Finite Group Databasebeta

← C2×#11738577 #14226655

Subgroups — C2×#11738577 #14226655

Order 1792.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic, complemented
2C2×#90271C211yesmaximal, characteristic
3C2×C7×#477C211yesmaximal, characteristic, complemented
4C2×#89856C211yesmaximal, characteristic, complemented
5C2×#89803C211yesmaximal, characteristic, complemented
6#11738577C2116yesmaximal, direct product factor
7C2×#89649C212yesmaximal, complemented
8C2×#89658C212yesmaximal
9C2×#89613C211yesmaximal, characteristic, complemented
10C2×#89609C211yesmaximal, characteristic, complemented
11C2×#89570C212yesmaximal, complemented
12C2×#89588C212yesmaximal, complemented
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6510C2211yescharacteristic
16C2×#6509—21no—
17C2×#6509C2211yescharacteristic
18C2×Q8×Dic7—21no—
19C22×#1144C2212yes—
20C22×#1132C2212yes—
21C22×#1131C2212yes—
22#90419—24no—
23#90419C2218yescomplemented
24#90271—24no—
25#90271C2218yes—
26C2×#6461—21no—
27C2×#6460C2212yes—
28C2×#6458C2211yescharacteristic
29C2×#6457—21no—
30C2×#6455—21no—
31C2×#6453—21no—
32C2×#6453C2211yescharacteristic
33C2×#6450C2212yes—
34C2×C4×D28—21no—
35C2×C4×D28C2211yescharacteristic
36C2×#6435—21no—
37C2×C42×D7C2211yescharacteristic
38C2×#6432C2211yescharacteristic
39C7×#477C2218yescomplemented
40C2×C7×#107C2212yes—
41C2×C7×C4⋊C8C2211yescharacteristic
42C2×C7×#103C2212yes—
43C2×#6390C2212yes—
44C2×Dic7⋊C8C2212yes—
45#89856C2218yescomplemented
46#89803C2218yescomplemented
47C2×#6384C2212yes—
48C2×#6383C2212yes—
49C2×#6382C2212yes—
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89658C22116yes—
52#89649C22116yescomplemented
53#89613C2218yescomplemented
54#89609C2218yescomplemented
55#89588C22116yescomplemented
56#89570C22116yescomplemented
57C2×#3252—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8C2311yescharacteristic
60C22×D28—21no—
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7C2311yescharacteristic
66C22×Dic14C2312yes—
67C22×C7×C8C2312yes—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8C2314yes—
70C4×C7×Q8—22no—
71C4×C7×Q8C2314yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78#6510—22no—
79#6510C2314yes—
80#6509—28no—
81#6509—22no—
82#6509C2314yes—
83Q8×Dic7—28no—
84C2×#1144—44no—
85C2×#1144C2318yes—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—22no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—22no—
100C2×C4×Dic7—22no—
101C2×C4×Dic7D411yescharacteristic
102C2×C4×Dic7D411yescharacteristic
103C2×C4×Dic7C2311yescharacteristic
104C22×C7⋊C8C2312yes—
105C2×#1132—44no—
106C2×#1132C2318yes—
107C2×#1131—44no—
108C2×#1131C2318yes—
109#6461—28no—
110#6460—24no—
111#6460C2318yes—
112#6458—22no—
113#6458C2314yes—
114#6457—28no—
115#6455—28no—
116#6453—28no—
117#6453—22no—
118#6453C2314yes—
119#6450—24no—
120#6450C2318yes—
121C4×D28—28no—
122C4×D28—22no—
123C4×D28C2314yes—
124#6435—28no—
125C42×D7D414yes—
126C42×D7C2314yes—
127#6432—22no—
128#6432C2314yes—
129C7×#107C2318yes—
130C7×C4⋊C8C2314yes—
131C7×#103C2318yes—
132#6390C2318yes—
133Dic7⋊C8C2318yes—
134#6384C2318yes—
135#6383C2318yes—
136#6382C2318yes—
137C4⋊(C7⋊C8)C2314yes—
138C2×#533—71no—
139C2×#529—71no—
140#3252—716no—
141C2×#479—72no—
142C2×#477D711yescharacteristic, complemented
143C2×#476—71no—
144C2×#470—72no—
145C2×#468—72no—
146C2×#464—72no—
147C2×#462—71no—
148C2×#461—71no—
149C2×#423—71no—
150C23×D7—21no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—24no—
153C2×C7×Q8—21no—
154C2×C7×Q8—22no—
155C2×C7×Q8—22no—
156C2×C7×Q8—44no—
157C2×C7×Q8C2412yes—
158C22×C4×C7—22no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—22no—
165C22×Dic7—21no—
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C2×D28—22no—
170C2×D28—22no—
171C2×D28—24no—
172C2×D28—21no—
173C2×D28—44no—
174C2×D28C2412yes—
175C2×C4×D7—24no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—24no—
179C2×C4×D7—42no—
180C2×C4×D7—44no—
181C2×C4×D7C2×D412yes—
182C2×C4×D7C2×D414yes—
183C2×C4×D7C2×D414yes—
184C2×C4×D7C2412yes—
185C2×Dic14—24no—
186C2×Dic14—24no—
187C2×Dic14—22no—
188C2×Dic14C2414yes—
189C2×C7×C8—24no—
190C2×C7×C8C2414yes—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4C2414yes—
196C7×C4⋊C4C2412yes—
197C42×C7—24no—
198C42×C7C2×D412yes—
199C42×C7C2412yes—
200#1144—48no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—28no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—24no—
205(C2×D7)⋊C4—24no—
206(C2×D7)⋊C4—24no—
207C4⋊Dic7—24no—
208C4⋊Dic7—24no—
209C4⋊Dic7—21no—
210C4⋊Dic7—24no—
211C4⋊Dic7—28no—
212C4⋊Dic7C2412yes—
213Dic7⋊C4—24no—
214Dic7⋊C4—24no—
215Dic7⋊C4—28no—
216C4×Dic7—28no—
217C4×Dic7C2×D412yes—
218C4×Dic7C2×D414yes—
219C4×Dic7C2×D414yes—
220C4×Dic7C2412yes—
221C2×C7⋊C8—24no—
222C2×C7⋊C8C2414yes—
223#1132—48no—
224#1131—48no—
225C22×Q16—72no—
226C22×SD16—72no—
227C22×M16—72no—
228#533—78no—
229#533—144no—
230C2×C4⋊Q8—71no—
231C2×C4⋊Q8C2×D711yescharacteristic
232#529—78no—
233#529—144no—
234C2×#126—141no—
235C2×#125—71no—
236C2×#125—141no—
237C2×#124—71no—
238C2×#123—141no—
239C2×C22⋊Q8—71no—
240C2×C22⋊Q8—72no—
241C2×C22⋊Q8—141no—
242C2×#121—71no—
243C2×C4×Q8—72no—
244C2×C4×Q8—141no—
245C2×C4×Q8C2×D711yescharacteristic
246C2×C4×D4—71no—
247C2×C4×D4—141no—
248C2×#117—141no—
249C2×#117—141no—
250C22×C42—71no—
251#479—716no—
252#477C2×D718yescomplemented
253#476—78no—
254#470—716no—
255#468—716no—
256#464—716no—
257#462—78no—
258#461—78no—
259C2×#107C2×D712yes—
260C2×#106—72no—
261#423—78no—
262C2×C4⋊C8—72no—
263C2×C4⋊C8—71no—
264C2×C4⋊C8C2×D711yescharacteristic
265C2×#103—72no—
266C2×#103—72no—
267C2×#103C2×D712yes—
268C2×#102—72no—
269C23×C7C22×D411yescharacteristic
270C22×D7—22no—
271C22×D7—22no—
272C22×D7—22no—
273C22×D7—22no—
274C22×D7—42no—
275C22×D7—42no—
276C22×D7—42no—
277C22×D7C22×D412yes—
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—48no—
281C7×Q8—42no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—42no—
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×C7H(5,𝔽2)12yes—
294C2×C4×C72−1+412yes—
295C2×C4×C7C2511yescharacteristic
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—24no—
300C2×Dic7—22no—
301C2×Dic7—44no—
302C2×Dic7—42no—
303C2×Dic7C22×D412yes—
304C2×Dic7C22×D412yes—
305C2×Dic7C22×D412yes—
306D28—22no—
307D28—22no—
308D28—48no—
309D28—42no—
310C4×D7—24no—
311C4×D7—28no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—44no—
315C4×D7—48no—
316Dic14—24no—
317Dic14—24no—
318Dic14—44no—
319C7×C8—24no—
320C7⋊C8—24no—
321C22×Q8—21no—
322C22×Q8—72no—
323C22×Q8C22×D711yescharacteristic
324C22×D4—71no—
325C22×D4—141no—
326C23×C4—71no—
327C23×C4—71no—
328C23×C4—71no—
329C23×C4—141no—
330C2×Q16—78no—
331C2×Q16—284no—
332C2×SD16—78no—
333C2×SD16—284no—
334C2×M16—78no—
335C2×M16—284no—
336C22×C8—72no—
337C22×C8C22×D712yes—
338C4⋊Q8—22no—
339C4⋊Q8—74no—
340C4⋊Q8—142no—
341C4⋊Q8C22×D714yes—
342#126—148no—
343#125—74no—
344#125—142no—
345#125—148no—
346#124—74no—
347#124—142no—
348#123—148no—
349C22⋊Q8—74no—
350C22⋊Q8—78no—
351C22⋊Q8—144no—
352C22⋊Q8—142no—
353C22⋊Q8—148no—
354#121—74no—
355#121—142no—
356C4×Q8—22no—
357C4×Q8—78no—
358C4×Q8—144no—
359C4×Q8—148no—
360C4×Q8C22×D714yes—
361C4×D4—74no—
362C4×D4—148no—
363C4×D4—142no—
364#117—148no—
365#117—148no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4C22×D712yes—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C22⋊C4—141no—
379C2×C22⋊C4—141no—
380C2×C22⋊C4—141no—
381C2×C22⋊C4—142no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C42—21no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—71no—
388C2×C42—74no—
389C2×C42—74no—
390C2×C42—142no—
391C2×C42C22×D711yescharacteristic
392#107C22×D718yes—
393#106—78no—
394C4⋊C8—78no—
395C4⋊C8—74no—
396C4⋊C8C22×D714yes—
397#103—78no—
398#103—78no—
399#103C22×D718yes—
400#102—78no—
401C22×C7#53712yes—
402C22×C7C2×#13712yes—
403C22×C7#57512yes—
404C22×C7C23×D411yescharacteristic
405C2×D7—22no—
406C2×D7—22no—
407C2×D7—22no—
408C2×D7—44no—
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
412C2×D7—84no—
413C4×C7—22no—
414C4×C7—42no—
415C4×C7—42no—
416C4×C7—44no—
417C4×C7#53712yes—
418C4×C7C2×#13714yes—
419C4×C7C2×H(5,𝔽2)11yescharacteristic
420C4×C7C2×2−1+411yescharacteristic
421Dic7—22no—
422Dic7—22no—
423Dic7—22no—
424Dic7—44no—
425Dic7—42no—
426C24—71no—
427C24—141no—
428C2×Q8—21no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—44no—
433C2×Q8—74no—
434C2×Q8—142no—
435C2×Q8—144no—
436C2×Q8—144no—
437C2×Q8C23×D712yes—
438C2×D4—72no—
439C2×D4—144no—
440C2×D4—141no—
441C2×D4—142no—
442C2×D4—142no—
443C2×D4—284no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—74no—
451C22×C4—72no—
452C22×C4—72no—
453C22×C4—74no—
454C22×C4—144no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—144no—
460C22×C4—282no—
461C22×C4—284no—
462C22×C4D4×D711yescharacteristic
463C22×C4D4×D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465Q16—288no—
466SD16—288no—
467M16—288no—
468C2×C8—24no—
469C2×C8—74no—
470C2×C8—144no—
471C2×C8C23×D714yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—72no—
477C4⋊C4—144no—
478C4⋊C4—144no—
479C4⋊C4—148no—
480C4⋊C4—144no—
481C4⋊C4—148no—
482C4⋊C4—141no—
483C4⋊C4—144no—
484C4⋊C4—144no—
485C4⋊C4C23×D714yes—
486C4⋊C4C23×D712yes—
487C22⋊C4—144no—
488C22⋊C4—144no—
489C22⋊C4—144no—
490C22⋊C4—148no—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C42—24no—
494C42—74no—
495C42—72no—
496C42—74no—
497C42—72no—
498C42—148no—
499C42D4×D712yes—
500C42C23×D712yes—
501C2×C7#325214yescomplemented
502C2×C7C2×#53711yescharacteristic
503C2×C7C22×#13711yescharacteristic
504C2×C7C2×#57511yescharacteristic
505D7—42no—
506D7—82no—
507C23—72no—
508C23—142no—
509C23—142no—
510C23—142no—
511C23—142no—
512C23—282no—
513C23—282no—
514C23—282no—
515C23C2×D4×D711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—48no—
519Q8—42no—
520Q8—144no—
521Q8—144no—
522Q8—284no—
523D4—142no—
524D4—142no—
525D4—288no—
526D4—282no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—72no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—148no—
541C2×C4—144no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—284no—
547C2×C4—282no—
548C2×C4—284no—
549C2×C4—288no—
550C2×C4#648114yes—
551C2×C4C2×D4×D712yes—
552C2×C4C2×D4×D712yes—
553C2×C4C2×D4×D711yescharacteristic
554C2×C4#655112yes—
555C2×C4#655312yes—
556C2×C4C24×D711yescharacteristic
557C8—24no—
558C8—144no—
559C7C2×#325211yescharacteristic, complemented, Sylow
560C22—142no—
561C22—142no—
562C22—142no—
563C22—284no—
564C22—282no—
565C22—282no—
566C22—282no—
567C22—564no—
568C22#9032012yes—
569C22C2×#648112yes—
570C22#9046412yes—
571C22C22×D4×D711yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—142no—
577C4—142no—
578C4—142no—
579C4—284no—
580C4—282no—
581C4#9032012yes—
582C4C2×#648114yes—
583C4C2×#655111yescharacteristic
584C4C2×#655311yescharacteristic
585C2—282no—
586C2—562no—
587C2#1173857714yesdirect product factor
588C2C2×#9032011yescharacteristic
589C2C22×#648111yescharacteristic
590C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#477C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
hypercenterC2×#477D7
hypocenterC7C2×#3252
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#90419C2semidirect
3. C2×C7×#477C2semidirect
4. C2×#89856C2semidirect
5. C2×#89803C2semidirect
6. #11738577C2direct (N × H)
7. C2×#89649C2semidirect
9. C2×#89613C2semidirect
10. C2×#89609C2semidirect
11. C2×#89570C2semidirect
12. C2×#89588C2semidirect
23. #90419C22semidirect
39. C7×#477C22semidirect
45. #89856C22semidirect
46. #89803C22semidirect
52. #89649C22semidirect
53. #89613C22semidirect
54. #89609C22semidirect
55. #89588C22semidirect
56. #89570C22semidirect
142. C2×#477D7semidirect
252. #477C2×D7semidirect
501. C2×C7#3252semidirect
559. C7C2×#3252semidirect
587. C2#11738577direct (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×#902712
C2×C7×#4772
C2×#898562
C2×#898032
#117385772
C2×#896492
C2×#896582
C2×#896132
C2×#896092
C2×#895702
C2×#895882
C2×#32527