented
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×D4D412yes—
38C22×C7×D4D414yes—
39C22×C7×D4C2312yes—
40C23×C4×C7—22no—
41C23×C4×C7D411yescharacteristic
42C23×Dic7—21no—
43C23×Dic7—21no—
44C23×Dic7—22no—
45C23×Dic7—21no—
46C23×Dic7—22no—
47C23×Dic7—22no—
48C23×Dic7—24no—
49C23×Dic7—22no—
50C23×Dic7C2×C414yescomplemented
51C23×Dic7D414yes—
52C23×Dic7C2312yes—
53C2×C7×C22⋊C4—24no—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4C2314yes—
57C2×C22⋊Dic7—22no—
58C2×C22⋊Dic7—24no—
59C2×C22⋊Dic7—28no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—22no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7C2×C414yescomplemented
69C2×C22⋊Dic7D414yes—
70C2×C22⋊Dic7C2314yes—
71C2×C22⋊Dic7C2312yes—
72#6498—216no—
73#6498—28no—
74#6498C2×C4116yescomplemented
75D4×Dic7—216no—
76D4×Dic7—216no—
77#6489—24no—
78#6489—24no—
79C2×C4⋊Dic7—22no—
80C2×Dic7⋊C4—28no—
81C2×Dic7⋊C4—24no—
82C2×C4×Dic7—22no—
83C7×#99—28no—
84C7⋊#95—216no—
85C22×#123—71no—
86C2×#518—71no—
87C22×#99D711yescharacteristic
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—48no—
122C22×C4×C7—44no—
123C22×C4×C7C22⋊C414yes—
124C22×C4×C7C2×D412yes—
125C22×Dic7—24no—
126C22×Dic7—24no—
127C22×Dic7—24no—
128C22×Dic7—28no—
129C22×Dic7—28no—
130C22×Dic7—24no—
131C22×Dic7—24no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—24no—
136C22×Dic7—28no—
137C22×Dic7—24no—
138C22×Dic7—44no—
139C22×Dic7—41no—
140C22×Dic7—42no—
141C22×Dic7—41no—
142C22×Dic7—44no—
143C22×Dic7—44no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C22×Dic7—48no—
147C22×Dic7—84no—
148C22×Dic7C22×C414yescomplemented
149C22×Dic7C2×D418yes—
150C7×C22⋊C4—24no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—44no—
153C22⋊Dic7—28no—
154C22⋊Dic7—24no—
155C22⋊Dic7—216no—
156C22⋊Dic7—24no—
157C22⋊Dic7—24no—
158C22⋊Dic7—24no—
159C22⋊Dic7—24no—
160C22⋊Dic7—22no—
161C22⋊Dic7C22×C414yescomplemented
162C4⋊Dic7—28no—
163Dic7⋊C4—216no—
164C4×Dic7—28no—
165C23×D4C2×D711yescharacteristic
166C24×C4—71no—
167C2×#123—78no—
168C2×#123—74no—
169C2×#123—142no—
170C2×#123—144no—
171#518—74no—
172#518—144no—
173#518—142no—
174C2×C4×D4—144no—
175C2×#117—141no—
176C22×C4⋊C4—141no—
177C22×C22⋊C4—71no—
178C22×C22⋊C4—72no—
179C22×C22⋊C4—141no—
180C22×C22⋊C4—141no—
181C22×C22⋊C4C2×D712yes—
182C2×#99—24no—
183C2×#99C2×D714yes—
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—42no—
196C23×C7—44no—
197C23×C7—44no—
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×D411yescharacteristic
219C23×C7C22×D412yes—
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—48no—
230C2×C4×C7—44no—
231C2×C4×C7—88no—
232C2×C4×C7C2×C22⋊C414yes—
233C2×Dic7—24no—
234C2×Dic7—24no—
235C2×Dic7—28no—
236C2×Dic7—28no—
237C2×Dic7—48no—
238C2×Dic7—44no—
239C2×Dic7—42no—
240C2×Dic7—44no—
241C2×Dic7—42no—
242C2×Dic7—44no—
243C2×Dic7—48no—
244C2×Dic7—44no—
245C2×Dic7—48no—
246C2×Dic7—48no—
247C2×Dic7—84no—
248C2×Dic7—88no—
249C25D2811yescharacteristic
250C25C22×D711yescharacteristic
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4C4×D714yes—
255C22×D4D2812yes—
256C22×D4C22⋊D714yes—
257C22×D4C22×D712yes—
258C23×C4—22no—
259C23×C4—74no—
260C23×C4—72no—
261C23×C4—74no—
262C23×C4—141no—
263C23×C4—141no—
264C23×C4—141no—
265C23×C4—142no—
266C23×C4—142no—
267C23×C4—144no—
268C23×C4—142no—
269C23×C4—142no—
270C23×C4D2811yescharacteristic
271#123—716no—
272#123—1416no—
273#123—148no—
274C4×D4—1416no—
275C4×D4—1416no—
276#117—144no—
277#117—144no—
278C2×C4⋊C4—142no—
279C2×C4⋊C4—148no—
280C2×C4⋊C4—144no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—74no—
285C2×C22⋊C4—72no—
286C2×C22⋊C4—74no—
287C2×C22⋊C4—74no—
288C2×C22⋊C4—142no—
289C2×C22⋊C4—144no—
290C2×C22⋊C4—148no—
291C2×C22⋊C4—142no—
292C2×C22⋊C4—142no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—144no—
295C2×C22⋊C4—144no—
296C2×C22⋊C4—142no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4C22×D714yes—
300C2×C42—142no—
301#99—28no—
302#95—1416no—
303C22×C7—24no—
304C22×C7—28no—
305C22×C7—24no—
306C22×C7—28no—
307C22×C7—24no—
308C22×C7—22no—
309C22×C7—24no—
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—84no—
335Dic7—44no—
336Dic7—48no—
337Dic7—44no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—44no—
342C24—44no—
343C24(C2×D7)⋊C414yes—
344C24C2×C4×D712yes—
345C24C2×D2811yescharacteristic
346C24C2×D2812yes—
347C24C2×D2811yescharacteristic
348C24D4×D712yes—
349C24D4×D712yes—
350C24#115914yes—
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×D4(C2×D7)⋊C418yes—
362C2×D4C2×C4×D714yes—
363C2×D4C2×C22⋊D714yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—74no—
371C22×C4—78no—
372C22×C4—148no—
373C22×C4—148no—
374C22×C4—144no—
375C22×C4—144no—
376C22×C4—144no—
377C22×C4—144no—
378C22×C4—142no—
379C22×C4—142no—
380C22×C4—144no—
381C22×C4—144no—
382C22×C4—142no—
383C22×C4—148no—
384C22×C4—144no—
385C22×C4—284no—
386C22×C4—281no—
387C22×C4—282no—
388C22×C4—281no—
389C22×C4—282no—
390C22×C4—282no—
391C22×C4—284no—
392C22×C4—284no—
393C22×C4—288no—
394C22×C4—564no—
395C22×C4(C2×D7)⋊C414yes—
396C22×C4C2×D2812yes—
397C4⋊C4—148no—
398C4⋊C4—1416no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—44no—
402C22⋊C4—74no—
403C22⋊C4—148no—
404C22⋊C4—144no—
405C22⋊C4—144no—
406C22⋊C4—144no—
407C22⋊C4—1416no—
408C22⋊C4—144no—
409C22⋊C4—144no—
410C22⋊C4—142no—
411C42—148no—
412C2×C7—21no—
413C2×C7—22no—
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—28no—
423C23—24no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—48no—
432C23—44no—
433C23—42no—
434C23—48no—
435C23—44no—
436C23—44no—
437C23—82no—
438C23—84no—
439C23(C22⋊D7)⋊C414yes—
440C23(C22⋊D7)⋊C414yes—
441C23#644511yescharacteristic
442C23#644512yes—
443C23#644511yescharacteristic
444C23#644912yes—
445C23#644912yes—
446C23C2×(C2×D7)⋊C412yes—
447C23C2×(C2×D7)⋊C414yes—
448C23C2×(C2×D7)⋊C412yes—
449C23#650214yes—
450C23#650214yes—
451C23C22×C4×D711yescharacteristic
452C23C22×D2811yescharacteristic
453C23C2×D4×D712yes—
454C23C2×#115912yes—
455C23C22×C22⋊D711yescharacteristic
456D4—28no—
457D4—28no—
458D4—216no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—88no—
466C2×C4—148no—
467C2×C4—148no—
468C2×C4—144no—
469C2×C4—144no—
470C2×C4—284no—
471C2×C4—288no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—284no—
475C2×C4—284no—
476C2×C4—288no—
477C2×C4—284no—
478C2×C4—288no—
479C2×C4—288no—
480C2×C4—564no—
481C2×C4—568no—
482C2×C4C2×(C2×D7)⋊C414yes—
483C7C2×#189011yescharacteristic, complemented, Sylow
484C22—24no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—28no—
489C22—24no—
490C22—24no—
491C22—28no—
492C22—24no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—44no—
501C22—44no—
502C22—48no—
503C22—82no—
504C22—84no—
505C22#8951414yes—
506C22#8973012yes—
507C22#8973014yes—
508C22#8973012yes—
509C22C2×(C22⋊D7)⋊C412yes—
510C22C2×#644511yescharacteristic
511C22C2×#644911yescharacteristic
512C22C22×(C2×D7)⋊C411yescharacteristic
513C22C2×#650212yes—
514C4—24no—
515C4—84no—
516C4—284no—
517C4—288no—
518C4—284no—
519C2—22no—
520C2—21no—
521C2—21no—
522C2—22no—
523C2—22no—
524C2—24no—
525C2—44no—
526C2#1172777714yesdirect product factor
527C2C2×#8951412yes—
528C2C2×#8973011yescharacteristic
subgroupG/H
centerC23#6445
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC22×C7×#99C2
socleC23×C7#120
prime radicalC23×C7×D4C22
cosocleC24×C7C24
hypercenterC22×#99D7
hypocenterC7C2×#1890
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
4. C2×#89715C2semidirect
5. C2×#89322C2semidirect
6. #11727777C2direct (N × H)
19. C2×#6498C4semidirect
28. #89715C22semidirect
29. #89322C22semidirect
50. C23×Dic7C2×C4semidirect
68. C2×C22⋊Dic7C2×C4semidirect
74. #6498C2×C4semidirect
148. C22×Dic7C22×C4semidirect
161. C22⋊Dic7C22×C4semidirect
419. C2×C7#1890semidirect
483. C7C2×#1890semidirect
526. C2#11727777direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64982
C2×#904892
C22×C7×#992
C2×#897152
C2×#893222
#117277772
C2×#18907