Finite Group Databasebeta

← C2×#11731381 #13525569

Subgroups — C2×#11731381 #13525569

Order 1792.

3614 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊D7C211yesmaximal, characteristic
3C22×#6460C211yesmaximal, characteristic, complemented
4C22×#6450C211yesmaximal, characteristic, complemented
5C2×C7×#380C211yesmaximal, characteristic, complemented
6C2×#89759C211yesmaximal, characteristic, complemented
7C2×#89748C211yesmaximal, characteristic, complemented
8C2×#89720C212yesmaximal, complemented
9C2×#89729C212yesmaximal
10#11731381C2116yesmaximal, direct product factor
11C2×#89448C212yesmaximal, complemented
12C2×#89419C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C23×C22⋊D7C2211yescharacteristic
15C23×C4×D7C2211yescharacteristic
16C23×Dic14C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C22⋊Dic7C2212yes—
22C2×C4×C22⋊D7—28no—
23C2×C4×C22⋊D7C2218yes—
24C22×(C2×D7)⋊C4C2211yescharacteristic
25C22×(C2×D7)⋊C4C2212yes—
26C22×(C2×D7)⋊C4C2212yes—
27C22×C4⋊Dic7C2211yescharacteristic
28C22×C4⋊Dic7C2212yes—
29C2×#6487—28no—
30C2×#6487C2218yescomplemented
31C22×Dic7⋊C4C2211yescharacteristic
32C22×Dic7⋊C4C2212yes—
33C22×Dic7⋊C4C2212yes—
34C22×Dic7⋊C4C2212yes—
35C22×Dic7⋊C4C2212yes—
36C22×C4×Dic7C2212yes—
37C22×C4×Dic7C2211yescharacteristic
38C2×#6460—28no—
39C2×#6460C2218yescomplemented
40C2×#6450—28no—
41C2×#6450C2218yescomplemented
42C7×#380C2218yescomplemented
43C2×C7×#95C2212yes—
44C2×C7⋊#95C2212yes—
45#89759C2218yescomplemented
46#89748C2218yescomplemented
47#89729C22116yes—
48#89720C22116yescomplemented
49#89448C22116yescomplemented
50#89419C22116yescomplemented
51C2×#2538—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C23×C4×C7—24no—
55C23×C4×C7—24no—
56C23×C4×C7—24no—
57C23×C4×C7C2312yes—
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2312yes—
60C23×C4×C7C2314yes—
61C23×C4×C7C2311yescharacteristic
62C22×C22⋊D7—24no—
63C22×C22⋊D7—24no—
64C22×C22⋊D7Q818yescomplemented
65C22×C22⋊D7C2314yes—
66C23×Dic7C2312yes—
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2312yes—
70C23×Dic7C2311yescharacteristic
71C22×C4×D7—24no—
72C22×C4×D7—24no—
73C22×C4×D7—24no—
74C22×C4×D7C2314yes—
75C22×Dic14—24no—
76C22×Dic14—24no—
77C22×Dic14D418yescomplemented
78C22×Dic14C2314yes—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4C2314yes—
82C2×C7×C4⋊C4C2314yes—
83C2×C7×C22⋊C4—44no—
84C2×C7×C22⋊C4C2318yes—
85C2×C22⋊Dic7—28no—
86C2×C22⋊Dic7—24no—
87C2×C22⋊Dic7C2318yes—
88C2×C22⋊Dic7C2314yes—
89C4×C22⋊D7—232no—
90C2×(C2×D7)⋊C4—28no—
91C2×(C2×D7)⋊C4—24no—
92C2×(C2×D7)⋊C4—44no—
93C2×(C2×D7)⋊C4C2318yes—
94C2×(C2×D7)⋊C4C2314yes—
95C2×(C2×D7)⋊C4C2318yes—
96C2×C4⋊Dic7—24no—
97C2×C4⋊Dic7—28no—
98C2×C4⋊Dic7C2318yes—
99C2×C4⋊Dic7C2314yes—
100#6487—232no—
101C2×Dic7⋊C4—28no—
102C2×Dic7⋊C4—28no—
103C2×Dic7⋊C4—24no—
104C2×Dic7⋊C4—28no—
105C2×Dic7⋊C4—44no—
106C2×Dic7⋊C4C2314yes—
107C2×Dic7⋊C4C2318yes—
108C2×Dic7⋊C4C2318yes—
109C2×Dic7⋊C4C2318yes—
110C2×Dic7⋊C4C2318yes—
111C2×C4×Dic7—28no—
112C2×C4×Dic7D418yes—
113C2×C4×Dic7C2318yes—
114C2×C4×Dic7C2314yes—
115#6460—232no—
116#6450—232no—
117C7×#95C2318yes—
118C7⋊#95C2318yes—
119C22×C22⋊Q8—71no—
120C22×C22⋊Q8—71no—
121C22×C4×Q8—71no—
122C22×C4×D4—71no—
123#2538—716no—
124C2×#396—72no—
125C2×#396—72no—
126C2×#390—72no—
127C2×#389—72no—
128C2×#384—71no—
129C2×#380—71no—
130C2×#380D711yescharacteristic, complemented
131C24×C7—22no—
132C24×C7—22no—
133C24×C7—22no—
134C24×C7—22no—
135C24×C7—22no—
136C24×C7—22no—
137C24×C7C2×Q814yes—
138C24×C7C2412yes—
139C24×C7C2411yescharacteristic
140C23×D7—22no—
141C23×D7—22no—
142C23×D7—22no—
143C23×D7—22no—
144C23×D7—22no—
145C23×D7—22no—
146C23×D7C2×Q814yes—
147C23×D7C2412yes—
148C22×C4×C7—24no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—22no—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—24no—
157C22×C4×C7—28no—
158C22×C4×C7—28no—
159C22×C4×C7—28no—
160C22×C4×C7—416no—
161C22×C4×C7C2×D414yes—
162C22×C4×C7C2×D414yes—
163C22×C4×C7#4114yes—
164C22×C4×C7C2414yes—
165C22×C4×C7C2412yes—
166C22×C4×C7C2414yes—
167C22×C4×C7C2412yes—
168C2×C22⋊D7—232no—
169C2×C22⋊D7—28no—
170C2×C22⋊D7—28no—
171C2×C22⋊D7C2×Q8116yescomplemented
172C22×Dic7—24no—
173C22×Dic7—24no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—24no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—24no—
180C22×Dic7—22no—
181C22×Dic7—24no—
182C22×Dic7C2×D414yes—
183C22×Dic7C2×D418yes—
184C22×Dic7C2×D414yes—
185C22×Dic7C2×Q814yes—
186C22×Dic7#4114yes—
187C22×Dic7C2412yes—
188C22×Dic7C2414yes—
189C22×Dic7C2412yes—
190C22×Dic7C2414yes—
191C22×Dic7C2412yes—
192C2×C4×D7—28no—
193C2×C4×D7—28no—
194C2×C4×D7—28no—
195C2×C4×D7—416no—
196C2×Dic14—232no—
197C2×Dic14—28no—
198C2×Dic14—28no—
199C2×Dic14C2×D4116yescomplemented
200C7×C4⋊C4—28no—
201C7×C4⋊C4—28no—
202C7×C22⋊C4—48no—
203C22⋊Dic7—216no—
204C22⋊Dic7—28no—
205(C2×D7)⋊C4—216no—
206(C2×D7)⋊C4—28no—
207(C2×D7)⋊C4—48no—
208C4⋊Dic7—28no—
209C4⋊Dic7—216no—
210Dic7⋊C4—216no—
211Dic7⋊C4—216no—
212Dic7⋊C4—28no—
213Dic7⋊C4—216no—
214Dic7⋊C4—48no—
215C4×Dic7—216no—
216C4×Dic7C2×D4116yes—
217C23×Q8—71no—
218C23×D4—71no—
219C24×C4—71no—
220C24×C4C2×D711yescharacteristic
221C2×C22⋊Q8—78no—
222C2×C22⋊Q8—78no—
223C2×C22⋊Q8—148no—
224C2×C22⋊Q8—148no—
225C2×C4×Q8—78no—
226C2×C4×Q8—148no—
227C2×C4×D4—78no—
228C2×C4×D4—148no—
229C22×C4⋊C4—72no—
230C22×C4⋊C4—72no—
231C22×C4⋊C4—72no—
232C22×C4⋊C4—71no—
233C22×C4⋊C4—71no—
234C22×C4⋊C4—72no—
235C22×C4⋊C4—72no—
236C22×C4⋊C4C2×D711yescharacteristic
237C22×C4⋊C4C2×D711yescharacteristic
238C22×C22⋊C4—71no—
239C22×C22⋊C4—72no—
240C22×C22⋊C4—71no—
241C22×C22⋊C4—72no—
242C22×C22⋊C4—72no—
243C22×C22⋊C4C2×D712yes—
244C22×C42—71no—
245C22×C42—72no—
246#396—716no—
247#396—716no—
248#390—716no—
249#389—716no—
250#384—78no—
251#380—78no—
252#380C2×D718yescomplemented
253C2×#95—72no—
254C2×#95C2×D712yes—
255C23×C7—28no—
256C23×C7—28no—
257C23×C7—28no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—44no—
265C23×C7—44no—
266C23×C7—44no—
267C23×C7—44no—
268C23×C7C22×D412yes—
269C23×C7C22×D412yes—
270C23×C7C22×Q814yes—
271C23×C7C22×Q812yes—
272C23×C7C2×#4112yes—
273C23×C7C2×#4112yes—
274C23×C7H(5,𝔽2)12yes—
275C23×C72−1+412yes—
276C23×C7C2511yescharacteristic
277C22×D7—28no—
278C22×D7—28no—
279C22×D7—28no—
280C22×D7—22no—
281C22×D7—22no—
282C22×D7—22no—
283C22×D7—22no—
284C22×D7—22no—
285C22×D7—22no—
286C22×D7—44no—
287C22×D7—44no—
288C22×D7—44no—
289C22×D7—44no—
290C22×D7C22×Q814yes—
291C2×C4×C7—24no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—24no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—24no—
299C2×C4×C7—24no—
300C2×C4×C7—44no—
301C2×C4×C7—48no—
302C2×C4×C7—48no—
303C2×C4×C7—416no—
304C2×C4×C7#121116yes—
305C2×C4×C7C22×D414yes—
306C2×C4×C7C22×D414yes—
307C2×C4×C7C2×#4114yes—
308C22⋊D7—232no—
309C2×Dic7—216no—
310C2×Dic7—28no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—24no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—24no—
319C2×Dic7—22no—
320C2×Dic7—24no—
321C2×Dic7—44no—
322C2×Dic7—48no—
323C2×Dic7C22⋊Q8116yescomplemented
324C2×Dic7C22×D414yes—
325C2×Dic7C22×D418yes—
326C2×Dic7C22×D414yes—
327C2×Dic7C22×Q814yes—
328C2×Dic7C2×#4114yes—
329C4×D7—416no—
330Dic14—232no—
331C25—71no—
332C25C22×D711yescharacteristic
333C22×Q8—78no—
334C22×Q8—74no—
335C22×Q8—144no—
336C22×Q8—144no—
337C22×D4—78no—
338C22×D4—74no—
339C22×D4—144no—
340C22×D4—144no—
341C23×C4—24no—
342C23×C4—24no—
343C23×C4—24no—
344C23×C4—71no—
345C23×C4—71no—
346C23×C4—72no—
347C23×C4—72no—
348C23×C4—71no—
349C23×C4—74no—
350C23×C4—144no—
351C23×C4—144no—
352C23×C4—144no—
353C23×C4C22×D714yes—
354C23×C4C22×D712yes—
355C23×C4C22×D712yes—
356C23×C4C22×D711yescharacteristic
357C23×C4C22×D711yescharacteristic
358C22⋊Q8—1432no—
359C22⋊Q8—1432no—
360C4×Q8—1432no—
361C4×D4—1432no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—74no—
365C2×C4⋊C4—78no—
366C2×C4⋊C4—78no—
367C2×C4⋊C4—74no—
368C2×C4⋊C4—78no—
369C2×C4⋊C4—78no—
370C2×C4⋊C4—78no—
371C2×C4⋊C4—144no—
372C2×C4⋊C4—148no—
373C2×C4⋊C4—148no—
374C2×C4⋊C4—148no—
375C2×C4⋊C4—148no—
376C2×C4⋊C4—144no—
377C2×C4⋊C4—284no—
378C2×C4⋊C4C22×D714yes—
379C2×C4⋊C4C22×D714yes—
380C2×C22⋊C4—44no—
381C2×C22⋊C4—74no—
382C2×C22⋊C4—78no—
383C2×C22⋊C4—74no—
384C2×C22⋊C4—78no—
385C2×C22⋊C4—78no—
386C2×C22⋊C4—144no—
387C2×C22⋊C4—148no—
388C2×C22⋊C4—144no—
389C2×C22⋊C4—148no—
390C2×C22⋊C4—284no—
391C2×C22⋊C4C22×D718yes—
392C2×C42—78no—
393C2×C42—74no—
394C2×C42—78no—
395C2×C42—148no—
396#95—78no—
397#95C22×D718yes—
398C22×C7—24no—
399C22×C7—24no—
400C22×C7—24no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—416no—
406C22×C7C2×#12114yes—
407C22×C7C2×C22⋊Q814yes—
408C22×C7#53714yes—
409C22×C7D4×Q814yes—
410C22×C7#55014yes—
411C22×C7#55414yes—
412C22×C7#55614yes—
413C22×C7C23×D411yescharacteristic
414C22×C7C23×D411yescharacteristic
415C22×C7C23×Q811yescharacteristic
416C22×C7C22×#4111yescharacteristic
417C22×C7C22×#4111yescharacteristic
418C22×C7C2×H(5,𝔽2)11yescharacteristic
419C22×C7C2×2−1+411yescharacteristic
420C2×D7—24no—
421C2×D7—24no—
422C2×D7—24no—
423C2×D7—42no—
424C2×D7—42no—
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—416no—
428C4×C7—42no—
429C4×C7—44no—
430C4×C7—44no—
431C4×C7C2×#12118yes—
432Dic7—28no—
433Dic7—24no—
434Dic7—42no—
435Dic7—44no—
436Dic7C2×C22⋊Q818yescomplemented
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—74no—
444C24—72no—
445C24—142no—
446C24—142no—
447C24—142no—
448C24—142no—
449C24—142no—
450C24—142no—
451C24Q8×D714yes—
452C24C23×D712yes—
453C24C23×D711yescharacteristic
454C2×Q8—716no—
455C2×Q8—1432no—
456C2×Q8—148no—
457C2×Q8—148no—
458C2×D4—716no—
459C2×D4—148no—
460C2×D4—1432no—
461C2×D4—148no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—416no—
475C22×C4—74no—
476C22×C4—74no—
477C22×C4—74no—
478C22×C4—72no—
479C22×C4—74no—
480C22×C4—78no—
481C22×C4—72no—
482C22×C4—74no—
483C22×C4—72no—
484C22×C4—74no—
485C22×C4—144no—
486C22×C4—144no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—144no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—144no—
494C22×C4—144no—
495C22×C4—148no—
496C22×C4—148no—
497C22×C4—148no—
498C22×C4—2816no—
499C22×C4#115714yes—
500C22×C4D4×D714yes—
501C22×C4D4×D714yes—
502C22×C4C23×D714yes—
503C22×C4C23×D712yes—
504C22×C4C23×D714yes—
505C22×C4C23×D712yes—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—1416no—
509C4⋊C4—148no—
510C4⋊C4—1416no—
511C4⋊C4—1416no—
512C4⋊C4—1416no—
513C4⋊C4—148no—
514C4⋊C4—288no—
515C22⋊C4—48no—
516C22⋊C4—148no—
517C22⋊C4—1416no—
518C22⋊C4—148no—
519C22⋊C4—1416no—
520C22⋊C4—288no—
521C42—716no—
522C42—1416no—
523C2×C7—44no—
524C2×C7#253818yescomplemented
525C2×C7C22×#12111yescharacteristic
526C2×C7C22×C22⋊Q811yescharacteristic
527C2×C7C2×#53711yescharacteristic
528C2×C7C2×D4×Q811yescharacteristic
529C2×C7C2×#55011yescharacteristic
530C2×C7C2×#55411yescharacteristic
531C2×C7C2×#55611yescharacteristic
532D7—44no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—74no—
547C23—148no—
548C23—148no—
549C23—148no—
550C23—142no—
551C23—142no—
552C23—142no—
553C23—142no—
554C23—142no—
555C23—142no—
556C23—284no—
557C23—284no—
558C23—284no—
559C23—284no—
560C23C2×#115712yes—
561C23C2×D4×D712yes—
562C23C2×D4×D712yes—
563C23#654812yes—
564C23C2×Q8×D714yes—
565C23C2×Q8×D712yes—
566C23D7×#4112yes—
567C23#655412yes—
568C23C24×D711yescharacteristic
569Q8—1432no—
570D4—1432no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—416no—
584C2×C4—716no—
585C2×C4—74no—
586C2×C4—74no—
587C2×C4—74no—
588C2×C4—74no—
589C2×C4—78no—
590C2×C4—1416no—
591C2×C4—148no—
592C2×C4—144no—
593C2×C4—144no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—144no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—144no—
601C2×C4—144no—
602C2×C4—284no—
603C2×C4—288no—
604C2×C4—2816no—
605C2×C4#6447116yes—
606C2×C4C2×#115714yes—
607C2×C4C2×D4×D714yes—
608C2×C4C2×D4×D714yes—
609C7C2×#253811yescharacteristic, complemented, Sylow
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—416no—
618C22—144no—
619C22—144no—
620C22—144no—
621C22—282no—
622C22—282no—
623C22—282no—
624C22—282no—
625C22—2816no—
626C22C2×#644714yes—
627C22#9020214yes—
628C22#9022414yes—
629C22#9029014yes—
630C22#9029914yes—
631C22D7×C22⋊Q814yes—
632C22#9032314yes—
633C22C22×#115711yescharacteristic
634C22C22×D4×D711yescharacteristic
635C22C22×D4×D711yescharacteristic
636C22C2×#654811yescharacteristic
637C22C22×Q8×D711yescharacteristic
638C22C2×D7×#4111yescharacteristic
639C22C2×#655411yescharacteristic
640C4—44no—
641C4—44no—
642C4—42no—
643C4—78no—
644C4—144no—
645C4—148no—
646C4—284no—
647C4—282no—
648C4C2×#644718yes—
649C2—44no—
650C2—284no—
651C2#1173138118yesdirect product factor
652C2C22×#644711yescharacteristic
653C2C2×#9020211yescharacteristic
654C2C2×#9022411yescharacteristic
655C2C2×#9029011yescharacteristic
656C2C2×#9029911yescharacteristic
657C2C2×#9032311yescharacteristic
658C2C2×D7×C22⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#380C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC2×#380D7
hypocenterC7C2×#2538
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#6487C2semidirect
3. C22×#6460C2semidirect
4. C22×#6450C2semidirect
5. C2×C7×#380C2semidirect
6. C2×#89759C2semidirect
7. C2×#89748C2semidirect
8. C2×#89720C2semidirect
10. #11731381C2direct (N × H)
11. C2×#89448C2semidirect
12. C2×#89419C2semidirect
30. C2×#6487C22semidirect
39. C2×#6460C22semidirect
41. C2×#6450C22semidirect
42. C7×#380C22semidirect
45. #89759C22semidirect
46. #89748C22semidirect
48. #89720C22semidirect
49. #89448C22semidirect
50. #89419C22semidirect
64. C22×C22⋊D7Q8semidirect
77. C22×Dic14D4semidirect
130. C2×#380D7semidirect
171. C2×C22⋊D7C2×Q8semidirect
199. C2×Dic14C2×D4semidirect
252. #380C2×D7semidirect
323. C2×Dic7C22⋊Q8semidirect
436. Dic7C2×C22⋊Q8semidirect
524. C2×C7#2538semidirect
609. C7C2×#2538semidirect
651. C2#11731381direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C22×C4×C22⋊D72
C22×#64602
C22×#64502
C2×C7×#3802
C2×#897592
C2×#897482
C2×#897202
C2×#897292
#117313812
C2×#894482
C2×#894192
C2×#25387