Finite Group Databasebeta

← C2×#11737647 #14225051

Subgroups — C2×#11737647 #14225051

Order 1792.

2242 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6510C211yesmaximal, characteristic
2C2×#90313C211yesmaximal, characteristic, complemented
3C2×C7×#451C211yesmaximal, characteristic, complemented
4C2×#90000C211yesmaximal, characteristic, complemented
5C2×#89962C212yesmaximal, complemented
6C2×#89957C212yesmaximal, complemented
7C2×#89818C211yesmaximal, characteristic
8#11737647C2116yesmaximal, direct product factor
9C2×#89570C212yesmaximal, complemented
10C2×#89567C212yesmaximal, complemented
11C2×#89503C211yesmaximal, characteristic
12C2×#89491C211yesmaximal, characteristic, complemented
13C23×C7×Q8C2211yescharacteristic
14C23×D28C2211yescharacteristic
15C22×C7×Q16C2212yescomplemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×#6510—21no—
18C2×#6510—21no—
19C2×#6510—24no—
20C2×#6510C2218yes—
21C2×#6510C2212yes—
22C2×#6510C2212yes—
23C2×Q8×Dic7C2211yescharacteristic, complemented
24C2×#6507C2211yescharacteristic, complemented
25C22×#1144C2212yescomplemented
26C22×#1143C2212yes—
27C22×(C2×D7)⋊C4—21no—
28C22×(C2×D7)⋊C4—21no—
29C2×#6487C2211yescharacteristic
30C22×C4×Dic7C2211yescharacteristic
31C22×#1132C2212yes—
32#90313—24no—
33#90313C2218yescomplemented
34C2×#6452—21no—
35C2×#6451C2211yescharacteristic, complemented
36C2×#6450—21no—
37C2×#6450C2211yescharacteristic, complemented
38C2×#6441—21no—
39C2×#6440—21no—
40C7×#451C2218yescomplemented
41C2×C7×#103C2212yescomplemented
42C2×C7×C22⋊C8C2211yescharacteristic
43#90000C2218yescomplemented
44C2×#6409C2212yescomplemented
45C2×C22⋊(C7⋊C8)C2211yescharacteristic
46#89962C22116yescomplemented
47#89957C22116yescomplemented
48C2×#6395C2212yes—
49C2×#6389C2212yescomplemented
50C2×Dic7⋊C8C2212yescomplemented
51#89818C2218yes—
52C2×#6383C2212yes—
53#89570C22116yescomplemented
54#89567C22116yescomplemented
55#89503C2218yes—
56#89491C2218yescomplemented
57C2×#3215—71nomaximal, Sylow
58C24×D7—21no—
59C22×C7×Q8—21no—
60C22×C7×Q8—22no—
61C22×C7×Q8—44no—
62C22×C7×Q8C2312yes—
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2314yes—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C23×Dic7D411yescharacteristic
68C23×Dic7D411yescharacteristic
69C22×D28—21no—
70C22×D28—22no—
71C22×D28—44no—
72C22×D28C2312yes—
73C22×D28C2314yes—
74C22×Dic14—21no—
75C22×Dic14C2311yescharacteristic
76C2×C7×Q16—44no—
77C2×C7×Q16C2318yescomplemented
78C22×C7×C8C2312yes—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8C2314yes—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—21no—
84C2×C22⋊Dic7—21no—
85#6510—28no—
86#6510—24no—
87#6510—24no—
88#6510—28no—
89#6510C2318yes—
90#6510C2318yes—
91Q8×Dic7—22no—
92Q8×Dic7C2314yescomplemented
93#6507—22no—
94#6507C2314yescomplemented
95C2×#1144—44no—
96C2×#1144C2318yescomplemented
97C2×#1143—44no—
98C2×#1143C2318yes—
99C2×(C2×D7)⋊C4—28no—
100C2×(C2×D7)⋊C4—22no—
101C2×(C2×D7)⋊C4—22no—
102C2×(C2×D7)⋊C4—22no—
103C2×(C2×D7)⋊C4—22no—
104C2×(C2×D7)⋊C4—28no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7C2311yescharacteristic
107#6487—22no—
108#6487C2314yes—
109C2×Dic7⋊C4—21no—
110C2×Dic7⋊C4—21no—
111C2×Dic7⋊C4—21no—
112C2×Dic7⋊C4—21no—
113C2×Dic7⋊C4—21no—
114C2×C4×Dic7—21no—
115C2×C4×Dic7—21no—
116C2×C4×Dic7D411yescharacteristic
117C2×C4×Dic7D411yescharacteristic
118C2×C4×Dic7D414yes—
119C2×C4×Dic7C2311yescharacteristic
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2314yes—
122C22×C7⋊C8C2312yes—
123C2×#1132—44no—
124C2×#1132C2318yes—
125#6452—28no—
126#6451—22no—
127#6451C2314yescomplemented
128#6450—28no—
129#6450—22no—
130#6450C2314yescomplemented
131#6441—28no—
132#6440—28no—
133C7×#103C2318yescomplemented
134C7×C22⋊C8C2314yes—
135#6409C2318yescomplemented
136C22⋊(C7⋊C8)C2314yes—
137#6395C2318yes—
138#6389C2318yescomplemented
139Dic7⋊C8C2318yescomplemented
140#6383C2318yes—
141C2×#533—71no—
142C22×#124—71no—
143#3215—716no—
144C2×#497—72no—
145C2×#489—72no—
146C2×#464—72no—
147C2×#464—72no—
148C2×#451—71no—
149C2×#451D711yescharacteristic, complemented
150C2×#450—71no—
151C2×#450—71no—
152C2×#432—71no—
153C24×C7C2×D411yescharacteristic
154C23×D7—24no—
155C23×D7—22no—
156C23×D7—44no—
157C23×D7—44no—
158C23×D7—44no—
159C2×C7×Q8—21no—
160C2×C7×Q8—24no—
161C2×C7×Q8—22no—
162C2×C7×Q8—24no—
163C2×C7×Q8—24no—
164C2×C7×Q8—24no—
165C2×C7×Q8—48no—
166C2×C7×Q8—44no—
167C2×C7×Q8—48no—
168C2×C7×Q8C2414yes—
169C2×C7×Q8C2412yes—
170C22×C4×C7—24no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—24no—
174C22×C4×C7—22no—
175C22×C4×C7—44no—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2412yes—
179C22×C4×C7C2411yescharacteristic
180C22×Dic7—24no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—42no—
184C22×Dic7C2×D414yes—
185C22×Dic7C2×D411yescharacteristic
186C22×Dic7C2×D411yescharacteristic
187C22×Dic7C2×D411yescharacteristic
188C22×Dic7C2×D411yescharacteristic
189C22×Dic7C2×D414yes—
190C2×D28—24no—
191C2×D28—24no—
192C2×D28—24no—
193C2×D28—22no—
194C2×D28—48no—
195C2×D28—44no—
196C2×D28—48no—
197C2×D28C2414yes—
198C2×Dic14—24no—
199C2×Dic14—21no—
200C2×Dic14—24no—
201C2×Dic14—44no—
202C2×Dic14C2412yes—
203C7×Q16—48no—
204C2×C7×C8—24no—
205C2×C7×C8C2414yes—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—24no—
208C7×C4⋊C4C2412yes—
209C7×C22⋊C4—24no—
210C22⋊Dic7—24no—
211#1144—48no—
212#1143—48no—
213(C2×D7)⋊C4—28no—
214(C2×D7)⋊C4—28no—
215(C2×D7)⋊C4—28no—
216(C2×D7)⋊C4—28no—
217C4⋊Dic7—21no—
218C4⋊Dic7—24no—
219C4⋊Dic7C2412yes—
220Dic7⋊C4—24no—
221Dic7⋊C4—24no—
222Dic7⋊C4—24no—
223Dic7⋊C4—24no—
224Dic7⋊C4—24no—
225C4×Dic7—24no—
226C4×Dic7—24no—
227C4×Dic7C2×D414yes—
228C4×Dic7C2×D412yes—
229C4×Dic7C2×D412yes—
230C4×Dic7C2×D414yes—
231C4×Dic7C2412yes—
232C4×Dic7C2412yes—
233C2×C7⋊C8—24no—
234C2×C7⋊C8C2414yes—
235#1132—48no—
236C23×Q8C2×D711yescharacteristic
237C23×D4—71no—
238C22×Q16—72no—
239C22×Q16C2×D712yescomplemented
240C22×SD16—72no—
241C22×SD16—72no—
242#533—78no—
243#533—144no—
244C2×C4⋊Q8—71no—
245C2×C4⋊Q8—71no—
246C2×#125—141no—
247C2×#124—72no—
248C2×#124—78no—
249C2×#124—72no—
250C2×#124—141no—
251C2×#124—141no—
252C2×#124—144no—
253C2×C22⋊Q8—71no—
254C2×C22⋊Q8—141no—
255C2×C22⋊Q8C2×D711yescharacteristic
256C2×C4×Q8—71no—
257C2×C4×Q8—71no—
258C2×C4×Q8—141no—
259C2×#117—141no—
260C22×C22⋊C4—141no—
261C22×C22⋊C4—141no—
262C22×C42—71no—
263#497—716no—
264#489—716no—
265#464—716no—
266#464—716no—
267#451—78no—
268#451C2×D718yescomplemented
269#450—78no—
270#450—78no—
271#432—78no—
272C2×C4⋊C8—72no—
273C2×#103—72no—
274C2×#103C2×D712yescomplemented
275C2×#102—72no—
276C2×#102—72no—
277C2×C22⋊C8—71no—
278C2×C22⋊C8C2×D711yescharacteristic
279C2×#97—72no—
280C23×C7—22no—
281C23×C7—22no—
282C23×C7#13714yes—
283C23×C7C22×D412yes—
284C23×C7C22×D411yescharacteristic
285C22×D7—24no—
286C22×D7—48no—
287C22×D7—48no—
288C22×D7—44no—
289C22×D7—48no—
290C22×D7—44no—
291C22×D7—44no—
292C22×D7—816no—
293C7×Q8—24no—
294C7×Q8—24no—
295C7×Q8—24no—
296C7×Q8—48no—
297C7×Q8—42no—
298C7×Q8—44no—
299C2×C4×C7—22no—
300C2×C4×C7—24no—
301C2×C4×C7—24no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—24no—
305C2×C4×C7—48no—
306C2×C4×C7—44no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7C22×D412yes—
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7H(5,𝔽2)14yes—
312C2×C4×C7C2511yescharacteristic
313C2×Dic7—28no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—44no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7C22×D412yes—
324C2×Dic7C22×D412yes—
325C2×Dic7C22×D412yes—
326C2×Dic7C22×D412yes—
327D28—24no—
328D28—24no—
329D28—48no—
330D28—44no—
331Dic14—24no—
332Dic14—48no—
333Dic14—42no—
334C7×C8—24no—
335C7⋊C8—24no—
336C25—141no—
337C22×Q8—22no—
338C22×Q8—21no—
339C22×Q8—44no—
340C22×Q8—71no—
341C22×Q8—141no—
342C22×Q8C22×D711yescharacteristic
343C22×Q8C22×D712yes—
344C22×Q8C22×D714yes—
345C22×D4—72no—
346C22×D4—74no—
347C22×D4—142no—
348C22×D4—141no—
349C22×D4—284no—
350C23×C4—21no—
351C23×C4—71no—
352C23×C4—71no—
353C23×C4C22×D711yescharacteristic
354C2×Q16—44no—
355C2×Q16—78no—
356C2×Q16—284no—
357C2×Q16C22×D718yescomplemented
358C2×SD16—78no—
359C2×SD16—78no—
360C2×SD16—284no—
361C2×SD16—284no—
362C22×C8—72no—
363C22×C8C22×D712yes—
364C4⋊Q8—74no—
365C4⋊Q8—74no—
366C4⋊Q8—142no—
367C4⋊Q8—142no—
368#125—148no—
369#124—78no—
370#124—78no—
371#124—148no—
372#124—144no—
373#124—148no—
374#124—144no—
375C22⋊Q8—22no—
376C22⋊Q8—74no—
377C22⋊Q8—148no—
378C22⋊Q8—142no—
379C22⋊Q8C22×D714yes—
380C4×Q8—74no—
381C4×Q8—74no—
382C4×Q8—142no—
383C4×Q8—148no—
384C4×Q8—142no—
385#117—148no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—148no—
397C2×C22⋊C4—148no—
398C2×C22⋊C4—142no—
399C2×C22⋊C4—142no—
400C2×C22⋊C4—142no—
401C2×C22⋊C4—142no—
402C2×C22⋊C4—141no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—74no—
406C2×C42—71no—
407C2×C42—71no—
408C2×C42—74no—
409C2×C42—141no—
410C2×C42—141no—
411C4⋊C8—78no—
412#103—78no—
413#103C22×D718yescomplemented
414#102—78no—
415#102—78no—
416C22⋊C8—74no—
417C22⋊C8C22×D714yes—
418#97—78no—
419C22×C7—22no—
420C22×C7—28no—
421C22×C7—22no—
422C22×C7#53512yes—
423C22×C7C2×#13714yes—
424C22×C7C2×#13712yes—
425C22×C7#57512yes—
426C22×C7C23×D411yescharacteristic
427C2×D7—44no—
428C2×D7—44no—
429C2×D7—44no—
430C2×D7—816no—
431C2×D7—88no—
432C4×C7—22no—
433C4×C7—44no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7#53512yes—
437C4×C7C2×H(5,𝔽2)12yes—
438Dic7—22no—
439Dic7—22no—
440Dic7—22no—
441Dic7—22no—
442Dic7—42no—
443Dic7—42no—
444C24—144no—
445C24—142no—
446C24—284no—
447C24—284no—
448C24—284no—
449C24D4×D711yescharacteristic
450C2×Q8—21no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—48no—
457C2×Q8—44no—
458C2×Q8—48no—
459C2×Q8—72no—
460C2×Q8—144no—
461C2×Q8—141no—
462C2×Q8—144no—
463C2×Q8—284no—
464C2×Q8C23×D714yes—
465C2×Q8C23×D712yes—
466C2×D4—74no—
467C2×D4—144no—
468C2×D4—142no—
469C2×D4—144no—
470C2×D4—144no—
471C2×D4—288no—
472C2×D4—284no—
473C2×D4—288no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—74no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—74no—
486C22×C4—144no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4—282no—
490C22×C4D4×D712yes—
491C22×C4D4×D711yescharacteristic
492C22×C4C23×D712yes—
493C22×C4C23×D711yescharacteristic
494Q16—48no—
495Q16—288no—
496SD16—288no—
497SD16—288no—
498C2×C8—24no—
499C2×C8—74no—
500C2×C8—144no—
501C2×C8C23×D714yes—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—72no—
505C4⋊C4—144no—
506C4⋊C4—144no—
507C4⋊C4—144no—
508C4⋊C4—144no—
509C4⋊C4—144no—
510C4⋊C4—144no—
511C4⋊C4—141no—
512C4⋊C4C23×D712yes—
513C22⋊C4—24no—
514C22⋊C4—148no—
515C22⋊C4—148no—
516C22⋊C4—148no—
517C22⋊C4—148no—
518C22⋊C4—144no—
519C42—72no—
520C42—74no—
521C42—74no—
522C42—72no—
523C42—72no—
524C42—72no—
525C42—144no—
526C42—144no—
527C2×C7—24no—
528C2×C7#321514yescomplemented
529C2×C7C2×#53511yescharacteristic
530C2×C7C22×#13711yescharacteristic
531C2×C7C2×#57511yescharacteristic
532D7—84no—
533C23—22no—
534C23—22no—
535C23—144no—
536C23—288no—
537C23—288no—
538C23—288no—
539C23—284no—
540C23—284no—
541C23—284no—
542C23—5616no—
543C23#648114yes—
544C23C2×D4×D712yes—
545C23C2×D4×D711yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—48no—
550Q8—42no—
551Q8—44no—
552Q8—144no—
553Q8—288no—
554Q8—282no—
555D4—144no—
556D4—144no—
557D4—288no—
558D4—284no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—72no—
570C2×C4—72no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—148no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—284no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4C2×D4×D712yes—
584C2×C4C2×D4×D711yescharacteristic
585C2×C4#654814yes—
586C2×C4C24×D711yescharacteristic
587C8—24no—
588C8—144no—
589C7C2×#321511yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—28no—
593C22—284no—
594C22—284no—
595C22—284no—
596C22—5616no—
597C22—568no—
598C22#9028312yes—
599C22C2×#648114yes—
600C22C2×#648112yes—
601C22#9046812yes—
602C22C22×D4×D711yescharacteristic
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—142no—
608C4—142no—
609C4—142no—
610C4—142no—
611C4—282no—
612C4—282no—
613C4#9028312yes—
614C4C2×#654812yes—
615C2—24no—
616C2—564no—
617C2#1173764714yesdirect product factor
618C2C2×#9028311yescharacteristic
619C2C22×#648111yescharacteristic
620C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#451C2
socleC23×C7C22×D4
prime radicalC23×D28C22
hypercenterC2×#451D7
hypocenterC7C2×#3215
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C2×#90313C2semidirect
3. C2×C7×#451C2semidirect
4. C2×#90000C2semidirect
5. C2×#89962C2semidirect
6. C2×#89957C2semidirect
8. #11737647C2direct (N × H)
9. C2×#89570C2semidirect
10. C2×#89567C2semidirect
12. C2×#89491C2semidirect
15. C22×C7×Q16C22semidirect
23. C2×Q8×Dic7C22semidirect
24. C2×#6507C22semidirect
25. C22×#1144C22semidirect
33. #90313C22semidirect
35. C2×#6451C22semidirect
37. C2×#6450C22semidirect
40. C7×#451C22semidirect
41. C2×C7×#103C22semidirect
43. #90000C22semidirect
44. C2×#6409C22semidirect
46. #89962C22semidirect
47. #89957C22semidirect
49. C2×#6389C22semidirect
50. C2×Dic7⋊C8C22semidirect
53. #89570C22semidirect
54. #89567C22semidirect
56. #89491C22semidirect
77. C2×C7×Q16C23semidirect
92. Q8×Dic7C23semidirect
94. #6507C23semidirect
96. C2×#1144C23semidirect
127. #6451C23semidirect
130. #6450C23semidirect
133. C7×#103C23semidirect
135. #6409C23semidirect
138. #6389C23semidirect
139. Dic7⋊C8C23semidirect
149. C2×#451D7semidirect
239. C22×Q16C2×D7semidirect
268. #451C2×D7semidirect
274. C2×#103C2×D7semidirect
357. C2×Q16C22×D7semidirect
413. #103C22×D7semidirect
528. C2×C7#3215semidirect
589. C7C2×#3215semidirect
617. C2#11737647direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#65102
C2×#903132
C2×C7×#4512
C2×#900002
C2×#899622
C2×#899572
C2×#898182
#117376472
C2×#895702
C2×#895672
C2×#895032
C2×#894912
C2×#32157