Finite Group Databasebeta

← C2×#11739890 #14221944

Subgroups — C2×#11739890 #14221944

Order 1792.

2150 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×Q8C211yesmaximal, characteristic, complemented
2C2×#90170C211yesmaximal, characteristic
3C2×#90000C211yesmaximal, characteristic, complemented
4C2×#89999C211yesmaximal, characteristic
5#11739890C2116yesmaximal, direct product factor
6C2×#89800C212yesmaximal, complemented
7C2×#89798C212yesmaximal, complemented
8C2×#89794C212yesmaximal, complemented
9C2×#89802C212yesmaximal, complemented
10C2×#89742C211yesmaximal, characteristic, complemented
11C2×#89740C211yesmaximal, characteristic
12C2×#89698C211yesmaximal, characteristic, complemented
13C23×C7×Q8C2211yescharacteristic
14C2×C4×C7×Q8—21no—
15C2×C4×C7×Q8—21no—
16C2×C4×C7×Q8—24no—
17C2×C4×C7×Q8C2212yes—
18C2×C4×C7×Q8C2212yes—
19C2×C4×C7×Q8C2218yescomplemented
20C22×C7×C4⋊C4—21no—
21C22×C7×C4⋊C4C2211yescharacteristic
22C22×C42×C7—21no—
23C22×C42×C7C2211yescharacteristic
24C22×#1144C2212yes—
25C22×#1143C2212yes—
26C2×#6493C2211yescharacteristic
27C2×#6492—21no—
28C2×C4×C22⋊D7—21no—
29C2×#6487C2211yescharacteristic
30#90170—24no—
31#90170C2218yes—
32C2×#6439—21no—
33C2×#6438C2211yescharacteristic
34C2×C4×D28C2211yescharacteristic
35C2×#6435—21no—
36C2×#6433C2211yescharacteristic
37C2×C4×Dic14C2211yescharacteristic
38#90000C2218yescomplemented
39#89999C2218yes—
40C2×#6409C2212yes—
41C2×#6409C2212yes—
42C2×C22⋊(C7⋊C8)C2211yescharacteristic
43C2×C22⋊(C7⋊C8)C2211yescharacteristic
44#89802C22116yescomplemented
45#89800C22116yescomplemented
46#89798C22116yescomplemented
47#89794C22116yescomplemented
48#89742C2218yescomplemented
49C2×#6384C2212yes—
50#89740C2218yes—
51C2×#6383C2212yes—
52C2×#6382C2212yes—
53C2×#6381C2212yes—
54#89698C2218yescomplemented
55C2×C4⋊(C7⋊C8)C2212yes—
56C2×#6377C2212yes—
57C2×#3117—71nomaximal, Sylow
58C22×C7×Q8—28no—
59C22×C7×Q8—21no—
60C22×C7×Q8—22no—
61C22×C7×Q8C2314yes—
62C22×C7×Q8C2312yes—
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7D411yescharacteristic
66C23×C4×C7D411yescharacteristic
67C23×C4×C7C2311yescharacteristic
68C22×C22⋊D7—21no—
69C22×D28C2311yescharacteristic
70C22×C4×D7—21no—
71C22×Dic14C2311yescharacteristic
72C4×C7×Q8—24no—
73C4×C7×Q8—24no—
74C4×C7×Q8—28no—
75C4×C7×Q8—28no—
76C4×C7×Q8C2318yes—
77C4×C7×Q8C2318yes—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—28no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2314yes—
85C2×C42×C7—22no—
86C2×C42×C7—22no—
87C2×C42×C7—28no—
88C2×C42×C7D411yescharacteristic
89C2×C42×C7D411yescharacteristic
90C2×C42×C7D414yes—
91C2×C42×C7C2311yescharacteristic
92C2×C42×C7C2311yescharacteristic
93C2×C42×C7C2314yes—
94C2×C22⋊Dic7—21no—
95C2×#1144—28no—
96C2×#1144C2318yes—
97C2×#1143—28no—
98C2×#1143C2318yes—
99#6493—22no—
100#6493C2314yes—
101#6492—28no—
102C4×C22⋊D7—28no—
103C2×(C2×D7)⋊C4—21no—
104C2×(C2×D7)⋊C4—21no—
105C2×(C2×D7)⋊C4—21no—
106C2×(C2×D7)⋊C4—21no—
107C2×C4⋊Dic7C2311yescharacteristic
108C2×C4⋊Dic7C2311yescharacteristic
109#6487—22no—
110#6487C2314yes—
111C2×Dic7⋊C4—21no—
112C2×Dic7⋊C4—21no—
113C2×Dic7⋊C4—21no—
114C2×Dic7⋊C4—21no—
115C2×C4×Dic7—21no—
116C22×C7⋊C8C2312yes—
117C22×C7⋊C8C2312yes—
118#6439—28no—
119#6438—22no—
120#6438C2314yes—
121C4×D28—22no—
122C4×D28C2314yes—
123#6435—28no—
124#6433—22no—
125#6433C2314yes—
126C4×Dic14—22no—
127C4×Dic14C2314yes—
128#6409C2318yes—
129#6409C2318yes—
130C22⋊(C7⋊C8)C2314yes—
131C22⋊(C7⋊C8)C2314yes—
132#6384C2318yes—
133#6383C2318yes—
134#6382C2318yes—
135#6381C2318yes—
136C4⋊(C7⋊C8)C2318yes—
137#6377C2318yes—
138C2×#529—71no—
139C22×C4×Q8D711yescharacteristic, complemented
140#3117—716no—
141C2×#484—71no—
142C2×#481—71no—
143C2×#479—72no—
144C2×#464—72no—
145C2×#451—71no—
146C2×#448—71no—
147C2×#441—72no—
148C2×#440—72no—
149C2×#432—71no—
150C24×C7C2×D411yescharacteristic
151C23×D7—21no—
152C2×C7×Q8—216no—
153C2×C7×Q8—28no—
154C2×C7×Q8—216no—
155C2×C7×Q8—24no—
156C2×C7×Q8—22no—
157C2×C7×Q8—24no—
158C2×C7×Q8#4118yes—
159C2×C7×Q8C2414yes—
160C22×C4×C7—28no—
161C22×C4×C7—24no—
162C22×C4×C7—24no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—24no—
166C22×C4×C7—22no—
167C22×C4×C7—42no—
168C22×C4×C7—44no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7C2×D414yes—
177C22×C4×C7C2412yes—
178C22×C4×C7C2411yescharacteristic
179C2×C22⋊D7—24no—
180C2×C22⋊D7—44no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184C2×D28—24no—
185C2×D28—21no—
186C2×D28C2412yes—
187C2×C4×D7—24no—
188C2×C4×D7—44no—
189C2×Dic14—24no—
190C2×Dic14—21no—
191C2×Dic14C2412yes—
192C7×C4⋊C4—28no—
193C7×C4⋊C4—28no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4C2414yes—
197C42×C7—28no—
198C42×C7—28no—
199C42×C7C2×D412yes—
200C42×C7C2×D412yes—
201C42×C7C2×D414yes—
202C42×C7C2×D414yes—
203C42×C7C2412yes—
204C42×C7C2412yes—
205C22⋊Dic7—24no—
206#1144—216no—
207#1143—216no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—24no—
210(C2×D7)⋊C4—24no—
211(C2×D7)⋊C4—24no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7C2412yes—
215C4⋊Dic7C2412yes—
216Dic7⋊C4—24no—
217Dic7⋊C4—24no—
218Dic7⋊C4—24no—
219Dic7⋊C4—24no—
220C4×Dic7—24no—
221C2×C7⋊C8—24no—
222C2×C7⋊C8—24no—
223C2×C7⋊C8C2414yes—
224C2×C7⋊C8C2414yes—
225C23×Q8C2×D711yescharacteristic
226C22×Q16—72no—
227C22×SD16—72no—
228#529—78no—
229#529—144no—
230C2×#126—141no—
231C2×#125—71no—
232C2×#124—71no—
233C2×#123—141no—
234C2×C22⋊Q8—71no—
235C2×#121—71no—
236C2×C4×Q8—21no—
237C2×C4×Q8—21no—
238C2×C4×Q8—24no—
239C2×C4×Q8—71no—
240C2×C4×Q8C2×D712yes—
241C2×C4×Q8C2×D712yes—
242C2×C4×Q8C2×D718yescomplemented
243C2×C4×D4—71no—
244C2×C4×D4—141no—
245C2×#117—141no—
246C22×C4⋊C4—21no—
247C22×C4⋊C4C2×D711yescharacteristic
248C22×C42—21no—
249C22×C42C2×D711yescharacteristic
250#484—78no—
251#481—78no—
252#479—716no—
253#464—716no—
254#451—78no—
255#448—78no—
256#441—716no—
257#440—716no—
258#432—78no—
259C2×#107—72no—
260C2×#106—72no—
261C2×C4⋊C8—72no—
262C2×#103—72no—
263C2×#103—72no—
264C2×#103—72no—
265C2×#102—72no—
266C2×C22⋊C8—71no—
267C2×C22⋊C8—71no—
268C2×#97—72no—
269C23×C7—22no—
270C23×C7—22no—
271C23×C7#13714yes—
272C23×C7C22×D412yes—
273C23×C7C22×D411yescharacteristic
274C22×D7—22no—
275C22×D7—42no—
276C22×D7—42no—
277C22×D7—42no—
278C7×Q8—216no—
279C7×Q8—28no—
280C7×Q8—24no—
281C7×Q8C2×#4118yes—
282C2×C4×C7—216no—
283C2×C4×C7—28no—
284C2×C4×C7—22no—
285C2×C4×C7—28no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—24no—
294C2×C4×C7—44no—
295C2×C4×C7—48no—
296C2×C4×C7—44no—
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D412yes—
300C2×C4×C7C22×D412yes—
301C2×C4×C7C22×D412yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C2×#4114yes—
304C2×C4×C7C2511yescharacteristic
305C22⋊D7—48no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312D28—24no—
313D28—42no—
314C4×D7—48no—
315Dic14—24no—
316Dic14—42no—
317C7⋊C8—24no—
318C7⋊C8—24no—
319C22×Q8—28no—
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—71no—
323C22×Q8C22×D714yes—
324C22×Q8C22×D712yes—
325C22×D4—71no—
326C22×D4—141no—
327C23×C4—21no—
328C23×C4—21no—
329C23×C4—141no—
330C23×C4C22⋊D711yescharacteristic
331C23×C4C22⋊D711yescharacteristic
332C23×C4C22×D711yescharacteristic
333C2×Q16—78no—
334C2×Q16—148no—
335C2×SD16—78no—
336C2×SD16—148no—
337C22×C8—72no—
338C22×C8—72no—
339#126—148no—
340#125—74no—
341#125—142no—
342#124—74no—
343#124—142no—
344#123—148no—
345C22⋊Q8—74no—
346C22⋊Q8—142no—
347#121—74no—
348#121—142no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—28no—
352C4×Q8—28no—
353C4×Q8—74no—
354C4×Q8—142no—
355C4×Q8C22×D718yes—
356C4×Q8C22×D718yes—
357C4×D4—74no—
358C4×D4—148no—
359C4×D4—142no—
360#117—148no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4C22×D712yes—
373C2×C4⋊C4C22×D714yes—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—28no—
382C2×C42—141no—
383C2×C42C22⋊D711yescharacteristic
384C2×C42C22⋊D711yescharacteristic
385C2×C42C22⋊D714yes—
386C2×C42C22×D711yescharacteristic
387C2×C42C22×D711yescharacteristic
388C2×C42C22×D714yes—
389#107—78no—
390#106—78no—
391C4⋊C8—78no—
392#103—78no—
393#103—78no—
394#103—78no—
395#102—78no—
396C22⋊C8—74no—
397C22⋊C8—74no—
398#97—78no—
399C22×C7—28no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7#52512yes—
403C22×C7C2×#13714yes—
404C22×C7C2×#13712yes—
405C22×C7#57512yes—
406C22×C7C23×D411yescharacteristic
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—84no—
411C4×C7—28no—
412C4×C7—22no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—44no—
418C4×C7#52512yes—
419C4×C7C22×#4112yes—
420Dic7—42no—
421Dic7—42no—
422Dic7—42no—
423C24—141no—
424C24C2×C22⋊D711yescharacteristic
425C2×Q8—216no—
426C2×Q8—28no—
427C2×Q8—216no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—72no—
432C2×Q8—141no—
433C2×Q8—144no—
434C2×Q8#115718yes—
435C2×Q8C23×D714yes—
436C2×D4—72no—
437C2×D4—144no—
438C2×D4—144no—
439C2×D4—141no—
440C2×D4—284no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—144no—
454C22×C4—284no—
455C22×C4C2×C22⋊D714yes—
456C22×C4C2×C22⋊D712yes—
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D714yes—
463C22×C4C23×D712yes—
464C22×C4C23×D711yescharacteristic
465Q16—1416no—
466SD16—1416no—
467C2×C8—74no—
468C2×C8—74no—
469C2×C8—144no—
470C2×C8—144no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—72no—
476C4⋊C4—72no—
477C4⋊C4—144no—
478C4⋊C4—144no—
479C4⋊C4—144no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4C23×D714yes—
484C22⋊C4—144no—
485C22⋊C4—144no—
486C22⋊C4—144no—
487C22⋊C4—144no—
488C22⋊C4—144no—
489C42—28no—
490C42—28no—
491C42—144no—
492C42C2×C22⋊D714yes—
493C42C2×C22⋊D712yes—
494C42C2×C22⋊D712yes—
495C42C2×C22⋊D714yes—
496C42C23×D712yes—
497C42C23×D712yes—
498C2×C7—24no—
499C2×C7#311714yescomplemented
500C2×C7C2×#52511yescharacteristic
501C2×C7C22×#13711yescharacteristic
502C2×C7C2×#57511yescharacteristic
503D7—82no—
504C23—22no—
505C23—22no—
506C23—142no—
507C23—282no—
508C23—282no—
509C23—282no—
510C23#650514yes—
511C23C22×C22⋊D712yes—
512C23C22×C22⋊D711yescharacteristic
513Q8—216no—
514Q8—28no—
515Q8—24no—
516Q8—144no—
517Q8—282no—
518Q8C2×#115718yes—
519D4—144no—
520D4—288no—
521D4—282no—
522C2×C4—216no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—288no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4C2×#115714yes—
545C2×C4C22×C22⋊D712yes—
546C2×C4C22×C22⋊D712yes—
547C2×C4C22×C22⋊D712yes—
548C2×C4C22×C22⋊D712yes—
549C2×C4C22×C22⋊D712yes—
550C2×C4C22×C22⋊D711yescharacteristic
551C2×C4C24×D711yescharacteristic
552C8—144no—
553C8—144no—
554C7C2×#311711yescharacteristic, complemented, Sylow
555C22—22no—
556C22—22no—
557C22—28no—
558C22—282no—
559C22—282no—
560C22—282no—
561C22—564no—
562C22#9048212yes—
563C22C2×#650514yes—
564C22C2×#650512yes—
565C22#9051312yes—
566C22C23×C22⋊D711yescharacteristic
567C4—28no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—44no—
574C4—282no—
575C4—282no—
576C4—282no—
577C4#9048212yes—
578C4C22×#115712yes—
579C2—24no—
580C2—562no—
581C2#1173989014yesdirect product factor
582C2C2×#9048211yescharacteristic
583C2C22×#650511yescharacteristic
584C2C2×#9051311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C4×C7×Q8C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC22×C4×Q8D7
hypocenterC7C2×#3117
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C4×C7×Q8C2semidirect
3. C2×#90000C2semidirect
5. #11739890C2direct (N × H)
6. C2×#89800C2semidirect
7. C2×#89798C2semidirect
8. C2×#89794C2semidirect
9. C2×#89802C2semidirect
10. C2×#89742C2semidirect
12. C2×#89698C2semidirect
19. C2×C4×C7×Q8C22semidirect
38. #90000C22semidirect
44. #89802C22semidirect
45. #89800C22semidirect
46. #89798C22semidirect
47. #89794C22semidirect
48. #89742C22semidirect
54. #89698C22semidirect
139. C22×C4×Q8D7semidirect
242. C2×C4×Q8C2×D7semidirect
499. C2×C7#3117semidirect
554. C7C2×#3117semidirect
581. C2#11739890direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×Q82
C2×#901702
C2×#900002
C2×#899992
#117398902
C2×#898002
C2×#897982
C2×#897942
C2×#898022
C2×#897422
C2×#897402
C2×#896982
C2×#31177