Finite Group Databasebeta

← C2×#11736316 #14221333

Subgroups — C2×#11736316 #14221333

Order 1792.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic7C211yesmaximal, characteristic
2C22×#6453C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic
4C2×#90004C211yesmaximal, characteristic
5C2×#89893C211yesmaximal, characteristic, complemented
6C2×#89890C211yesmaximal, characteristic, complemented
7C2×#89736C211yesmaximal, characteristic, complemented
8#11736316C2116yesmaximal, direct product factor
9C2×#89568C212yesmaximal, complemented
10C2×#89571C212yesmaximal, complemented
11C2×#89535C212yesmaximal, complemented
12C2×#89538C212yesmaximal, complemented
13C22×C7×#41C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16#41×Dic7—24no—
17#41×Dic7C2218yes—
18C2×Q8×Dic7C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×D4×Dic7—21no—
21C2×D4×Dic7—21no—
22C2×D4×Dic7C2211yescharacteristic
23C2×D4×Dic7C2211yescharacteristic
24C2×#6489—21no—
25C2×#6489C2211yescharacteristic
26C22×C4⋊Dic7—21no—
27C22×C4⋊Dic7C2211yescharacteristic
28C22×Dic7⋊C4—21no—
29C22×C4×Dic7—21no—
30C22×C4×Dic7C2211yescharacteristic
31C22×#1136C2211yescharacteristic
32C2×#6453—24no—
33C2×#6453—21no—
34C2×#6453—21no—
35C2×#6453C2212yes—
36C2×#6453C2212yes—
37C2×#6453C2218yescomplemented
38C7×#417C2218yes—
39C2×C7×#103C2212yes—
40C2×C7×#102C2212yes—
41#90004C2218yes—
42C2×#6409C2212yes—
43C2×#6406C2212yes—
44#89893C2218yescomplemented
45#89890C2218yescomplemented
46C2×#6392C2212yes—
47C2×#6391C2212yes—
48C2×Dic7⋊C8C2212yes—
49C2×Dic7⋊C8C2212yes—
50#89736C2218yescomplemented
51C2×#6382C2212yes—
52C2×#6381C2212yes—
53#89571C22116yescomplemented
54#89568C22116yescomplemented
55#89538C22116yescomplemented
56#89535C22116yescomplemented
57C2×#3080—71nomaximal, Sylow
58C2×C7×#41—22no—
59C2×C7×#41—24no—
60C2×C7×#41Q818yes—
61C2×C7×#41C2314yes—
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C23×Dic7—21no—
69C23×Dic7—21no—
70C23×Dic7—21no—
71C23×Dic7D411yescharacteristic
72C23×Dic7D411yescharacteristic
73C2×C7×M16—24no—
74C2×C7×M16C2314yes—
75C22×C7×C8C2312yes—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2314yes—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83Q8×Dic7—22no—
84Q8×Dic7—22no—
85Q8×Dic7C2314yes—
86Q8×Dic7C2314yes—
87D4×Dic7—28no—
88D4×Dic7—28no—
89D4×Dic7—22no—
90D4×Dic7—22no—
91D4×Dic7C2314yes—
92D4×Dic7C2314yes—
93#6489—22no—
94#6489—28no—
95#6489C2314yes—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—22no—
98C2×C4⋊Dic7—22no—
99C2×C4⋊Dic7—22no—
100C2×C4⋊Dic7—28no—
101C2×C4⋊Dic7—21no—
102C2×C4⋊Dic7—21no—
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4⋊Dic7C2312yes—
105C2×C4⋊Dic7C2311yescharacteristic
106C2×C4⋊Dic7C2314yes—
107C2×Dic7⋊C4—22no—
108C2×Dic7⋊C4—22no—
109C2×Dic7⋊C4—28no—
110C2×C4×Dic7—21no—
111C2×C4×Dic7—21no—
112C2×C4×Dic7—21no—
113C2×C4×Dic7—28no—
114C2×C4×Dic7D411yescharacteristic
115C2×C4×Dic7D411yescharacteristic
116C2×C4×Dic7D414yes—
117C2×C4×Dic7C2311yescharacteristic
118C2×C4×Dic7C2311yescharacteristic
119C2×C4×Dic7C2314yes—
120C2×#1136—24no—
121C2×#1136C2314yes—
122C22×C7⋊C8C2312yes—
123#6453—24no—
124#6453—24no—
125#6453—28no—
126#6453—28no—
127#6453C2318yes—
128#6453C2318yes—
129C7×#103C2318yes—
130C7×#102C2318yes—
131#6409C2318yes—
132#6406C2318yes—
133#6392C2318yes—
134#6391C2318yes—
135Dic7⋊C8C2318yes—
136Dic7⋊C8C2318yes—
137#6382C2318yes—
138#6381C2318yes—
139C22×#125—71no—
140C2×C4×#41—71no—
141#3080—716no—
142C2×#479—72no—
143C2×#479—72no—
144C2×#478—72no—
145C2×#478—72no—
146C2×#428—71no—
147C2×#428—71no—
148C2×#423—71no—
149C2×#417—71no—
150C2×#417D711yescharacteristic
151C24×C7—21no—
152C24×C7C2×D411yescharacteristic
153C7×#41—216no—
154C7×#41—28no—
155C7×#41C2×Q8116yes—
156C2×C7×Q8—21no—
157C2×C7×Q8—22no—
158C2×C7×Q8C2×Q814yes—
159C2×C7×Q8C2412yes—
160C2×C7×D4—28no—
161C2×C7×D4—24no—
162C2×C7×D4—21no—
163C2×C7×D4—22no—
164C2×C7×D4C2×Q814yes—
165C2×C7×D4C2412yes—
166C22×C4×C7—21no—
167C22×C4×C7—28no—
168C22×C4×C7—24no—
169C22×C4×C7—24no—
170C22×C4×C7—22no—
171C22×C4×C7—44no—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×Q814yes—
175C22×C4×C7#4114yes—
176C22×C4×C7C2411yescharacteristic
177C22×C4×C7C2412yes—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—24no—
181C22×Dic7—22no—
182C22×Dic7—24no—
183C22×Dic7—24no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—44no—
187C22×Dic7—44no—
188C22×Dic7—44no—
189C22×Dic7C2×D411yescharacteristic
190C22×Dic7C2×D411yescharacteristic
191C22×Dic7C2×D411yescharacteristic
192C22×Dic7C2×D411yescharacteristic
193C22×Dic7C2×D414yes—
194C22×Dic7C2×D414yes—
195C7×M16—28no—
196C2×C7×C8—24no—
197C2×C7×C8C2414yes—
198C7×C4⋊C4—24no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4C2414yes—
201C22⋊Dic7—24no—
202C22⋊Dic7—24no—
203C22⋊Dic7—24no—
204C4⋊Dic7—24no—
205C4⋊Dic7—22no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—28no—
209C4⋊Dic7—28no—
210C4⋊Dic7—24no—
211C4⋊Dic7—24no—
212C4⋊Dic7C2414yes—
213C4⋊Dic7C2412yes—
214C4⋊Dic7C2412yes—
215Dic7⋊C4—28no—
216Dic7⋊C4—28no—
217C4×Dic7—24no—
218C4×Dic7—24no—
219C4×Dic7—24no—
220C4×Dic7C2×D414yes—
221C4×Dic7C2×D414yes—
222C4×Dic7C2×D412yes—
223C4×Dic7C2×D412yes—
224C4×Dic7C2412yes—
225C4×Dic7C2412yes—
226#1136—28no—
227C2×C7⋊C8—24no—
228C2×C7⋊C8C2414yes—
229C22×#41C2×D711yescharacteristic
230C22×M16—71no—
231C22×M16C2×D711yescharacteristic
232C2×#125—72no—
233C2×#125—72no—
234C2×#125—78no—
235C2×#125—141no—
236C2×#125—144no—
237C2×#125—141no—
238C4×#41—78no—
239C4×#41—144no—
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×D4—71no—
243C2×C4×D4—71no—
244C2×C4×D4—141no—
245C2×C4×D4—141no—
246C2×#117—71no—
247C2×#117—141no—
248C22×C4⋊C4—71no—
249C22×C4⋊C4—141no—
250C22×C4⋊C4—141no—
251C22×C4⋊C4C2×D711yescharacteristic
252C22×C42—71no—
253C22×C42—141no—
254#479—716no—
255#479—716no—
256#478—716no—
257#478—716no—
258#428—78no—
259#428—78no—
260C2×#107—72no—
261C2×#107—72no—
262C2×#106—72no—
263C2×#106—72no—
264#423—78no—
265C2×C4⋊C8—72no—
266C2×C4⋊C8—72no—
267#417—78no—
268#417C2×D718yes—
269C2×#103—72no—
270C2×#103C2×D712yes—
271C2×#102—72no—
272C2×#102C2×D712yes—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—24no—
276C23×C7—22no—
277C23×C7—42no—
278C23×C7—42no—
279C23×C7C22⋊Q814yes—
280C23×C7C22×D412yes—
281C23×C7C22×D411yescharacteristic
282C7×Q8—24no—
283C7×Q8—22no—
284C7×Q8C22×Q814yes—
285C7×D4—216no—
286C7×D4—24no—
287C7×D4—22no—
288C7×D4C22×Q814yes—
289C2×C4×C7—24no—
290C2×C4×C7—22no—
291C2×C4×C7—216no—
292C2×C4×C7—24no—
293C2×C4×C7—48no—
294C2×C4×C7—44no—
295C2×C4×C7C22⋊Q814yes—
296C2×C4×C7C22⋊Q818yes—
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C22×Q812yes—
300C2×C4×C7C22×Q814yes—
301C2×C4×C7C2×#4112yes—
302C2×C4×C7C2×#4114yes—
303C2×C4×C7C2511yescharacteristic
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—44no—
312C2×Dic7—44no—
313C2×Dic7—48no—
314C2×Dic7—48no—
315C2×Dic7—48no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—44no—
319C2×Dic7C22×D412yes—
320C2×Dic7C22×D412yes—
321C2×Dic7C22×D412yes—
322C2×Dic7C22×D412yes—
323C7×C8—24no—
324C7⋊C8—24no—
325C2×#41—22no—
326C2×#41—24no—
327C2×#41Dic1418yes—
328C2×#41C22×D714yes—
329C22×Q8C22×D711yescharacteristic
330C22×D4—21no—
331C22×D4C22×D711yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—141no—
337C23×C4—141no—
338C23×C4—141no—
339C23×C4C22×D711yescharacteristic
340C2×M16—24no—
341C2×M16—74no—
342C2×M16—144no—
343C2×M16C22×D714yes—
344C22×C8—72no—
345C22×C8C22×D712yes—
346#125—78no—
347#125—78no—
348#125—148no—
349#125—148no—
350#125—144no—
351#125—144no—
352C4×Q8—74no—
353C4×Q8—74no—
354C4×Q8—142no—
355C4×Q8—142no—
356C4×D4—74no—
357C4×D4—74no—
358C4×D4—142no—
359C4×D4—142no—
360C4×D4—148no—
361C4×D4—148no—
362#117—74no—
363#117—148no—
364#117—142no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—74no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—148no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—148no—
375C2×C4⋊C4—142no—
376C2×C4⋊C4—142no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4C22×D712yes—
382C2×C4⋊C4C22×D714yes—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C42—71no—
387C2×C42—74no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—74no—
392C2×C42—148no—
393C2×C42—141no—
394C2×C42—141no—
395C2×C42—141no—
396#107—78no—
397#107—78no—
398#106—78no—
399#106—78no—
400C4⋊C8—78no—
401C4⋊C8—78no—
402#103—78no—
403#103C22×D718yes—
404#102—78no—
405#102C22×D718yes—
406C22×C7—28no—
407C22×C7—22no—
408C22×C7—22no—
409C22×C7—24no—
410C22×C7—48no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7C2×C22⋊Q814yes—
414C22×C7C2×C22⋊Q812yes—
415C22×C7#57512yes—
416C22×C7#57512yes—
417C22×C7C23×D411yescharacteristic
418C4×C7—24no—
419C4×C7—44no—
420C4×C7C2×C22⋊Q814yes—
421C4×C7C2×C22⋊Q812yes—
422C4×C7C23×Q811yescharacteristic
423C4×C7C22×#4111yescharacteristic
424Dic7—22no—
425Dic7—22no—
426Dic7—22no—
427Dic7—22no—
428Dic7—42no—
429Dic7—42no—
430Dic7—44no—
431C24—21no—
432C24D4×D711yescharacteristic
433#41—216no—
434#41—28no—
435#41C2×Dic14116yes—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8C2×Dic1414yes—
439C2×Q8C23×D712yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4C2×Dic1414yes—
445C2×D4C23×D712yes—
446C22×C4—21no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—44no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—74no—
457C22×C4—74no—
458C22×C4—144no—
459C22×C4—144no—
460C22×C4—144no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—142no—
464C22×C4—282no—
465C22×C4—282no—
466C22×C4—284no—
467C22×C4—284no—
468C22×C4—284no—
469C22×C4C2×Dic1414yes—
470C22×C4D4×D712yes—
471C22×C4D4×D711yescharacteristic
472C22×C4#115914yes—
473C22×C4C23×D712yes—
474C22×C4C23×D711yescharacteristic
475M16—28no—
476M16—148no—
477C2×C8—24no—
478C2×C8—74no—
479C2×C8—144no—
480C2×C8C23×D714yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—74no—
484C4⋊C4—72no—
485C4⋊C4—72no—
486C4⋊C4—148no—
487C4⋊C4—148no—
488C4⋊C4—148no—
489C4⋊C4—148no—
490C4⋊C4—144no—
491C4⋊C4—144no—
492C4⋊C4—144no—
493C4⋊C4—142no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4C23×D714yes—
497C22⋊C4—144no—
498C22⋊C4—144no—
499C22⋊C4—144no—
500C42—74no—
501C42—74no—
502C42—72no—
503C42—72no—
504C42—72no—
505C42—72no—
506C42—144no—
507C42—144no—
508C42—144no—
509C2×C7—24no—
510C2×C7—44no—
511C2×C7#308014yescomplemented
512C2×C7C22×C22⋊Q811yescharacteristic
513C2×C7C2×#57511yescharacteristic
514C2×C7C2×#57511yescharacteristic
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23#644114yes—
522C23C2×D4×D712yes—
523C23C2×D4×D711yescharacteristic
524Q8—24no—
525Q8—22no—
526Q8C22×Dic1414yes—
527D4—216no—
528D4—24no—
529D4—22no—
530D4C22×Dic1414yes—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—216no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—72no—
540C2×C4—72no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—284no—
549C2×C4—284no—
550C2×C4—288no—
551C2×C4—288no—
552C2×C4—288no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—284no—
556C2×C4#644114yes—
557C2×C4#644118yes—
558C2×C4C22×Dic1412yes—
559C2×C4C22×Dic1414yes—
560C2×C4C2×D4×D712yes—
561C2×C4C2×D4×D711yescharacteristic
562C2×C4C2×#115912yes—
563C2×C4C2×#115914yes—
564C2×C4C24×D711yescharacteristic
565C8—24no—
566C8—144no—
567C7C2×#308011yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—28no—
571C22—24no—
572C22—42no—
573C22—42no—
574C22—48no—
575C22C2×#644114yes—
576C22C2×#644112yes—
577C22#9046412yes—
578C22#9046812yes—
579C22C22×D4×D711yescharacteristic
580C4—24no—
581C4—44no—
582C4—142no—
583C4—142no—
584C4—142no—
585C4—142no—
586C4—282no—
587C4—282no—
588C4—284no—
589C4C2×#644114yes—
590C4C2×#644112yes—
591C4C23×Dic1411yescharacteristic
592C4C22×#115911yescharacteristic
593C2—24no—
594C2—44no—
595C2#1173631614yesdirect product factor
596C2C22×#644111yescharacteristic
597C2C2×#9046411yescharacteristic
598C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×#417D7
hypocenterC7C2×#3080
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#6453C2semidirect
5. C2×#89893C2semidirect
6. C2×#89890C2semidirect
7. C2×#89736C2semidirect
8. #11736316C2direct (N × H)
9. C2×#89568C2semidirect
10. C2×#89571C2semidirect
11. C2×#89535C2semidirect
12. C2×#89538C2semidirect
37. C2×#6453C22semidirect
44. #89893C22semidirect
45. #89890C22semidirect
50. #89736C22semidirect
53. #89571C22semidirect
54. #89568C22semidirect
55. #89538C22semidirect
56. #89535C22semidirect
511. C2×C7#3080semidirect
567. C7C2×#3080semidirect
595. C2#11736316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×Dic72
C22×#64532
C2×C7×#4172
C2×#900042
C2×#898932
C2×#898902
C2×#897362
#117363162
C2×#895682
C2×#895712
C2×#895352
C2×#895382
C2×#30807