Finite Group Databasebeta

← C2×#11738950 #14226007

Subgroups — C2×#11738950 #14226007

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×#90316C211yesmaximal, characteristic
2C2×#90210C211yesmaximal, characteristic
3C2×C7×#485C211yesmaximal, characteristic
4C2×#89816C211yesmaximal, characteristic
5C2×#89768C211yesmaximal, characteristic
6#11738950C2116yesmaximal, direct product factor
7C2×#89647C211yesmaximal, characteristic, complemented
8C2×#89648C211yesmaximal, characteristic, complemented
9C2×#89629C211yesmaximal, characteristic, complemented
10C2×#89630C211yesmaximal, characteristic, complemented
11C2×#89569C211yesmaximal, characteristic, complemented
12C2×#89564C211yesmaximal, characteristic, complemented
13C2×#89571C211yesmaximal, characteristic, complemented
14C2×#89563C211yesmaximal, characteristic, complemented
15C2×#89493C211yesmaximal, characteristic
16C2×#89492C211yesmaximal, characteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×#6507C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C2×#6487C2211yescharacteristic
24#90316—24no—
25#90316C2218yes—
26#90210—24no—
27#90210C2218yes—
28C2×#6453—21no—
29C2×#6453C2211yescharacteristic
30C2×#6453C2211yescharacteristic
31C2×#6453C2211yescharacteristic
32C2×#6452—21no—
33C2×#6451C2211yescharacteristic
34C2×#6451C2211yescharacteristic
35C2×#6450C2211yescharacteristic
36C2×#6442—21no—
37C2×#6442—21no—
38C2×#6441—21no—
39C2×#6441—21no—
40C2×#6433—21no—
41C2×#6432—21no—
42C7×#485C2218yes—
43C2×C7×#107C2211yescharacteristic
44C2×C7×#106C2211yescharacteristic
45C2×C7×#103C2211yescharacteristic
46C2×C7×#103C2211yescharacteristic
47C2×C7×C22⋊C8C2211yescharacteristic
48C2×#6409C2211yescharacteristic
49C2×#6409C2211yescharacteristic
50C2×C22⋊(C7⋊C8)C2211yescharacteristic
51C2×#6391C2211yescharacteristic
52C2×#6391C2211yescharacteristic
53C2×#6390C2211yescharacteristic
54C2×#6390C2211yescharacteristic
55C2×#6389C2211yescharacteristic
56C2×Dic7⋊C8C2211yescharacteristic
57C2×Dic7⋊C8C2211yescharacteristic
58C2×C8×Dic7C2211yescharacteristic
59#89816C2218yes—
60#89768C2218yes—
61C2×#6384C2211yescharacteristic
62C2×#6384C2211yescharacteristic
63C2×#6382C2211yescharacteristic
64C2×#6381C2211yescharacteristic
65C2×#6381C2211yescharacteristic
66C2×#6381C2211yescharacteristic
67#89648C2218yescomplemented
68#89647C2218yescomplemented
69#89630C2218yescomplemented
70#89629C2218yescomplemented
71#89571C2218yescomplemented
72#89569C2218yescomplemented
73#89564C2218yescomplemented
74#89563C2218yescomplemented
75#89493C2218yes—
76#89492C2218yes—
77C2×#3242—71nomaximal, Sylow
78C22×C7×Q8C2311yescharacteristic
79C23×C4×C7C2311yescharacteristic
80C23×Dic7—21no—
81C22×Dic14—21no—
82C22×Dic14—21no—
83C22×Dic14C2311yescharacteristic
84C22×C7×C8C2311yescharacteristic
85C22×C7×C8C2311yescharacteristic
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8C2314yes—
88C7×#117—22no—
89C7×#117C2314yes—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7—21no—
97C2×C22⋊Dic7—21no—
98C2×C22⋊Dic7D411yescharacteristic
99C2×C22⋊Dic7D411yescharacteristic
100Q8×Dic7—22no—
101Q8×Dic7C2314yes—
102#6507—22no—
103#6507C2314yes—
104#6489D414yes—
105#6489C2314yes—
106C2×C4⋊Dic7—21no—
107C2×C4⋊Dic7—22no—
108C2×C4⋊Dic7—21no—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7—21no—
111C2×C4⋊Dic7—21no—
112C2×C4⋊Dic7D411yescharacteristic
113C2×C4⋊Dic7D411yescharacteristic
114C2×C4⋊Dic7C2311yescharacteristic
115C2×C4⋊Dic7C2311yescharacteristic
116C2×C4⋊Dic7C2311yescharacteristic
117C2×C4⋊Dic7C2314yes—
118#6487—22no—
119#6487C2314yes—
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×Dic7⋊C4—21no—
129C2×C4×Dic7—21no—
130C2×C4×Dic7—21no—
131C2×C4×Dic7C2311yescharacteristic
132C2×C4×Dic7C2311yescharacteristic
133C22×C7⋊C8C2311yescharacteristic
134C22×C7⋊C8C2311yescharacteristic
135#6453—28no—
136#6453—22no—
137#6453—22no—
138#6453—22no—
139#6453C2314yes—
140#6453C2314yes—
141#6453C2314yes—
142#6452—28no—
143#6451—22no—
144#6451—22no—
145#6451C2314yes—
146#6451C2314yes—
147#6450—22no—
148#6450C2314yes—
149#6442—28no—
150#6442—28no—
151#6441—28no—
152#6441—28no—
153#6433—28no—
154#6432—28no—
155C7×#107C2314yes—
156C7×#106C2314yes—
157C7×#103C2314yes—
158C7×#103C2314yes—
159C7×C22⋊C8C2314yes—
160#6409C2314yes—
161#6409C2314yes—
162C22⋊(C7⋊C8)C2314yes—
163#6391C2314yes—
164#6391C2314yes—
165#6390C2314yes—
166#6390C2314yes—
167#6389C2314yes—
168Dic7⋊C8C2314yes—
169Dic7⋊C8C2314yes—
170C8×Dic7C2314yes—
171#6384C2314yes—
172#6384C2314yes—
173#6382C2314yes—
174#6381C2314yes—
175#6381C2314yes—
176#6381C2314yes—
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×#485D711yescharacteristic
187C2×#483—71no—
188C2×#479—71no—
189C2×#479—71no—
190C2×#477—71no—
191C2×#475—71no—
192C2×#433—71no—
193C24×C7C2×D411yescharacteristic
194C2×C7×Q8—21no—
195C2×C7×Q8—22no—
196C2×C7×Q8—22no—
197C2×C7×Q8C2412yes—
198C22×C4×C7—22no—
199C22×C4×C7—22no—
200C22×C4×C7—21no—
201C22×C4×C7—21no—
202C22×C4×C7—21no—
203C22×C4×C7—21no—
204C22×C4×C7C2×D411yescharacteristic
205C22×C4×C7C2×D412yes—
206C22×C4×C7C2412yes—
207C22×C4×C7C2411yescharacteristic
208C22×Dic7—21no—
209C22×Dic7—21no—
210C22×Dic7—21no—
211C22×Dic7—21no—
212C22×Dic7—24no—
213C22×Dic7—82no—
214C22×Dic7C2×D411yescharacteristic
215C22×Dic7C2×D411yescharacteristic
216C22×Dic7C2×D411yescharacteristic
217C22×Dic7C2×D411yescharacteristic
218C2×Dic14—22no—
219C2×Dic14—22no—
220C2×Dic14—21no—
221C2×Dic14—24no—
222C2×Dic14—24no—
223C2×Dic14—82no—
224C2×Dic14—82no—
225C2×Dic14C2412yes—
226C2×C7×C8—22no—
227C2×C7×C8—22no—
228C2×C7×C8C2412yes—
229C2×C7×C8C2412yes—
230C7×C4⋊C4—21no—
231C7×C4⋊C4—21no—
232C7×C4⋊C4—24no—
233C7×C4⋊C4—21no—
234C7×C4⋊C4C2412yes—
235C7×C4⋊C4C2412yes—
236C7×C4⋊C4C2412yes—
237C7×C22⋊C4—24no—
238C7×C22⋊C4—24no—
239C42×C7—24no—
240C22⋊Dic7—24no—
241C22⋊Dic7C2×D414yes—
242C22⋊Dic7C2×D414yes—
243C4⋊Dic7—24no—
244C4⋊Dic7—21no—
245C4⋊Dic7—21no—
246C4⋊Dic7—21no—
247C4⋊Dic7—24no—
248C4⋊Dic7—24no—
249C4⋊Dic7—24no—
250C4⋊Dic7—24no—
251C4⋊Dic7C2×D414yes—
252C4⋊Dic7C2×D414yes—
253C4⋊Dic7C2412yes—
254C4⋊Dic7C2412yes—
255C4⋊Dic7C2412yes—
256Dic7⋊C4—24no—
257Dic7⋊C4—24no—
258Dic7⋊C4—24no—
259Dic7⋊C4—24no—
260Dic7⋊C4—24no—
261Dic7⋊C4—24no—
262Dic7⋊C4—24no—
263Dic7⋊C4—24no—
264Dic7⋊C4—24no—
265C4×Dic7—24no—
266C4×Dic7—24no—
267C4×Dic7C2×D412yes—
268C4×Dic7C2×D412yes—
269C4×Dic7C2412yes—
270C4×Dic7C2412yes—
271C2×C7⋊C8—22no—
272C2×C7⋊C8—22no—
273C2×C7⋊C8C2412yes—
274C2×C7⋊C8C2412yes—
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—141no—
293C2×C22⋊Q8—141no—
294C2×C22⋊Q8C2×D711yescharacteristic
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#485C2×D718yes—
306#484—78no—
307#483—78no—
308#479—78no—
309#479—78no—
310#477—78no—
311#475—78no—
312#433—78no—
313C2×#107—71no—
314C2×#107—71no—
315C2×#107—71no—
316C2×#107C2×D711yescharacteristic
317C2×#106—71no—
318C2×#106—71no—
319C2×#106—71no—
320C2×#106C2×D711yescharacteristic
321C2×C4⋊C8—71no—
322C2×C4⋊C8—71no—
323C2×#103—71no—
324C2×#103—71no—
325C2×#103—71no—
326C2×#103—71no—
327C2×#103—71no—
328C2×#103—71no—
329C2×#103C2×D711yescharacteristic
330C2×#103C2×D711yescharacteristic
331C2×C22⋊C8—71no—
332C2×C22⋊C8C2×D711yescharacteristic
333C2×#97—71no—
334C2×C4×C8—71no—
335C23×C7—22no—
336C23×C7—22no—
337C23×C7—22no—
338C23×C7C22×D412yes—
339C23×C7C22×D411yescharacteristic
340C7×Q8—22no—
341C7×Q8—22no—
342C7×Q8—42no—
343C2×C4×C7—22no—
344C2×C4×C7—24no—
345C2×C4×C7—22no—
346C2×C4×C7—22no—
347C2×C4×C7—22no—
348C2×C4×C7—22no—
349C2×C4×C7—22no—
350C2×C4×C7—22no—
351C2×C4×C7—42no—
352C2×C4×C7—42no—
353C2×C4×C7—42no—
354C2×C4×C7—42no—
355C2×C4×C7C22×D412yes—
356C2×C4×C7C22×D411yescharacteristic
357C2×C4×C72−1+412yes—
358C2×C4×C72−1+412yes—
359C2×C4×C7C2511yescharacteristic
360C2×Dic7—22no—
361C2×Dic7—22no—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—22no—
365C2×Dic7—22no—
366C2×Dic7—22no—
367C2×Dic7—22no—
368C2×Dic7—42no—
369C2×Dic7—42no—
370C2×Dic7—42no—
371C2×Dic7—42no—
372C2×Dic7—84no—
373C2×Dic7C22×D412yes—
374C2×Dic7C22×D412yes—
375C2×Dic7C22×D412yes—
376C2×Dic7C22×D412yes—
377Dic14—22no—
378Dic14—22no—
379Dic14—42no—
380Dic14—84no—
381Dic14—84no—
382C7×C8—22no—
383C7×C8—22no—
384C7⋊C8—22no—
385C7⋊C8—22no—
386C22×Q8—71no—
387C22×Q8—141no—
388C22×Q8—141no—
389C22×Q8C22×D711yescharacteristic
390C23×C4—141no—
391C23×C4C22×D711yescharacteristic
392C22×C8—71no—
393C22×C8—71no—
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—142no—
410#125—142no—
411#125—142no—
412#125—148no—
413#125—148no—
414C22⋊Q8—22no—
415C22⋊Q8—74no—
416C22⋊Q8—148no—
417C22⋊Q8—148no—
418C22⋊Q8—142no—
419C22⋊Q8C22×D714yes—
420C4×Q8—74no—
421C4×Q8—74no—
422C4×Q8—142no—
423C4×Q8—142no—
424#117—22no—
425#117—74no—
426#117—74no—
427#117C22×D714yes—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—71no—
430C2×C4⋊C4—71no—
431C2×C4⋊C4—71no—
432C2×C4⋊C4—71no—
433C2×C4⋊C4—74no—
434C2×C4⋊C4—71no—
435C2×C4⋊C4—141no—
436C2×C4⋊C4—141no—
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—142no—
449C2×C4⋊C4—141no—
450C2×C4⋊C4C22×D711yescharacteristic
451C2×C4⋊C4C22×D711yescharacteristic
452C2×C4⋊C4C22×D711yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—71no—
456C2×C22⋊C4—71no—
457C2×C22⋊C4—141no—
458C2×C42—21no—
459C2×C42—71no—
460C2×C42—71no—
461C2×C42—141no—
462C2×C42—141no—
463#107—74no—
464#107—74no—
465#107—74no—
466#107C22×D714yes—
467#106—74no—
468#106—74no—
469#106—74no—
470#106C22×D714yes—
471C4⋊C8—74no—
472C4⋊C8—74no—
473#103—74no—
474#103—74no—
475#103—74no—
476#103—74no—
477#103—74no—
478#103—74no—
479#103C22×D714yes—
480#103C22×D714yes—
481C22⋊C8—74no—
482C22⋊C8C22×D714yes—
483#97—74no—
484C4×C8—74no—
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—42no—
495C4×C7—42no—
496C4×C7—42no—
497C4×C7—42no—
498C4×C7#53612yes—
499C4×C7C2×2−1+411yescharacteristic
500C4×C7C2×2−1+411yescharacteristic
501Dic7—22no—
502Dic7—22no—
503Dic7—22no—
504Dic7—22no—
505Dic7—42no—
506Dic7—42no—
507Dic7—42no—
508Dic7—42no—
509C24D4×D711yescharacteristic
510C2×Q8—21no—
511C2×Q8—22no—
512C2×Q8—22no—
513C2×Q8—72no—
514C2×Q8—144no—
515C2×Q8—144no—
516C2×Q8—141no—
517C2×Q8—142no—
518C2×Q8—142no—
519C2×Q8—562no—
520C2×Q8—562no—
521C2×Q8C23×D712yes—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—71no—
529C22×C4—71no—
530C22×C4—71no—
531C22×C4—71no—
532C22×C4—141no—
533C22×C4—141no—
534C22×C4—141no—
535C22×C4—141no—
536C22×C4—144no—
537C22×C4—562no—
538C22×C4D4×D712yes—
539C22×C4D4×D711yescharacteristic
540C22×C4C23×D712yes—
541C22×C4C23×D711yescharacteristic
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—72no—
545C2×C8—72no—
546C2×C8—142no—
547C2×C8—142no—
548C2×C8C23×D712yes—
549C2×C8C23×D712yes—
550C4⋊C4—24no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—74no—
555C4⋊C4—74no—
556C4⋊C4—72no—
557C4⋊C4—72no—
558C4⋊C4—72no—
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—144no—
568C4⋊C4—144no—
569C4⋊C4—144no—
570C4⋊C4—144no—
571C4⋊C4—144no—
572C4⋊C4—144no—
573C4⋊C4—141no—
574C4⋊C4—141no—
575C4⋊C4—141no—
576C4⋊C4C23×D712yes—
577C4⋊C4C23×D712yes—
578C4⋊C4C23×D712yes—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—74no—
582C22⋊C4—74no—
583C22⋊C4—144no—
584C42—24no—
585C42—72no—
586C42—72no—
587C42—72no—
588C42—72no—
589C42—144no—
590C42—144no—
591C2×C7—42no—
592C2×C7#324214yescomplemented
593C2×C7C2×#53611yescharacteristic
594C2×C7C2×#57711yescharacteristic
595C2×C7C2×#57811yescharacteristic
596C23—22no—
597C23—22no—
598C23—22no—
599C23C2×D4×D712yes—
600C23C2×D4×D711yescharacteristic
601Q8—22no—
602Q8—22no—
603Q8—42no—
604Q8—142no—
605Q8—142no—
606Q8—282no—
607Q8—564no—
608Q8—564no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—72no—
622C2×C4—72no—
623C2×C4—72no—
624C2×C4—72no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—142no—
631C2×C4—142no—
632C2×C4—142no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—564no—
638C2×C4C2×D4×D712yes—
639C2×C4C2×D4×D711yescharacteristic
640C2×C4#655412yes—
641C2×C4#655412yes—
642C2×C4C24×D711yescharacteristic
643C8—22no—
644C8—22no—
645C8—142no—
646C8—142no—
647C7C2×#324211yescharacteristic, complemented, Sylow
648C22—22no—
649C22—22no—
650C22—22no—
651C22—44no—
652C22#9033912yes—
653C22#9046112yes—
654C22#9047012yes—
655C22C22×D4×D711yescharacteristic
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—142no—
662C4—142no—
663C4—142no—
664C4—142no—
665C4—282no—
666C4—282no—
667C4—282no—
668C4—282no—
669C4#9033912yes—
670C4C2×#655411yescharacteristic
671C4C2×#655411yescharacteristic
672C2—42no—
673C2#1173895014yesdirect product factor
674C2C2×#9033911yescharacteristic
675C2C2×#9046111yescharacteristic
676C2C2×#9047011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#485C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#485D7
hypocenterC7C2×#3242
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
6. #11738950C2direct (N × H)
7. C2×#89647C2semidirect
8. C2×#89648C2semidirect
9. C2×#89629C2semidirect
10. C2×#89630C2semidirect
11. C2×#89569C2semidirect
12. C2×#89564C2semidirect
13. C2×#89571C2semidirect
14. C2×#89563C2semidirect
67. #89648C22semidirect
68. #89647C22semidirect
69. #89630C22semidirect
70. #89629C22semidirect
71. #89571C22semidirect
72. #89569C22semidirect
73. #89564C22semidirect
74. #89563C22semidirect
592. C2×C7#3242semidirect
647. C7C2×#3242semidirect
673. C2#11738950direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#903162
C2×#902102
C2×C7×#4852
C2×#898162
C2×#897682
#117389502
C2×#896472
C2×#896482
C2×#896292
C2×#896302
C2×#895692
C2×#895642
C2×#895712
C2×#895632
C2×#894932
C2×#894922
C2×#32427