Finite Group Databasebeta

← C2×#11736607 #14221382

Subgroups — C2×#11736607 #14221382

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×#90195C211yesmaximal, characteristic
2C2×#90191C211yesmaximal, characteristic, complemented
3C22×C7×#107C211yesmaximal, characteristic, complemented
4C2×#89885C211yesmaximal, characteristic
5C2×#89874C211yesmaximal, characteristic, complemented
6C2×#89738C211yesmaximal, characteristic
7C2×#89736C211yesmaximal, characteristic, complemented
8C2×#89649C212yesmaximal, complemented
9C2×#89647C212yesmaximal, complemented
10#11736607C2116yesmaximal, direct product factor
11C2×#89663C212yesmaximal, complemented
12C2×#89662C212yesmaximal, complemented
13C22×#1157C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×C4×C22⋊D7—22no—
18C2×#6489C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C2×#6487—22no—
21C22×#1136C2211yescharacteristic
22#90195—24no—
23#90195C2218yes—
24C2×#6456C2212yes—
25C2×#6455—21no—
26C2×D7×C4⋊C4—21no—
27#90191—24no—
28#90191C2218yescomplemented
29C2×#6453—21no—
30C2×#6453C2212yes—
31C2×#6451—21no—
32C2×#6451C2212yes—
33C2×#6450C2212yes—
34C2×C7×#107C2212yes—
35C2×C7×#107C2212yes—
36C2×C7×#107C2218yescomplemented
37#89885C2218yes—
38C2×#6395C2211yescharacteristic
39C2×#6395C2211yescharacteristic
40C2×#6390C2211yescharacteristic
41C2×#6390C2211yescharacteristic
42#89874C2218yescomplemented
43C2×Dic7⋊C8C2212yes—
44C2×Dic7⋊C8C2212yes—
45C2×#6384C2212yes—
46#89738C2218yes—
47C2×#6383C2212yes—
48#89736C2218yescomplemented
49C2×#6382C2212yes—
50C2×#6381C2212yes—
51#89663C22116yescomplemented
52#89662C22116yescomplemented
53#89649C22116yescomplemented
54#89647C22116yescomplemented
55C2×#3083—71nomaximal, Sylow
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C22×C22⋊D7—21no—
60C2×#1157—22no—
61C2×#1157—24no—
62C2×#1157Q818yescomplemented
63C2×#1157C2314yes—
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×C4×D7—22no—
67C22×Dic14—22no—
68C22×Dic14C2311yescharacteristic
69C22×C7×C8—24no—
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2314yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2314yes—
80C2×C7×C4⋊C4C2314yes—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7D412yes—
83C4×C22⋊D7—216no—
84C2×(C2×D7)⋊C4—22no—
85#6489—22no—
86#6489D414yes—
87#6489C2314yes—
88#6489C2314yes—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7D411yescharacteristic
91C2×C4⋊Dic7D411yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94#6487—216no—
95C2×Dic7⋊C4—22no—
96C2×Dic7⋊C4—22no—
97C2×Dic7⋊C4—22no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—22no—
100C2×C4×Dic7C2312yes—
101C2×#1136—24no—
102C2×#1136C2314yes—
103C22×C7⋊C8C2312yes—
104#6456—24no—
105#6456C2318yes—
106#6455—28no—
107D7×C4⋊C4—28no—
108#6453—28no—
109#6453—24no—
110#6453C2318yes—
111#6451—28no—
112#6451—24no—
113#6451C2318yes—
114#6450—24no—
115#6450C2318yes—
116C7×#107C2318yes—
117C7×#107C2318yes—
118#6395C2314yes—
119#6395C2314yes—
120#6390C2314yes—
121#6390C2314yes—
122Dic7⋊C8C2318yes—
123Dic7⋊C8C2318yes—
124#6384C2318yes—
125#6383C2318yes—
126#6382C2318yes—
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×#107D711yescharacteristic, complemented
137C2×#424—71no—
138C2×#417—71no—
139C2×#416—71no—
140C24×C7C2×D411yescharacteristic
141C23×D7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7—82no—
147C22×C4×C7—82no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×Q814yes—
151C22×C4×C7#4114yes—
152C22×C4×C7C2411yescharacteristic
153C22×C4×C7C2412yes—
154C2×C22⋊D7—28no—
155C2×C22⋊D7—24no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7C2×D412yes—
160C22×Dic7C2×D412yes—
161#1157—216no—
162#1157—28no—
163#1157C2×Q8116yescomplemented
164C2×D28—21no—
165C2×D28—22no—
166C2×D28C2×Q814yes—
167C2×D28C2412yes—
168C2×C4×D7—28no—
169C2×C4×D7—24no—
170C2×C4×D7—28no—
171C2×C4×D7—84no—
172C2×Dic14—21no—
173C2×Dic14—22no—
174C2×Dic14—28no—
175C2×Dic14—84no—
176C2×Dic14C2×Q814yes—
177C2×Dic14C2412yes—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—28no—
181C2×C7×C8C2412yes—
182C2×C7×C8C2412yes—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4C2414yes—
188C7×C4⋊C4C2414yes—
189C22⋊Dic7—24no—
190C22⋊Dic7C2×D418yes—
191(C2×D7)⋊C4—28no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—28no—
195C4⋊Dic7C2×D414yes—
196C4⋊Dic7C2×D414yes—
197C4⋊Dic7C2412yes—
198C4⋊Dic7C2412yes—
199Dic7⋊C4—28no—
200Dic7⋊C4—28no—
201Dic7⋊C4—28no—
202C4×Dic7—24no—
203C4×Dic7—28no—
204C4×Dic7C2×D414yes—
205C4×Dic7C2414yes—
206#1136—28no—
207C2×C7⋊C8—24no—
208C2×C7⋊C8C2414yes—
209C22×#41—71no—
210C22×M16—71no—
211C23×C8C2×D711yescharacteristic
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—141no—
228C22×C4⋊C4C2×D711yescharacteristic
229C22×C4⋊C4C2×D711yescharacteristic
230#479—716no—
231#478—716no—
232#477—716no—
233#474—716no—
234#428—78no—
235C2×#107—72no—
236C2×#107C2×D712yes—
237C2×#107C2×D712yes—
238C2×#107C2×D718yescomplemented
239C2×#106—72no—
240#424—78no—
241C2×C4⋊C8—72no—
242C2×C4⋊C8—72no—
243#417—78no—
244#416—78no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—72no—
248C2×#102—71no—
249C2×#102—71no—
250C2×#102—72no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7C22⋊Q814yes—
254C23×C7C22×D412yes—
255C23×C7C22×D411yescharacteristic
256C22×D7—24no—
257C22×D7—22no—
258C22×D7—42no—
259C22×D7—42no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—44no—
263C2×C4×C7—44no—
264C2×C4×C7—84no—
265C2×C4×C7—84no—
266C2×C4×C7C22⋊Q814yes—
267C2×C4×C7C22⋊Q818yes—
268C2×C4×C7C22×D412yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×Q812yes—
271C2×C4×C7C22×Q814yes—
272C2×C4×C7C2×#4112yes—
273C2×C4×C7C2×#4114yes—
274C2×C4×C7C2511yescharacteristic
275C22⋊D7—216no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7C22×D414yes—
285C2×Dic7C22×D414yes—
286D28—24no—
287D28—22no—
288D28C22×Q814yes—
289C4×D7—216no—
290C4×D7—88no—
291Dic14—24no—
292Dic14—22no—
293Dic14—88no—
294Dic14C22×Q814yes—
295C7×C8—22no—
296C7×C8—22no—
297C7⋊C8—24no—
298C2×#41—78no—
299C2×#41—74no—
300C2×#41—144no—
301C2×#41—142no—
302C22×Q8—71no—
303C22×Q8—142no—
304C22×D4—71no—
305C22×D4—141no—
306C23×C4—21no—
307C23×C4—21no—
308C23×C4—142no—
309C23×C4—141no—
310C23×C4C22×D711yescharacteristic
311C2×M16—74no—
312C2×M16—144no—
313C22×C8—24no—
314C22×C8—72no—
315C22×C8C22×D711yescharacteristic
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D714yes—
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—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
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⋊C4C22×D712yes—
349C2×C4⋊C4C22×D714yes—
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#107C22×D718yes—
360#107C22×D718yes—
361#106—78no—
362C4⋊C8—78no—
363C4⋊C8—78no—
364#103—74no—
365#103—74no—
366#103—78no—
367#102—74no—
368#102—74no—
369#102—78no—
370C22×C7—28no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7C2×C22⋊Q814yes—
374C22×C7C2×C22⋊Q812yes—
375C22×C7#57612yes—
376C22×C7#57812yes—
377C22×C7C23×D411yescharacteristic
378C2×D7—24no—
379C2×D7—48no—
380C2×D7—42no—
381C2×D7—42no—
382C4×C7—44no—
383C4×C7—44no—
384C4×C7C2×C22⋊Q812yes—
385C4×C7C2×C22⋊Q814yes—
386C4×C7C23×Q811yescharacteristic
387C4×C7C22×#4111yescharacteristic
388Dic7—24no—
389Dic7—24no—
390Dic7—24no—
391Dic7—42no—
392Dic7—42no—
393C24—141no—
394C24D4×D711yescharacteristic
395#41—716no—
396#41—1416no—
397#41—148no—
398C2×Q8—74no—
399C2×Q8—72no—
400C2×Q8—148no—
401C2×Q8—142no—
402C2×Q8—141no—
403C2×Q8—564no—
404C2×D4—74no—
405C2×D4—72no—
406C2×D4—148no—
407C2×D4—144no—
408C2×D4—142no—
409C2×D4—141no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—72no—
415C22×C4—72no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—148no—
419C22×C4—141no—
420C22×C4—148no—
421C22×C4—144no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—564no—
425C22×C4#115714yes—
426C22×C4D4×D712yes—
427C22×C4D4×D711yescharacteristic
428C22×C4Q8×D714yes—
429C22×C4C23×D712yes—
430C22×C4C23×D711yescharacteristic
431M16—148no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—28no—
435C2×C8—74no—
436C2×C8—144no—
437C2×C8C23×D712yes—
438C2×C8C23×D712yes—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—74no—
444C4⋊C4—74no—
445C4⋊C4—72no—
446C4⋊C4—72no—
447C4⋊C4—148no—
448C4⋊C4—148no—
449C4⋊C4—148no—
450C4⋊C4—148no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4C23×D714yes—
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#308314yescomplemented
464C2×C7C22×C22⋊Q811yescharacteristic
465C2×C7C2×#57611yescharacteristic
466C2×C7C2×#57811yescharacteristic
467D7—44no—
468C23—22no—
469C23—22no—
470C23—144no—
471C23—142no—
472C23—282no—
473C23—282no—
474C23#645914yes—
475C23C2×D4×D712yes—
476C23C2×D4×D711yescharacteristic
477Q8—74no—
478Q8—144no—
479Q8—142no—
480Q8—568no—
481D4—74no—
482D4—1416no—
483D4—144no—
484D4—142no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—74no—
490C2×C4—74no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—144no—
494C2×C4—144no—
495C2×C4—144no—
496C2×C4—142no—
497C2×C4—1416no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—568no—
503C2×C4#645914yes—
504C2×C4#645918yes—
505C2×C4C2×#115712yes—
506C2×C4C2×#115714yes—
507C2×C4C2×D4×D711yescharacteristic
508C2×C4C2×D4×D712yes—
509C2×C4C2×Q8×D712yes—
510C2×C4C2×Q8×D714yes—
511C2×C4C24×D711yescharacteristic
512C8—22no—
513C8—22no—
514C8—144no—
515C7C2×#308311yescharacteristic, complemented, Sylow
516C22—28no—
517C22—22no—
518C22—22no—
519C22—144no—
520C22—288no—
521C22—282no—
522C22—282no—
523C22C2×#645914yes—
524C22C2×#645912yes—
525C22#9044812yes—
526C22#9045712yes—
527C22C22×D4×D711yescharacteristic
528C4—44no—
529C4—44no—
530C4—144no—
531C4—144no—
532C4—144no—
533C4—282no—
534C4—282no—
535C4C2×#645914yes—
536C4C2×#645912yes—
537C4C22×#115711yescharacteristic
538C4C22×Q8×D711yescharacteristic
539C2—24no—
540C2—284no—
541C2#1173660714yesdirect product factor
542C2C22×#645911yescharacteristic
543C2C2×#9044811yescharacteristic
544C2C2×#9045711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C7×#107C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
hypercenterC22×#107D7
hypocenterC7C2×#3083
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C2×#90191C2semidirect
3. C22×C7×#107C2semidirect
5. C2×#89874C2semidirect
7. C2×#89736C2semidirect
8. C2×#89649C2semidirect
9. C2×#89647C2semidirect
10. #11736607C2direct (N × H)
11. C2×#89663C2semidirect
12. C2×#89662C2semidirect
28. #90191C22semidirect
36. C2×C7×#107C22semidirect
42. #89874C22semidirect
48. #89736C22semidirect
51. #89663C22semidirect
52. #89662C22semidirect
53. #89649C22semidirect
54. #89647C22semidirect
62. C2×#1157Q8semidirect
136. C22×#107D7semidirect
163. #1157C2×Q8semidirect
238. C2×#107C2×D7semidirect
463. C2×C7#3083semidirect
515. C7C2×#3083semidirect
541. C2#11736607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#901952
C2×#901912
C22×C7×#1072
C2×#898852
C2×#898742
C2×#897382
C2×#897362
C2×#896492
C2×#896472
#117366072
C2×#896632
C2×#896622
C2×#30837