Finite Group Databasebeta

← C2×#11737021 #14234271

Subgroups — C2×#11737021 #14234271

Order 1792.

2082 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90234C211yesmaximal, characteristic, complemented
2C2×D4×Dic14C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic
4C2×#89924C212yesmaximal
5C2×#89780C211yesmaximal, characteristic, complemented
6C2×#89779C211yesmaximal, characteristic, complemented
7#11737021C2116yesmaximal, direct product factor
8C2×#89648C212yesmaximal, complemented
9C2×#89535C212yesmaximal, complemented
10C2×#89533C212yesmaximal, complemented
11C2×#89451C211yesmaximal, characteristic, complemented
12C2×#89452C211yesmaximal, characteristic, complemented
13C23×Dic14—21no—
14C22×C7×D8C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2212yes—
18C2×D4×Dic7C2212yes—
19C22×C4⋊Dic7—21no—
20C2×#6487—21no—
21C2×#6487—21no—
22#90234—24no—
23#90234C2218yescomplemented
24D4×Dic14—24no—
25D4×Dic14C2218yescomplemented
26C2×#6453C2212yes—
27C2×#6451C2212yes—
28C2×#6441—22no—
29C2×#6441—22no—
30C2×#6432C2211yescharacteristic
31C2×#6432C2211yescharacteristic
32C2×C4×Dic14C2211yescharacteristic
33C4×C7×D8C2218yes—
34C2×C7×#107C2211yescharacteristic
35C2×C7×#102C2211yescharacteristic
36C2×C7×#102C2211yescharacteristic
37C2×C4×C7×C8C2211yescharacteristic
38C2×#6406C2212yes—
39C2×#6406C2212yes—
40#89924C22116yes—
41C2×#6391C2212yes—
42C2×#6391C2212yes—
43C2×#6389C2212yes—
44C2×Dic7⋊C8C2212yes—
45#89780C2218yescomplemented
46#89779C2218yescomplemented
47C2×#6382C2212yes—
48C2×#6381C2212yes—
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89648C22116yescomplemented
51#89535C22116yescomplemented
52#89533C22116yescomplemented
53#89452C2218yescomplemented
54#89451C2218yescomplemented
55C2×#3384—71nomaximal, Sylow
56C22×C7×D4C2311yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×Dic7—22no—
61C23×Dic7—22no—
62C22×Dic14—21no—
63C22×Dic14—22no—
64C22×Dic14—21no—
65C22×Dic14—28no—
66C22×Dic14—48no—
67C22×Dic14D411yescharacteristic
68C2×C7×D8Q818yescomplemented
69C2×C7×D8C2314yes—
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4C2314yes—
75C4×C7×D4C2314yes—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—22no—
82C2×C22⋊Dic7—22no—
83D4×Dic7—24no—
84D4×Dic7—24no—
85D4×Dic7C2318yes—
86D4×Dic7C2318yes—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—28no—
91C2×C4⋊Dic7D411yescharacteristic
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94#6487—28no—
95#6487—28no—
96C2×Dic7⋊C4—22no—
97C2×Dic7⋊C4—22no—
98C2×Dic7⋊C4D412yes—
99C2×C4×Dic7C2312yes—
100C22×C7⋊C8C2312yes—
101#6453—24no—
102#6453C2318yes—
103#6451—24no—
104#6451C2318yes—
105#6441—216no—
106#6441—216no—
107#6432—22no—
108#6432—22no—
109#6432C2314yes—
110#6432C2314yes—
111C4×Dic14D414yes—
112C4×Dic14C2314yes—
113C7×#107C2314yes—
114C7×#102C2314yes—
115C7×#102C2314yes—
116C4×C7×C8C2314yes—
117#6406C2318yes—
118#6406C2318yes—
119#6391C2318yes—
120#6391C2318yes—
121#6389C2318yes—
122Dic7⋊C8C2318yes—
123#6382C2318yes—
124#6381C2318yes—
125C4⋊(C7⋊C8)C2314yes—
126C2×#554—71no—
127C2×D4×Q8—71no—
128#3384—716no—
129C2×#501—72no—
130C2×#498—71no—
131C2×#478—72no—
132C2×#476—72no—
133C2×#476—71no—
134C2×#474—71no—
135C2×#446—71no—
136C2×#442—72no—
137C2×C4×D8D711yescharacteristic
138C24×C7—21no—
139C24×C7—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4C2×Q814yes—
145C2×C7×D4C2×Q814yescomplemented
146C2×C7×D4C2412yes—
147C2×C7×D4C2412yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—44no—
153C22×C4×C7—44no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—28no—
160C22×Dic7—28no—
161C22×Dic7—48no—
162C22×Dic7—84no—
163C22×Dic7C2×D412yes—
164C22×Dic7C2×D412yes—
165C2×Dic14—24no—
166C2×Dic14—24no—
167C2×Dic14—28no—
168C2×Dic14—24no—
169C2×Dic14—432no—
170C2×Dic14—48no—
171C2×Dic14—44no—
172C2×Dic14—82no—
173C2×Dic14C2×D414yes—
174C7×D8C2×Q8116yescomplemented
175C2×C7×C8—22no—
176C2×C7×C8C2×Q814yescomplemented
177C2×C7×C8C2412yes—
178C2×C7×C8C2412yes—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C2412yes—
182C7×C4⋊C4C2412yes—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C42×C7C2×D412yes—
186C42×C7C2412yes—
187C22⋊Dic7—28no—
188C22⋊Dic7—28no—
189C4⋊Dic7—22no—
190C4⋊Dic7—22no—
191C4⋊Dic7—24no—
192C4⋊Dic7—24no—
193C4⋊Dic7—28no—
194C4⋊Dic7C2×D414yes—
195C4⋊Dic7C2414yes—
196C4⋊Dic7C2414yes—
197Dic7⋊C4—28no—
198Dic7⋊C4—28no—
199Dic7⋊C4C2×D418yes—
200C4×Dic7C2×D414yes—
201C4×Dic7C2414yes—
202C2×C7⋊C8—24no—
203C2×C7⋊C8C2414yes—
204C23×Q8—141no—
205C22×D8C2×D711yescharacteristic
206#554—78no—
207#554—144no—
208D4×Q8—78no—
209D4×Q8—144no—
210C2×C4⋊Q8—72no—
211C2×C4⋊Q8—71no—
212C2×C4⋊Q8—71no—
213C2×#125—72no—
214C2×C22⋊Q8—142no—
215C2×C22⋊Q8—142no—
216C2×C22⋊Q8—141no—
217C2×C22⋊Q8—141no—
218C2×C4×Q8—71no—
219C2×C4×D4—72no—
220C2×C4×D4—72no—
221C2×C4×D4C2×D711yescharacteristic
222C2×C4×D4C2×D711yescharacteristic
223C22×C4⋊C4—141no—
224#501—716no—
225#498—78no—
226#478—716no—
227#476—78no—
228#476—716no—
229#474—78no—
230#446—78no—
231#442—716no—
232C4×D8C2×D718yes—
233C2×#107—72no—
234C2×#107C2×D711yescharacteristic
235C2×#106—72no—
236C2×#106—72no—
237C2×#106—72no—
238C2×C4⋊C8—72no—
239C2×C4⋊C8—71no—
240C2×#102—72no—
241C2×#102—72no—
242C2×#102C2×D711yescharacteristic
243C2×#102C2×D711yescharacteristic
244C2×#97—72no—
245C2×C4×C8C2×D711yescharacteristic
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7C22×D411yescharacteristic
255C7×D4—24no—
256C7×D4—24no—
257C7×D4—22no—
258C7×D4—22no—
259C7×D4C22×Q814yes—
260C7×D4C22×Q814yescomplemented
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—48no—
267C2×C4×C7—48no—
268C2×C4×C7AGL(1,ℤ/8)14yes—
269C2×C4×C7C22×D412yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D412yes—
272C2×C4×C7C22×Q812yes—
273C2×C4×C72−1+412yes—
274C2×C4×C7C2511yescharacteristic
275C2×Dic7—24no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—416no—
280C2×Dic7—44no—
281C2×Dic7—44no—
282C2×Dic7—88no—
283C2×Dic7C22×D414yes—
284C2×Dic7C22×D414yes—
285Dic14—28no—
286Dic14—416no—
287Dic14—48no—
288Dic14—84no—
289C7×C8—22no—
290C7×C8C22×Q814yescomplemented
291C7⋊C8—24no—
292C22×Q8—71no—
293C22×Q8—142no—
294C22×Q8—148no—
295C22×Q8—141no—
296C22×Q8—141no—
297C22×Q8—288no—
298C22×D4C22×D711yescharacteristic
299C22×D4C22×D711yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—142no—
303C23×C4—142no—
304C2×D8Dic1418yescomplemented
305C2×D8C22×D714yes—
306C22×C8—72no—
307C22×C8C22×D711yescharacteristic
308C22×C8C22×D711yescharacteristic
309C4⋊Q8—78no—
310C4⋊Q8—74no—
311C4⋊Q8—74no—
312C4⋊Q8—144no—
313C4⋊Q8—142no—
314C4⋊Q8—142no—
315#125—78no—
316#125—144no—
317C22⋊Q8—1416no—
318C22⋊Q8—1416no—
319C22⋊Q8—148no—
320C22⋊Q8—148no—
321C4×Q8—74no—
322C4×Q8—74no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—78no—
326C4×D4—78no—
327C4×D4—144no—
328C4×D4—144no—
329C4×D4C22×D714yes—
330C4×D4C22×D714yes—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—72no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—148no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—142no—
347C2×C42—72no—
348C2×C42C22×D711yescharacteristic
349#107—78no—
350#107C22×D714yes—
351#106—78no—
352#106—78no—
353#106—78no—
354C4⋊C8—78no—
355C4⋊C8—74no—
356#102—78no—
357#102—78no—
358#102C22×D714yes—
359#102C22×D714yes—
360#97—78no—
361C4×C8C22×D714yes—
362C22×C7—24no—
363C22×C7—24no—
364C22×C7—48no—
365C22×C7—48no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7D4×Q812yes—
371C22×C7C2×AGL(1,ℤ/8)12yes—
372C22×C7#57712yes—
373C22×C7C23×D411yescharacteristic
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—42no—
377C4×C7D4×Q812yes—
378C4×C7C2×AGL(1,ℤ/8)14yes—
379C4×C7C23×Q811yescharacteristic
380C4×C7C2×2−1+411yescharacteristic
381Dic7—24no—
382Dic7—24no—
383Dic7—44no—
384Dic7—44no—
385C24—21no—
386C24—21no—
387C2×Q8—74no—
388C2×Q8—148no—
389C2×Q8—144no—
390C2×Q8—144no—
391C2×Q8—144no—
392C2×Q8—288no—
393C2×Q8—2832no—
394C2×Q8—284no—
395C2×Q8—562no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4C2×Dic1414yes—
401C2×D4C2×Dic1414yescomplemented
402C2×D4C23×D712yes—
403C2×D4C23×D712yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—72no—
411C22×C4—72no—
412C22×C4—148no—
413C22×C4—148no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—288no—
417C22×C4—564no—
418C22×C4D4×D711yescharacteristic
419C22×C4D4×D711yescharacteristic
420C22×C4C23×D711yescharacteristic
421D8C2×Dic14116yescomplemented
422C2×C8—22no—
423C2×C8—74no—
424C2×C8—144no—
425C2×C8C2×Dic1414yescomplemented
426C2×C8C23×D712yes—
427C2×C8C23×D712yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—78no—
431C4⋊C4—74no—
432C4⋊C4—74no—
433C4⋊C4—74no—
434C4⋊C4—148no—
435C4⋊C4—148no—
436C4⋊C4—148no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—144no—
440C4⋊C4—144no—
441C4⋊C4C23×D712yes—
442C4⋊C4C23×D712yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—148no—
446C22⋊C4—148no—
447C42—74no—
448C42—74no—
449C42D4×D712yes—
450C42C23×D712yes—
451C2×C7—44no—
452C2×C7—44no—
453C2×C7#338414yescomplemented
454C2×C7C2×D4×Q811yescharacteristic
455C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
456C2×C7C2×#57711yescharacteristic
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C2×D4×D711yescharacteristic
466Q8—148no—
467Q8—2816no—
468Q8—288no—
469Q8—564no—
470D4—24no—
471D4—24no—
472D4—22no—
473D4—22no—
474D4C22×Dic1414yes—
475D4C22×Dic1414yescomplemented
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—74no—
484C2×C4—74no—
485C2×C4—144no—
486C2×C4—144no—
487C2×C4—144no—
488C2×C4—144no—
489C2×C4—2816no—
490C2×C4—284no—
491C2×C4—284no—
492C2×C4—568no—
493C2×C4#647414yes—
494C2×C4C22×Dic1412yes—
495C2×C4C2×D4×D712yes—
496C2×C4C2×D4×D712yes—
497C2×C4C2×D4×D711yescharacteristic
498C2×C4#655412yes—
499C2×C4C24×D711yescharacteristic
500C8—22no—
501C8—144no—
502C8C22×Dic1414yescomplemented
503C7C2×#338411yescharacteristic, complemented, Sylow
504C22—24no—
505C22—24no—
506C22—48no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22D4×Dic1412yes—
513C22C2×#647412yes—
514C22#9046112yes—
515C22C22×D4×D711yescharacteristic
516C4—22no—
517C4—42no—
518C4—42no—
519C4—144no—
520C4—144no—
521C4—284no—
522C4—284no—
523C4D4×Dic1412yes—
524C4C2×#647414yes—
525C4C23×Dic1411yescharacteristic
526C4C2×#655411yescharacteristic
527C2—44no—
528C2—44no—
529C2#1173702114yesdirect product factor
530C2C2×D4×Dic1411yescharacteristic
531C2C22×#647411yescharacteristic
532C2C2×#9046111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C4×C7×D8C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
hypercenterC2×C4×D8D7
hypocenterC7C2×#3384
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#90234C2semidirect
2. C2×D4×Dic14C2semidirect
5. C2×#89780C2semidirect
6. C2×#89779C2semidirect
7. #11737021C2direct (N × H)
8. C2×#89648C2semidirect
9. C2×#89535C2semidirect
10. C2×#89533C2semidirect
11. C2×#89451C2semidirect
12. C2×#89452C2semidirect
23. #90234C22semidirect
25. D4×Dic14C22semidirect
45. #89780C22semidirect
46. #89779C22semidirect
50. #89648C22semidirect
51. #89535C22semidirect
52. #89533C22semidirect
53. #89452C22semidirect
54. #89451C22semidirect
68. C2×C7×D8Q8semidirect
145. C2×C7×D4C2×Q8semidirect
174. C7×D8C2×Q8semidirect
176. C2×C7×C8C2×Q8semidirect
260. C7×D4C22×Q8semidirect
290. C7×C8C22×Q8semidirect
304. C2×D8Dic14semidirect
401. C2×D4C2×Dic14semidirect
421. D8C2×Dic14semidirect
425. C2×C8C2×Dic14semidirect
453. C2×C7#3384semidirect
475. D4C22×Dic14semidirect
502. C8C22×Dic14semidirect
503. C7C2×#3384semidirect
529. C2#11737021direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902342
C2×D4×Dic142
C2×C4×C7×D82
C2×#899242
C2×#897802
C2×#897792
#117370212
C2×#896482
C2×#895352
C2×#895332
C2×#894512
C2×#894522
C2×#33847