Finite Group Databasebeta

← C2×#11736323 #14223083

Subgroups — C2×#11736323 #14223083

Order 1792.

2398 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90523C211yesmaximal, characteristic
2C22×#6451C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C2×#90004C211yesmaximal, characteristic, complemented
5C2×#89905C211yesmaximal, characteristic, complemented
6C2×M16×Dic7C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic, complemented
8#11736323C2116yesmaximal, direct product factor
9C2×#89567C212yesmaximal, complemented
10C2×#89564C212yesmaximal, complemented
11C2×#89537C212yesmaximal, complemented
12C2×#89532C212yesmaximal, complemented
13C22×C7×#41C2211yescharacteristic
14C22×#1157C2211yescharacteristic
15C23×Dic14—21no—
16C22×C7×M16C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18#90523—24no—
19#90523C2218yes—
20C2×#6510C2211yescharacteristic
21C2×#6507C2211yescharacteristic
22C2×#6503C2211yescharacteristic
23C2×#6502—21no—
24C2×#6502—21no—
25C2×#6499C2211yescharacteristic
26C2×C4×C22⋊D7—21no—
27C2×C4×C22⋊D7—21no—
28C22×C4⋊Dic7C2211yescharacteristic
29C22×Dic7⋊C4—21no—
30C22×C4×Dic7C2211yescharacteristic
31C22×#1136C2211yescharacteristic
32C2×#6451—24no—
33C2×#6451—21no—
34C2×#6451—21no—
35C2×#6451C2212yes—
36C2×#6451C2212yes—
37C2×#6451C2218yescomplemented
38C7×#417C2218yescomplemented
39C2×C7×#103C2212yes—
40C2×C7×#102C2212yes—
41#90004C2218yescomplemented
42C2×#6409C2212yes—
43C2×#6406C2212yes—
44#89905C2218yescomplemented
45M16×Dic7C2218yescomplemented
46C2×#6395C2212yes—
47C2×#6390C2212yes—
48C2×#6389C2212yes—
49C2×C8×Dic7C2212yes—
50C2×#6384C2212yes—
51#89738C2218yescomplemented
52C2×#6383C2212yes—
53#89567C22116yescomplemented
54#89564C22116yescomplemented
55#89537C22116yescomplemented
56#89532C22116yescomplemented
57C2×#3141—71nomaximal, Sylow
58C2×C7×#41—22no—
59C2×C7×#41—24no—
60C2×C7×#41—24no—
61C2×C7×#41C2314yes—
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C22×C22⋊D7—21no—
69C22×C22⋊D7—21no—
70C22×C22⋊D7—21no—
71C23×Dic7—21no—
72C23×Dic7D411yescharacteristic
73C23×Dic7D411yescharacteristic
74C2×#1157—22no—
75C2×#1157—24no—
76C2×#1157—24no—
77C2×#1157C2314yes—
78C22×D28C2311yescharacteristic
79C22×C4×D7—21no—
80C22×Dic14—22no—
81C22×Dic14—22no—
82C22×Dic14—28no—
83C22×Dic14—48no—
84C22×Dic14C2311yescharacteristic
85C2×C7×M16—24no—
86C2×C7×M16C2314yes—
87C22×C7×C8C2312yes—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C4⋊C4C2314yes—
92C2×C22⋊Dic7—21no—
93C2×C22⋊Dic7—21no—
94#6510—22no—
95#6510C2314yes—
96#6507—22no—
97#6507C2314yes—
98#6503—22no—
99#6503C2314yes—
100#6502—28no—
101#6502—28no—
102#6499—22no—
103#6499C2314yes—
104C4×C22⋊D7—28no—
105C4×C22⋊D7—28no—
106C2×(C2×D7)⋊C4—21no—
107C2×(C2×D7)⋊C4—21no—
108C2×C4⋊Dic7—21no—
109C2×C4⋊Dic7—22no—
110C2×C4⋊Dic7C2312yes—
111C2×C4⋊Dic7C2314yes—
112C2×Dic7⋊C4—22no—
113C2×Dic7⋊C4—22no—
114C2×Dic7⋊C4—28no—
115C2×Dic7⋊C4—21no—
116C2×Dic7⋊C4—21no—
117C2×C4×Dic7D411yescharacteristic
118C2×C4×Dic7D411yescharacteristic
119C2×C4×Dic7D414yes—
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2311yescharacteristic
122C2×C4×Dic7C2314yes—
123C2×#1136—24no—
124C2×#1136C2314yes—
125C22×C7⋊C8C2312yes—
126#6451—24no—
127#6451—24no—
128#6451—28no—
129#6451—28no—
130#6451C2318yes—
131#6451C2318yes—
132C7×#103C2318yes—
133C7×#102C2318yes—
134#6409C2318yes—
135#6406C2318yes—
136#6395C2318yes—
137#6390C2318yes—
138#6389C2318yes—
139C8×Dic7C2318yes—
140#6384C2318yes—
141#6383C2318yes—
142C2×#532—71no—
143C22×C4⋊Q8—71no—
144#3141—716no—
145C2×#489—72no—
146C2×#489—72no—
147C2×#487—72no—
148C2×#486—72no—
149C2×#417—71no—
150C2×#417—71no—
151C2×#417—71no—
152C2×#417D711yescharacteristic, complemented
153C2×C4×M16—71no—
154C24×C7—21no—
155C24×C7C2×D411yescharacteristic
156C23×D7—21no—
157C7×#41—28no—
158C7×#41—28no—
159C7×#41—48no—
160C2×C7×Q8—21no—
161C2×C7×Q8—22no—
162C2×C7×Q8—22no—
163C2×C7×Q8C2412yes—
164C2×C7×D4—24no—
165C2×C7×D4—21no—
166C2×C7×D4—22no—
167C2×C7×D4—22no—
168C2×C7×D4—44no—
169C2×C7×D4C2412yes—
170C22×C4×C7—21no—
171C22×C4×C7—24no—
172C22×C4×C7—22no—
173C22×C4×C7—24no—
174C22×C4×C7—44no—
175C22×C4×C7—44no—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7#4114yes—
179C22×C4×C7#4114yes—
180C22×C4×C7C2411yescharacteristic
181C22×C4×C7C2412yes—
182C2×C22⋊D7—24no—
183C2×C22⋊D7—24no—
184C2×C22⋊D7—24no—
185C2×C22⋊D7—44no—
186C2×C22⋊D7—44no—
187C2×C22⋊D7—44no—
188C22×Dic7—24no—
189C22×Dic7—24no—
190C22×Dic7—24no—
191C22×Dic7—22no—
192C22×Dic7—21no—
193C22×Dic7—44no—
194C22×Dic7C2×D411yescharacteristic
195C22×Dic7C2×D414yes—
196C22×Dic7C2×D411yescharacteristic
197C22×Dic7C2×D414yes—
198C22×Dic7C2×D411yescharacteristic
199C22×Dic7C2×D411yescharacteristic
200#1157—28no—
201#1157—28no—
202#1157—48no—
203C2×D28—22no—
204C2×D28—22no—
205C2×D28—21no—
206C2×D28C2412yes—
207C2×C4×D7—24no—
208C2×C4×D7—44no—
209C2×Dic14—22no—
210C2×Dic14—22no—
211C2×Dic14—21no—
212C2×Dic14—28no—
213C2×Dic14—28no—
214C2×Dic14—48no—
215C2×Dic14—432no—
216C2×Dic14—48no—
217C2×Dic14C2412yes—
218C7×M16—28no—
219C2×C7×C8—24no—
220C2×C7×C8C2414yes—
221C7×C4⋊C4—22no—
222C7×C4⋊C4—24no—
223C7×C4⋊C4C2414yes—
224C22⋊Dic7—24no—
225C22⋊Dic7—24no—
226(C2×D7)⋊C4—24no—
227(C2×D7)⋊C4—24no—
228C4⋊Dic7—22no—
229C4⋊Dic7—24no—
230C4⋊Dic7C2414yes—
231Dic7⋊C4—28no—
232Dic7⋊C4—28no—
233Dic7⋊C4—24no—
234Dic7⋊C4—24no—
235C4×Dic7C2×D414yes—
236C4×Dic7C2×D414yes—
237C4×Dic7C2×D412yes—
238C4×Dic7C2×D412yes—
239C4×Dic7C2412yes—
240C4×Dic7C2412yes—
241#1136—28no—
242C2×C7⋊C8—24no—
243C2×C7⋊C8C2414yes—
244C22×#41—71no—
245C22×#41C2×D711yescharacteristic
246C23×Q8—141no—
247C22×M16—71no—
248C22×M16C2×D711yescharacteristic
249#532—78no—
250#532—144no—
251C2×C4⋊Q8—72no—
252C2×C4⋊Q8—72no—
253C2×C4⋊Q8—78no—
254C2×C4⋊Q8—71no—
255C2×C4⋊Q8—141no—
256C2×C4⋊Q8—141no—
257C2×C4⋊Q8—144no—
258C2×#127—71no—
259C2×#124—71no—
260C2×#124—71no—
261C2×#121—141no—
262C2×#121—141no—
263C2×C4×D4—141no—
264C2×C4×D4—141no—
265C22×C4⋊C4—71no—
266C22×C4⋊C4—141no—
267C22×C4⋊C4C2×D711yescharacteristic
268C22×C42—71no—
269#489—716no—
270#489—716no—
271#487—716no—
272#486—716no—
273#417—78no—
274#417—78no—
275#417—78no—
276#417C2×D718yescomplemented
277C2×#103—72no—
278C2×#103—72no—
279C2×#103—72no—
280C2×#103C2×D712yes—
281C2×#102—72no—
282C2×#102—72no—
283C2×#102—72no—
284C2×#102C2×D712yes—
285C4×M16—78no—
286C2×#97—72no—
287C2×C4×C8—72no—
288C23×C7—22no—
289C23×C7—22no—
290C23×C7—22no—
291C23×C7—42no—
292C23×C7—42no—
293C23×C7—42no—
294C23×C7#12414yes—
295C23×C7C22×D412yes—
296C23×C7C22×D411yescharacteristic
297C22×D7—22no—
298C22×D7—42no—
299C22×D7—42no—
300C22×D7—42no—
301C7×Q8—22no—
302C7×Q8—22no—
303C7×Q8—42no—
304C7×D4—22no—
305C7×D4—22no—
306C7×D4—48no—
307C7×D4—42no—
308C2×C4×C7—22no—
309C2×C4×C7—24no—
310C2×C4×C7—48no—
311C2×C4×C7—42no—
312C2×C4×C7—44no—
313C2×C4×C7—48no—
314C2×C4×C7#12414yes—
315C2×C4×C7#12418yes—
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D412yes—
318C2×C4×C7C2×#4112yes—
319C2×C4×C7C2×#4114yes—
320C2×C4×C7C2×#4112yes—
321C2×C4×C7C2×#4114yes—
322C2×C4×C7C2511yescharacteristic
323C22⋊D7—48no—
324C22⋊D7—48no—
325C22⋊D7—48no—
326C2×Dic7—22no—
327C2×Dic7—22no—
328C2×Dic7—28no—
329C2×Dic7—28no—
330C2×Dic7—24no—
331C2×Dic7—22no—
332C2×Dic7—48no—
333C2×Dic7—42no—
334C2×Dic7—44no—
335C2×Dic7AGL(1,ℤ/8)14yes—
336C2×Dic7#13714yes—
337C2×Dic7C22×D412yes—
338C2×Dic7C22×D412yes—
339C2×Dic7C22×D412yes—
340C2×Dic7C22×D412yes—
341D28—22no—
342D28—22no—
343D28—42no—
344C4×D7—48no—
345Dic14—22no—
346Dic14—22no—
347Dic14—416no—
348Dic14—42no—
349Dic14—416no—
350C7×C8—24no—
351C7⋊C8—24no—
352C2×#41—22no—
353C2×#41—24no—
354C2×#41—24no—
355C2×#41—74no—
356C2×#41—142no—
357C2×#41—144no—
358C2×#41—144no—
359C2×#41C22×D714yes—
360C22×Q8—71no—
361C22×Q8—148no—
362C22×Q8—142no—
363C22×Q8—142no—
364C22×Q8—288no—
365C22×Q8C22×D711yescharacteristic
366C22×D4—21no—
367C22×D4—71no—
368C22×D4—141no—
369C22×D4—141no—
370C22×D4—141no—
371C22×D4C22×D711yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—71no—
375C23×C4—71no—
376C23×C4—141no—
377C23×C4—141no—
378C23×C4C22×D711yescharacteristic
379C2×M16—24no—
380C2×M16—74no—
381C2×M16—144no—
382C2×M16C22×D714yes—
383C22×C8—72no—
384C22×C8C22×D712yes—
385C4⋊Q8—78no—
386C4⋊Q8—78no—
387C4⋊Q8—74no—
388C4⋊Q8—148no—
389C4⋊Q8—148no—
390C4⋊Q8—144no—
391C4⋊Q8—144no—
392C4⋊Q8—142no—
393#127—74no—
394#127—142no—
395#124—74no—
396#124—74no—
397#124—142no—
398#124—142no—
399#121—148no—
400#121—148no—
401C4×D4—148no—
402C4×D4—148no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—72no—
406C2×C4⋊C4—74no—
407C2×C4⋊C4—148no—
408C2×C4⋊C4—142no—
409C2×C4⋊C4—142no—
410C2×C4⋊C4—141no—
411C2×C4⋊C4—141no—
412C2×C4⋊C4—142no—
413C2×C4⋊C4—141no—
414C2×C4⋊C4C22×D712yes—
415C2×C4⋊C4C22×D714yes—
416C2×C22⋊C4—141no—
417C2×C22⋊C4—141no—
418C2×C22⋊C4—141no—
419C2×C22⋊C4—141no—
420C2×C42—71no—
421C2×C42—71no—
422C2×C42—74no—
423C2×C42—71no—
424C2×C42—71no—
425C2×C42—74no—
426#103—78no—
427#103—78no—
428#103—78no—
429#103C22×D718yes—
430#102—78no—
431#102—78no—
432#102—78no—
433#102C22×D718yes—
434#97—78no—
435C4×C8—78no—
436C22×C7—28no—
437C22×C7—22no—
438C22×C7—22no—
439C22×C7—42no—
440C22×C7—42no—
441C22×C7—42no—
442C22×C7—84no—
443C22×C7C2×#12414yes—
444C22×C7C2×#12412yes—
445C22×C7C2×AGL(1,ℤ/8)12yes—
446C22×C7C2×#13712yes—
447C22×C7C23×D411yescharacteristic
448C2×D7—42no—
449C2×D7—42no—
450C2×D7—42no—
451C2×D7—84no—
452C4×C7—42no—
453C4×C7—44no—
454C4×C7C2×#12412yes—
455C4×C7C2×#12414yes—
456C4×C7C22×#4111yescharacteristic
457C4×C7C22×#4111yescharacteristic
458Dic7—22no—
459Dic7—22no—
460Dic7—42no—
461Dic7—44no—
462Dic7C2×AGL(1,ℤ/8)14yes—
463Dic7C2×#13714yes—
464C24—21no—
465C24—141no—
466C24D4×D711yescharacteristic
467#41—28no—
468#41—28no—
469#41—48no—
470#41—148no—
471#41—148no—
472#41—288no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—72no—
477C2×Q8—148no—
478C2×Q8—148no—
479C2×Q8—141no—
480C2×Q8—142no—
481C2×Q8—142no—
482C2×Q8—288no—
483C2×Q8—2832no—
484C2×Q8—288no—
485C2×Q8C23×D712yes—
486C2×D4—24no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—44no—
491C2×D4—72no—
492C2×D4—144no—
493C2×D4—144no—
494C2×D4—144no—
495C2×D4—142no—
496C2×D4—141no—
497C2×D4—142no—
498C2×D4—284no—
499C2×D4—284no—
500C2×D4—284no—
501C2×D4C23×D712yes—
502C22×C4—24no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—71no—
509C22×C4—71no—
510C22×C4—74no—
511C22×C4—74no—
512C22×C4—71no—
513C22×C4—71no—
514C22×C4—144no—
515C22×C4—144no—
516C22×C4—144no—
517C22×C4—141no—
518C22×C4—142no—
519C22×C4—144no—
520C22×C4—284no—
521C22×C4—284no—
522C22×C4#115714yes—
523C22×C4D4×D712yes—
524C22×C4D4×D711yescharacteristic
525C22×C4#115914yes—
526C22×C4C23×D712yes—
527C22×C4C23×D711yescharacteristic
528M16—28no—
529M16—148no—
530C2×C8—24no—
531C2×C8—74no—
532C2×C8—144no—
533C2×C8C23×D714yes—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—74no—
537C4⋊C4—148no—
538C4⋊C4—148no—
539C4⋊C4—144no—
540C4⋊C4—144no—
541C4⋊C4—144no—
542C4⋊C4—142no—
543C4⋊C4C23×D714yes—
544C22⋊C4—144no—
545C22⋊C4—144no—
546C22⋊C4—144no—
547C22⋊C4—144no—
548C42—74no—
549C42—74no—
550C42—72no—
551C42—72no—
552C42—72no—
553C42—72no—
554C2×C7—24no—
555C2×C7—82no—
556C2×C7#314114yescomplemented
557C2×C7C22×#12411yescharacteristic
558C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
559C2×C7C22×#13711yescharacteristic
560D7—82no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—142no—
568C23—282no—
569C23—282no—
570C23—282no—
571C23#644814yes—
572C23C2×D4×D712yes—
573C23C2×D4×D711yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—42no—
577Q8—142no—
578Q8—142no—
579Q8—2816no—
580Q8—2816no—
581Q8—282no—
582D4—22no—
583D4—22no—
584D4—48no—
585D4—42no—
586D4—142no—
587D4—142no—
588D4—288no—
589D4—288no—
590D4—288no—
591D4—282no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—74no—
599C2×C4—74no—
600C2×C4—72no—
601C2×C4—72no—
602C2×C4—72no—
603C2×C4—72no—
604C2×C4—148no—
605C2×C4—148no—
606C2×C4—142no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—144no—
610C2×C4—288no—
611C2×C4—282no—
612C2×C4—284no—
613C2×C4—288no—
614C2×C4#644814yes—
615C2×C4#644818yes—
616C2×C4C2×#115712yes—
617C2×C4C2×#115714yes—
618C2×C4C2×D4×D712yes—
619C2×C4C2×D4×D711yescharacteristic
620C2×C4C2×#115912yes—
621C2×C4C2×#115914yes—
622C2×C4C24×D711yescharacteristic
623C8—24no—
624C8—144no—
625C7C2×#314111yescharacteristic, complemented, Sylow
626C22—22no—
627C22—28no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—282no—
634C22—282no—
635C22—282no—
636C22—564no—
637C22C2×#644814yes—
638C22C2×#644812yes—
639C22D7×AGL(1,ℤ/8)12yes—
640C22D7×#13712yes—
641C22C22×D4×D711yescharacteristic
642C4—44no—
643C4—42no—
644C4—74no—
645C4—74no—
646C4—142no—
647C4—142no—
648C4—284no—
649C4—282no—
650C4C2×#644814yes—
651C4C2×#644812yes—
652C4C22×#115711yescharacteristic
653C4C22×#115911yescharacteristic
654C2—24no—
655C2—82no—
656C2—562no—
657C2#1173632314yesdirect product factor
658C2C22×#644811yescharacteristic
659C2C2×D7×AGL(1,ℤ/8)11yescharacteristic
660C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC2×#90523C2
hypercenterC2×#417D7
hypocenterC7C2×#3141
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#6451C2semidirect
3. C2×C7×#417C2semidirect
4. C2×#90004C2semidirect
5. C2×#89905C2semidirect
6. C2×M16×Dic7C2semidirect
7. C2×#89738C2semidirect
8. #11736323C2direct (N × H)
9. C2×#89567C2semidirect
10. C2×#89564C2semidirect
11. C2×#89537C2semidirect
12. C2×#89532C2semidirect
37. C2×#6451C22semidirect
38. C7×#417C22semidirect
41. #90004C22semidirect
44. #89905C22semidirect
45. M16×Dic7C22semidirect
51. #89738C22semidirect
53. #89567C22semidirect
54. #89564C22semidirect
55. #89537C22semidirect
56. #89532C22semidirect
152. C2×#417D7semidirect
276. #417C2×D7semidirect
556. C2×C7#3141semidirect
625. C7C2×#3141semidirect
657. C2#11736323direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#905232
C22×#64512
C2×C7×#4172
C2×#900042
C2×#899052
C2×M16×Dic72
C2×#897382
#117363232
C2×#895672
C2×#895642
C2×#895372
C2×#895322
C2×#31417