Finite Group Databasebeta

← C2×#11731471 #13523315

Subgroups — C2×#11731471 #13523315

Order 1792.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D7C211yesmaximal, characteristic
2C22×#6489C211yesmaximal, characteristic, complemented
3C2×C7×#380C211yesmaximal, characteristic, complemented
4C2×#89759C211yesmaximal, characteristic, complemented
5C2×#89759C211yesmaximal, characteristic, complemented
6C2×#89757C211yesmaximal, characteristic, complemented
7C2×#89757C211yesmaximal, characteristic, complemented
8C2×#89725C212yesmaximal, complemented
9C2×#89733C212yesmaximal
10#11731471C2116yesmaximal, direct product factor
11C2×#89442C212yesmaximal, complemented
12C2×#89425C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C23×C22⋊D7C2211yescharacteristic
15C23×C4×D7C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2212yes—
19C22×C22⋊Dic7C2212yes—
20C22×C22⋊Dic7C2211yescharacteristic
21C2×C4×C22⋊D7—28no—
22C2×C4×C22⋊D7C2218yes—
23C22×(C2×D7)⋊C4C2211yescharacteristic
24C22×(C2×D7)⋊C4C2212yes—
25C22×(C2×D7)⋊C4C2212yes—
26C2×#6489—28no—
27C2×#6489C2218yescomplemented
28C22×C4⋊Dic7C2211yescharacteristic
29C22×C4⋊Dic7C2211yescharacteristic
30C22×C4⋊Dic7C2212yes—
31C22×C4⋊Dic7C2212yes—
32C22×Dic7⋊C4C2211yescharacteristic
33C22×C4×Dic7C2212yes—
34C22×C4×Dic7C2211yescharacteristic
35C7×#380C2218yescomplemented
36C2×C7×#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39C2×C7⋊#95C2212yes—
40C2×C7⋊#95C2212yes—
41#89759C2218yescomplemented
42#89759C2218yescomplemented
43#89757C2218yescomplemented
44#89757C2218yescomplemented
45#89733C22116yes—
46#89725C22116yescomplemented
47#89442C22116yescomplemented
48#89425C22116yescomplemented
49C2×#2485—71nomaximal, Sylow
50C25×C7C2311yescharacteristic
51C24×D7C2311yescharacteristic
52C23×C4×C7—24no—
53C23×C4×C7—24no—
54C23×C4×C7D418yes—
55C23×C4×C7C2312yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—24no—
61C22×C22⋊D7—24no—
62C22×C22⋊D7—24no—
63C22×C22⋊D7C2314yes—
64C23×Dic7C2312yes—
65C23×Dic7C2312yes—
66C23×Dic7C2311yescharacteristic
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C22×C4×D7—24no—
70C22×C4×D7—24no—
71C22×C4×D7—24no—
72C22×C4×D7C2314yes—
73C2×C7×C4⋊C4—24no—
74C2×C7×C4⋊C4—24no—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C4⋊C4C2314yes—
77C2×C7×C22⋊C4—44no—
78C2×C7×C22⋊C4C2318yes—
79C2×C22⋊Dic7—28no—
80C2×C22⋊Dic7—24no—
81C2×C22⋊Dic7C2318yes—
82C2×C22⋊Dic7C2314yes—
83C4×C22⋊D7—232no—
84C2×(C2×D7)⋊C4—28no—
85C2×(C2×D7)⋊C4—24no—
86C2×(C2×D7)⋊C4—44no—
87C2×(C2×D7)⋊C4C2314yes—
88C2×(C2×D7)⋊C4C2318yes—
89C2×(C2×D7)⋊C4C2318yes—
90#6489—232no—
91C2×C4⋊Dic7—24no—
92C2×C4⋊Dic7—24no—
93C2×C4⋊Dic7—28no—
94C2×C4⋊Dic7—44no—
95C2×C4⋊Dic7C2318yes—
96C2×C4⋊Dic7C2314yes—
97C2×C4⋊Dic7C2314yes—
98C2×C4⋊Dic7C2318yes—
99C2×Dic7⋊C4—24no—
100C2×Dic7⋊C4C2314yes—
101C2×C4×Dic7—28no—
102C2×C4×Dic7—24no—
103C2×C4×Dic7C2318yes—
104C2×C4×Dic7C2314yes—
105C7×#95C2318yes—
106C7⋊#95C2318yes—
107C7⋊#95C2318yes—
108C7⋊#95C2318yes—
109C7⋊#95C2318yes—
110C22×C4×D4—71no—
111C22×#117—71no—
112#2485—716no—
113C2×#397—72no—
114C2×#394—72no—
115C2×#389—71no—
116C2×#389—71no—
117C2×#388—72no—
118C2×#387—72no—
119C2×#380—71no—
120C2×#380—71no—
121C2×#380D711yescharacteristic, complemented
122C24×C7—22no—
123C24×C7—22no—
124C24×C7—22no—
125C24×C7—22no—
126C24×C7—22no—
127C24×C7—22no—
128C24×C7C2×D414yes—
129C24×C7C2412yes—
130C24×C7C2411yescharacteristic
131C23×D7—22no—
132C23×D7—22no—
133C23×D7—22no—
134C23×D7—22no—
135C23×D7—22no—
136C23×D7—22no—
137C23×D7#4114yes—
138C23×D7C2412yes—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—232no—
142C22×C4×C7—28no—
143C22×C4×C7—28no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D4116yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7#4114yes—
154C22×C4×C7#4114yes—
155C22×C4×C7C2414yes—
156C22×C4×C7C2414yes—
157C22×C4×C7C2412yes—
158C22×C4×C7C2412yes—
159C2×C22⋊D7—232no—
160C2×C22⋊D7—28no—
161C2×C22⋊D7—28no—
162C2×C22⋊D7—28no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—24no—
170C22×Dic7—24no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—24no—
175C22×Dic7—24no—
176C22×Dic7—22no—
177C22×Dic7#4114yes—
178C22×Dic7C2412yes—
179C22×Dic7C2414yes—
180C22×Dic7C2414yes—
181C22×Dic7C2412yes—
182C22×Dic7C2412yes—
183C2×C4×D7—28no—
184C2×C4×D7—28no—
185C2×C4×D7—28no—
186C2×C4×D7—416no—
187C7×C4⋊C4—28no—
188C7×C4⋊C4—28no—
189C7×C22⋊C4—48no—
190C22⋊Dic7—216no—
191C22⋊Dic7—28no—
192(C2×D7)⋊C4—216no—
193(C2×D7)⋊C4—28no—
194(C2×D7)⋊C4—48no—
195C4⋊Dic7—28no—
196C4⋊Dic7—28no—
197C4⋊Dic7—216no—
198C4⋊Dic7—48no—
199Dic7⋊C4—28no—
200C4×Dic7—216no—
201C4×Dic7—28no—
202C23×D4—71no—
203C24×C4—71no—
204C24×C4C2×D711yescharacteristic
205C2×C4×D4—78no—
206C2×C4×D4—148no—
207C2×#117—78no—
208C2×#117—148no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—71no—
212C22×C4⋊C4—71no—
213C22×C4⋊C4—72no—
214C22×C4⋊C4C2×D711yescharacteristic
215C22×C4⋊C4C2×D711yescharacteristic
216C22×C22⋊C4—72no—
217C22×C22⋊C4—71no—
218C22×C22⋊C4—72no—
219C22×C22⋊C4—71no—
220C22×C22⋊C4—72no—
221C22×C22⋊C4C2×D712yes—
222C22×C42—72no—
223C22×C42—71no—
224#397—716no—
225#394—716no—
226#389—78no—
227#389—78no—
228#388—716no—
229#387—716no—
230#380—78no—
231#380—78no—
232#380C2×D718yescomplemented
233C2×#95—72no—
234C2×#95—72no—
235C2×#95—72no—
236C2×#95—72no—
237C2×#95C2×D712yes—
238C23×C7—28no—
239C23×C7—28no—
240C23×C7—28no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—44no—
248C23×C7—44no—
249C23×C7—44no—
250C23×C7—44no—
251C23×C7C22×D414yes—
252C23×C7C22×D412yes—
253C23×C7C2×#4112yes—
254C23×C7C2×#4112yes—
255C23×C7C2×#4112yes—
256C23×C7C2×#4112yes—
257C23×C7H(5,𝔽2)12yes—
258C23×C72−1+412yes—
259C23×C7C2511yescharacteristic
260C22×D7—28no—
261C22×D7—28no—
262C22×D7—28no—
263C22×D7—22no—
264C22×D7—22no—
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—44no—
270C22×D7—44no—
271C22×D7—44no—
272C22×D7—44no—
273C22×D7C2×#4114yes—
274C2×C4×C7—28no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—232no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—48no—
285C2×C4×C7—48no—
286C2×C4×C7#123116yes—
287C2×C4×C7C22×D414yes—
288C2×C4×C7C22×D414yes—
289C2×C4×C7C2×#4114yes—
290C2×C4×C7C2×#4114yes—
291C22⋊D7—232no—
292C2×Dic7—28no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—22no—
307C2×Dic7—44no—
308C2×Dic7—48no—
309C2×Dic7—48no—
310C2×Dic7—44no—
311C2×Dic7C2×#4114yes—
312C4×D7—416no—
313C25—71no—
314C25C22×D711yescharacteristic
315C22×D4—74no—
316C22×D4—144no—
317C22×D4—144no—
318C22×D4—144no—
319C23×C4—24no—
320C23×C4—24no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—72no—
324C23×C4—72no—
325C23×C4—71no—
326C23×C4—74no—
327C23×C4—144no—
328C23×C4—144no—
329C23×C4—144no—
330C23×C4D2818yes—
331C23×C4C22×D712yes—
332C23×C4C22×D712yes—
333C23×C4C22×D714yes—
334C23×C4C22×D711yescharacteristic
335C23×C4C22×D711yescharacteristic
336C4×D4—1432no—
337#117—1432no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—78no—
344C2×C4⋊C4—78no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—284no—
350C2×C4⋊C4C22×D714yes—
351C2×C4⋊C4C22×D714yes—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—78no—
354C2×C22⋊C4—74no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—78no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—148no—
359C2×C22⋊C4—148no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—284no—
363C2×C22⋊C4C22×D718yes—
364C2×C42—78no—
365C2×C42—74no—
366C2×C42—148no—
367C2×C42—144no—
368#95—78no—
369#95—78no—
370#95—78no—
371#95—78no—
372#95C22×D718yes—
373C22×C7—24no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—416no—
381C22×C7C2×#12314yes—
382C22×C7#52514yes—
383C22×C7#53714yes—
384C22×C7#55214yes—
385C22×C7#55314yes—
386C22×C7#55514yes—
387C22×C7#55614yes—
388C22×C7C23×D411yescharacteristic
389C22×C7C22×#4111yescharacteristic
390C22×C7C22×#4111yescharacteristic
391C22×C7C22×#4111yescharacteristic
392C22×C7C22×#4111yescharacteristic
393C22×C7C2×H(5,𝔽2)11yescharacteristic
394C22×C7C2×2−1+411yescharacteristic
395C2×D7—24no—
396C2×D7—24no—
397C2×D7—24no—
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—42no—
401C2×D7—42no—
402C2×D7—416no—
403C4×C7—24no—
404C4×C7—44no—
405C4×C7—44no—
406C4×C7C2×#12318yes—
407Dic7—24no—
408Dic7—42no—
409Dic7—44no—
410Dic7—44no—
411Dic7—42no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—74no—
419C24—72no—
420C24—142no—
421C24—142no—
422C24—142no—
423C24—142no—
424C24—142no—
425C24—142no—
426C24C2×D2814yes—
427C24C23×D712yes—
428C24C23×D711yescharacteristic
429C2×D4—148no—
430C2×D4—148no—
431C2×D4—1432no—
432C2×D4—148no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—232no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—74no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—72no—
448C22×C4—74no—
449C22×C4—74no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—144no—
454C22×C4—144no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—144no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—144no—
462C22×C4—144no—
463C22×C4—144no—
464C22×C4—148no—
465C22×C4—148no—
466C22×C4—148no—
467C22×C4—2816no—
468C22×C4C2×D2814yes—
469C22×C4C2×D28116yes—
470C22×C4C2×D2814yes—
471C22×C4#115914yes—
472C22×C4#115914yes—
473C22×C4C23×D714yes—
474C22×C4C23×D714yes—
475C22×C4C23×D712yes—
476C22×C4C23×D712yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—1416no—
480C4⋊C4—148no—
481C4⋊C4—148no—
482C4⋊C4—148no—
483C4⋊C4—288no—
484C22⋊C4—48no—
485C22⋊C4—1416no—
486C22⋊C4—1416no—
487C22⋊C4—148no—
488C22⋊C4—148no—
489C22⋊C4—288no—
490C42—1416no—
491C42—148no—
492C2×C7—44no—
493C2×C7#248518yescomplemented
494C2×C7C22×#12311yescharacteristic
495C2×C7C2×#52511yescharacteristic
496C2×C7C2×#53711yescharacteristic
497C2×C7C2×#55211yescharacteristic
498C2×C7C2×#55311yescharacteristic
499C2×C7C2×#55511yescharacteristic
500C2×C7C2×#55611yescharacteristic
501D7—44no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—74no—
516C23—148no—
517C23—148no—
518C23—148no—
519C23—142no—
520C23—142no—
521C23—142no—
522C23—142no—
523C23—142no—
524C23—142no—
525C23—284no—
526C23—284no—
527C23—284no—
528C23—284no—
529C23C22×D2814yes—
530C23C22×D2812yes—
531C23C2×#115912yes—
532C23C2×#115912yes—
533C23#654812yes—
534C23#655112yes—
535C23D7×#4112yes—
536C23D7×#4112yes—
537C23C24×D711yescharacteristic
538D4—1432no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—232no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—74no—
552C2×C4—148no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—144no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—144no—
567C2×C4—284no—
568C2×C4—284no—
569C2×C4—288no—
570C2×C4—288no—
571C2×C4—2816no—
572C2×C4#6449116yes—
573C2×C4C22×D2814yes—
574C2×C4C22×D2814yes—
575C2×C4C2×#115914yes—
576C2×C4C2×#115914yes—
577C7C2×#248511yescharacteristic, complemented, Sylow
578C22—24no—
579C22—24no—
580C22—24no—
581C22—416no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—144no—
587C22—144no—
588C22—144no—
589C22—2816no—
590C22—282no—
591C22—282no—
592C22—282no—
593C22—282no—
594C22C2×#644914yes—
595C22#9020114yes—
596C22#9021514yes—
597C22#9030514yes—
598C22#9030714yes—
599C22#9033014yes—
600C22#9033214yes—
601C22C23×D2811yescharacteristic
602C22C22×#115911yescharacteristic
603C22C22×#115911yescharacteristic
604C22C2×#654811yescharacteristic
605C22C2×#655111yescharacteristic
606C22C2×D7×#4111yescharacteristic
607C22C2×D7×#4111yescharacteristic
608C4—24no—
609C4—44no—
610C4—44no—
611C4—144no—
612C4—282no—
613C4—282no—
614C4—284no—
615C4—284no—
616C4C2×#644918yes—
617C2—44no—
618C2—284no—
619C2#1173147118yesdirect product factor
620C2C22×#644911yescharacteristic
621C2C2×#9020111yescharacteristic
622C2C2×#9021511yescharacteristic
623C2C2×#9030711yescharacteristic
624C2C2×#9030511yescharacteristic
625C2C2×#9033211yescharacteristic
626C2C2×#9033011yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#380C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC2×#380D7
hypocenterC7C2×#2485
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#6489C2semidirect
3. C2×C7×#380C2semidirect
4. C2×#89759C2semidirect
5. C2×#89759C2semidirect
6. C2×#89757C2semidirect
7. C2×#89757C2semidirect
8. C2×#89725C2semidirect
10. #11731471C2direct (N × H)
11. C2×#89442C2semidirect
12. C2×#89425C2semidirect
27. C2×#6489C22semidirect
35. C7×#380C22semidirect
41. #89759C22semidirect
42. #89759C22semidirect
43. #89757C22semidirect
44. #89757C22semidirect
46. #89725C22semidirect
47. #89442C22semidirect
48. #89425C22semidirect
121. C2×#380D7semidirect
232. #380C2×D7semidirect
493. C2×C7#2485semidirect
577. C7C2×#2485semidirect
619. C2#11731471direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D72
C22×#64892
C2×C7×#3802
C2×#897592
C2×#897572
C2×#897252
C2×#897332
#117314712
C2×#894422
C2×#894252
C2×#24857