Finite Group Databasebeta

← C2×#11733791 #13525564

Subgroups — C2×#11733791 #13525564

Order 1792.

3614 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic7C211yesmaximal, characteristic
2C22×#6441C212yesmaximal, complemented
3C22×C4×Dic14C211yesmaximal, characteristic, complemented
4C2×C7×#390C211yesmaximal, characteristic
5C2×#89993C211yesmaximal, characteristic
6C2×#89757C211yesmaximal, characteristic, complemented
7C2×#89748C211yesmaximal, characteristic, complemented
8C2×#89720C212yesmaximal, complemented
9C2×#89720C212yesmaximal, complemented
10C2×#89714C212yesmaximal, complemented
11C2×#89700C211yesmaximal, characteristic, complemented
12#11733791C2116yesmaximal, direct product factor
13C23×C7×D4C2211yescharacteristic
14C24×Dic7C2212yes—
15C23×Dic14C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C42×C7C2211yescharacteristic
20C22×C22⋊Dic7C2212yes—
21C22×C22⋊Dic7C2212yes—
22C22×C22⋊Dic7C2212yes—
23C2×D4×Dic7—28no—
24C2×D4×Dic7C2218yes—
25C22×C4⋊Dic7C2211yescharacteristic
26C22×C4⋊Dic7C2212yes—
27C22×C4⋊Dic7C2211yescharacteristic
28C22×C4⋊Dic7C2211yescharacteristic
29C22×Dic7⋊C4C2211yescharacteristic
30C22×Dic7⋊C4C2211yescharacteristic
31C22×Dic7⋊C4C2212yes—
32C22×Dic7⋊C4C2212yes—
33C22×Dic7⋊C4C2212yes—
34C22×C4×Dic7C2211yescharacteristic
35C22×C4×Dic7C2211yescharacteristic
36C2×#6441—216no—
37C2×#6441C22116yescomplemented
38C2×C4×Dic14—28no—
39C2×C4×Dic14C2218yescomplemented
40C7×#390C2218yes—
41#89993C2218yes—
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44#89757C2218yescomplemented
45#89748C2218yescomplemented
46#89720C22116yescomplemented
47#89720C22116yescomplemented
48#89714C22116yescomplemented
49#89700C2218yescomplemented
50C2×#2538—71nomaximal, Sylow
51C25×C7C2312yes—
52C22×C7×D4—24no—
53C22×C7×D4—24no—
54C22×C7×D4Q818yescomplemented
55C22×C7×D4C2314yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—28no—
61C23×Dic7—28no—
62C23×Dic7—28no—
63C23×Dic7C2318yes—
64C23×Dic7C2311yescharacteristic
65C23×Dic7C2312yes—
66C23×Dic7C2312yes—
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2311yescharacteristic
70C22×Dic14—24no—
71C22×Dic14—24no—
72C22×Dic14D418yescomplemented
73C22×Dic14C2314yes—
74C2×C7×C4⋊C4—24no—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C22⋊C4—28no—
77C2×C7×C22⋊C4—44no—
78C2×C7×C22⋊C4C2318yes—
79C2×C7×C22⋊C4C2318yes—
80C2×C42×C7—24no—
81C2×C42×C7C2314yes—
82C2×C22⋊Dic7—28no—
83C2×C22⋊Dic7—28no—
84C2×C22⋊Dic7—44no—
85C2×C22⋊Dic7C2318yes—
86C2×C22⋊Dic7C2318yes—
87C2×C22⋊Dic7C2318yes—
88D4×Dic7—232no—
89C2×C4⋊Dic7—24no—
90C2×C4⋊Dic7—24no—
91C2×C4⋊Dic7—24no—
92C2×C4⋊Dic7—28no—
93C2×C4⋊Dic7C2314yes—
94C2×C4⋊Dic7C2318yes—
95C2×C4⋊Dic7C2314yes—
96C2×C4⋊Dic7C2314yes—
97C2×Dic7⋊C4—28no—
98C2×Dic7⋊C4—24no—
99C2×Dic7⋊C4—24no—
100C2×Dic7⋊C4—28no—
101C2×Dic7⋊C4—44no—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2318yes—
105C2×Dic7⋊C4C2318yes—
106C2×Dic7⋊C4C2318yes—
107C2×C4×Dic7—24no—
108C2×C4×Dic7D418yes—
109C2×C4×Dic7C2314yes—
110C2×C4×Dic7C2314yes—
111#6441—264no—
112C4×Dic14—232no—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C22×C22⋊Q8—72no—
116C22×C4×Q8—71no—
117C22×C4×D4—71no—
118#2538—716no—
119C2×#396—72no—
120C2×#396—72no—
121C2×#390—71no—
122C2×#390D711yescharacteristic
123C2×#389—71no—
124C2×#389—71no—
125C2×#384—71no—
126C2×#380—72no—
127C24×C7—24no—
128C24×C7—24no—
129C24×C7—24no—
130C24×C7—24no—
131C24×C7—24no—
132C24×C7—24no—
133C24×C7C2×Q818yes—
134C24×C7C2414yes—
135C24×C7C2411yescharacteristic
136C2×C7×D4—232no—
137C2×C7×D4—28no—
138C2×C7×D4—28no—
139C2×C7×D4C2×Q8116yescomplemented
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7C2×D414yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×Q814yes—
151C22×C4×C7#4114yes—
152C22×C4×C7C2412yes—
153C22×C4×C7C2414yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—24no—
162C22×Dic7—24no—
163C22×Dic7—24no—
164C22×Dic7—24no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—216no—
170C22×Dic7—216no—
171C22×Dic7—216no—
172C22×Dic7—432no—
173C22×Dic7C2×D414yes—
174C22×Dic7C2×D414yes—
175C22×Dic7C2×D414yes—
176C22×Dic7C2×D414yes—
177C22×Dic7#4114yes—
178C22×Dic7C2412yes—
179C22×Dic7C2414yes—
180C22×Dic7C2414yes—
181C22×Dic7C2412yes—
182C22×Dic7C2412yes—
183C22×Dic7C2412yes—
184C2×Dic14—232no—
185C2×Dic14—28no—
186C2×Dic14—28no—
187C2×Dic14C2×D4116yescomplemented
188C7×C4⋊C4—28no—
189C7×C22⋊C4—216no—
190C7×C22⋊C4—48no—
191C42×C7—28no—
192C22⋊Dic7—216no—
193C22⋊Dic7—216no—
194C22⋊Dic7—48no—
195C4⋊Dic7—28no—
196C4⋊Dic7—28no—
197C4⋊Dic7—28no—
198C4⋊Dic7—216no—
199Dic7⋊C4—216no—
200Dic7⋊C4—28no—
201Dic7⋊C4—28no—
202Dic7⋊C4—216no—
203Dic7⋊C4—48no—
204C4×Dic7—28no—
205C4×Dic7C2×D4116yes—
206C23×Q8—71no—
207C23×D4C2×D711yescharacteristic
208C24×C4—72no—
209C2×C22⋊Q8—716no—
210C2×C22⋊Q8—1416no—
211C2×C4×Q8—78no—
212C2×C4×Q8—148no—
213C2×C4×D4—78no—
214C2×C4×D4—148no—
215C22×C4⋊C4—71no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—71no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4—72no—
222C22×C4⋊C4—71no—
223C22×C4⋊C4—72no—
224C22×C4⋊C4C2×D711yescharacteristic
225C22×C22⋊C4—72no—
226C22×C22⋊C4—72no—
227C22×C22⋊C4—72no—
228C22×C22⋊C4C2×D712yes—
229C22×C22⋊C4C2×D712yes—
230C22×C42—71no—
231C22×C42—71no—
232C22×C42C2×D711yescharacteristic
233#396—716no—
234#396—716no—
235#390—78no—
236#390C2×D718yes—
237#389—78no—
238#389—78no—
239#384—78no—
240#380—716no—
241C2×#95—72no—
242C2×#95—72no—
243C23×C7—216no—
244C23×C7—216no—
245C23×C7—216no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—24no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—48no—
253C23×C7—48no—
254C23×C7—48no—
255C23×C7—48no—
256C23×C7C22×D412yes—
257C23×C7C22×D412yes—
258C23×C7C22×Q818yes—
259C23×C7C22×Q812yes—
260C23×C7C2×#4112yes—
261C23×C7C2×#4112yes—
262C23×C7H(5,𝔽2)12yes—
263C23×C72−1+412yes—
264C23×C7C2511yescharacteristic
265C7×D4—232no—
266C2×C4×C7—28no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—44no—
276C2×C4×C7—48no—
277C2×C4×C7C22⋊Q8116yescomplemented
278C2×C4×C7C22×D414yes—
279C2×C4×C7C22×D414yes—
280C2×C4×C7C22×Q814yes—
281C2×C4×C7C2×#4114yes—
282C2×Dic7—28no—
283C2×Dic7—28no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—48no—
298C2×Dic7—44no—
299C2×Dic7—48no—
300C2×Dic7—432no—
301C2×Dic7#121116yes—
302C2×Dic7C22×D414yes—
303C2×Dic7C22×D414yes—
304C2×Dic7C22×D414yes—
305C2×Dic7C22×D414yes—
306C2×Dic7C2×#4114yes—
307Dic14—232no—
308C25C22×D712yes—
309C22×Q8—78no—
310C22×Q8—74no—
311C22×Q8—144no—
312C22×Q8—144no—
313C22×D4—24no—
314C22×D4—24no—
315C22×D4Dic1418yescomplemented
316C22×D4C22×D714yes—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—78no—
321C23×C4—72no—
322C23×C4—71no—
323C23×C4—72no—
324C23×C4—148no—
325C23×C4—148no—
326C23×C4—148no—
327C23×C4C22×D712yes—
328C23×C4C22×D711yescharacteristic
329C23×C4C22×D711yescharacteristic
330C23×C4C22×D711yescharacteristic
331C22⋊Q8—1464no—
332C4×Q8—1432no—
333C4×D4—1432no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—74no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—78no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—78no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—148no—
348C2×C4⋊C4—148no—
349C2×C4⋊C4—144no—
350C2×C4⋊C4—148no—
351C2×C4⋊C4—144no—
352C2×C4⋊C4—284no—
353C2×C4⋊C4C22×D714yes—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—78no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—78no—
359C2×C22⋊C4—148no—
360C2×C22⋊C4—148no—
361C2×C22⋊C4—284no—
362C2×C22⋊C4C22×D718yes—
363C2×C22⋊C4C22×D718yes—
364C2×C42—24no—
365C2×C42—78no—
366C2×C42—74no—
367C2×C42—74no—
368C2×C42—144no—
369C2×C42C22×D714yes—
370#95—78no—
371#95—78no—
372C22×C7—28no—
373C22×C7—28no—
374C22×C7—28no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—44no—
378C22×C7—44no—
379C22×C7—432no—
380C22×C7C2×#12114yes—
381C22×C7C2×C22⋊Q814yes—
382C22×C7#53714yes—
383C22×C7D4×Q814yes—
384C22×C7#55014yes—
385C22×C7#55414yes—
386C22×C7#55614yes—
387C22×C7C23×D411yescharacteristic
388C22×C7C23×D411yescharacteristic
389C22×C7C23×Q811yescharacteristic
390C22×C7C22×#4111yescharacteristic
391C22×C7C22×#4111yescharacteristic
392C22×C7C2×H(5,𝔽2)11yescharacteristic
393C22×C7C2×2−1+411yescharacteristic
394C4×C7—24no—
395C4×C7—42no—
396C4×C7—44no—
397C4×C7C2×C22⋊Q818yescomplemented
398Dic7—24no—
399Dic7—24no—
400Dic7—44no—
401Dic7—42no—
402Dic7—44no—
403Dic7C2×#12118yes—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24—24no—
409C24—24no—
410C24C2×Dic1418yes—
411C24C23×D714yes—
412C24C23×D711yescharacteristic
413C2×Q8—716no—
414C2×Q8—148no—
415C2×Q8—1432no—
416C2×Q8—148no—
417C2×D4—232no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4C2×Dic14116yescomplemented
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—74no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—72no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—72no—
438C22×C4—74no—
439C22×C4—74no—
440C22×C4—1416no—
441C22×C4—1416no—
442C22×C4—1416no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—142no—
453C22×C4—144no—
454C22×C4—144no—
455C22×C4—142no—
456C22×C4—2832no—
457C22×C4C2×Dic1414yes—
458C22×C4D4×D714yes—
459C22×C4D4×D714yes—
460C22×C4#115914yes—
461C22×C4C23×D712yes—
462C22×C4C23×D714yes—
463C22×C4C23×D712yes—
464C22×C4C23×D712yes—
465C4⋊C4—28no—
466C4⋊C4—148no—
467C4⋊C4—148no—
468C4⋊C4—1416no—
469C4⋊C4—148no—
470C4⋊C4—1416no—
471C4⋊C4—1416no—
472C4⋊C4—148no—
473C4⋊C4—148no—
474C4⋊C4—288no—
475C22⋊C4—216no—
476C22⋊C4—48no—
477C22⋊C4—1416no—
478C22⋊C4—1416no—
479C22⋊C4—288no—
480C42—28no—
481C42—716no—
482C42—148no—
483C2×C7—48no—
484C2×C7#253818yescomplemented
485C2×C7C22×#12111yescharacteristic
486C2×C7C22×C22⋊Q811yescharacteristic
487C2×C7C2×#53711yescharacteristic
488C2×C7C2×D4×Q811yescharacteristic
489C2×C7C2×#55011yescharacteristic
490C2×C7C2×#55411yescharacteristic
491C2×C7C2×#55611yescharacteristic
492C23—216no—
493C23—216no—
494C23—216no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23C22×Dic1418yes—
506C23C22×Dic1412yes—
507C23C2×D4×D712yes—
508C23C2×D4×D712yes—
509C23C2×#115912yes—
510C23#654812yes—
511C23D7×#4112yes—
512C23#655412yes—
513C23C24×D711yescharacteristic
514Q8—1432no—
515D4—232no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—716no—
528C2×C4—74no—
529C2×C4—74no—
530C2×C4—74no—
531C2×C4—74no—
532C2×C4—74no—
533C2×C4—148no—
534C2×C4—148no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—144no—
541C2×C4—144no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—144no—
546C2×C4—144no—
547C2×C4—142no—
548C2×C4—2832no—
549C2×C4—288no—
550C2×C4—288no—
551C2×C4—284no—
552C2×C4#6441116yescomplemented
553C2×C4C22×Dic1414yes—
554C2×C4C2×D4×D714yes—
555C2×C4C2×D4×D714yes—
556C2×C4C2×#115914yes—
557C7C2×#253811yescharacteristic, complemented, Sylow
558C22—28no—
559C22—28no—
560C22—28no—
561C22—44no—
562C22—432no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22C2×#644114yes—
567C22D4×Dic1414yes—
568C22#9023014yes—
569C22D7×#12114yes—
570C22#9029914yes—
571C22#9036214yes—
572C22#9040914yes—
573C22C23×Dic1411yescharacteristic
574C22C22×D4×D711yescharacteristic
575C22C22×D4×D711yescharacteristic
576C22C22×#115911yescharacteristic
577C22C2×#654811yescharacteristic
578C22C2×D7×#4111yescharacteristic
579C22C2×#655411yescharacteristic
580C4—24no—
581C4—44no—
582C4—42no—
583C4—78no—
584C4—144no—
585C4—144no—
586C4—284no—
587C4—284no—
588C4—282no—
589C4C2×#644118yescomplemented
590C2—48no—
591C2#1173379118yesdirect product factor
592C2C22×#644111yescharacteristic
593C2C2×D4×Dic1411yescharacteristic
594C2C2×#9023011yescharacteristic
595C2C2×D7×#12111yescharacteristic
596C2C2×#9029911yescharacteristic
597C2C2×#9036211yescharacteristic
598C2C2×#9040911yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#390C2
socleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7C2×#2538
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#6441C2semidirect
3. C22×C4×Dic14C2semidirect
6. C2×#89757C2semidirect
7. C2×#89748C2semidirect
8. C2×#89720C2semidirect
9. C2×#89720C2semidirect
10. C2×#89714C2semidirect
11. C2×#89700C2semidirect
12. #11733791C2direct (N × H)
37. C2×#6441C22semidirect
39. C2×C4×Dic14C22semidirect
44. #89757C22semidirect
45. #89748C22semidirect
46. #89720C22semidirect
47. #89720C22semidirect
48. #89714C22semidirect
49. #89700C22semidirect
54. C22×C7×D4Q8semidirect
72. C22×Dic14D4semidirect
139. C2×C7×D4C2×Q8semidirect
187. C2×Dic14C2×D4semidirect
277. C2×C4×C7C22⋊Q8semidirect
315. C22×D4Dic14semidirect
397. C4×C7C2×C22⋊Q8semidirect
420. C2×D4C2×Dic14semidirect
484. C2×C7#2538semidirect
552. C2×C4#6441semidirect
557. C7C2×#2538semidirect
589. C4C2×#6441semidirect
591. C2#11733791direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic72
C22×#64412
C22×C4×Dic142
C2×C7×#3902
C2×#899932
C2×#897572
C2×#897482
C2×#897202
C2×#897142
C2×#897002
#117337912
C2×#25387