Finite Group Databasebeta

← C2×#11740601 #14226652

Subgroups — C2×#11740601 #14226652

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×C7×#529C211yesmaximal, characteristic
2C2×#90161C211yesmaximal, characteristic, complemented
3#11740601C2116yesmaximal, direct product factor
4C2×#89835C212yesmaximal, complemented
5C2×#89826C212yesmaximal, complemented
6C2×#89824C212yesmaximal, complemented
7C2×#89814C212yesmaximal
8C2×#89803C211yesmaximal, characteristic, complemented
9C2×#89793C211yesmaximal, characteristic, complemented
10C2×#89791C211yesmaximal, characteristic, complemented
11C2×#89780C211yesmaximal, characteristic, complemented
12C2×#89696C211yesmaximal, characteristic, complemented
13C7×#529—24no—
14C7×#529C2218yes—
15C2×C7×#126—21no—
16C2×C7×#125C2211yescharacteristic
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117—21no—
25C22×C42×C7C2211yescharacteristic
26C22×#1144C2212yes—
27C22×#1142C2212yes—
28C2×#6489—21no—
29C2×#6487—21no—
30C2×#6487C2212yes—
31C22×#1136C2212yes—
32#90161—24no—
33#90161C2218yescomplemented
34C2×#6433—21no—
35C2×#6432C2211yescharacteristic
36C2×#6432C2211yescharacteristic
37C2×C4×Dic14—21no—
38C2×C4×Dic14C2212yes—
39C2×#6409C2212yes—
40C2×#6406C2212yes—
41#89835C22116yescomplemented
42#89826C22116yescomplemented
43#89824C22116yescomplemented
44#89814C22116yes—
45#89803C2218yescomplemented
46#89793C2218yescomplemented
47#89791C2218yescomplemented
48#89780C2218yescomplemented
49C2×#6384C2212yes—
50C2×#6384C2212yes—
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53#89696C2218yescomplemented
54C2×C4⋊(C7⋊C8)C2212yes—
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56C2×C4⋊(C7⋊C8)C2211yescharacteristic
57C2×#3252—71nomaximal, Sylow
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7D411yescharacteristic
63C23×C4×C7D411yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C22×Dic14—21no—
66C22×Dic14C2312yes—
67C7×#126—28no—
68C7×#125—22no—
69C7×#125C2314yes—
70C7×#124—22no—
71C7×#124C2314yes—
72C7×#123—28no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8C2314yes—
75C7×#121—22no—
76C7×#121C2314yes—
77C4×C7×Q8—22no—
78C4×C7×Q8C2314yes—
79C4×C7×D4—28no—
80C4×C7×D4—22no—
81C4×C7×D4C2314yes—
82C7×#117—28no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C42×C7—21no—
95C2×C42×C7D411yescharacteristic
96C2×C42×C7D411yescharacteristic
97C2×C42×C7D414yes—
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2311yescharacteristic
100C2×C42×C7C2314yes—
101C2×C22⋊Dic7—22no—
102C2×#1144—44no—
103C2×#1144C2318yes—
104C2×#1142—44no—
105C2×#1142C2318yes—
106#6489—28no—
107C2×C4⋊Dic7—21no—
108C2×C4⋊Dic7—21no—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7C2312yes—
111#6487—24no—
112#6487—28no—
113#6487C2318yes—
114C2×Dic7⋊C4—22no—
115C2×Dic7⋊C4—22no—
116C2×C4×Dic7—22no—
117C2×#1136—44no—
118C2×#1136C2318yes—
119C22×C7⋊C8C2312yes—
120C22×C7⋊C8C2312yes—
121#6433—28no—
122#6432—22no—
123#6432—22no—
124#6432C2314yes—
125#6432C2314yes—
126C4×Dic14—28no—
127C4×Dic14—24no—
128C4×Dic14C2318yes—
129#6409C2318yes—
130#6406C2318yes—
131#6384C2318yes—
132#6384C2318yes—
133#6382C2318yes—
134#6381C2318yes—
135C4⋊(C7⋊C8)C2314yes—
136C4⋊(C7⋊C8)C2314yes—
137C4⋊(C7⋊C8)C2318yes—
138C2×#533—71no—
139C2×#529D711yescharacteristic
140#3252—716no—
141C2×#479—72no—
142C2×#477—71no—
143C2×#476—71no—
144C2×#470—72no—
145C2×#468—72no—
146C2×#464—72no—
147C2×#462—71no—
148C2×#461—71no—
149C2×#423—71no—
150C24×C7—21no—
151C24×C7C2×D411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8—22no—
154C2×C7×Q8—22no—
155C2×C7×Q8C2412yes—
156C2×C7×D4—24no—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—44no—
161C2×C7×D4C2412yes—
162C22×C4×C7—24no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—24no—
169C22×C4×C7—42no—
170C22×C4×C7—44no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D414yes—
177C22×C4×C7C2×D414yes—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2412yes—
180C22×C4×C7C2411yescharacteristic
181C22×Dic7—22no—
182C22×Dic7—22no—
183C2×Dic14—24no—
184C2×Dic14—24no—
185C2×Dic14—22no—
186C2×Dic14—24no—
187C2×Dic14—44no—
188C2×Dic14C2414yes—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4C2412yes—
196C7×C4⋊C4C2412yes—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C42×C7—24no—
203C42×C7C2×D412yes—
204C42×C7C2×D414yes—
205C42×C7C2×D412yes—
206C42×C7C2×D414yes—
207C42×C7C2412yes—
208C42×C7C2412yes—
209C22⋊Dic7—28no—
210#1144—48no—
211#1142—48no—
212C4⋊Dic7—22no—
213C4⋊Dic7—24no—
214C4⋊Dic7—24no—
215C4⋊Dic7—24no—
216C4⋊Dic7C2414yes—
217Dic7⋊C4—28no—
218Dic7⋊C4—28no—
219C4×Dic7—28no—
220#1136—48no—
221C2×C7⋊C8—24no—
222C2×C7⋊C8—24no—
223C2×C7⋊C8C2414yes—
224C2×C7⋊C8C2414yes—
225C22×Q16—72no—
226C22×SD16—72no—
227C22×M16—72no—
228#533—78no—
229#533—144no—
230C2×C4⋊Q8—71no—
231C2×C4⋊Q8—71no—
232#529—24no—
233#529C2×D718yes—
234C2×#126—21no—
235C2×#125—141no—
236C2×#125C2×D711yescharacteristic
237C2×#124C2×D711yescharacteristic
238C2×#123—21no—
239C2×C22⋊Q8—72no—
240C2×C22⋊Q8—141no—
241C2×C22⋊Q8C2×D711yescharacteristic
242C2×#121C2×D711yescharacteristic
243C2×C4×Q8—72no—
244C2×C4×Q8—141no—
245C2×C4×Q8C2×D711yescharacteristic
246C2×C4×D4—21no—
247C2×C4×D4C2×D711yescharacteristic
248C2×#117—21no—
249C2×#117—141no—
250C22×C42C2×D711yescharacteristic
251#479—716no—
252#477—78no—
253#476—78no—
254#470—716no—
255#468—716no—
256#464—716no—
257#462—78no—
258#461—78no—
259C2×#107—72no—
260C2×#106—72no—
261#423—78no—
262C2×C4⋊C8—71no—
263C2×C4⋊C8—72no—
264C2×C4⋊C8—71no—
265C2×#103—72no—
266C2×#103—72no—
267C2×#103—72no—
268C2×#102—72no—
269C23×C7—22no—
270C23×C7—22no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—42no—
274C23×C7—42no—
275C23×C7—42no—
276C23×C7C22×D412yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—42no—
281C7×D4—22no—
282C7×D4—22no—
283C7×D4—48no—
284C7×D4—42no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—24no—
290C2×C4×C7—28no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—44no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—48no—
301C2×C4×C7#13714yes—
302C2×C4×C7C22×D412yes—
303C2×C4×C7C22×D412yes—
304C2×C4×C7C22×D412yes—
305C2×C4×C7C22×D412yes—
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C22×D412yes—
308C2×C4×C7H(5,𝔽2)12yes—
309C2×C4×C72−1+412yes—
310C2×C4×C7C2511yescharacteristic
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—44no—
314C2×Dic7—44no—
315Dic14—24no—
316Dic14—24no—
317Dic14—48no—
318Dic14—44no—
319C7⋊C8—24no—
320C7⋊C8—24no—
321C22×Q8—72no—
322C22×Q8—141no—
323C22×Q8C22×D711yescharacteristic
324C22×D4—21no—
325C22×D4C22×D711yescharacteristic
326C23×C4—21no—
327C23×C4C22⋊D711yescharacteristic
328C23×C4C22⋊D711yescharacteristic
329C23×C4C22×D711yescharacteristic
330C2×Q16—78no—
331C2×Q16—284no—
332C2×SD16—78no—
333C2×SD16—284no—
334C2×M16—78no—
335C2×M16—284no—
336C22×C8—72no—
337C22×C8—72no—
338C4⋊Q8—74no—
339C4⋊Q8—74no—
340C4⋊Q8—142no—
341C4⋊Q8—142no—
342#126—28no—
343#125—22no—
344#125—148no—
345#125C22×D714yes—
346#124—22no—
347#124C22×D714yes—
348#123—28no—
349C22⋊Q8—22no—
350C22⋊Q8—78no—
351C22⋊Q8—144no—
352C22⋊Q8—148no—
353C22⋊Q8C22×D714yes—
354#121—22no—
355#121C22×D714yes—
356C4×Q8—22no—
357C4×Q8—78no—
358C4×Q8—144no—
359C4×Q8—148no—
360C4×Q8C22×D714yes—
361C4×D4—28no—
362C4×D4—22no—
363C4×D4C22×D714yes—
364#117—28no—
365#117—148no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—72no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—142no—
384C2×C42—21no—
385C2×C42—142no—
386C2×C42C22⋊D711yescharacteristic
387C2×C42C22⋊D711yescharacteristic
388C2×C42C22⋊D714yes—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D714yes—
392#107—78no—
393#106—78no—
394C4⋊C8—74no—
395C4⋊C8—74no—
396C4⋊C8—78no—
397#103—78no—
398#103—78no—
399#103—78no—
400#102—78no—
401C22×C7—22no—
402C22×C7—22no—
403C22×C7—22no—
404C22×C7—44no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—84no—
409C22×C7#53712yes—
410C22×C7C2×#13712yes—
411C22×C7#57512yes—
412C22×C7C23×D411yescharacteristic
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—42no—
420C4×C7#53712yes—
421C4×C7C2×#13714yes—
422C4×C7C2×H(5,𝔽2)11yescharacteristic
423C4×C7C2×2−1+411yescharacteristic
424Dic7—44no—
425Dic7—44no—
426C24—21no—
427C24C2×C22⋊D711yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—74no—
432C2×Q8—144no—
433C2×Q8—142no—
434C2×Q8—144no—
435C2×Q8—144no—
436C2×Q8—284no—
437C2×Q8C23×D712yes—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—44no—
443C2×D4C23×D712yes—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4C2×C22⋊D712yes—
456C22×C4C2×C22⋊D714yes—
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D714yes—
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C23×D712yes—
464C22×C4C23×D711yescharacteristic
465Q16—288no—
466SD16—288no—
467M16—288no—
468C2×C8—74no—
469C2×C8—74no—
470C2×C8—144no—
471C2×C8—144no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—74no—
479C4⋊C4—148no—
480C4⋊C4—148no—
481C4⋊C4—142no—
482C4⋊C4—144no—
483C4⋊C4—144no—
484C4⋊C4—144no—
485C4⋊C4C23×D712yes—
486C4⋊C4C23×D712yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—148no—
493C42—24no—
494C42—148no—
495C42C2×C22⋊D714yes—
496C42C2×C22⋊D712yes—
497C42C2×C22⋊D712yes—
498C42C2×C22⋊D714yes—
499C42C23×D712yes—
500C42C23×D712yes—
501C2×C7—42no—
502C2×C7—82no—
503C2×C7#325214yescomplemented
504C2×C7C2×#53711yescharacteristic
505C2×C7C22×#13711yescharacteristic
506C2×C7C2×#57511yescharacteristic
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C22×C22⋊D712yes—
515C23C22×C22⋊D711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—42no—
519Q8—144no—
520Q8—144no—
521Q8—288no—
522Q8—284no—
523D4—22no—
524D4—22no—
525D4—48no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—144no—
544C2×C4—144no—
545C2×C4—284no—
546C2×C4—284no—
547C2×C4#651414yes—
548C2×C4#655312yes—
549C2×C4#655412yes—
550C2×C4C22×C22⋊D712yes—
551C2×C4C22×C22⋊D712yes—
552C2×C4C22×C22⋊D712yes—
553C2×C4C22×C22⋊D712yes—
554C2×C4C22×C22⋊D712yes—
555C2×C4C22×C22⋊D711yescharacteristic
556C2×C4C24×D711yescharacteristic
557C8—144no—
558C8—144no—
559C7C2×#325211yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—84no—
568C22#9051312yes—
569C22C2×#651412yes—
570C22#9052412yes—
571C22C23×C22⋊D711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—284no—
580C4—284no—
581C4C2×#651414yes—
582C4#9052412yes—
583C4C2×#655311yescharacteristic
584C4C2×#655411yescharacteristic
585C2—42no—
586C2—82no—
587C2#1174060114yesdirect product factor
588C2C2×#9051311yescharacteristic
589C2C22×#651411yescharacteristic
590C2C2×#9052411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#529C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#529D7
hypocenterC7C2×#3252
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#90161C2semidirect
3. #11740601C2direct (N × H)
4. C2×#89835C2semidirect
5. C2×#89826C2semidirect
6. C2×#89824C2semidirect
8. C2×#89803C2semidirect
9. C2×#89793C2semidirect
10. C2×#89791C2semidirect
11. C2×#89780C2semidirect
12. C2×#89696C2semidirect
33. #90161C22semidirect
41. #89835C22semidirect
42. #89826C22semidirect
43. #89824C22semidirect
45. #89803C22semidirect
46. #89793C22semidirect
47. #89791C22semidirect
48. #89780C22semidirect
53. #89696C22semidirect
503. C2×C7#3252semidirect
559. C7C2×#3252semidirect
587. C2#11740601direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5292
C2×#901612
#117406012
C2×#898352
C2×#898262
C2×#898242
C2×#898142
C2×#898032
C2×#897932
C2×#897912
C2×#897802
C2×#896962
C2×#32527