Finite Group Databasebeta

← C2×#11736358 #14220634

Subgroups — C2×#11736358 #14220634

Order 1792.

2810 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90523C211yesmaximal, characteristic
2C22×#6514C211yesmaximal, characteristic, complemented
3C22×#6472C211yesmaximal, characteristic, complemented
4C22×#6450C211yesmaximal, characteristic, complemented
5C2×C7×#417C211yesmaximal, characteristic, complemented
6C2×#89890C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic, complemented
8#11736358C2116yesmaximal, direct product factor
9C2×#89570C212yesmaximal, complemented
10C2×#89565C212yesmaximal, complemented
11C2×#89539C212yesmaximal, complemented
12C2×#89534C212yesmaximal, complemented
13C22×C7×#41C2211yescharacteristic
14C22×#1157C2211yescharacteristic
15C23×Dic14C2211yescharacteristic
16C22×C7×M16C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18#90523—24no—
19#90523C2218yes—
20C2×#6514—28no—
21C2×#6514C2218yescomplemented
22C2×#6510C2211yescharacteristic
23C2×#6507C2211yescharacteristic
24C22×#1144C2212yes—
25C2×#6503C2211yescharacteristic
26C2×#6502—21no—
27C2×#6502—21no—
28C2×#6499C2211yescharacteristic
29C22×#1142C2212yes—
30C2×C4×C22⋊D7—21no—
31C2×C4×C22⋊D7—21no—
32C22×Dic7⋊C4—21no—
33C22×C4×Dic7—21no—
34C22×C4×Dic7C2211yescharacteristic
35C22×#1136C2211yescharacteristic
36C2×#6472—28no—
37C2×#6472C2218yescomplemented
38C22×Dic28C2212yes—
39C22×#1132C2212yes—
40C2×#6450—24no—
41C2×#6450—21no—
42C2×#6450—21no—
43C2×#6450C2212yes—
44C2×#6450C2212yes—
45C2×#6450C2218yescomplemented
46C7×#417C2218yescomplemented
47C2×C7×#103C2212yes—
48C2×C7×#102C2212yes—
49#89890C2218yescomplemented
50C2×Dic7⋊C8C2212yes—
51C2×Dic7⋊C8C2212yes—
52C2×#6384C2212yes—
53#89738C2218yescomplemented
54C2×#6383C2212yes—
55#89570C22116yescomplemented
56#89565C22116yescomplemented
57#89539C22116yescomplemented
58#89534C22116yescomplemented
59C2×#3037—71nomaximal, Sylow
60C2×C7×#41—22no—
61C2×C7×#41—24no—
62C2×C7×#41—24no—
63C2×C7×#41C2314yes—
64C22×C7×Q8C2311yescharacteristic
65C22×C7×D4—21no—
66C22×C7×D4C2311yescharacteristic
67C23×C4×C7—21no—
68C23×C4×C7—21no—
69C23×C4×C7C2311yescharacteristic
70C22×C22⋊D7—21no—
71C22×C22⋊D7—21no—
72C22×C22⋊D7—21no—
73C23×Dic7—21no—
74C23×Dic7D411yescharacteristic
75C23×Dic7D411yescharacteristic
76C2×#1157—22no—
77C2×#1157—24no—
78C2×#1157—24no—
79C2×#1157C2314yes—
80C22×D28C2311yescharacteristic
81C22×C4×D7—21no—
82C22×Dic14—21no—
83C22×Dic14—22no—
84C22×Dic14—24no—
85C22×Dic14D418yescomplemented
86C22×Dic14C2311yescharacteristic
87C22×Dic14C2312yes—
88C22×Dic14C2314yes—
89C2×C7×M16—24no—
90C2×C7×M16C2314yes—
91C22×C7×C8C2312yes—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2314yes—
96C2×C22⋊Dic7—21no—
97C2×C22⋊Dic7—21no—
98#6514—232no—
99#6510—22no—
100#6510C2314yes—
101#6507—22no—
102#6507C2314yes—
103C2×#1144—28no—
104C2×#1144C2318yes—
105#6503—22no—
106#6503C2314yes—
107#6502—28no—
108#6502—28no—
109#6499—22no—
110#6499C2314yes—
111C2×#1142—28no—
112C2×#1142C2318yes—
113C4×C22⋊D7—28no—
114C4×C22⋊D7—28no—
115C2×(C2×D7)⋊C4—21no—
116C2×(C2×D7)⋊C4—21no—
117C2×Dic7⋊C4—22no—
118C2×Dic7⋊C4—22no—
119C2×Dic7⋊C4—28no—
120C2×Dic7⋊C4—21no—
121C2×Dic7⋊C4—21no—
122C2×C4×Dic7—22no—
123C2×C4×Dic7—22no—
124C2×C4×Dic7—28no—
125C2×C4×Dic7D411yescharacteristic
126C2×C4×Dic7D411yescharacteristic
127C2×C4×Dic7D414yes—
128C2×C4×Dic7C2311yescharacteristic
129C2×C4×Dic7C2311yescharacteristic
130C2×C4×Dic7C2314yes—
131C2×#1136—24no—
132C2×#1136C2314yes—
133C22×C7⋊C8C2312yes—
134#6472—232no—
135C2×Dic28—28no—
136C2×Dic28C2318yes—
137C2×#1132—28no—
138C2×#1132C2318yes—
139#6450—24no—
140#6450—24no—
141#6450—28no—
142#6450—28no—
143#6450C2318yes—
144#6450C2318yes—
145C7×#103C2318yes—
146C7×#102C2318yes—
147Dic7⋊C8C2318yes—
148Dic7⋊C8C2318yes—
149#6384C2318yes—
150#6383C2318yes—
151C22×#137—71no—
152C22×#137—71no—
153C2×#532—71no—
154C22×C4×Q8—71no—
155#3037—716no—
156C2×#464—72no—
157C2×#464—72no—
158C2×#462—72no—
159C2×#460—72no—
160C2×#423—71no—
161C2×#417—71no—
162C2×#417D711yescharacteristic, complemented
163C24×C7—21no—
164C24×C7C2×D411yescharacteristic
165C23×D7—21no—
166C7×#41—28no—
167C7×#41—28no—
168C7×#41—48no—
169C2×C7×Q8—21no—
170C2×C7×Q8—22no—
171C2×C7×Q8—22no—
172C2×C7×Q8C2412yes—
173C2×C7×D4—24no—
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—22no—
177C2×C7×D4—44no—
178C2×C7×D4C2412yes—
179C22×C4×C7—24no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—24no—
183C22×C4×C7—44no—
184C22×C4×C7—44no—
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D414yes—
188C22×C4×C7#4114yes—
189C22×C4×C7C2411yescharacteristic
190C22×C4×C7C2412yes—
191C2×C22⋊D7—24no—
192C2×C22⋊D7—24no—
193C2×C22⋊D7—24no—
194C2×C22⋊D7—44no—
195C2×C22⋊D7—44no—
196C2×C22⋊D7—44no—
197C22×Dic7—28no—
198C22×Dic7—22no—
199C22×Dic7—24no—
200C22×Dic7—24no—
201C22×Dic7—21no—
202C22×Dic7—42no—
203C22×Dic7C2×D411yescharacteristic
204C22×Dic7C2×D414yes—
205C22×Dic7C2×D411yescharacteristic
206C22×Dic7C2×D411yescharacteristic
207C22×Dic7C2×D414yes—
208C22×Dic7C2×D411yescharacteristic
209#1157—28no—
210#1157—28no—
211#1157—48no—
212C2×D28—22no—
213C2×D28—22no—
214C2×D28—21no—
215C2×D28C2412yes—
216C2×C4×D7—24no—
217C2×C4×D7—44no—
218C2×Dic14—216no—
219C2×Dic14—28no—
220C2×Dic14—22no—
221C2×Dic14—24no—
222C2×Dic14—24no—
223C2×Dic14—28no—
224C2×Dic14—22no—
225C2×Dic14—22no—
226C2×Dic14—21no—
227C2×Dic14C2×D418yes—
228C2×Dic14C2×D4116yescomplemented
229C2×Dic14C2412yes—
230C2×Dic14C2414yes—
231C7×M16—28no—
232C2×C7×C8—24no—
233C2×C7×C8C2414yes—
234C7×C4⋊C4—22no—
235C7×C4⋊C4—24no—
236C7×C4⋊C4C2414yes—
237C22⋊Dic7—24no—
238C22⋊Dic7—24no—
239#1144—216no—
240#1142—216no—
241(C2×D7)⋊C4—24no—
242(C2×D7)⋊C4—24no—
243Dic7⋊C4—28no—
244Dic7⋊C4—28no—
245Dic7⋊C4—24no—
246Dic7⋊C4—24no—
247C4×Dic7—28no—
248C4×Dic7—28no—
249C4×Dic7C2×D414yes—
250C4×Dic7C2×D414yes—
251C4×Dic7C2×D412yes—
252C4×Dic7C2×D412yes—
253C4×Dic7C2412yes—
254C4×Dic7C2412yes—
255#1136—28no—
256C2×C7⋊C8—24no—
257C2×C7⋊C8C2414yes—
258Dic28—216no—
259#1132—216no—
260C22×#41—71no—
261C22×#41C2×D711yescharacteristic
262C23×Q8—71no—
263C2×#137—78no—
264C2×#137—78no—
265C2×#137—148no—
266C2×#137—148no—
267C22×Q16—72no—
268C22×Q16—72no—
269C22×SD16—72no—
270C22×SD16—72no—
271C22×M16—71no—
272C22×M16C2×D711yescharacteristic
273#532—78no—
274#532—144no—
275C2×C4⋊Q8—71no—
276C2×#127—71no—
277C2×#124—71no—
278C2×#124—71no—
279C2×#121—141no—
280C2×#121—141no—
281C2×C4×Q8—72no—
282C2×C4×Q8—72no—
283C2×C4×Q8—78no—
284C2×C4×Q8—141no—
285C2×C4×Q8—141no—
286C2×C4×Q8—144no—
287C2×C4×D4—141no—
288C2×C4×D4—141no—
289C22×C4⋊C4—141no—
290C22×C4⋊C4C2×D711yescharacteristic
291C22×C42—71no—
292C22×C42—141no—
293#464—716no—
294#464—716no—
295#462—716no—
296#460—716no—
297#423—78no—
298C2×C4⋊C8—72no—
299C2×C4⋊C8—72no—
300#417—78no—
301#417C2×D718yescomplemented
302C2×#103—72no—
303C2×#103C2×D712yes—
304C2×#102—72no—
305C2×#102C2×D712yes—
306C23×C7—22no—
307C23×C7—22no—
308C23×C7—22no—
309C23×C7—42no—
310C23×C7—42no—
311C23×C7—42no—
312C23×C7#12114yes—
313C23×C7C22×D412yes—
314C23×C7C22×D411yescharacteristic
315C22×D7—22no—
316C22×D7—42no—
317C22×D7—42no—
318C22×D7—42no—
319C7×Q8—22no—
320C7×Q8—22no—
321C7×Q8—42no—
322C7×D4—22no—
323C7×D4—22no—
324C7×D4—48no—
325C7×D4—42no—
326C2×C4×C7—24no—
327C2×C4×C7—22no—
328C2×C4×C7—48no—
329C2×C4×C7—42no—
330C2×C4×C7—44no—
331C2×C4×C7—48no—
332C2×C4×C7#12114yes—
333C2×C4×C7#12118yes—
334C2×C4×C7C22×D411yescharacteristic
335C2×C4×C7C22×D412yes—
336C2×C4×C7C22×D412yes—
337C2×C4×C7C22×D414yes—
338C2×C4×C7C2×#4112yes—
339C2×C4×C7C2×#4114yes—
340C2×C4×C7C2511yescharacteristic
341C22⋊D7—48no—
342C22⋊D7—48no—
343C22⋊D7—48no—
344C2×Dic7—28no—
345C2×Dic7—216no—
346C2×Dic7—22no—
347C2×Dic7—22no—
348C2×Dic7—22no—
349C2×Dic7—24no—
350C2×Dic7—28no—
351C2×Dic7—22no—
352C2×Dic7—44no—
353C2×Dic7—42no—
354C2×Dic7#13714yescomplemented
355C2×Dic7C22×D412yes—
356C2×Dic7C22×D412yes—
357C2×Dic7C22×D412yes—
358C2×Dic7C22×D412yes—
359D28—22no—
360D28—22no—
361D28—42no—
362C4×D7—48no—
363Dic14—28no—
364Dic14—216no—
365Dic14—24no—
366Dic14—22no—
367Dic14—22no—
368Dic14—42no—
369Dic14C22×D418yes—
370C7×C8—24no—
371C7⋊C8—24no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—24no—
375C2×#41—74no—
376C2×#41—142no—
377C2×#41—144no—
378C2×#41—144no—
379C2×#41C22×D714yes—
380C22×Q8—78no—
381C22×Q8—72no—
382C22×Q8—74no—
383C22×Q8—71no—
384C22×Q8—144no—
385C22×Q8—142no—
386C22×Q8—141no—
387C22×Q8C22×D711yescharacteristic
388C22×D4—21no—
389C22×D4—71no—
390C22×D4—141no—
391C22×D4—141no—
392C22×D4—141no—
393C22×D4C22×D711yescharacteristic
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—71no—
397C23×C4—71no—
398C23×C4—141no—
399C23×C4—141no—
400C23×C4C22×D711yescharacteristic
401#137—1432no—
402#137—1432no—
403C2×Q16—78no—
404C2×Q16—78no—
405C2×Q16—148no—
406C2×Q16—148no—
407C2×SD16—78no—
408C2×SD16—78no—
409C2×SD16—148no—
410C2×SD16—148no—
411C2×M16—24no—
412C2×M16—74no—
413C2×M16—144no—
414C2×M16C22×D714yes—
415C22×C8—72no—
416C22×C8C22×D712yes—
417C4⋊Q8—74no—
418C4⋊Q8—142no—
419#127—74no—
420#127—142no—
421#124—74no—
422#124—74no—
423#124—142no—
424#124—142no—
425#121—148no—
426#121—148no—
427C4×Q8—78no—
428C4×Q8—78no—
429C4×Q8—148no—
430C4×Q8—148no—
431C4×Q8—144no—
432C4×Q8—144no—
433C4×D4—148no—
434C4×D4—148no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—148no—
438C2×C4⋊C4—142no—
439C2×C4⋊C4—142no—
440C2×C4⋊C4—141no—
441C2×C4⋊C4—141no—
442C2×C4⋊C4C22×D712yes—
443C2×C4⋊C4C22×D714yes—
444C2×C22⋊C4—141no—
445C2×C22⋊C4—141no—
446C2×C22⋊C4—141no—
447C2×C22⋊C4—141no—
448C2×C42—71no—
449C2×C42—71no—
450C2×C42—74no—
451C2×C42—71no—
452C2×C42—71no—
453C2×C42—74no—
454C2×C42—148no—
455C2×C42—142no—
456C2×C42—142no—
457C4⋊C8—78no—
458C4⋊C8—78no—
459#103—78no—
460#103C22×D718yes—
461#102—78no—
462#102C22×D718yes—
463C22×C7—28no—
464C22×C7—22no—
465C22×C7—22no—
466C22×C7—42no—
467C22×C7—42no—
468C22×C7—42no—
469C22×C7—84no—
470C22×C7C2×#12114yes—
471C22×C7C2×#12112yes—
472C22×C7C2×#13712yes—
473C22×C7#57512yes—
474C22×C7C23×D411yescharacteristic
475C2×D7—42no—
476C2×D7—42no—
477C2×D7—42no—
478C2×D7—84no—
479C4×C7—42no—
480C4×C7—44no—
481C4×C7C2×#12112yes—
482C4×C7C2×#12114yes—
483C4×C7C23×D411yescharacteristic
484C4×C7C22×#4111yescharacteristic
485Dic7—28no—
486Dic7—22no—
487Dic7—22no—
488Dic7—22no—
489Dic7—42no—
490Dic7C2×#13714yescomplemented
491C24—21no—
492C24—141no—
493C24D4×D711yescharacteristic
494#41—28no—
495#41—28no—
496#41—48no—
497#41—148no—
498#41—148no—
499#41—288no—
500C2×Q8—21no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—78no—
504C2×Q8—716no—
505C2×Q8—74no—
506C2×Q8—72no—
507C2×Q8—148no—
508C2×Q8—1416no—
509C2×Q8—144no—
510C2×Q8—148no—
511C2×Q8—144no—
512C2×Q8—142no—
513C2×Q8—141no—
514C2×Q8—142no—
515C2×Q8—142no—
516C2×Q8C23×D712yes—
517C2×D4—24no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—44no—
522C2×D4—72no—
523C2×D4—144no—
524C2×D4—144no—
525C2×D4—144no—
526C2×D4—142no—
527C2×D4—142no—
528C2×D4—141no—
529C2×D4—284no—
530C2×D4—284no—
531C2×D4—284no—
532C2×D4C23×D712yes—
533C22×C4—24no—
534C22×C4—22no—
535C22×C4—21no—
536C22×C4—24no—
537C22×C4—44no—
538C22×C4—44no—
539C22×C4—71no—
540C22×C4—74no—
541C22×C4—71no—
542C22×C4—71no—
543C22×C4—71no—
544C22×C4—74no—
545C22×C4—144no—
546C22×C4—148no—
547C22×C4—144no—
548C22×C4—142no—
549C22×C4—141no—
550C22×C4—144no—
551C22×C4—282no—
552C22×C4—284no—
553C22×C4#115714yes—
554C22×C4D4×D712yes—
555C22×C4D4×D711yescharacteristic
556C22×C4D4×D714yes—
557C22×C4C23×D712yes—
558C22×C4C23×D711yescharacteristic
559Q16—1416no—
560Q16—1416no—
561SD16—1416no—
562SD16—1416no—
563M16—28no—
564M16—148no—
565C2×C8—24no—
566C2×C8—74no—
567C2×C8—144no—
568C2×C8C23×D714yes—
569C4⋊C4—24no—
570C4⋊C4—22no—
571C4⋊C4—148no—
572C4⋊C4—148no—
573C4⋊C4—144no—
574C4⋊C4—144no—
575C4⋊C4C23×D714yes—
576C22⋊C4—144no—
577C22⋊C4—144no—
578C22⋊C4—144no—
579C22⋊C4—144no—
580C42—74no—
581C42—74no—
582C42—72no—
583C42—72no—
584C42—72no—
585C42—72no—
586C42—148no—
587C42—148no—
588C2×C7—24no—
589C2×C7—82no—
590C2×C7#303714yescomplemented
591C2×C7C22×#12111yescharacteristic
592C2×C7C22×#13711yescharacteristic
593C2×C7C2×#57511yescharacteristic
594D7—82no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—142no—
602C23—282no—
603C23—282no—
604C23—282no—
605C23#644714yes—
606C23C2×D4×D712yes—
607C23C2×D4×D711yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—42no—
611Q8—78no—
612Q8—1416no—
613Q8—148no—
614Q8—144no—
615Q8—142no—
616Q8—142no—
617Q8—282no—
618D4—22no—
619D4—22no—
620D4—48no—
621D4—42no—
622D4—142no—
623D4—142no—
624D4—288no—
625D4—288no—
626D4—288no—
627D4—282no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—74no—
635C2×C4—72no—
636C2×C4—72no—
637C2×C4—72no—
638C2×C4—72no—
639C2×C4—148no—
640C2×C4—1416no—
641C2×C4—148no—
642C2×C4—142no—
643C2×C4—142no—
644C2×C4—142no—
645C2×C4—144no—
646C2×C4—142no—
647C2×C4—284no—
648C2×C4—282no—
649C2×C4—288no—
650C2×C4#644714yes—
651C2×C4#644718yes—
652C2×C4C2×#115712yes—
653C2×C4C2×#115714yes—
654C2×C4C2×D4×D712yes—
655C2×C4C2×D4×D711yescharacteristic
656C2×C4C2×D4×D712yes—
657C2×C4C2×D4×D714yes—
658C2×C4C24×D711yescharacteristic
659C8—24no—
660C8—144no—
661C7C2×#303711yescharacteristic, complemented, Sylow
662C22—22no—
663C22—22no—
664C22—28no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—84no—
669C22—282no—
670C22—282no—
671C22—282no—
672C22—564no—
673C22C2×#644714yes—
674C22C2×#644712yes—
675C22#9046412yes—
676C22D7×#13712yes—
677C22C22×D4×D711yescharacteristic
678C4—44no—
679C4—42no—
680C4—74no—
681C4—142no—
682C4—142no—
683C4—142no—
684C4—148no—
685C4—282no—
686C4C2×#644714yes—
687C4C2×#644712yes—
688C4C22×#115711yescharacteristic
689C4C22×D4×D711yescharacteristic
690C2—24no—
691C2—82no—
692C2—562no—
693C2#1173635814yesdirect product factor
694C2C22×#644711yescharacteristic
695C2C2×#9046411yescharacteristic
696C2C2×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×#3037
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×#6514C2semidirect
3. C22×#6472C2semidirect
4. C22×#6450C2semidirect
5. C2×C7×#417C2semidirect
6. C2×#89890C2semidirect
7. C2×#89738C2semidirect
8. #11736358C2direct (N × H)
9. C2×#89570C2semidirect
10. C2×#89565C2semidirect
11. C2×#89539C2semidirect
12. C2×#89534C2semidirect
21. C2×#6514C22semidirect
37. C2×#6472C22semidirect
45. C2×#6450C22semidirect
46. C7×#417C22semidirect
49. #89890C22semidirect
53. #89738C22semidirect
55. #89570C22semidirect
56. #89565C22semidirect
57. #89539C22semidirect
58. #89534C22semidirect
85. C22×Dic14D4semidirect
162. C2×#417D7semidirect
228. C2×Dic14C2×D4semidirect
301. #417C2×D7semidirect
354. C2×Dic7#137semidirect
490. Dic7C2×#137semidirect
590. C2×C7#3037semidirect
661. C7C2×#3037semidirect
693. C2#11736358direct (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×#65142
C22×#64722
C22×#64502
C2×C7×#4172
C2×#898902
C2×#897382
#117363582
C2×#895702
C2×#895652
C2×#895392
C2×#895342
C2×#30377