Finite Group Databasebeta

← C2×#11738443 #14222001

Subgroups — C2×#11738443 #14222001

Order 1792.

2118 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C2×C4×Q8×D7C211yesmaximal, characteristic
3C2×C7×#475C211yesmaximal, characteristic, complemented
4C2×#89866C211yesmaximal, characteristic, complemented
5C2×#89803C211yesmaximal, characteristic, complemented
6#11738443C2116yesmaximal, direct product factor
7C2×#89634C212yesmaximal
8C2×#89625C212yesmaximal, complemented
9C2×#89621C211yesmaximal, characteristic, complemented
10C2×#89607C211yesmaximal, characteristic
11C2×#89580C212yesmaximal
12C2×#89561C212yesmaximal, complemented
13C22×Q8×D7C2211yescharacteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C4×C7×Q8C2211yescharacteristic
16C2×#6509—21no—
17C2×Q8×Dic7—21no—
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1144C2212yes—
20C22×Dic28C2212yes—
21#90419—24no—
22#90419C2218yes—
23C4×Q8×D7—24no—
24C4×Q8×D7C2218yes—
25C2×#6460C2212yes—
26C2×#6455—21no—
27C2×D7×C4⋊C4—21no—
28C2×D7×C4⋊C4C2211yescharacteristic
29C2×#6453—21no—
30C2×#6450—21no—
31C2×#6450C2212yes—
32C2×#6450C2211yescharacteristic
33C2×C42×D7—21no—
34C2×C42×D7C2211yescharacteristic
35C2×#6432C2211yescharacteristic
36C2×C4×Dic14—21no—
37C2×C4×Dic14C2211yescharacteristic
38C7×#475C2218yescomplemented
39C2×C7×#106C2212yes—
40C2×C7×C4⋊C8C2211yescharacteristic
41C2×C7×#103C2212yes—
42C2×(C2×D7)⋊C8C2212yes—
43C2×#6390C2212yes—
44C2×#6389C2212yes—
45#89866C2218yescomplemented
46#89803C2218yescomplemented
47C2×#6384C2212yes—
48C2×#6384C2212yes—
49C2×#6382C2212yes—
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89634C22116yes—
52#89625C22116yescomplemented
53#89621C2218yescomplemented
54#89607C2218yes—
55#89580C22116yes—
56#89561C22116yescomplemented
57C2×#3113—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8C2311yescharacteristic
60C2×Q8×D7—24no—
61C2×Q8×D7—22no—
62C2×Q8×D7—24no—
63C2×Q8×D7C2314yes—
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C22×C4×D7D411yescharacteristic
67C22×C4×D7D411yescharacteristic
68C22×C4×D7C2311yescharacteristic
69C22×Dic14—21no—
70C22×Dic14C2312yes—
71C22×Dic14C2311yescharacteristic
72C22×C7×C8C2312yes—
73C7×C4⋊Q8—22no—
74C7×C4⋊Q8C2314yes—
75C4×C7×Q8—22no—
76C4×C7×Q8C2314yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83#6509—28no—
84Q8×Dic7—28no—
85Q8×Dic7—22no—
86Q8×Dic7C2314yes—
87C2×#1144—28no—
88C2×#1144C2318yes—
89C2×(C2×D7)⋊C4—22no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—22no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C2×C4×Dic7D411yescharacteristic
104C2×C4×Dic7D411yescharacteristic
105C2×C4×Dic7C2311yescharacteristic
106C22×C7⋊C8C2312yes—
107C2×Dic28—28no—
108C2×Dic28C2318yes—
109#6460—24no—
110#6460C2318yes—
111#6455—28no—
112D7×C4⋊C4—28no—
113D7×C4⋊C4—22no—
114D7×C4⋊C4C2314yes—
115#6453—28no—
116#6450—24no—
117#6450—28no—
118#6450—22no—
119#6450C2314yes—
120#6450C2318yes—
121C42×D7—28no—
122C42×D7D414yes—
123C42×D7C2314yes—
124#6432—22no—
125#6432C2314yes—
126C4×Dic14—22no—
127C4×Dic14—28no—
128C4×Dic14C2314yes—
129C7×#106C2318yes—
130C7×C4⋊C8C2314yes—
131C7×#103C2318yes—
132(C2×D7)⋊C8C2318yes—
133#6390C2318yes—
134#6389C2318yes—
135#6384C2318yes—
136#6384C2318yes—
137#6382C2318yes—
138C4⋊(C7⋊C8)C2314yes—
139C2×#533—71no—
140C22×C4×Q8—71no—
141#3113—716no—
142C2×#483—72no—
143C2×#475—71no—
144C2×#475D711yescharacteristic, complemented
145C2×#462—71no—
146C2×#462—71no—
147C2×#451—72no—
148C2×#441—72no—
149C2×#441—72no—
150C2×#432—71no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8—24no—
154C2×C7×Q8—22no—
155C2×C7×Q8—44no—
156C2×C7×Q8#4114yes—
157C2×C7×Q8C2412yes—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—22no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170Q8×D7—216no—
171Q8×D7—28no—
172Q8×D7—28no—
173C2×C4×D7—28no—
174C2×C4×D7—24no—
175C2×C4×D7—24no—
176C2×C4×D7—24no—
177C2×C4×D7—22no—
178C2×C4×D7—22no—
179C2×C4×D7—24no—
180C2×C4×D7—44no—
181C2×C4×D7C2×D414yes—
182C2×C4×D7C2×D412yes—
183C2×C4×D7C2×D414yes—
184C2×C4×D7C2412yes—
185C2×Dic14—28no—
186C2×Dic14—21no—
187C2×Dic14—22no—
188C2×Dic14—24no—
189C2×Dic14—24no—
190C2×Dic14—24no—
191C2×Dic14—22no—
192C2×Dic14#4114yes—
193C2×Dic14C2414yes—
194C2×Dic14C2412yes—
195C2×C7×C8—24no—
196C2×C7×C8C2414yes—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—24no—
201C7×C4⋊C4C2414yes—
202C7×C4⋊C4C2412yes—
203C42×C7—24no—
204C42×C7C2×D412yes—
205C42×C7C2412yes—
206#1144—216no—
207(C2×D7)⋊C4—28no—
208C4⋊Dic7—21no—
209C4⋊Dic7—24no—
210C4⋊Dic7—24no—
211C4⋊Dic7—24no—
212C4⋊Dic7—28no—
213C4⋊Dic7C2412yes—
214Dic7⋊C4—24no—
215Dic7⋊C4—24no—
216Dic7⋊C4—24no—
217Dic7⋊C4—28no—
218C4×Dic7—24no—
219C4×Dic7—24no—
220C4×Dic7—24no—
221C4×Dic7—28no—
222C4×Dic7C2×D414yes—
223C4×Dic7C2×D412yes—
224C4×Dic7C2×D414yes—
225C4×Dic7C2412yes—
226C2×C7⋊C8—24no—
227C2×C7⋊C8C2414yes—
228Dic28—216no—
229C23×Q8—71no—
230C22×Q16—72no—
231C22×Q16—72no—
232#533—78no—
233#533—144no—
234C2×C4⋊Q8—71no—
235C2×C4⋊Q8C2×D711yescharacteristic
236C2×#125—141no—
237C2×C22⋊Q8—72no—
238C2×C22⋊Q8—141no—
239C2×C4×Q8—71no—
240C2×C4×Q8—78no—
241C2×C4×Q8—71no—
242C2×C4×Q8—72no—
243C2×C4×Q8—71no—
244C2×C4×Q8—141no—
245C2×C4×Q8—141no—
246C2×C4×Q8—144no—
247C2×C4×Q8—141no—
248C2×C4×Q8C2×D711yescharacteristic
249C2×#117—141no—
250C22×C4⋊C4—71no—
251C22×C4⋊C4—141no—
252C22×C42—71no—
253C22×C42—141no—
254#483—716no—
255#475—78no—
256#475C2×D718yescomplemented
257#462—78no—
258#462—78no—
259#451—716no—
260#441—716no—
261#441—716no—
262#432—78no—
263C2×#106—72no—
264C2×#106C2×D712yes—
265C2×C4⋊C8—71no—
266C2×C4⋊C8C2×D711yescharacteristic
267C2×#103—72no—
268C2×#103—72no—
269C2×#103—72no—
270C2×#103C2×D712yes—
271C2×C22⋊C8—72no—
272C2×#97—72no—
273C23×C7C22×D411yescharacteristic
274C22×D7—22no—
275C22×D7—22no—
276C22×D7#13714yes—
277C22×D7C22×D412yes—
278C7×Q8—24no—
279C7×Q8—22no—
280C7×Q8—48no—
281C7×Q8C2×#4114yes—
282C2×C4×C7—24no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—24no—
287C2×C4×C7—42no—
288C2×C4×C7—44no—
289C2×C4×C7#13714yes—
290C2×C4×C7C22×D412yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D412yes—
293C2×C4×C7C2×#4112yes—
294C2×C4×C7C2×#4112yes—
295C2×C4×C7C2511yescharacteristic
296C2×Dic7—22no—
297C2×Dic7—24no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—24no—
303C2×Dic7—44no—
304C2×Dic7—42no—
305C2×Dic7C22×D412yes—
306C2×Dic7C22×D412yes—
307C2×Dic7C22×D412yes—
308C4×D7—216no—
309C4×D7—28no—
310C4×D7—24no—
311C4×D7—28no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—48no—
315Dic14—216no—
316Dic14—24no—
317Dic14—22no—
318Dic14—24no—
319Dic14—24no—
320Dic14—44no—
321Dic14C2×#4114yes—
322C7×C8—24no—
323C7⋊C8—24no—
324C22×Q8—21no—
325C22×Q8—74no—
326C22×Q8—71no—
327C22×Q8—72no—
328C22×Q8—141no—
329C22×Q8—142no—
330C22×Q8—144no—
331C22×Q8—144no—
332C22×Q8C22×D711yescharacteristic
333C23×C4—71no—
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—141no—
337C23×C4—141no—
338C2×Q16—78no—
339C2×Q16—78no—
340C2×Q16—148no—
341C2×Q16—148no—
342C22×C8—72no—
343C22×C8C22×D712yes—
344C4⋊Q8—22no—
345C4⋊Q8—74no—
346C4⋊Q8—142no—
347C4⋊Q8C22×D714yes—
348#125—148no—
349C22⋊Q8—78no—
350C22⋊Q8—144no—
351C22⋊Q8—148no—
352C4×Q8—22no—
353C4×Q8—74no—
354C4×Q8—74no—
355C4×Q8—78no—
356C4×Q8—74no—
357C4×Q8—148no—
358C4×Q8—142no—
359C4×Q8—148no—
360C4×Q8—144no—
361C4×Q8—142no—
362C4×Q8—148no—
363C4×Q8—142no—
364C4×Q8C22×D714yes—
365#117—148no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—74no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—148no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—142no—
376C2×C4⋊C4—142no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D712yes—
382C2×C22⋊C4—142no—
383C2×C42—21no—
384C2×C42—71no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—74no—
388C2×C42—74no—
389C2×C42—148no—
390C2×C42—142no—
391C2×C42—141no—
392C2×C42—141no—
393C2×C42—141no—
394C2×C42C22×D711yescharacteristic
395#106—78no—
396#106C22×D718yes—
397C4⋊C8—74no—
398C4⋊C8C22×D714yes—
399#103—78no—
400#103—78no—
401#103—78no—
402#103C22×D718yes—
403C22⋊C8—78no—
404#97—78no—
405C22×C7#52512yes—
406C22×C7C2×#13712yes—
407C22×C7C2×#13712yes—
408C22×C7C23×D411yescharacteristic
409C2×D7—28no—
410C2×D7—22no—
411C2×D7—22no—
412C2×D7C2×#13714yes—
413C4×C7—24no—
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—44no—
417C4×C7#52512yes—
418C4×C7C2×#13714yes—
419C4×C7C22×#4111yescharacteristic
420C4×C7C22×#4111yescharacteristic
421Dic7—22no—
422Dic7—24no—
423Dic7—22no—
424Dic7—22no—
425Dic7—44no—
426Dic7—42no—
427C24—71no—
428C2×Q8—21no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—44no—
432C2×Q8—74no—
433C2×Q8—72no—
434C2×Q8—74no—
435C2×Q8—144no—
436C2×Q8—148no—
437C2×Q8—148no—
438C2×Q8—1416no—
439C2×Q8—148no—
440C2×Q8—141no—
441C2×Q8—142no—
442C2×Q8—142no—
443C2×Q8—144no—
444C2×Q8—144no—
445C2×Q8#116114yes—
446C2×Q8C23×D712yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—74no—
454C22×C4—74no—
455C22×C4—72no—
456C22×C4—72no—
457C22×C4—144no—
458C22×C4—144no—
459C22×C4—148no—
460C22×C4—144no—
461C22×C4—141no—
462C22×C4—144no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—141no—
466C22×C4—142no—
467C22×C4—284no—
468C22×C4D4×D711yescharacteristic
469C22×C4D4×D711yescharacteristic
470C22×C4C23×D711yescharacteristic
471Q16—1416no—
472Q16—1416no—
473C2×C8—24no—
474C2×C8—74no—
475C2×C8—144no—
476C2×C8C23×D714yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—72no—
482C4⋊C4—144no—
483C4⋊C4—144no—
484C4⋊C4—148no—
485C4⋊C4—144no—
486C4⋊C4—148no—
487C4⋊C4—144no—
488C4⋊C4—141no—
489C4⋊C4—144no—
490C4⋊C4—144no—
491C4⋊C4C23×D712yes—
492C4⋊C4C23×D714yes—
493C22⋊C4—148no—
494C42—24no—
495C42—72no—
496C42—74no—
497C42—74no—
498C42—72no—
499C42—144no—
500C42—144no—
501C42—148no—
502C42—144no—
503C42D4×D712yes—
504C42C23×D712yes—
505C2×C7#311314yescomplemented
506C2×C7C2×#52511yescharacteristic
507C2×C7C22×#13711yescharacteristic
508C2×C7C22×#13711yescharacteristic
509D7—24no—
510C23—74no—
511C23—72no—
512C23—142no—
513C23—142no—
514C23C2×D4×D711yescharacteristic
515Q8—24no—
516Q8—22no—
517Q8—48no—
518Q8—74no—
519Q8—1416no—
520Q8—144no—
521Q8—142no—
522Q8—144no—
523Q8—144no—
524Q8—284no—
525Q8C2×#116114yes—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—148no—
537C2×C4—1416no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—144no—
542C2×C4—142no—
543C2×C4—148no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—144no—
549C2×C4—288no—
550C2×C4—282no—
551C2×C4—284no—
552C2×C4#647814yes—
553C2×C4C2×D4×D712yes—
554C2×C4C2×D4×D712yes—
555C2×C4C2×D4×D711yescharacteristic
556C2×C4C2×#116112yes—
557C2×C4D7×#4112yes—
558C2×C4C24×D711yescharacteristic
559C8—24no—
560C8—144no—
561C7C2×#311311yescharacteristic, complemented, Sylow
562C22—74no—
563C22—142no—
564C22—148no—
565C22—142no—
566C22#9031812yes—
567C22C2×#647812yes—
568C22D7×#13712yes—
569C22C22×D4×D711yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—44no—
574C4—142no—
575C4—142no—
576C4—144no—
577C4—142no—
578C4—282no—
579C4—284no—
580C4#9031812yes—
581C4C2×#647814yes—
582C4C22×#116111yescharacteristic
583C4C2×D7×#4111yescharacteristic
584C2—144no—
585C2#1173844314yesdirect product factor
586C2C2×#9031811yescharacteristic
587C2C22×#647811yescharacteristic
588C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#475C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#475D7
hypocenterC7C2×#3113
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. C2×C7×#475C2semidirect
4. C2×#89866C2semidirect
5. C2×#89803C2semidirect
6. #11738443C2direct (N × H)
8. C2×#89625C2semidirect
9. C2×#89621C2semidirect
12. C2×#89561C2semidirect
38. C7×#475C22semidirect
45. #89866C22semidirect
46. #89803C22semidirect
52. #89625C22semidirect
53. #89621C22semidirect
56. #89561C22semidirect
144. C2×#475D7semidirect
256. #475C2×D7semidirect
505. C2×C7#3113semidirect
561. C7C2×#3113semidirect
585. C2#11738443direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904192
C2×C4×Q8×D72
C2×C7×#4752
C2×#898662
C2×#898032
#117384432
C2×#896342
C2×#896252
C2×#896212
C2×#896072
C2×#895802
C2×#895612
C2×#31137