Finite Group Databasebeta

← C2×#11736808 #14226003

Subgroups — C2×#11736808 #14226003

Order 1792.

1678 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90210C211yesmaximal, characteristic
2C2×#90209C211yesmaximal, characteristic
3C2×C7×#433C211yesmaximal, characteristic
4#11736808C2116yesmaximal, direct product factor
5C2×#89601C211yesmaximal, characteristic, complemented
6C2×#89601C211yesmaximal, characteristic, complemented
7C2×#89622C211yesmaximal, characteristic, complemented
8C2×#89623C211yesmaximal, characteristic, complemented
9C2×#89494C211yesmaximal, characteristic
10C2×#89495C211yesmaximal, characteristic
11C2×#89495C211yesmaximal, characteristic
12C2×#89493C211yesmaximal, characteristic
13C2×#89475C211yesmaximal, characteristic, complemented
14C2×#89474C211yesmaximal, characteristic, complemented
15C2×#89453C211yesmaximal, characteristic, complemented
16C2×#89455C211yesmaximal, characteristic, complemented
17C2×C7×#117C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C2×#6487C2211yescharacteristic
22#90210—24no—
23#90210C2218yes—
24#90209—24no—
25#90209C2218yes—
26C2×#6453—21no—
27C2×#6452—21no—
28C2×#6452—21no—
29C2×#6451—21no—
30C2×#6442—21no—
31C2×#6442—21no—
32C2×#6441—21no—
33C2×#6441—21no—
34C2×#6433C2211yescharacteristic
35C2×#6433C2211yescharacteristic
36C2×#6433C2211yescharacteristic
37C2×#6432C2211yescharacteristic
38C2×#6432C2211yescharacteristic
39C2×#6432C2211yescharacteristic
40C2×C4×Dic14C2211yescharacteristic
41C2×C4×Dic14C2211yescharacteristic
42C7×#433C2218yes—
43C2×C7×C4⋊C8C2211yescharacteristic
44C2×C7×C4⋊C8C2211yescharacteristic
45C2×C7×C22⋊C8C2211yescharacteristic
46C2×C7×C22⋊C8C2211yescharacteristic
47C2×C7×#97C2211yescharacteristic
48C2×C4×C7×C8C2211yescharacteristic
49C2×#6392C2211yescharacteristic
50C2×#6392C2211yescharacteristic
51C2×#6392C2211yescharacteristic
52C2×#6392C2211yescharacteristic
53C2×#6391C2211yescharacteristic
54C2×#6391C2211yescharacteristic
55C2×#6391C2211yescharacteristic
56C2×#6391C2211yescharacteristic
57C2×#6390C2211yescharacteristic
58C2×#6390C2211yescharacteristic
59C2×#6390C2211yescharacteristic
60C2×#6390C2211yescharacteristic
61C2×#6390C2211yescharacteristic
62C2×#6390C2211yescharacteristic
63C2×#6390C2211yescharacteristic
64C2×#6390C2211yescharacteristic
65#89623C2218yescomplemented
66#89622C2218yescomplemented
67#89601C2218yescomplemented
68#89601C2218yescomplemented
69#89495C2218yes—
70#89495C2218yes—
71#89494C2218yes—
72#89493C2218yes—
73#89475C2218yescomplemented
74#89474C2218yescomplemented
75#89455C2218yescomplemented
76#89453C2218yescomplemented
77C2×#3242—71nomaximal, Sylow
78C23×C4×C7C2311yescharacteristic
79C23×Dic7—21no—
80C22×Dic14—21no—
81C22×Dic14—21no—
82C22×Dic14C2311yescharacteristic
83C22×Dic14C2311yescharacteristic
84C22×C7×C8C2311yescharacteristic
85C22×C7×C8C2311yescharacteristic
86C22×C7×C8C2311yescharacteristic
87C22×C7×C8C2311yescharacteristic
88C7×#117D414yes—
89C7×#117C2314yes—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C22⋊C4D411yescharacteristic
93C2×C7×C22⋊C4D411yescharacteristic
94C2×C42×C7C2311yescharacteristic
95C2×C42×C7C2311yescharacteristic
96C2×C22⋊Dic7—21no—
97C2×C22⋊Dic7—21no—
98C2×C22⋊Dic7—21no—
99#6489—22no—
100#6489C2314yes—
101C2×C4⋊Dic7—21no—
102C2×C4⋊Dic7—21no—
103C2×C4⋊Dic7—21no—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7—22no—
107C2×C4⋊Dic7C2311yescharacteristic
108C2×C4⋊Dic7C2311yescharacteristic
109C2×C4⋊Dic7C2311yescharacteristic
110C2×C4⋊Dic7C2311yescharacteristic
111C2×C4⋊Dic7C2311yescharacteristic
112C2×C4⋊Dic7C2311yescharacteristic
113C2×C4⋊Dic7C2314yes—
114#6487—22no—
115#6487—22no—
116#6487C2314yes—
117#6487C2314yes—
118C2×Dic7⋊C4—21no—
119C2×Dic7⋊C4—21no—
120C2×Dic7⋊C4—21no—
121C2×Dic7⋊C4—21no—
122C2×Dic7⋊C4—21no—
123C2×Dic7⋊C4—21no—
124C2×Dic7⋊C4—21no—
125C2×Dic7⋊C4—21no—
126C2×Dic7⋊C4—21no—
127C2×Dic7⋊C4—21no—
128C2×C4×Dic7—21no—
129C2×C4×Dic7—21no—
130C2×C4×Dic7—21no—
131#6453—28no—
132#6452—28no—
133#6452—28no—
134#6451—28no—
135#6442—28no—
136#6442—28no—
137#6441—28no—
138#6441—28no—
139#6433—22no—
140#6433—22no—
141#6433—22no—
142#6433C2314yes—
143#6433C2314yes—
144#6433C2314yes—
145#6432—22no—
146#6432—22no—
147#6432—22no—
148#6432C2314yes—
149#6432C2314yes—
150#6432C2314yes—
151C4×Dic14—22no—
152C4×Dic14—22no—
153C4×Dic14C2314yes—
154C4×Dic14C2314yes—
155C7×C4⋊C8C2314yes—
156C7×C4⋊C8C2314yes—
157C7×C22⋊C8C2314yes—
158C7×C22⋊C8C2314yes—
159C7×#97C2314yes—
160C4×C7×C8C2314yes—
161#6392C2314yes—
162#6392C2314yes—
163#6392C2314yes—
164#6392C2314yes—
165#6391C2314yes—
166#6391C2314yes—
167#6391C2314yes—
168#6391C2314yes—
169#6390C2314yes—
170#6390C2314yes—
171#6390C2314yes—
172#6390C2314yes—
173#6390C2314yes—
174#6390C2314yes—
175#6390C2314yes—
176#6390C2314yes—
177C2×#544—71no—
178C2×#541—71no—
179#3242—716no—
180C2×#501—71no—
181C2×#499—71no—
182C2×#491—71no—
183C2×#487—71no—
184C2×#484—71no—
185C2×#485—71no—
186C2×#485—71no—
187C2×#483—71no—
188C2×#479—71no—
189C2×#479—71no—
190C2×#477—71no—
191C2×#475—71no—
192C2×#433D711yescharacteristic
193C24×C7C2×D411yescharacteristic
194C22×C4×C7—22no—
195C22×C4×C7—22no—
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2×D411yescharacteristic
198C22×C4×C7C2×D411yescharacteristic
199C22×C4×C7C2×D411yescharacteristic
200C22×C4×C7C2×D411yescharacteristic
201C22×C4×C7C2×D412yes—
202C22×C4×C7C2412yes—
203C22×C4×C7C2411yescharacteristic
204C22×Dic7—21no—
205C22×Dic7—21no—
206C22×Dic7—21no—
207C22×Dic7—21no—
208C22×Dic7—21no—
209C22×Dic7—21no—
210C22×Dic7—21no—
211C22×Dic7—21no—
212C22×Dic7—24no—
213C22×Dic7—82no—
214C2×Dic14—22no—
215C2×Dic14—22no—
216C2×Dic14—22no—
217C2×Dic14—22no—
218C2×Dic14—21no—
219C2×Dic14—21no—
220C2×Dic14—24no—
221C2×Dic14—24no—
222C2×Dic14—82no—
223C2×Dic14—82no—
224C2×Dic14C2412yes—
225C2×Dic14C2412yes—
226C2×C7×C8—22no—
227C2×C7×C8—22no—
228C2×C7×C8—22no—
229C2×C7×C8—22no—
230C2×C7×C8C2412yes—
231C2×C7×C8C2412yes—
232C2×C7×C8C2412yes—
233C2×C7×C8C2412yes—
234C7×C4⋊C4C2×D414yes—
235C7×C4⋊C4C2×D414yes—
236C7×C22⋊C4C2×D414yes—
237C7×C22⋊C4C2×D414yes—
238C42×C7C2×D412yes—
239C42×C7C2×D412yes—
240C42×C7C2412yes—
241C42×C7C2412yes—
242C22⋊Dic7—24no—
243C22⋊Dic7—24no—
244C22⋊Dic7—24no—
245C4⋊Dic7—21no—
246C4⋊Dic7—21no—
247C4⋊Dic7—21no—
248C4⋊Dic7—21no—
249C4⋊Dic7—21no—
250C4⋊Dic7—24no—
251C4⋊Dic7—24no—
252C4⋊Dic7—24no—
253C4⋊Dic7—24no—
254C4⋊Dic7—21no—
255C4⋊Dic7—24no—
256C4⋊Dic7C2412yes—
257C4⋊Dic7C2412yes—
258C4⋊Dic7C2412yes—
259C4⋊Dic7C2412yes—
260C4⋊Dic7C2412yes—
261C4⋊Dic7C2412yes—
262Dic7⋊C4—24no—
263Dic7⋊C4—24no—
264Dic7⋊C4—24no—
265Dic7⋊C4—24no—
266Dic7⋊C4—24no—
267Dic7⋊C4—24no—
268Dic7⋊C4—24no—
269Dic7⋊C4—24no—
270Dic7⋊C4—24no—
271Dic7⋊C4—24no—
272C4×Dic7—24no—
273C4×Dic7—24no—
274C4×Dic7—24no—
275#544—78no—
276#544—144no—
277#541—78no—
278#541—144no—
279C2×C4⋊Q8—71no—
280C2×C4⋊Q8—71no—
281C2×C4⋊Q8—71no—
282C2×C4⋊Q8—141no—
283C2×#126—141no—
284C2×#126—141no—
285C2×#125—71no—
286C2×#125—71no—
287C2×#125—71no—
288C2×#125—141no—
289C2×#125—141no—
290C2×#125—141no—
291C2×C22⋊Q8—71no—
292C2×C22⋊Q8—71no—
293C2×C22⋊Q8—141no—
294C2×C22⋊Q8—141no—
295C2×C4×Q8—71no—
296C2×C4×Q8—71no—
297C2×#117—71no—
298C2×#117C2×D711yescharacteristic
299C22×C4⋊C4—71no—
300#501—78no—
301#499—78no—
302#491—78no—
303#487—78no—
304#485—78no—
305#485—78no—
306#484—78no—
307#483—78no—
308#479—78no—
309#479—78no—
310#477—78no—
311#475—78no—
312#433C2×D718yes—
313C2×#107—71no—
314C2×#107—71no—
315C2×#107—71no—
316C2×#107—71no—
317C2×#106—71no—
318C2×#106—71no—
319C2×#106—71no—
320C2×#106—71no—
321C2×C4⋊C8C2×D711yescharacteristic
322C2×C4⋊C8C2×D711yescharacteristic
323C2×#103—71no—
324C2×#103—71no—
325C2×#103—71no—
326C2×#103—71no—
327C2×#103—71no—
328C2×#103—71no—
329C2×#103—71no—
330C2×#103—71no—
331C2×C22⋊C8C2×D711yescharacteristic
332C2×C22⋊C8C2×D711yescharacteristic
333C2×#97C2×D711yescharacteristic
334C2×C4×C8C2×D711yescharacteristic
335C23×C7—22no—
336C23×C7—22no—
337C23×C7—22no—
338C23×C7C22×D412yes—
339C23×C7C22×D411yescharacteristic
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—22no—
344C2×C4×C7—22no—
345C2×C4×C7—24no—
346C2×C4×C7—22no—
347C2×C4×C7—22no—
348C2×C4×C7C22×D412yes—
349C2×C4×C7C22×D412yes—
350C2×C4×C7C22×D412yes—
351C2×C4×C7C22×D412yes—
352C2×C4×C7C22×D412yes—
353C2×C4×C7C22×D411yescharacteristic
354C2×C4×C72−1+412yes—
355C2×C4×C72−1+412yes—
356C2×C4×C7C2511yescharacteristic
357C2×Dic7—22no—
358C2×Dic7—22no—
359C2×Dic7—22no—
360C2×Dic7—22no—
361C2×Dic7—22no—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—22no—
365C2×Dic7—42no—
366C2×Dic7—42no—
367C2×Dic7—42no—
368C2×Dic7—42no—
369C2×Dic7—42no—
370C2×Dic7—42no—
371C2×Dic7—42no—
372C2×Dic7—42no—
373C2×Dic7—84no—
374Dic14—22no—
375Dic14—22no—
376Dic14—22no—
377Dic14—22no—
378Dic14—42no—
379Dic14—42no—
380Dic14—84no—
381Dic14—84no—
382C7×C8—22no—
383C7×C8—22no—
384C7×C8—22no—
385C7×C8—22no—
386C22×Q8—71no—
387C22×Q8—71no—
388C22×Q8—141no—
389C22×Q8—141no—
390C23×C4—141no—
391C23×C4C22×D711yescharacteristic
392C22×C8C22×D711yescharacteristic
393C22×C8C22×D711yescharacteristic
394C22×C8C22×D711yescharacteristic
395C22×C8C22×D711yescharacteristic
396C4⋊Q8—74no—
397C4⋊Q8—74no—
398C4⋊Q8—74no—
399C4⋊Q8—148no—
400C4⋊Q8—142no—
401C4⋊Q8—142no—
402C4⋊Q8—142no—
403#126—148no—
404#126—148no—
405#125—74no—
406#125—74no—
407#125—74no—
408#125—148no—
409#125—148no—
410#125—148no—
411#125—142no—
412#125—142no—
413#125—142no—
414C22⋊Q8—74no—
415C22⋊Q8—74no—
416C22⋊Q8—142no—
417C22⋊Q8—142no—
418C22⋊Q8—148no—
419C22⋊Q8—148no—
420C4×Q8—74no—
421C4×Q8—74no—
422C4×Q8—142no—
423C4×Q8—142no—
424#117—74no—
425#117—142no—
426#117D2814yes—
427#117C22×D714yes—
428C2×C4⋊C4—71no—
429C2×C4⋊C4—71no—
430C2×C4⋊C4—74no—
431C2×C4⋊C4—71no—
432C2×C4⋊C4—71no—
433C2×C4⋊C4—71no—
434C2×C4⋊C4—71no—
435C2×C4⋊C4—141no—
436C2×C4⋊C4—142no—
437C2×C4⋊C4—141no—
438C2×C4⋊C4—141no—
439C2×C4⋊C4—141no—
440C2×C4⋊C4—141no—
441C2×C4⋊C4—141no—
442C2×C4⋊C4—141no—
443C2×C4⋊C4—141no—
444C2×C4⋊C4—141no—
445C2×C4⋊C4—141no—
446C2×C4⋊C4—141no—
447C2×C4⋊C4—141no—
448C2×C4⋊C4—141no—
449C2×C4⋊C4—141no—
450C2×C4⋊C4—141no—
451C2×C4⋊C4D2811yescharacteristic
452C2×C4⋊C4D2811yescharacteristic
453C2×C22⋊C4—141no—
454C2×C22⋊C4—141no—
455C2×C22⋊C4—141no—
456C2×C22⋊C4D2811yescharacteristic
457C2×C22⋊C4D2811yescharacteristic
458C2×C42—141no—
459C2×C42—141no—
460C2×C42—141no—
461C2×C42C22×D711yescharacteristic
462C2×C42C22×D711yescharacteristic
463#107—74no—
464#107—74no—
465#107—74no—
466#107—74no—
467#106—74no—
468#106—74no—
469#106—74no—
470#106—74no—
471C4⋊C8C22×D714yes—
472C4⋊C8C22×D714yes—
473#103—74no—
474#103—74no—
475#103—74no—
476#103—74no—
477#103—74no—
478#103—74no—
479#103—74no—
480#103—74no—
481C22⋊C8C22×D714yes—
482C22⋊C8C22×D714yes—
483#97C22×D714yes—
484C4×C8C22×D714yes—
485C22×C7—22no—
486C22×C7—22no—
487C22×C7—22no—
488C22×C7—44no—
489C22×C7#53612yes—
490C22×C7#57712yes—
491C22×C7#57812yes—
492C22×C7C23×D411yescharacteristic
493C4×C7—22no—
494C4×C7—22no—
495C4×C7—22no—
496C4×C7—22no—
497C4×C7—22no—
498C4×C7#53612yes—
499C4×C7C2×2−1+411yescharacteristic
500C4×C7C2×2−1+411yescharacteristic
501Dic7—42no—
502Dic7—42no—
503Dic7—42no—
504Dic7—42no—
505Dic7—42no—
506Dic7—42no—
507Dic7—42no—
508Dic7—42no—
509C24C2×D2811yescharacteristic
510C2×Q8—72no—
511C2×Q8—72no—
512C2×Q8—141no—
513C2×Q8—141no—
514C2×Q8—144no—
515C2×Q8—144no—
516C2×Q8—142no—
517C2×Q8—142no—
518C2×Q8—142no—
519C2×Q8—142no—
520C2×Q8—562no—
521C2×Q8—562no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—141no—
525C22×C4—141no—
526C22×C4—141no—
527C22×C4—141no—
528C22×C4—141no—
529C22×C4—141no—
530C22×C4—141no—
531C22×C4—141no—
532C22×C4—144no—
533C22×C4—562no—
534C22×C4C2×D2812yes—
535C22×C4C2×D2811yescharacteristic
536C22×C4C2×D2811yescharacteristic
537C22×C4C2×D2811yescharacteristic
538C22×C4C2×D2811yescharacteristic
539C22×C4C2×D2811yescharacteristic
540C22×C4C23×D712yes—
541C22×C4C23×D711yescharacteristic
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8C23×D712yes—
547C2×C8C23×D712yes—
548C2×C8C23×D712yes—
549C2×C8C23×D712yes—
550C4⋊C4—72no—
551C4⋊C4—72no—
552C4⋊C4—72no—
553C4⋊C4—72no—
554C4⋊C4—72no—
555C4⋊C4—72no—
556C4⋊C4—144no—
557C4⋊C4—144no—
558C4⋊C4—144no—
559C4⋊C4—144no—
560C4⋊C4—144no—
561C4⋊C4—144no—
562C4⋊C4—144no—
563C4⋊C4—144no—
564C4⋊C4—144no—
565C4⋊C4—144no—
566C4⋊C4—144no—
567C4⋊C4—141no—
568C4⋊C4—141no—
569C4⋊C4—141no—
570C4⋊C4—141no—
571C4⋊C4—141no—
572C4⋊C4—141no—
573C4⋊C4—144no—
574C4⋊C4—144no—
575C4⋊C4—144no—
576C4⋊C4—144no—
577C4⋊C4C2×D2814yes—
578C4⋊C4C2×D2814yes—
579C22⋊C4—144no—
580C22⋊C4—144no—
581C22⋊C4—144no—
582C22⋊C4C2×D2814yes—
583C22⋊C4C2×D2814yes—
584C42—144no—
585C42—144no—
586C42—144no—
587C42C2×D2812yes—
588C42C2×D2812yes—
589C42C23×D712yes—
590C42C23×D712yes—
591C2×C7—42no—
592C2×C7#324214yescomplemented
593C2×C7C2×#53611yescharacteristic
594C2×C7C2×#57711yescharacteristic
595C2×C7C2×#57811yescharacteristic
596C23—22no—
597C23—22no—
598C23—22no—
599C23C22×D2812yes—
600C23C22×D2811yescharacteristic
601Q8—142no—
602Q8—142no—
603Q8—142no—
604Q8—142no—
605Q8—282no—
606Q8—282no—
607Q8—564no—
608Q8—564no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—142no—
618C2×C4—142no—
619C2×C4—142no—
620C2×C4—142no—
621C2×C4—142no—
622C2×C4—142no—
623C2×C4—142no—
624C2×C4—142no—
625C2×C4—282no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—564no—
634C2×C4C22×D2812yes—
635C2×C4C22×D2812yes—
636C2×C4C22×D2812yes—
637C2×C4C22×D2812yes—
638C2×C4C22×D2812yes—
639C2×C4C22×D2811yescharacteristic
640C2×C4#655412yes—
641C2×C4#655412yes—
642C2×C4C24×D711yescharacteristic
643C8—22no—
644C8—22no—
645C8—22no—
646C8—22no—
647C7C2×#324211yescharacteristic, complemented, Sylow
648C22—22no—
649C22—22no—
650C22—22no—
651C22—44no—
652C22#9021712yes—
653C22#9044212yes—
654C22#9044412yes—
655C22C23×D2811yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—282no—
662C4—282no—
663C4—282no—
664C4—282no—
665C4—282no—
666C4—282no—
667C4—282no—
668C4—282no—
669C4#9021712yes—
670C4C2×#655411yescharacteristic
671C4C2×#655411yescharacteristic
672C2—42no—
673C2#1173680814yesdirect product factor
674C2C2×#9021711yescharacteristic
675C2C2×#9044211yescharacteristic
676C2C2×#9044411yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#433C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#433D7
hypocenterC7C2×#3242
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
4. #11736808C2direct (N × H)
5. C2×#89601C2semidirect
6. C2×#89601C2semidirect
7. C2×#89622C2semidirect
8. C2×#89623C2semidirect
13. C2×#89475C2semidirect
14. C2×#89474C2semidirect
15. C2×#89453C2semidirect
16. C2×#89455C2semidirect
65. #89623C22semidirect
66. #89622C22semidirect
67. #89601C22semidirect
68. #89601C22semidirect
73. #89475C22semidirect
74. #89474C22semidirect
75. #89455C22semidirect
76. #89453C22semidirect
592. C2×C7#3242semidirect
647. C7C2×#3242semidirect
673. C2#11736808direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902102
C2×#902092
C2×C7×#4332
#117368082
C2×#896012
C2×#896222
C2×#896232
C2×#894942
C2×#894952
C2×#894932
C2×#894752
C2×#894742
C2×#894532
C2×#894552
C2×#32427