Finite Group Databasebeta

← C2×#11731267 #13513012

Subgroups — C2×#11731267 #13513012

Order 1792.

4990 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic7C211yesmaximal, characteristic
2C23×C22⋊Dic7C211yesmaximal, characteristic
3C22×#6440C211yesmaximal, characteristic, complemented
4C2×C7×#379C211yesmaximal, characteristic
5C2×#89717C212yesmaximal, complemented
6C2×#89716C212yesmaximal, complemented
7C2×#89718C212yesmaximal, complemented
8C2×#89715C211yesmaximal, characteristic, complemented
9C2×#89714C211yesmaximal, characteristic, complemented
10C2×C22⋊C4×Dic7C212yesmaximal, complemented
11C2×C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
12#11731267C2116yesmaximal, direct product factor
13C26×C7C2211yescharacteristic
14C24×Dic7C2212yes—
15C24×Dic7C2211yescharacteristic
16C24×Dic7C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C22⋊Dic7—28no—
22C22×C22⋊Dic7C4116yescomplemented
23C22×C22⋊Dic7C2218yes—
24C22×C22⋊Dic7C2211yescharacteristic
25C22×C22⋊Dic7C2212yes—
26C22×C22⋊Dic7C2211yescharacteristic
27C22×C22⋊Dic7C2211yescharacteristic
28C22×C22⋊Dic7C2218yes—
29C22×C22⋊Dic7C2211yescharacteristic
30C22×C22⋊Dic7C2211yescharacteristic
31C22×C22⋊Dic7C2211yescharacteristic
32C22×Dic7⋊C4C2212yescomplemented
33C22×Dic7⋊C4C2212yescomplemented
34C22×C4×Dic7C2212yescomplemented
35C22×C4×Dic7C2212yescomplemented
36C2×#6440—28no—
37C2×#6440C2218yescomplemented
38C7×#379C2218yes—
39C2×C7⋊#95C2212yescomplemented
40C2×C7⋊#95C2212yescomplemented
41C2×C7⋊#95C2212yescomplemented
42C2×C7⋊#95C2212yescomplemented
43#89718C22116yescomplemented
44#89717C22116yescomplemented
45#89716C22116yescomplemented
46#89715C2218yescomplemented
47#89714C2218yescomplemented
48C22⋊C4×Dic7C22116yescomplemented
49C22⋊C4×Dic7C2218yescomplemented
50C2×#2342—71nomaximal, Sylow
51C25×C7—24no—
52C25×C7—28no—
53C25×C7—24no—
54C25×C7—44no—
55C25×C7C2×C418yes—
56C25×C7C2311yescharacteristic
57C25×C7C2314yes—
58C25×C7C2311yescharacteristic
59C25×C7C2311yescharacteristic
60C23×C4×C7C2312yes—
61C23×C4×C7C2312yes—
62C23×Dic7—24no—
63C23×Dic7—24no—
64C23×Dic7—28no—
65C23×Dic7—28no—
66C23×Dic7—44no—
67C23×Dic7—44no—
68C23×Dic7C2×C418yescomplemented
69C23×Dic7C2×C418yescomplemented
70C23×Dic7C2312yes—
71C23×Dic7C2318yes—
72C23×Dic7C2311yescharacteristic
73C23×Dic7C2311yescharacteristic
74C23×Dic7C2311yescharacteristic
75C23×Dic7C2311yescharacteristic
76C23×Dic7C2314yes—
77C23×Dic7C2312yes—
78C23×Dic7C2314yes—
79C2×C7×C22⋊C4—24no—
80C2×C7×C22⋊C4—28no—
81C2×C7×C22⋊C4—24no—
82C2×C7×C22⋊C4—44no—
83C2×C7×C22⋊C4C2314yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C7×C22⋊C4C2318yes—
86C2×C7×C22⋊C4C2318yes—
87C2×C22⋊Dic7—24no—
88C2×C22⋊Dic7—28no—
89C2×C22⋊Dic7—232no—
90C2×C22⋊Dic7C2×C418yes—
91C2×C22⋊Dic7C2×C418yes—
92C2×C22⋊Dic7C2×C4164yescomplemented
93C2×C22⋊Dic7C2×C418yescomplemented
94C2×C22⋊Dic7C2×C418yescomplemented
95C2×C22⋊Dic7D418yes—
96C2×C22⋊Dic7C2314yes—
97C2×C22⋊Dic7C2318yes—
98C2×C22⋊Dic7C2314yes—
99C2×C22⋊Dic7C2314yes—
100C2×C22⋊Dic7C2314yes—
101C2×C22⋊Dic7C2314yes—
102C2×C22⋊Dic7C2314yes—
103C2×Dic7⋊C4—28no—
104C2×Dic7⋊C4—44no—
105C2×Dic7⋊C4C2318yescomplemented
106C2×Dic7⋊C4C2318yescomplemented
107C2×C4×Dic7—28no—
108C2×C4×Dic7—44no—
109C2×C4×Dic7C2318yescomplemented
110C2×C4×Dic7C2318yescomplemented
111#6440—232no—
112C7⋊#95C2318yescomplemented
113C7⋊#95C2318yescomplemented
114C7⋊#95C2318yescomplemented
115C7⋊#95C2318yescomplemented
116C22×#117—71no—
117C23×C22⋊C4—71no—
118C23×C22⋊C4—71no—
119#2342—716no—
120C2×#388—72no—
121C2×#388—72no—
122C2×#385—72no—
123C2×#381—71no—
124C2×#380—71no—
125C2×#379D711yescharacteristic
126C2×C4×C22⋊C4—71no—
127C2×C4×C22⋊C4—72no—
128C24×C7—22no—
129C24×C7—24no—
130C24×C7—22no—
131C24×C7—24no—
132C24×C7—24no—
133C24×C7—232no—
134C24×C7—232no—
135C24×C7—28no—
136C24×C7—22no—
137C24×C7—216no—
138C24×C7—28no—
139C24×C7—24no—
140C24×C7—22no—
141C24×C7—24no—
142C24×C7—416no—
143C24×C7—48no—
144C24×C7—432no—
145C24×C7—432no—
146C24×C7C22×C414yes—
147C24×C7C22×C4116yes—
148C24×C7C22×C414yes—
149C24×C7C22×C414yes—
150C24×C7C2×D414yes—
151C24×C7#4118yes—
152C24×C7#4114yes—
153C24×C7C2412yes—
154C24×C7C2412yes—
155C24×C7C2412yes—
156C24×C7C2411yescharacteristic
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7—24no—
163C22×C4×C7C2414yes—
164C22×C4×C7C2414yes—
165C22×Dic7—232no—
166C22×Dic7—28no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—28no—
172C22×Dic7—24no—
173C22×Dic7—24no—
174C22×Dic7—24no—
175C22×Dic7—24no—
176C22×Dic7—24no—
177C22×Dic7—216no—
178C22×Dic7—24no—
179C22×Dic7—24no—
180C22×Dic7—216no—
181C22×Dic7—416no—
182C22×Dic7—48no—
183C22×Dic7—48no—
184C22×Dic7—432no—
185C22×Dic7C22×C414yes—
186C22×Dic7C22×C414yes—
187C22×Dic7C22×C414yes—
188C22×Dic7C22×C414yes—
189C22×Dic7C22×C4116yescomplemented
190C22×Dic7C22×C4116yescomplemented
191C22×Dic7C2×D418yes—
192C22×Dic7C2414yes—
193C22×Dic7C2412yes—
194C22×Dic7C2412yes—
195C22×Dic7C2412yes—
196C22×Dic7C2412yes—
197C22×Dic7C2414yes—
198C7×C22⋊C4—28no—
199C7×C22⋊C4—216no—
200C7×C22⋊C4—28no—
201C7×C22⋊C4—48no—
202C22⋊Dic7—28no—
203C22⋊Dic7—216no—
204C22⋊Dic7C22×C4116yes—
205C22⋊Dic7C22×C4116yes—
206C22⋊Dic7C22×C4116yescomplemented
207C22⋊Dic7C22×C4116yescomplemented
208C22⋊Dic7C2×D4116yes—
209Dic7⋊C4—216no—
210Dic7⋊C4—48no—
211C4×Dic7—216no—
212C4×Dic7—48no—
213C26C2×D711yescharacteristic
214C24×C4—71no—
215C24×C4—71no—
216C24×C4—72no—
217C2×#117—78no—
218C2×#117—148no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—72no—
221C22×C22⋊C4—716no—
222C22×C22⋊C4—78no—
223C22×C22⋊C4—71no—
224C22×C22⋊C4—71no—
225C22×C22⋊C4—71no—
226C22×C22⋊C4—71no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4—78no—
229C22×C22⋊C4—71no—
230C22×C22⋊C4—72no—
231C22×C22⋊C4—148no—
232C22×C22⋊C4C2×D711yescharacteristic
233C22×C22⋊C4C2×D711yescharacteristic
234C22×C22⋊C4C2×D712yes—
235C22×C22⋊C4C2×D712yes—
236C22×C42—72no—
237C22×C42—72no—
238#388—716no—
239#388—716no—
240#385—716no—
241#381—78no—
242#380—78no—
243#379C2×D718yes—
244C4×C22⋊C4—716no—
245C4×C22⋊C4—78no—
246C2×#95—72no—
247C2×#95—72no—
248C2×#95—72no—
249C2×#95—72no—
250C23×C7—28no—
251C23×C7—28no—
252C23×C7—28no—
253C23×C7—28no—
254C23×C7—216no—
255C23×C7—216no—
256C23×C7—216no—
257C23×C7—216no—
258C23×C7—22no—
259C23×C7—24no—
260C23×C7—22no—
261C23×C7—24no—
262C23×C7—24no—
263C23×C7—28no—
264C23×C7—232no—
265C23×C7—232no—
266C23×C7—28no—
267C23×C7—28no—
268C23×C7—22no—
269C23×C7—24no—
270C23×C7—22no—
271C23×C7—24no—
272C23×C7—416no—
273C23×C7—4128no—
274C23×C7—432no—
275C23×C7—432no—
276C23×C7—48no—
277C23×C7—48no—
278C23×C7#117116yescomplemented
279C23×C7C4×D4116yes—
280C23×C7C23×C414yes—
281C23×C7C23×C414yes—
282C23×C7C23×C414yes—
283C23×C7C23×C412yes—
284C23×C7C22×D414yes—
285C23×C7C22×D412yes—
286C23×C7C2×#4118yes—
287C23×C7C2×#4114yes—
288C23×C7C2×#4114yes—
289C23×C7C2×#4112yes—
290C23×C7H(5,𝔽2)14yes—
291C23×C7C2511yescharacteristic
292C2×C4×C7—24no—
293C2×C4×C7—24no—
294C2×C4×C7—24no—
295C2×C4×C7—24no—
296C2×C4×C7—24no—
297C2×C4×C7—24no—
298C2×C4×C7—48no—
299C2×C4×C7—48no—
300C2×Dic7—28no—
301C2×Dic7—28no—
302C2×Dic7—232no—
303C2×Dic7—28no—
304C2×Dic7—28no—
305C2×Dic7—216no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—416no—
318C2×Dic7—432no—
319C2×Dic7—48no—
320C2×Dic7C23×C414yes—
321C2×Dic7C23×C414yes—
322C2×Dic7C23×C414yes—
323C2×Dic7C23×C414yes—
324C2×Dic7C22×D418yes—
325C25—24no—
326C25—28no—
327C25—24no—
328C25—44no—
329C25C4×D718yes—
330C25C22×D711yescharacteristic
331C25C22×D714yes—
332C25C22×D711yescharacteristic
333C25C22×D711yescharacteristic
334C23×C4—78no—
335C23×C4—74no—
336C23×C4—71no—
337C23×C4—71no—
338C23×C4—72no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—78no—
342C23×C4—74no—
343C23×C4—72no—
344C23×C4—78no—
345C23×C4—148no—
346C23×C4—144no—
347C23×C4—144no—
348C23×C4—148no—
349C23×C4—284no—
350C23×C4—284no—
351C23×C4C22×D712yes—
352C23×C4C22×D712yes—
353#117—1432no—
354C2×C4⋊C4—78no—
355C2×C4⋊C4—78no—
356C2×C4⋊C4—148no—
357C2×C4⋊C4—284no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4—78no—
363C2×C22⋊C4—78no—
364C2×C22⋊C4—764no—
365C2×C22⋊C4—78no—
366C2×C22⋊C4—78no—
367C2×C22⋊C4—78no—
368C2×C22⋊C4—74no—
369C2×C22⋊C4—74no—
370C2×C22⋊C4—74no—
371C2×C22⋊C4—74no—
372C2×C22⋊C4—74no—
373C2×C22⋊C4—74no—
374C2×C22⋊C4—78no—
375C2×C22⋊C4—1432no—
376C2×C22⋊C4—148no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4C22×D714yes—
379C2×C22⋊C4C22×D714yes—
380C2×C22⋊C4C22×D718yes—
381C2×C22⋊C4C22×D718yes—
382C2×C42—78no—
383C2×C42—78no—
384C2×C42—148no—
385C2×C42—284no—
386#95—78no—
387#95—78no—
388#95—78no—
389#95—78no—
390C22×C7—232no—
391C22×C7—232no—
392C22×C7—24no—
393C22×C7—24no—
394C22×C7—24no—
395C22×C7—24no—
396C22×C7—28no—
397C22×C7—28no—
398C22×C7—28no—
399C22×C7—28no—
400C22×C7—24no—
401C22×C7—24no—
402C22×C7—24no—
403C22×C7—464no—
404C22×C7—416no—
405C22×C7—44no—
406C22×C7—44no—
407C22×C7C2×#11714yes—
408C22×C7C2×#11718yescomplemented
409C22×C7C2×C4×D418yes—
410C22×C7C2×C4×D414yes—
411C22×C7#51814yes—
412C22×C7#54618yes—
413C22×C7#55118yes—
414C22×C7C24×C411yescharacteristic
415C22×C7C23×D411yescharacteristic
416C22×C7C22×#4112yes—
417C22×C7C22×#4111yescharacteristic
418C22×C7C2×H(5,𝔽2)12yes—
419C4×C7—44no—
420C4×C7—44no—
421Dic7—24no—
422Dic7—24no—
423Dic7—24no—
424Dic7—24no—
425Dic7—28no—
426Dic7—44no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—24no—
431C24—24no—
432C24—232no—
433C24—232no—
434C24—28no—
435C24—22no—
436C24—216no—
437C24—28no—
438C24—24no—
439C24—22no—
440C24—24no—
441C24—416no—
442C24—48no—
443C24—432no—
444C24—432no—
445C24C2×C4×D714yes—
446C24C2×C4×D7116yes—
447C24C2×C4×D714yes—
448C24C2×C4×D714yes—
449C24D4×D714yes—
450C24#115918yes—
451C24#115914yes—
452C24C23×D712yes—
453C24C23×D712yes—
454C24C23×D712yes—
455C24C23×D711yescharacteristic
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—74no—
463C22×C4—74no—
464C22×C4—74no—
465C22×C4—74no—
466C22×C4—716no—
467C22×C4—78no—
468C22×C4—72no—
469C22×C4—72no—
470C22×C4—72no—
471C22×C4—72no—
472C22×C4—74no—
473C22×C4—716no—
474C22×C4—74no—
475C22×C4—1416no—
476C22×C4—1432no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—148no—
480C22×C4—148no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4—144no—
486C22×C4—144no—
487C22×C4—144no—
488C22×C4—1416no—
489C22×C4—144no—
490C22×C4—144no—
491C22×C4—2816no—
492C22×C4—288no—
493C22×C4—288no—
494C22×C4—2832no—
495C22×C4C23×D714yes—
496C22×C4C23×D714yes—
497C4⋊C4—1416no—
498C4⋊C4—288no—
499C22⋊C4—28no—
500C22⋊C4—216no—
501C22⋊C4—28no—
502C22⋊C4—48no—
503C22⋊C4—716no—
504C22⋊C4—716no—
505C22⋊C4—716no—
506C22⋊C4—716no—
507C22⋊C4—716no—
508C22⋊C4—1416no—
509C22⋊C4—148no—
510C42—1416no—
511C42—288no—
512C2×C7—28no—
513C2×C7—28no—
514C2×C7—44no—
515C2×C7#234218yescomplemented
516C2×C7C22×#11711yescharacteristic
517C2×C7C22×C4×D411yescharacteristic
518C2×C7C2×#51811yescharacteristic
519C2×C7C2×#54612yes—
520C2×C7C2×#55112yes—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—216no—
526C23—216no—
527C23—216no—
528C23—216no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—232no—
536C23—232no—
537C23—28no—
538C23—28no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—24no—
543C23—416no—
544C23—432no—
545C23—432no—
546C23—4128no—
547C23—48no—
548C23—48no—
549C23#6440116yescomplemented
550C23(C22⋊D7)⋊C4116yes—
551C23C22×C4×D714yes—
552C23C22×C4×D714yes—
553C23C22×C4×D714yes—
554C23C22×C4×D712yes—
555C23C2×D4×D714yes—
556C23C2×D4×D712yes—
557C23C2×#115918yes—
558C23C2×#115914yes—
559C23C2×#115914yes—
560C23C2×#115912yes—
561C23#654814yes—
562C23C24×D711yescharacteristic
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—74no—
572C2×C4—74no—
573C2×C4—74no—
574C2×C4—74no—
575C2×C4—78no—
576C2×C4—1432no—
577C2×C4—148no—
578C2×C4—148no—
579C2×C4—148no—
580C2×C4—148no—
581C2×C4—1416no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—144no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—144no—
588C2×C4—144no—
589C2×C4—144no—
590C2×C4—144no—
591C2×C4—144no—
592C2×C4—144no—
593C2×C4—2816no—
594C2×C4—2832no—
595C2×C4—288no—
596C7C2×#234211yescharacteristic, complemented, Sylow
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—232no—
606C22—232no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—464no—
611C22—44no—
612C22—44no—
613C22—416no—
614C22C2×#644014yes—
615C22C2×#644018yescomplemented
616C22C2×(C22⋊D7)⋊C418yes—
617C22C2×(C22⋊D7)⋊C414yes—
618C22#9017714yes—
619C22#9027818yes—
620C22#9028218yes—
621C22C23×C4×D711yescharacteristic
622C22C22×D4×D711yescharacteristic
623C22C22×#115911yescharacteristic
624C22C22×#115912yes—
625C22C2×#654812yes—
626C4—44no—
627C4—44no—
628C4—148no—
629C4—144no—
630C4—144no—
631C4—144no—
632C4—144no—
633C4—284no—
634C2—28no—
635C2—28no—
636C2—44no—
637C2#1173126718yesdirect product factor
638C2C22×#644011yescharacteristic
639C2C22×(C22⋊D7)⋊C411yescharacteristic
640C2C2×#9017711yescharacteristic
641C2C2×#9027812yes—
642C2C2×#9028212yes—
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC2×C7×#379C2
socleC24×C7C24
prime radicalC26×C7C22
cosocleC23×C7C25
hypercenterC2×#379D7
hypocenterC7C2×#2342
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
3. C22×#6440C2semidirect
5. C2×#89717C2semidirect
6. C2×#89716C2semidirect
7. C2×#89718C2semidirect
8. C2×#89715C2semidirect
9. C2×#89714C2semidirect
10. C2×C22⋊C4×Dic7C2semidirect
11. C2×C22⋊C4×Dic7C2semidirect
12. #11731267C2direct (N × H)
22. C22×C22⋊Dic7C4semidirect
32. C22×Dic7⋊C4C22semidirect
33. C22×Dic7⋊C4C22semidirect
34. C22×C4×Dic7C22semidirect
35. C22×C4×Dic7C22semidirect
37. C2×#6440C22semidirect
39. C2×C7⋊#95C22semidirect
40. C2×C7⋊#95C22semidirect
41. C2×C7⋊#95C22semidirect
42. C2×C7⋊#95C22semidirect
43. #89718C22semidirect
44. #89717C22semidirect
45. #89716C22semidirect
46. #89715C22semidirect
47. #89714C22semidirect
48. C22⋊C4×Dic7C22semidirect
49. C22⋊C4×Dic7C22semidirect
68. C23×Dic7C2×C4semidirect
69. C23×Dic7C2×C4semidirect
92. C2×C22⋊Dic7C2×C4semidirect
93. C2×C22⋊Dic7C2×C4semidirect
94. C2×C22⋊Dic7C2×C4semidirect
105. C2×Dic7⋊C4C23semidirect
106. C2×Dic7⋊C4C23semidirect
109. C2×C4×Dic7C23semidirect
110. C2×C4×Dic7C23semidirect
112. C7⋊#95C23semidirect
113. C7⋊#95C23semidirect
114. C7⋊#95C23semidirect
115. C7⋊#95C23semidirect
189. C22×Dic7C22×C4semidirect
190. C22×Dic7C22×C4semidirect
206. C22⋊Dic7C22×C4semidirect
207. C22⋊Dic7C22×C4semidirect
278. C23×C7#117semidirect
408. C22×C7C2×#117semidirect
515. C2×C7#2342semidirect
549. C23#6440semidirect
596. C7C2×#2342semidirect
615. C22C2×#6440semidirect
637. C2#11731267direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic72
C22×#64402
C2×C7×#3792
C2×#897172
C2×#897162
C2×#897182
C2×#897152
C2×#897142
C2×C22⋊C4×Dic72
#117312672
C2×#23427