Finite Group Databasebeta

← C2×#11736317 #14221381

Subgroups — C2×#11736317 #14221381

Order 1792.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90518C211yesmaximal, characteristic
2C2×#90191C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic
4C2×#90005C211yesmaximal, characteristic
5C2×#89893C211yesmaximal, characteristic, complemented
6C2×#89891C211yesmaximal, characteristic, complemented
7C22×#6381C211yesmaximal, characteristic, complemented
8#11736317C2116yesmaximal, direct product factor
9C2×#89571C212yesmaximal, complemented
10C2×#89566C212yesmaximal, complemented
11C2×#89535C212yesmaximal, complemented
12C2×#89531C212yesmaximal, complemented
13C22×C7×#41C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16#90518—24no—
17#90518C2218yes—
18C2×Q8×Dic7C2212yes—
19C2×D4×Dic7—22no—
20C2×D4×Dic7C2212yes—
21C2×#6489—21no—
22C2×#6489C2211yescharacteristic
23C2×#6489C2211yescharacteristic
24C22×C4⋊Dic7—21no—
25C22×C4⋊Dic7C2211yescharacteristic
26C2×#6487—22no—
27C23×C7⋊C8C2211yescharacteristic
28#90191—24no—
29#90191C2218yescomplemented
30C2×#6453—21no—
31C2×#6453C2212yes—
32C2×#6451—21no—
33C2×#6451C2212yes—
34C7×#417C2218yes—
35C2×C7×#103C2212yes—
36C2×C7×#102C2212yes—
37#90005C2218yes—
38C2×#6409C2211yescharacteristic
39C2×#6409C2211yescharacteristic
40C2×#6406C2211yescharacteristic
41C2×#6406C2211yescharacteristic
42#89893C2218yescomplemented
43#89891C2218yescomplemented
44C2×#6392C2212yes—
45C2×#6391C2212yes—
46C2×Dic7⋊C8C2212yes—
47C2×Dic7⋊C8C2212yes—
48C2×#6381C2212yes—
49C2×#6381C2212yes—
50C2×#6381C2218yescomplemented
51#89571C22116yescomplemented
52#89566C22116yescomplemented
53#89535C22116yescomplemented
54#89531C22116yescomplemented
55C2×#3083—71nomaximal, Sylow
56C2×C7×#41—22no—
57C2×C7×#41—24no—
58C2×C7×#41Q818yescomplemented
59C2×C7×#41C2314yes—
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7C2311yescharacteristic
66C23×Dic7—21no—
67C23×Dic7—22no—
68C22×Dic14—22no—
69C2×C7×M16—24no—
70C2×C7×M16C2314yes—
71C22×C7×C8C2312yes—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C4⋊C4C2314yes—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7D412yes—
79Q8×Dic7—24no—
80Q8×Dic7C2318yes—
81D4×Dic7—216no—
82D4×Dic7—24no—
83D4×Dic7C2318yes—
84#6489—22no—
85#6489—28no—
86#6489D414yes—
87#6489C2314yes—
88#6489C2314yes—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—22no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—28no—
94C2×C4⋊Dic7D411yescharacteristic
95C2×C4⋊Dic7D411yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2314yes—
100#6487—216no—
101C2×Dic7⋊C4—22no—
102C2×Dic7⋊C4—22no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—22no—
105C2×C4×Dic7C2312yes—
106C22×C7⋊C8—24no—
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2314yes—
110#6453—28no—
111#6453—24no—
112#6453C2318yes—
113#6451—28no—
114#6451—24no—
115#6451C2318yes—
116C7×#103C2318yes—
117C7×#102C2318yes—
118#6409C2314yes—
119#6409C2314yes—
120#6406C2314yes—
121#6406C2314yes—
122#6392C2318yes—
123#6391C2318yes—
124Dic7⋊C8C2318yes—
125Dic7⋊C8C2318yes—
126#6381C2318yes—
127#6381C2318yes—
128C2×#544—71no—
129C2×#520—71no—
130#3083—716no—
131C2×#479—72no—
132C2×#477—72no—
133C2×#478—72no—
134C2×#474—72no—
135C2×#428—71no—
136C22×#107—71no—
137C2×#424—71no—
138C2×#417D711yescharacteristic
139C2×#416—71no—
140C24×C7—21no—
141C24×C7C2×D411yescharacteristic
142C7×#41—216no—
143C7×#41—28no—
144C7×#41C2×Q8116yescomplemented
145C2×C7×Q8—21no—
146C2×C7×Q8—22no—
147C2×C7×Q8C2×Q814yes—
148C2×C7×Q8C2412yes—
149C2×C7×D4—28no—
150C2×C7×D4—24no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4C2×Q814yes—
154C2×C7×D4C2412yes—
155C22×C4×C7—21no—
156C22×C4×C7—28no—
157C22×C4×C7—24no—
158C22×C4×C7—22no—
159C22×C4×C7—24no—
160C22×C4×C7—82no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×Q814yes—
164C22×C4×C7#4114yes—
165C22×C4×C7C2411yescharacteristic
166C22×C4×C7C2412yes—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—22no—
170C22×Dic7—24no—
171C22×Dic7—28no—
172C22×Dic7—84no—
173C22×Dic7—82no—
174C22×Dic7C2×D412yes—
175C22×Dic7C2×D412yes—
176C2×Dic14—28no—
177C2×Dic14—84no—
178C7×M16—28no—
179C2×C7×C8—24no—
180C2×C7×C8C2414yes—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4C2414yes—
184C22⋊Dic7—24no—
185C22⋊Dic7—28no—
186C22⋊Dic7C2×D418yes—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—28no—
192C4⋊Dic7—28no—
193C4⋊Dic7C2×D414yes—
194C4⋊Dic7C2×D414yes—
195C4⋊Dic7C2412yes—
196C4⋊Dic7C2412yes—
197C4⋊Dic7C2414yes—
198Dic7⋊C4—28no—
199Dic7⋊C4—28no—
200C4×Dic7—24no—
201C4×Dic7—28no—
202C4×Dic7C2×D414yes—
203C4×Dic7C2414yes—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—28no—
207C2×C7⋊C8C2412yes—
208C2×C7⋊C8C2412yes—
209C22×#41C2×D711yescharacteristic
210C22×M16C2×D711yescharacteristic
211C23×C8—71no—
212#544—78no—
213#544—144no—
214C2×C4⋊Q8—72no—
215C2×C4⋊Q8—141no—
216C2×#125—72no—
217C2×#125—141no—
218C2×C22⋊Q8—142no—
219#520—78no—
220#520—144no—
221C2×C4×Q8—72no—
222C2×C4×D4—72no—
223C2×C4×D4—142no—
224C2×#117—71no—
225C2×#117—71no—
226C2×#117—141no—
227C22×C4⋊C4—71no—
228C22×C4⋊C4—141no—
229C22×C4⋊C4C2×D711yescharacteristic
230#479—716no—
231#478—716no—
232#477—716no—
233#474—716no—
234#428—78no—
235C2×#107—72no—
236C2×#107—72no—
237C2×#107—72no—
238C2×#107—78no—
239C2×#106—72no—
240#424—78no—
241C2×C4⋊C8—72no—
242C2×C4⋊C8—72no—
243#417C2×D718yes—
244#416—78no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103C2×D712yes—
248C2×#102—71no—
249C2×#102—71no—
250C2×#102C2×D712yes—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—24no—
254C23×C7—22no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7C22⋊Q814yes—
258C23×C7C22×D412yes—
259C23×C7C22×D411yescharacteristic
260C7×Q8—24no—
261C7×Q8—22no—
262C7×Q8C22×Q814yes—
263C7×D4—216no—
264C7×D4—24no—
265C7×D4—22no—
266C7×D4C22×Q814yes—
267C2×C4×C7—24no—
268C2×C4×C7—22no—
269C2×C4×C7—216no—
270C2×C4×C7—24no—
271C2×C4×C7—44no—
272C2×C4×C7—84no—
273C2×C4×C7C22⋊Q814yes—
274C2×C4×C7C22⋊Q818yes—
275C2×C4×C7C22×D412yes—
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C22×Q812yes—
278C2×C4×C7C22×Q814yes—
279C2×C4×C7C2×#4112yes—
280C2×C4×C7C2×#4114yes—
281C2×C4×C7C2511yescharacteristic
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—24no—
287C2×Dic7—42no—
288C2×Dic7—44no—
289C2×Dic7—42no—
290C2×Dic7—88no—
291C2×Dic7—84no—
292C2×Dic7C22×D414yes—
293C2×Dic7C22×D414yes—
294Dic14—88no—
295C7×C8—24no—
296C7⋊C8—22no—
297C7⋊C8—22no—
298C2×#41—22no—
299C2×#41—24no—
300C2×#41Dic1418yescomplemented
301C2×#41C22×D714yes—
302C22×Q8—142no—
303C22×Q8C22×D711yescharacteristic
304C22×D4—21no—
305C22×D4C22×D711yescharacteristic
306C23×C4—21no—
307C23×C4—21no—
308C23×C4—142no—
309C23×C4—141no—
310C23×C4C22×D711yescharacteristic
311C2×M16—24no—
312C2×M16C22×D714yes—
313C22×C8—71no—
314C22×C8—71no—
315C22×C8—74no—
316C22×C8—144no—
317C22×C8C22×D712yes—
318C4⋊Q8—78no—
319C4⋊Q8—144no—
320C4⋊Q8—148no—
321#125—78no—
322#125—144no—
323#125—148no—
324C22⋊Q8—1416no—
325C4×Q8—78no—
326C4×Q8—144no—
327C4×D4—78no—
328C4×D4—144no—
329C4×D4—1416no—
330#117—74no—
331#117—74no—
332#117—74no—
333#117—148no—
334#117—142no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4C22×D712yes—
351C2×C4⋊C4C22×D714yes—
352C2×C22⋊C4—72no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—141no—
355C2×C42—72no—
356C2×C42—142no—
357C2×C42—141no—
358#107—78no—
359#107—78no—
360#107—78no—
361#106—78no—
362C4⋊C8—78no—
363C4⋊C8—78no—
364#103—74no—
365#103—74no—
366#103C22×D718yes—
367#102—74no—
368#102—74no—
369#102C22×D718yes—
370C22×C7—28no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—24no—
374C22×C7—48no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7C2×C22⋊Q814yes—
378C22×C7C2×C22⋊Q812yes—
379C22×C7#57612yes—
380C22×C7#57812yes—
381C22×C7C23×D411yescharacteristic
382C4×C7—24no—
383C4×C7—44no—
384C4×C7C2×C22⋊Q814yes—
385C4×C7C2×C22⋊Q812yes—
386C4×C7C23×Q811yescharacteristic
387C4×C7C22×#4111yescharacteristic
388Dic7—24no—
389Dic7—24no—
390Dic7—42no—
391Dic7—44no—
392Dic7—42no—
393C24—21no—
394C24D4×D711yescharacteristic
395#41—216no—
396#41—28no—
397#41C2×Dic14116yescomplemented
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—148no—
401C2×Q8—564no—
402C2×Q8C2×Dic1414yes—
403C2×Q8C23×D712yes—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4C2×Dic1414yes—
409C2×D4C23×D712yes—
410C22×C4—21no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—72no—
416C22×C4—72no—
417C22×C4—82no—
418C22×C4—148no—
419C22×C4—141no—
420C22×C4—142no—
421C22×C4—141no—
422C22×C4—144no—
423C22×C4—564no—
424C22×C4—562no—
425C22×C4C2×Dic1414yes—
426C22×C4D4×D712yes—
427C22×C4D4×D711yescharacteristic
428C22×C4#115914yes—
429C22×C4C23×D712yes—
430C22×C4C23×D711yescharacteristic
431M16—28no—
432C2×C8—24no—
433C2×C8—72no—
434C2×C8—72no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—148no—
438C2×C8C23×D714yes—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—74no—
442C4⋊C4—74no—
443C4⋊C4—72no—
444C4⋊C4—72no—
445C4⋊C4—74no—
446C4⋊C4—148no—
447C4⋊C4—148no—
448C4⋊C4—148no—
449C4⋊C4—148no—
450C4⋊C4—144no—
451C4⋊C4—142no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4C23×D714yes—
455C22⋊C4—78no—
456C22⋊C4—148no—
457C22⋊C4—144no—
458C42—74no—
459C42—74no—
460C42—148no—
461C42—144no—
462C2×C7—24no—
463C2×C7—44no—
464C2×C7#308314yescomplemented
465C2×C7C22×C22⋊Q811yescharacteristic
466C2×C7C2×#57611yescharacteristic
467C2×C7C2×#57811yescharacteristic
468C23—22no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23#644114yes—
475C23C2×D4×D712yes—
476C23C2×D4×D711yescharacteristic
477Q8—24no—
478Q8—22no—
479Q8—568no—
480Q8C22×Dic1414yes—
481D4—216no—
482D4—24no—
483D4—22no—
484D4C22×Dic1414yes—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—216no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—74no—
491C2×C4—74no—
492C2×C4—84no—
493C2×C4—144no—
494C2×C4—144no—
495C2×C4—142no—
496C2×C4—144no—
497C2×C4—142no—
498C2×C4—284no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—568no—
502C2×C4—564no—
503C2×C4#644114yes—
504C2×C4#644118yes—
505C2×C4C22×Dic1412yes—
506C2×C4C22×Dic1414yes—
507C2×C4C2×D4×D711yescharacteristic
508C2×C4C2×D4×D712yes—
509C2×C4C2×#115912yes—
510C2×C4C2×#115914yes—
511C2×C4C24×D711yescharacteristic
512C8—24no—
513C8—142no—
514C8—142no—
515C7C2×#308311yescharacteristic, complemented, Sylow
516C22—22no—
517C22—22no—
518C22—28no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—48no—
523C22C2×#644114yes—
524C22C2×#644112yes—
525C22#9046512yes—
526C22#9047012yes—
527C22C22×D4×D711yescharacteristic
528C4—24no—
529C4—44no—
530C4—144no—
531C4—144no—
532C4—284no—
533C4—282no—
534C4—282no—
535C4C2×#644114yes—
536C4C2×#644112yes—
537C4C23×Dic1411yescharacteristic
538C4C22×#115911yescharacteristic
539C2—24no—
540C2—44no—
541C2#1173631714yesdirect product factor
542C2C22×#644111yescharacteristic
543C2C2×#9046511yescharacteristic
544C2C2×#9047011yescharacteristic
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×#3083
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. C2×#90191C2semidirect
5. C2×#89893C2semidirect
6. C2×#89891C2semidirect
7. C22×#6381C2semidirect
8. #11736317C2direct (N × H)
9. C2×#89571C2semidirect
10. C2×#89566C2semidirect
11. C2×#89535C2semidirect
12. C2×#89531C2semidirect
29. #90191C22semidirect
42. #89893C22semidirect
43. #89891C22semidirect
50. C2×#6381C22semidirect
51. #89571C22semidirect
52. #89566C22semidirect
53. #89535C22semidirect
54. #89531C22semidirect
58. C2×C7×#41Q8semidirect
144. C7×#41C2×Q8semidirect
300. C2×#41Dic14semidirect
397. #41C2×Dic14semidirect
464. C2×C7#3083semidirect
515. C7C2×#3083semidirect
541. C2#11736317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#905182
C2×#901912
C2×C7×#4172
C2×#900052
C2×#898932
C2×#898912
C22×#63812
#117363172
C2×#895712
C2×#895662
C2×#895352
C2×#895312
C2×#30837