Finite Group Databasebeta

← C2×#11729859 #13381668

Subgroups — C2×#11729859 #13381668

Order 1792.

1362 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#137C211yesmaximal, characteristic
2C2×#90006C211yesmaximal, characteristic
3C2×#90004C211yesmaximal, characteristic
4C22×#6410C211yesmaximal, characteristic, complemented
5C22×#6409C211yesmaximal, characteristic, complemented
6C22×#6396C211yesmaximal, characteristic, complemented
7#11729859C218yesmaximal, direct product factor
8C2×#89349C211yesmaximal, characteristic, complemented
9C22×C7×#41C2211yescharacteristic
10C23×C7×Q8C2211yescharacteristic
11C2×C7×#137—22no—
12C2×C7×#137—24no—
13C2×C7×#137—24no—
14C2×C7×#137C2214yes—
15C22×C7×Q16—21no—
16C22×C7×SD16—21no—
17C22×C7×M16C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C23×C7⋊C8C2211yescharacteristic
22#90006—22no—
23#90006C2214yes—
24#90004—22no—
25#90004C2214yes—
26C2×#6410—24no—
27C2×#6410C2214yescomplemented
28C2×#6409—21no—
29C2×#6409—22no—
30C2×#6409—22no—
31C2×#6409C2214yescomplemented
32C2×#6406—21no—
33C2×C22⋊(C7⋊C8)—22no—
34C2×#6396—24no—
35C2×#6396C2214yescomplemented
36#89349C2214yescomplemented
37C2×#2052—71nomaximal, Sylow
38C2×C7×#41—21no—
39C2×C7×#41—21no—
40C2×C7×#41—22no—
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41D412yes—
45C2×C7×#41C2312yes—
46C22×C7×Q8—22no—
47C22×C7×Q8—21no—
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8—21no—
51C22×C7×Q8—21no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8D414yes—
55C22×C7×Q8D411yescharacteristic
56C22×C7×Q8C2312yes—
57C22×C7×D4—21no—
58C22×C7×D4D411yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C7×#137—216no—
64C7×#137—28no—
65C7×#137—28no—
66C2×C7×Q16—28no—
67C2×C7×Q16—24no—
68C2×C7×SD16—28no—
69C2×C7×SD16—24no—
70C2×C7×M16—22no—
71C2×C7×M16—21no—
72C2×C7×M16D414yes—
73C2×C7×M16C2312yes—
74C22×C7×C8—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2312yes—
81C2×#1136—21no—
82C2×#1136—21no—
83C2×#1136—22no—
84C2×#1136—22no—
85C2×#1136—22no—
86C2×#1136—22no—
87C2×#1136C2312yes—
88C2×#1136C2312yes—
89C22×C7⋊C8—22no—
90C22×C7⋊C8—22no—
91C22×C7⋊C8—21no—
92C22×C7⋊C8—21no—
93C22×C7⋊C8D412yes—
94C22×C7⋊C8D412yes—
95C22×C7⋊C8C2312yes—
96#6410—28no—
97#6409—24no—
98#6409—28no—
99#6406—24no—
100C22⋊(C7⋊C8)—28no—
101#6396—28no—
102C22×#137D711yescharacteristic
103C22×#108—71no—
104C2×#417—71no—
105C22×#103—71no—
106C22×#101—71no—
107C2×#408—71no—
108#2052—78no—
109C2×#343—71no—
110C24×C7—21no—
111C24×C7#4111yescharacteristic
112C7×#41—24no—
113C7×#41—24no—
114C7×#41—22no—
115C7×#41—22no—
116C7×#41—22no—
117C7×#41—22no—
118C7×#41—42no—
119C7×#41—42no—
120C7×#41—42no—
121C7×#41—42no—
122C2×C7×Q8—21no—
123C2×C7×Q8—24no—
124C2×C7×Q8—22no—
125C2×C7×Q8—24no—
126C2×C7×Q8—22no—
127C2×C7×Q8—24no—
128C2×C7×Q8—24no—
129C2×C7×Q8—24no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8—41no—
136C2×C7×Q8—41no—
137C2×C7×Q8—42no—
138C2×C7×Q8—42no—
139C2×C7×Q8—42no—
140C2×C7×Q8—42no—
141C2×C7×Q8—42no—
142C2×C7×Q8C2×D412yes—
143C2×C7×Q8C2×D414yes—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—21no—
148C2×C7×D4—22no—
149C2×C7×D4—42no—
150C2×C7×D4—42no—
151C2×C7×D4C2×D412yes—
152C22×C4×C7—21no—
153C22×C4×C7—24no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—41no—
158C22×C4×C7—41no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7#4112yes—
169C22×C4×C7#4111yescharacteristic
170C22×C4×C7#4112yes—
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—22no—
173C22×Dic7—41no—
174C22×Dic7—41no—
175C22×Dic7—82no—
176C7×Q16—216no—
177C7×SD16—216no—
178C7×M16—24no—
179C7×M16—22no—
180C7×M16C2×D414yes—
181C2×C7×C8—24no—
182C2×C7×C8—22no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7—41no—
186C4⋊Dic7—41no—
187#1136—22no—
188#1136—22no—
189#1136—22no—
190#1136—22no—
191#1136—42no—
192#1136—42no—
193C2×C7⋊C8—24no—
194C2×C7⋊C8—24no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—42no—
200C2×C7⋊C8—42no—
201C2×C7⋊C8C2×D414yes—
202C22×#41C2×D711yescharacteristic
203C23×Q8C2×D711yescharacteristic
204C2×#137—22no—
205C2×#137—24no—
206C2×#137—24no—
207C2×#137C2×D714yes—
208C22×Q16—21no—
209C22×SD16—21no—
210C22×M16—71no—
211C22×M16—71no—
212C22×M16C2×D711yescharacteristic
213C23×C8—71no—
214C22×C4⋊C4—71no—
215C2×#108—74no—
216C2×#108—144no—
217#417—74no—
218#417—142no—
219C2×#103—74no—
220C2×#103—142no—
221C2×#103—142no—
222C2×#103—141no—
223C2×#102—141no—
224C2×#101—74no—
225C2×#101—144no—
226#408—74no—
227#408—142no—
228C2×C22⋊C8—142no—
229#343—74no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7#12111yescharacteristic
237C23×C7#12111yescharacteristic
238C23×C7#12312yes—
239C23×C7#12412yes—
240C23×C7C2×#4111yescharacteristic
241C7×Q8—24no—
242C7×Q8—22no—
243C7×Q8—24no—
244C7×Q8—22no—
245C7×Q8—42no—
246C7×Q8—42no—
247C7×Q8—44no—
248C7×Q8—44no—
249C7×Q8—42no—
250C7×D4—22no—
251C7×D4—22no—
252C7×D4—42no—
253C7×D4—42no—
254C7×D4—44no—
255C7×D4—42no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—44no—
262C2×C4×C7—44no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7#12012yes—
267C2×C4×C7#12012yes—
268C2×C4×C7#12112yes—
269C2×C4×C7#12112yes—
270C2×C4×C7#12112yes—
271C2×C4×C7#12112yes—
272C2×C4×C7#12111yescharacteristic
273C2×C4×C7#12111yescharacteristic
274C2×C4×C7#12312yes—
275C2×C4×C7#12312yes—
276C2×C4×C7#12312yes—
277C2×C4×C7#12412yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×Dic7—42no—
285C2×Dic7—82no—
286C2×Dic7—82no—
287C2×Dic7—82no—
288C7×C8—24no—
289C7⋊C8—24no—
290C7⋊C8—42no—
291C7⋊C8—42no—
292C2×#41—21no—
293C2×#41—21no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41C22⋊D712yes—
299C2×#41C22×D712yes—
300C22×Q8—22no—
301C22×Q8—21no—
302C22×Q8—21no—
303C22×Q8—21no—
304C22×Q8—22no—
305C22×Q8—21no—
306C22×Q8—22no—
307C22×Q8—21no—
308C22×Q8C22⋊D711yescharacteristic
309C22×Q8C22⋊D714yes—
310C22×Q8C22×D712yes—
311C22×D4—21no—
312C22×D4C22⋊D711yescharacteristic
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317#137—216no—
318#137—28no—
319#137—28no—
320C2×Q16—28no—
321C2×Q16—24no—
322C2×SD16—28no—
323C2×SD16—24no—
324C2×M16—22no—
325C2×M16—21no—
326C2×M16—72no—
327C2×M16—72no—
328C2×M16—141no—
329C2×M16—141no—
330C2×M16—142no—
331C2×M16—142no—
332C2×M16—142no—
333C2×M16—142no—
334C2×M16C22⋊D714yes—
335C2×M16C22×D712yes—
336C22×C8—21no—
337C22×C8—72no—
338C22×C8—72no—
339C22×C8—72no—
340C22×C8—142no—
341C22×C8—142no—
342C22×C8—141no—
343C22×C8—141no—
344C2×C4⋊C4—72no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350#108—148no—
351#103—148no—
352#103—144no—
353#102—144no—
354#101—148no—
355C22⋊C8—148no—
356C22×C7—24no—
357C22×C7—24no—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7—21no—
361C22×C7—21no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—84no—
366C22×C7#39412yes—
367C22×C7#39412yes—
368C22×C7#47112yes—
369C22×C7#47312yes—
370C22×C7C2×#12111yescharacteristic
371C22×C7C2×#12311yescharacteristic
372C22×C7C2×#12411yescharacteristic
373C4×C7—42no—
374C4×C7—44no—
375C4×C7#39412yes—
376C4×C7#39412yes—
377C4×C7C2×#12011yescharacteristic
378C4×C7C2×#12111yescharacteristic
379C4×C7C2×#12111yescharacteristic
380C4×C7C2×#12311yescharacteristic
381Dic7—82no—
382C24—21no—
383C24#115911yescharacteristic
384#41—24no—
385#41—24no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—42no—
391#41—42no—
392#41—42no—
393#41—42no—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8C2×C22⋊D714yes—
415C2×Q8C2×C22⋊D712yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×C22⋊D712yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—142no—
435C22×C4—281no—
436C22×C4—281no—
437C22×C4—562no—
438C22×C4D4×D711yescharacteristic
439C22×C4D4×D711yescharacteristic
440C22×C4#115912yes—
441C22×C4#115911yescharacteristic
442C22×C4#115912yes—
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D712yes—
446C22×C4C2×C22⋊D712yes—
447C22×C4C23×D711yescharacteristic
448Q16—216no—
449SD16—216no—
450M16—24no—
451M16—22no—
452M16—142no—
453M16—142no—
454M16—142no—
455M16—142no—
456M16—282no—
457M16—282no—
458M16C2×C22⋊D714yes—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—74no—
462C2×C8—144no—
463C2×C8—144no—
464C2×C8—142no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8—142no—
468C2×C8—282no—
469C2×C8—282no—
470C4⋊C4—142no—
471C4⋊C4—142no—
472C4⋊C4—281no—
473C4⋊C4—281no—
474C2×C7—21no—
475C2×C7—21no—
476C2×C7—22no—
477C2×C7—82no—
478C2×C7#205214yescomplemented
479C2×C7C2×#39411yescharacteristic
480C2×C7C2×#47111yescharacteristic
481C2×C7C2×#47311yescharacteristic
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#649812yes—
489C23#649912yes—
490C23#650211yescharacteristic
491C23#650211yescharacteristic
492C23C2×#115911yescharacteristic
493Q8—22no—
494Q8—24no—
495Q8—24no—
496Q8—22no—
497Q8—42no—
498Q8—42no—
499Q8—44no—
500Q8—44no—
501Q8—42no—
502D4—22no—
503D4—22no—
504D4—42no—
505D4—42no—
506D4—44no—
507D4—42no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—282no—
519C2×C4—562no—
520C2×C4—562no—
521C2×C4—562no—
522C2×C4#649812yes—
523C2×C4#649812yes—
524C2×C4#649812yes—
525C2×C4#649912yes—
526C2×C4#650112yes—
527C2×C4#650112yes—
528C2×C4#650212yes—
529C2×C4#650212yes—
530C2×C4#650211yescharacteristic
531C2×C4#650211yescharacteristic
532C2×C4#651612yes—
533C2×C4#651612yes—
534C2×C4C2×D4×D711yescharacteristic
535C2×C4C2×#115911yescharacteristic
536C2×C4C2×#115911yescharacteristic
537C2×C4C22×C22⋊D711yescharacteristic
538C2×C4C22×C22⋊D711yescharacteristic
539C2×C4C22×C22⋊D711yescharacteristic
540C8—24no—
541C8—144no—
542C8—282no—
543C8—282no—
544C7C2×#205211yescharacteristic, complemented, Sylow
545C22—24no—
546C22—24no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—84no—
555C22#8997512yes—
556C22#8997612yes—
557C22#8999412yes—
558C22#8999412yes—
559C22C2×#649811yescharacteristic
560C22C2×#649911yescharacteristic
561C22C2×#650211yescharacteristic
562C4—42no—
563C4—44no—
564C4—562no—
565C4#8999412yes—
566C4#8999412yes—
567C4C2×#649811yescharacteristic
568C4C2×#650111yescharacteristic
569C4C2×#650211yescharacteristic
570C4C2×#651611yescharacteristic
571C2—21no—
572C2—21no—
573C2—22no—
574C2—82no—
575C2#1172985914yesdirect product factor
576C2C2×#8997511yescharacteristic
577C2C2×#8997611yescharacteristic
578C2C2×#8999411yescharacteristic
subgroupG/H
centerC23#6502
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#137C2
socleC23×C7#121
prime radicalC22×C7×#41C22
hypercenterC22×#137D7
hypocenterC7C2×#2052
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
4. C22×#6410C2semidirect
5. C22×#6409C2semidirect
6. C22×#6396C2semidirect
7. #11729859C2direct (N × H)
8. C2×#89349C2semidirect
27. C2×#6410C22semidirect
31. C2×#6409C22semidirect
35. C2×#6396C22semidirect
36. #89349C22semidirect
478. C2×C7#2052semidirect
544. C7C2×#2052semidirect
575. C2#11729859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C2×#900062
C2×#900042
C22×#64102
C22×#64092
C22×#63962
#117298592
C2×#893492
C2×#20527