Finite Group Databasebeta

← C2×#11728018 #13381672

Subgroups — C2×#11728018 #13381672

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×#6505C211yesmaximal, characteristic
2C22×C7×#103C211yesmaximal, characteristic, complemented
3C22×#6410C211yesmaximal, characteristic, complemented
4C2×#89882C211yesmaximal, characteristic
5C22×#6393C211yesmaximal, characteristic, complemented
6C2×#89738C211yesmaximal, characteristic
7C2×#89325C211yesmaximal, characteristic, complemented
8#11728018C218yesmaximal, direct product factor
9C23×C7×Q8C2211yescharacteristic
10C22×#1157C2211yescharacteristic
11C23×C7×C8C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×#1144—21no—
14C2×#6505—24no—
15C2×#6505—22no—
16C2×#6505—24no—
17C2×#6505C2214yes—
18C22×#1143—21no—
19C22×#1136C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C2×C7×#103—22no—
23C2×C7×#103—22no—
24C2×C7×#103C2214yescomplemented
25C2×#6410—24no—
26C2×#6410C2214yescomplemented
27#89882—22no—
28#89882C2214yes—
29C2×(C2×D7)⋊C8—22no—
30C2×#6393—24no—
31C2×#6393C2214yescomplemented
32C2×#6384—21no—
33#89738—22no—
34#89738C2214yes—
35C2×#6383—21no—
36#89325C2214yescomplemented
37C2×#2052—71nomaximal, Sylow
38C22×C7×Q8—22no—
39C22×C7×Q8—21no—
40C22×C7×Q8—21no—
41C22×C7×Q8—21no—
42C22×C7×Q8—22no—
43C22×C7×Q8—21no—
44C22×C7×Q8—21no—
45C22×C7×Q8—22no—
46C22×C7×Q8D414yes—
47C22×C7×Q8C2312yes—
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—21no—
52C2×#1157—21no—
53C2×#1157—21no—
54C2×#1157—22no—
55C2×#1157—22no—
56C2×#1157—22no—
57C2×#1157—22no—
58C2×#1157D412yes—
59C2×#1157C2312yes—
60C22×D28D411yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14D411yescharacteristic
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8D412yes—
66C22×C7×C8D412yes—
67C22×C7×C8C2312yes—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2312yes—
74C2×#1144—28no—
75C2×#1144—24no—
76#6505—216no—
77#6505—28no—
78#6505—28no—
79C2×#1143—28no—
80C2×#1143—24no—
81C2×#1136—22no—
82C2×#1136—21no—
83C2×#1136—21no—
84C2×#1136—21no—
85C2×#1136—22no—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136D414yes—
90C2×#1136C2312yes—
91C2×#1136C2312yes—
92C2×#1136C2312yes—
93C22×C7⋊C8—21no—
94C22×C7⋊C8—21no—
95C22×C7⋊C8—21no—
96C7×#103—28no—
97#6410—28no—
98(C2×D7)⋊C8—28no—
99#6393—28no—
100#6384—24no—
101#6383—24no—
102C22×#137—71no—
103C22×#108—71no—
104C2×#417—71no—
105C22×#103D711yescharacteristic, complemented
106C22×#101—71no—
107C2×#408—71no—
108#2052—78no—
109C2×#343—71no—
110C24×C7#4111yescharacteristic
111C23×D7—21no—
112C2×C7×Q8—24no—
113C2×C7×Q8—22no—
114C2×C7×Q8—24no—
115C2×C7×Q8—24no—
116C2×C7×Q8—24no—
117C2×C7×Q8—24no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—41no—
123C2×C7×Q8—41no—
124C2×C7×Q8—42no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×Q8C2×D414yes—
130C22×C4×C7—24no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—41no—
134C22×C4×C7—41no—
135C22×C4×C7—42no—
136C22×C4×C7—41no—
137C22×C4×C7—41no—
138C22×C4×C7—82no—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7#4112yes—
146C22×C4×C7#4111yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C2×C22⋊D7—22no—
150C2×C22⋊D7—22no—
151C2×C22⋊D7—42no—
152C2×C22⋊D7—42no—
153C22×Dic7—21no—
154#1157—24no—
155#1157—24no—
156#1157—22no—
157#1157—22no—
158#1157—22no—
159#1157—22no—
160#1157—42no—
161#1157—42no—
162#1157—42no—
163#1157—42no—
164C2×D28—22no—
165C2×D28—22no—
166C2×D28—21no—
167C2×D28C2×D412yes—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C2×Dic14—22no—
173C2×Dic14—22no—
174C2×Dic14—21no—
175C2×Dic14C2×D412yes—
176C2×C7×C8—24no—
177C2×C7×C8—24no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8C2×D414yes—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—41no—
184C7×C4⋊C4—41no—
185#1144—216no—
186#1143—216no—
187#1136—24no—
188#1136—22no—
189#1136—22no—
190#1136—22no—
191#1136—22no—
192#1136—22no—
193#1136—42no—
194#1136—42no—
195#1136C2×D414yes—
196C2×C7⋊C8—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—42no—
201C2×C7⋊C8—42no—
202C22×#41—71no—
203C23×Q8C2×D711yescharacteristic
204C2×#137—74no—
205C2×#137—144no—
206C2×#137—142no—
207C2×#137—144no—
208C22×Q16—141no—
209C22×SD16—141no—
210C22×M16—71no—
211C22×M16—71no—
212C22×M16—71no—
213C23×C8C2×D711yescharacteristic
214C22×C4⋊C4C2×D711yescharacteristic
215C2×#108—74no—
216C2×#108—144no—
217#417—74no—
218#417—142no—
219C2×#103—22no—
220C2×#103—22no—
221C2×#103—141no—
222C2×#103C2×D714yescomplemented
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#12111yescharacteristic
233C23×C7#12111yescharacteristic
234C23×C7#12312yes—
235C23×C7#12412yes—
236C23×C7C2×#4111yescharacteristic
237C22×D7—22no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—42no—
241C7×Q8—24no—
242C7×Q8—24no—
243C7×Q8—42no—
244C7×Q8—42no—
245C7×Q8—44no—
246C7×Q8—44no—
247C2×C4×C7—24no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—44no—
251C2×C4×C7—44no—
252C2×C4×C7—42no—
253C2×C4×C7—82no—
254C2×C4×C7—82no—
255C2×C4×C7—82no—
256C2×C4×C7#12012yes—
257C2×C4×C7#12012yes—
258C2×C4×C7#12112yes—
259C2×C4×C7#12112yes—
260C2×C4×C7#12111yescharacteristic
261C2×C4×C7#12111yescharacteristic
262C2×C4×C7#12112yes—
263C2×C4×C7#12112yes—
264C2×C4×C7#12312yes—
265C2×C4×C7#12312yes—
266C2×C4×C7#12312yes—
267C2×C4×C7#12412yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C22⋊D7—42no—
275C22⋊D7—42no—
276C22⋊D7—44no—
277C2×Dic7—22no—
278C2×Dic7—42no—
279D28—22no—
280D28—22no—
281D28—42no—
282C4×D7—42no—
283C4×D7—42no—
284C4×D7—44no—
285Dic14—22no—
286Dic14—22no—
287Dic14—42no—
288C7×C8—24no—
289C7⋊C8—24no—
290C7⋊C8—42no—
291C7⋊C8—42no—
292C2×#41—72no—
293C2×#41—72no—
294C2×#41—141no—
295C2×#41—141no—
296C2×#41—142no—
297C2×#41—142no—
298C2×#41—142no—
299C2×#41—142no—
300C22×Q8—22no—
301C22×Q8—21no—
302C22×Q8—22no—
303C22×Q8—21no—
304C22×Q8—22no—
305C22×Q8—21no—
306C22×Q8—21no—
307C22×Q8—21no—
308C22×Q8—71no—
309C22×Q8D2814yes—
310C22×Q8C22×D712yes—
311C22×D4—71no—
312C22×D4—141no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317#137—148no—
318#137—1416no—
319#137—148no—
320C2×Q16—144no—
321C2×Q16—148no—
322C2×SD16—144no—
323C2×SD16—148no—
324C2×M16—74no—
325C2×M16—72no—
326C2×M16—72no—
327C2×M16—72no—
328C2×M16—142no—
329C2×M16—141no—
330C2×M16—141no—
331C2×M16—141no—
332C2×M16—142no—
333C2×M16—142no—
334C2×M16—142no—
335C2×M16—142no—
336C22×C8—22no—
337C22×C8—22no—
338C22×C8—141no—
339C22×C8—141no—
340C22×C8—141no—
341C22×C8C22⋊D712yes—
342C22×C8C22⋊D712yes—
343C22×C8C22×D712yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22×D712yes—
350#108—148no—
351#103—28no—
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#39412yes—
363C22×C7#39412yes—
364C22×C7#47112yes—
365C22×C7#47312yes—
366C22×C7C2×#12111yescharacteristic
367C22×C7C2×#12311yescharacteristic
368C22×C7C2×#12411yescharacteristic
369C2×D7—42no—
370C2×D7—42no—
371C2×D7—42no—
372C2×D7—84no—
373C4×C7—44no—
374C4×C7—82no—
375C4×C7#39412yes—
376C4×C7#39412yes—
377C4×C7C2×#12011yescharacteristic
378C4×C7C2×#12111yescharacteristic
379C4×C7C2×#12111yescharacteristic
380C4×C7C2×#12311yescharacteristic
381Dic7—42no—
382C24—141no—
383C24#115911yescharacteristic
384#41—144no—
385#41—144no—
386#41—142no—
387#41—142no—
388#41—142no—
389#41—142no—
390#41—282no—
391#41—282no—
392#41—282no—
393#41—282no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—72no—
412C2×Q8—141no—
413C2×Q8—142no—
414C2×Q8—142no—
415C2×Q8C2×D2814yes—
416C2×D4—72no—
417C2×D4—142no—
418C2×D4—142no—
419C2×D4—142no—
420C2×D4—142no—
421C2×D4—141no—
422C2×D4—282no—
423C2×D4—282no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—82no—
433C22×C4—141no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4C2×D2812yes—
439C22×C4#115712yes—
440C22×C4D4×D711yescharacteristic
441C22×C4D4×D711yescharacteristic
442C22×C4D4×D712yes—
443C22×C4#115912yes—
444C22×C4#115911yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C23×D711yescharacteristic
448Q16—1416no—
449SD16—1416no—
450M16—74no—
451M16—144no—
452M16—142no—
453M16—142no—
454M16—142no—
455M16—142no—
456M16—142no—
457M16—282no—
458M16—282no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—144no—
464C2×C8—142no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8—282no—
468C2×C8—282no—
469C2×C8C2×C22⋊D714yes—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C2×C7—21no—
475C2×C7—21no—
476C2×C7—22no—
477C2×C7#205214yescomplemented
478C2×C7C2×#39411yescharacteristic
479C2×C7C2×#47111yescharacteristic
480C2×C7C2×#47311yescharacteristic
481D7—82no—
482C23—22no—
483C23—22no—
484C23—142no—
485C23—282no—
486C23—282no—
487C23—282no—
488C23#644812yes—
489C23#644912yes—
490C23#650111yescharacteristic
491C23#650111yescharacteristic
492C23C2×#115911yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497Q8—44no—
498Q8—44no—
499Q8—142no—
500Q8—142no—
501Q8—282no—
502D4—142no—
503D4—142no—
504D4—282no—
505D4—282no—
506D4—284no—
507D4—282no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—142no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—284no—
522C2×C4#644512yes—
523C2×C4#644512yes—
524C2×C4#644612yes—
525C2×C4#644612yes—
526C2×C4#644812yes—
527C2×C4#644912yes—
528C2×C4#649312yes—
529C2×C4#649312yes—
530C2×C4#650111yescharacteristic
531C2×C4#650111yescharacteristic
532C2×C4#650212yes—
533C2×C4#650212yes—
534C2×C4C22×D2811yescharacteristic
535C2×C4C2×#115711yescharacteristic
536C2×C4C2×D4×D711yescharacteristic
537C2×C4C2×D4×D711yescharacteristic
538C2×C4C2×#115911yescharacteristic
539C2×C4C22×C22⋊D711yescharacteristic
540C8—24no—
541C8—144no—
542C8—282no—
543C8—282no—
544C7C2×#205211yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—24no—
548C22—24no—
549C22—21no—
550C22—21no—
551C22—282no—
552C22—282no—
553C22—282no—
554C22—564no—
555C22#8973312yes—
556C22#8973312yes—
557C22#8994912yes—
558C22#8996412yes—
559C22C2×#644811yescharacteristic
560C22C2×#644911yescharacteristic
561C22C2×#650111yescharacteristic
562C4—44no—
563C4—82no—
564C4—282no—
565C4#8973312yes—
566C4#8973312yes—
567C4C2×#644511yescharacteristic
568C4C2×#644611yescharacteristic
569C4C2×#649311yescharacteristic
570C4C2×#650211yescharacteristic
571C2—21no—
572C2—21no—
573C2—22no—
574C2—562no—
575C2#1172801814yesdirect product factor
576C2C2×#8973311yescharacteristic
577C2C2×#8994911yescharacteristic
578C2C2×#8996411yescharacteristic
subgroupG/H
centerC23#6501
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#103C2
socleC23×C7#121
prime radicalC22×#1157C22
hypercenterC22×#103D7
hypocenterC7C2×#2052
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×C7×#103C2semidirect
3. C22×#6410C2semidirect
5. C22×#6393C2semidirect
7. C2×#89325C2semidirect
8. #11728018C2direct (N × H)
24. C2×C7×#103C22semidirect
26. C2×#6410C22semidirect
31. C2×#6393C22semidirect
36. #89325C22semidirect
105. C22×#103D7semidirect
222. C2×#103C2×D7semidirect
477. C2×C7#2052semidirect
544. C7C2×#2052semidirect
575. C2#11728018direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#65052
C22×C7×#1032
C22×#64102
C2×#898822
C22×#63932
C2×#897382
C2×#893252
#117280182
C2×#20527