Finite Group Databasebeta

← C2×#11738900 #14221947

Subgroups — C2×#11738900 #14221947

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
1C2×#90315C211yesmaximal, characteristic
2C22×#6450C211yesmaximal, characteristic, complemented
3C2×C7×#484C211yesmaximal, characteristic, complemented
4C2×#89822C211yesmaximal, characteristic, complemented
5C2×#89740C211yesmaximal, characteristic
6#11738900C2116yesmaximal, direct product factor
7C2×#89649C212yesmaximal, complemented
8C2×#89661C212yesmaximal, complemented
9C2×#89570C212yesmaximal, complemented
10C2×#89561C212yesmaximal, complemented
11C2×#89510C211yesmaximal, characteristic
12C2×#89491C211yesmaximal, characteristic, complemented
13C23×Dic14C2211yescharacteristic
14C2×C7×C22⋊Q8C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6510C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C22×#1144C2212yes—
19C2×#6493C2211yescharacteristic
20C22×Dic7⋊C4—21no—
21C22×C4×Dic7—21no—
22C22×C4×Dic7C2211yescharacteristic
23C22×#1132C2212yes—
24#90315—24no—
25#90315C2218yes—
26C2×#6461—21no—
27C2×#6456C2211yescharacteristic
28C2×#6455—21no—
29C2×#6453C2211yescharacteristic
30C2×#6450—24no—
31C2×#6450—21no—
32C2×#6450—21no—
33C2×#6450C2212yes—
34C2×#6450C2212yes—
35C2×#6450C2218yescomplemented
36C2×#6449—21no—
37C2×(C22⋊D7)⋊C4—21no—
38C7×#484C2218yescomplemented
39C2×C7×#107C2212yes—
40C2×C7×#103C2212yes—
41C2×C7×C22⋊C8C2211yescharacteristic
42C2×C22⋊(C7⋊C8)C2211yescharacteristic
43C2×#6390C2212yes—
44C2×#6389C2212yes—
45C2×Dic7⋊C8C2212yes—
46#89822C2218yescomplemented
47C2×#6384C2212yes—
48#89740C2218yes—
49C2×#6383C2212yes—
50C2×#6382C2212yes—
51#89661C22116yescomplemented
52#89649C22116yescomplemented
53#89570C22116yescomplemented
54#89561C22116yescomplemented
55#89510C2218yes—
56#89491C2218yescomplemented
57C2×#3117—71nomaximal, Sylow
58C22×C7×Q8C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—21no—
62C23×Dic7—21no—
63C23×Dic7D411yescharacteristic
64C23×Dic7D411yescharacteristic
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C22×Dic14—21no—
68C22×Dic14—22no—
69C22×Dic14—28no—
70C22×Dic14C2314yes—
71C22×Dic14C2312yes—
72C22×C7×C8C2312yes—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8C2314yes—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2314yes—
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82#6510—22no—
83#6510C2314yes—
84Q8×Dic7—22no—
85Q8×Dic7C2314yes—
86C2×#1144—28no—
87C2×#1144C2318yes—
88#6493—22no—
89#6493C2314yes—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—22no—
99C2×Dic7⋊C4—22no—
100C2×Dic7⋊C4—28no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C2×C4×Dic7—22no—
104C2×C4×Dic7—28no—
105C2×C4×Dic7D411yescharacteristic
106C2×C4×Dic7D411yescharacteristic
107C2×C4×Dic7D414yes—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2314yes—
111C22×C7⋊C8C2312yes—
112C2×#1132—28no—
113C2×#1132C2318yes—
114#6461—28no—
115#6456—22no—
116#6456C2314yes—
117#6455—28no—
118#6453—22no—
119#6453C2314yes—
120#6450—24no—
121#6450—24no—
122#6450—28no—
123#6450—28no—
124#6450C2318yes—
125#6450C2318yes—
126#6449—28no—
127(C22⋊D7)⋊C4—28no—
128C7×#107C2318yes—
129C7×#103C2318yes—
130C7×C22⋊C8C2314yes—
131C22⋊(C7⋊C8)C2314yes—
132#6390C2318yes—
133#6389C2318yes—
134Dic7⋊C8C2318yes—
135#6384C2318yes—
136#6383C2318yes—
137#6382C2318yes—
138C2×#529—71no—
139C22×C4×Q8—71no—
140#3117—716no—
141C2×#484D711yescharacteristic, complemented
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—21no—
153C2×C7×Q8—24no—
154C2×C7×Q8C2412yes—
155C22×C4×C7—24no—
156C22×C4×C7—24no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—44no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2412yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—24no—
166C2×C22⋊D7—44no—
167C22×Dic7—28no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7—22no—
171C22×Dic7—21no—
172C22×Dic7—42no—
173C22×Dic7C2×D414yes—
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D414yes—
179C2×D28—24no—
180C2×D28—21no—
181C2×D28C2412yes—
182C2×C4×D7—24no—
183C2×C4×D7—44no—
184C2×Dic14—28no—
185C2×Dic14—216no—
186C2×Dic14—24no—
187C2×Dic14—22no—
188C2×Dic14—24no—
189C2×Dic14—216no—
190C2×Dic14#4118yes—
191C2×Dic14C2414yes—
192C2×C7×C8—24no—
193C2×C7×C8C2414yes—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4C2414yes—
199C7×C4⋊C4C2412yes—
200C7×C22⋊C4—24no—
201#1144—216no—
202(C2×D7)⋊C4—24no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—24no—
205(C2×D7)⋊C4—24no—
206C4⋊Dic7—21no—
207C4⋊Dic7—24no—
208C4⋊Dic7—24no—
209C4⋊Dic7C2412yes—
210Dic7⋊C4—24no—
211Dic7⋊C4—28no—
212Dic7⋊C4—28no—
213C4×Dic7—24no—
214C4×Dic7—28no—
215C4×Dic7—28no—
216C4×Dic7C2×D412yes—
217C4×Dic7C2×D414yes—
218C4×Dic7C2×D412yes—
219C4×Dic7C2×D414yes—
220C4×Dic7C2412yes—
221C4×Dic7C2412yes—
222C2×C7⋊C8—24no—
223C2×C7⋊C8C2414yes—
224#1132—216no—
225C23×Q8—71no—
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⋊Q8C2×D711yescharacteristic
235C2×#121—71no—
236C2×C4×Q8—71no—
237C2×C4×Q8—72no—
238C2×C4×Q8—78no—
239C2×C4×Q8—72no—
240C2×C4×Q8—141no—
241C2×C4×Q8—141no—
242C2×C4×Q8—144no—
243C2×C4×D4—71no—
244C2×C4×D4—141no—
245C2×#117—141no—
246C22×C4⋊C4—141no—
247C22×C4⋊C4C2×D711yescharacteristic
248C22×C42—71no—
249C22×C42—141no—
250#484C2×D718yescomplemented
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×#107C2×D712yes—
260C2×#106—72no—
261C2×C4⋊C8—72no—
262C2×#103—72no—
263C2×#103—72no—
264C2×#103C2×D712yes—
265C2×#102—72no—
266C2×C22⋊C8—71no—
267C2×C22⋊C8C2×D711yescharacteristic
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—24no—
279C7×Q8—42no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—48no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—44no—
290C2×C4×C7C22×D412yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C2×#4114yes—
293C2×C4×C7C2511yescharacteristic
294C22⋊D7—48no—
295C2×Dic7—216no—
296C2×Dic7—28no—
297C2×Dic7—28no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—24no—
303C2×Dic7—22no—
304C2×Dic7—44no—
305C2×Dic7—42no—
306C2×Dic7C22×D412yes—
307C2×Dic7C22×D412yes—
308C2×Dic7C22×D412yes—
309C2×Dic7C22×D412yes—
310D28—24no—
311D28—42no—
312C4×D7—48no—
313Dic14—216no—
314Dic14—28no—
315Dic14—24no—
316Dic14C2×#4118yes—
317C7×C8—24no—
318C7⋊C8—24no—
319C22×Q8—72no—
320C22×Q8—74no—
321C22×Q8—148no—
322C22×Q8—142no—
323C22×Q8—141no—
324C22×Q8C22×D711yescharacteristic
325C22×D4—71no—
326C22×D4—141no—
327C23×C4—21no—
328C23×C4—71no—
329C23×C4—71no—
330C23×C4—141no—
331C23×C4—141no—
332C23×C4C22×D711yescharacteristic
333C2×Q16—78no—
334C2×Q16—148no—
335C2×SD16—78no—
336C2×SD16—148no—
337C22×C8—72no—
338C22×C8C22×D712yes—
339#126—148no—
340#125—74no—
341#125—142no—
342#124—74no—
343#124—142no—
344#123—148no—
345C22⋊Q8—22no—
346C22⋊Q8C22×D714yes—
347#121—74no—
348#121—142no—
349C4×Q8—74no—
350C4×Q8—78no—
351C4×Q8—78no—
352C4×Q8—148no—
353C4×Q8—144no—
354C4×Q8—144no—
355C4×Q8—148no—
356C4×Q8—142no—
357C4×D4—74no—
358C4×D4—142no—
359C4×D4—148no—
360#117—148no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—148no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4C22×D712yes—
372C2×C4⋊C4C22×D714yes—
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—74no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—74no—
385C2×C42—141no—
386C2×C42—148no—
387C2×C42—142no—
388C2×C42—142no—
389#107C22×D718yes—
390#106—78no—
391C4⋊C8—78no—
392#103—78no—
393#103—78no—
394#103C22×D718yes—
395#102—78no—
396C22⋊C8—74no—
397C22⋊C8C22×D714yes—
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—22no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7—44no—
415C4×C7#52512yes—
416C4×C7C22×#4112yes—
417Dic7—28no—
418Dic7—22no—
419Dic7—22no—
420Dic7—22no—
421Dic7—22no—
422Dic7—42no—
423C24—141no—
424C24D4×D711yescharacteristic
425C2×Q8—21no—
426C2×Q8—24no—
427C2×Q8—78no—
428C2×Q8—74no—
429C2×Q8—1416no—
430C2×Q8—148no—
431C2×Q8—1416no—
432C2×Q8—144no—
433C2×Q8—142no—
434C2×Q8—144no—
435C2×Q8C23×D712yes—
436C2×D4—72no—
437C2×D4—144no—
438C2×D4—141no—
439C2×D4—144no—
440C2×D4—284no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—44no—
447C22×C4—74no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4—74no—
453C22×C4—144no—
454C22×C4—148no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—284no—
460C22×C4—282no—
461C22×C4D4×D712yes—
462C22×C4D4×D711yescharacteristic
463C22×C4C23×D712yes—
464C22×C4C23×D711yescharacteristic
465Q16—1416no—
466SD16—1416no—
467C2×C8—24no—
468C2×C8—74no—
469C2×C8—144no—
470C2×C8C23×D714yes—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—72no—
476C4⋊C4—148no—
477C4⋊C4—148no—
478C4⋊C4—144no—
479C4⋊C4—144no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4C23×D714yes—
483C4⋊C4C23×D712yes—
484C22⋊C4—24no—
485C22⋊C4—144no—
486C22⋊C4—144no—
487C22⋊C4—144no—
488C22⋊C4—144no—
489C42—74no—
490C42—72no—
491C42—72no—
492C42—74no—
493C42—72no—
494C42—72no—
495C42—148no—
496C42—148no—
497C42—144no—
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#648114yes—
511C23C2×D4×D712yes—
512C23C2×D4×D711yescharacteristic
513Q8—24no—
514Q8—42no—
515Q8—78no—
516Q8—1416no—
517Q8—148no—
518Q8—144no—
519D4—144no—
520D4—288no—
521D4—282no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—72no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—1416no—
537C2×C4—148no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—148no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—288no—
546C2×C4—284no—
547C2×C4—282no—
548C2×C4C2×D4×D712yes—
549C2×C4C2×D4×D711yescharacteristic
550C2×C4D7×#4114yes—
551C2×C4C24×D711yescharacteristic
552C8—24no—
553C8—144no—
554C7C2×#311711yescharacteristic, complemented, Sylow
555C22—22no—
556C22—28no—
557C22—22no—
558C22—282no—
559C22—282no—
560C22—282no—
561C22—564no—
562C22#9034712yes—
563C22C2×#648114yes—
564C22C2×#648112yes—
565C22#9046412yes—
566C22C22×D4×D711yescharacteristic
567C4—22no—
568C4—44no—
569C4—42no—
570C4—42no—
571C4—142no—
572C4—142no—
573C4—148no—
574C4—142no—
575C4—142no—
576C4—282no—
577C4#9034712yes—
578C4C2×D7×#4112yes—
579C2—24no—
580C2—562no—
581C2#1173890014yesdirect product factor
582C2C2×#9034711yescharacteristic
583C2C22×#648111yescharacteristic
584C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#484C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC2×#484D7
hypocenterC7C2×#3117
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#6450C2semidirect
3. C2×C7×#484C2semidirect
4. C2×#89822C2semidirect
6. #11738900C2direct (N × H)
7. C2×#89649C2semidirect
8. C2×#89661C2semidirect
9. C2×#89570C2semidirect
10. C2×#89561C2semidirect
12. C2×#89491C2semidirect
35. C2×#6450C22semidirect
38. C7×#484C22semidirect
46. #89822C22semidirect
51. #89661C22semidirect
52. #89649C22semidirect
53. #89570C22semidirect
54. #89561C22semidirect
56. #89491C22semidirect
141. C2×#484D7semidirect
250. #484C2×D7semidirect
499. C2×C7#3117semidirect
554. C7C2×#3117semidirect
581. C2#11738900direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#903152
C22×#64502
C2×C7×#4842
C2×#898222
C2×#897402
#117389002
C2×#896492
C2×#896612
C2×#895702
C2×#895612
C2×#895102
C2×#894912
C2×#31177