Finite Group Databasebeta

← C2×#11741071 #14232534

Subgroups — C2×#11741071 #14232534

Order 1792.

1866 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×#90233C211yesmaximal, characteristic
3#11741071C2116yesmaximal, direct product factor
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89851C211yesmaximal, characteristic, complemented
6C2×#89824C212yesmaximal, complemented
7C2×#89817C212yesmaximal, complemented
8C2×#89792C211yesmaximal, characteristic, complemented
9C2×C4×#1144C211yesmaximal, characteristic, complemented
10C2×#89784C211yesmaximal, characteristic
11C2×#89743C212yesmaximal, complemented
12C2×#89736C212yesmaximal, complemented
13C7×#554—24no—
14C7×#554C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4C2212yes—
23C22×#1144C2211yescharacteristic
24C2×#6499—21no—
25C2×#6489C2212yes—
26C2×#6487C2212yes—
27C22×#1136C2212yes—
28#90233—24no—
29#90233C2218yes—
30C2×#6450—21no—
31C2×#6442—22no—
32C2×#6432C2211yescharacteristic
33C2×C4×Dic14C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2212yes—
36C2×C22⋊(C7⋊C8)C2212yes—
37#89863C2218yescomplemented
38#89851C2218yescomplemented
39#89824C22116yescomplemented
40#89817C22116yescomplemented
41C4×#1144C2218yescomplemented
42#89792C2218yescomplemented
43#89784C2218yes—
44#89743C22116yescomplemented
45C2×#6384C2212yes—
46C2×#6384C2211yescharacteristic
47#89736C22116yescomplemented
48C2×#6382C2212yes—
49C2×#6382C2212yes—
50C2×#6382C2212yes—
51C2×#6381C2212yes—
52C2×#6381C2211yescharacteristic
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3356—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4D411yescharacteristic
59C23×C4×C7—22no—
60C23×C4×C7C2312yes—
61C22×Dic14—21no—
62C22×Dic14C2311yescharacteristic
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8C2314yes—
65C7×#125—28no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—216no—
68C7×C22⋊Q8C2318yes—
69C4×C7×Q8—22no—
70C4×C7×Q8C2314yes—
71C4×C7×D4—28no—
72C4×C7×D4D414yes—
73C4×C7×D4C2314yes—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2318yes—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4D412yes—
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C22⋊Dic7—22no—
88C2×C22⋊Dic7—22no—
89C2×#1144—24no—
90C2×#1144C2314yes—
91#6499—28no—
92#6489—24no—
93#6489C2318yes—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97#6487—24no—
98#6487C2318yes—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—22no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—22no—
103C2×#1136—28no—
104C2×#1136C2318yes—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2312yes—
107C22×C7⋊C8C2311yescharacteristic
108#6450—28no—
109#6442—216no—
110#6432—22no—
111#6432C2314yes—
112C4×Dic14—22no—
113C4×Dic14C2314yes—
114#6409C2318yes—
115#6409C2314yes—
116C22⋊(C7⋊C8)C2318yes—
117#6384C2318yes—
118#6384C2314yes—
119#6382C2318yes—
120#6382C2318yes—
121#6382C2318yes—
122#6381C2318yes—
123#6381C2314yes—
124C4⋊(C7⋊C8)C2314yes—
125C4×C7⋊C8C2314yes—
126C2×#556—71no—
127C2×#554D711yescharacteristic
128#3356—716no—
129C2×#498—71no—
130C2×#485—72no—
131C2×#483—72no—
132C2×#477—71no—
133C2×#475—71no—
134C2×#470—72no—
135C2×C4×Q16—71no—
136C2×#436—71no—
137C2×#428—72no—
138C24×C7C2×D412yes—
139C2×C7×Q8—21no—
140C2×C7×Q8—24no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—82no—
144C2×C7×Q8C2412yes—
145C2×C7×D4—24no—
146C2×C7×D4C2×D414yes—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—28no—
153C22×C4×C7—84no—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2414yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—21no—
166C2×Dic14—24no—
167C2×Dic14—44no—
168C2×Dic14C2412yes—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—28no—
175C7×C4⋊C4C2×D414yes—
176C7×C4⋊C4C2412yes—
177C7×C4⋊C4C2414yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4C2×D418yes—
180C42×C7—24no—
181C42×C7C2×D412yes—
182C42×C7C2412yes—
183C22⋊Dic7—28no—
184C22⋊Dic7—28no—
185#1144—28no—
186C4⋊Dic7—21no—
187C4⋊Dic7—22no—
188C4⋊Dic7—24no—
189C4⋊Dic7C2412yes—
190C4⋊Dic7C2414yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—28no—
193C4×Dic7—24no—
194C4×Dic7—28no—
195#1136—216no—
196C2×C7⋊C8—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8#4114yes—
199C2×C7⋊C8C2412yes—
200C2×C7⋊C8C2414yes—
201C2×C7⋊C8C2412yes—
202C22×Q16—71no—
203C22×M16—72no—
204#556—78no—
205#556—144no—
206#554—24no—
207#554C2×D718yes—
208C2×C4⋊Q8—71no—
209C2×C4⋊Q8C2×D711yescharacteristic
210C2×#126—142no—
211C2×#125—21no—
212C2×#124—141no—
213C2×C22⋊Q8—22no—
214C2×C22⋊Q8—72no—
215C2×C22⋊Q8C2×D712yes—
216C2×C4×Q8—71no—
217C2×C4×Q8—141no—
218C2×C4×Q8C2×D711yescharacteristic
219C2×C4×D4—21no—
220C2×C4×D4C2×D711yescharacteristic
221C2×#117—72no—
222C22×C4⋊C4C2×D712yes—
223#498—78no—
224#485—716no—
225#483—716no—
226#477—78no—
227#475—78no—
228#470—716no—
229C4×Q16—78no—
230#436—78no—
231#428—716no—
232C2×#107—72no—
233C2×#107—71no—
234C2×#106—72no—
235C2×#106—72no—
236C2×#106—72no—
237C2×C4⋊C8—71no—
238C2×#103—72no—
239C2×#103—71no—
240C2×#103—72no—
241C2×#103—71no—
242C2×C22⋊C8—72no—
243C2×C4×C8—71no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7C22×D414yes—
248C23×C7C22×D411yescharacteristic
249C7×Q8—22no—
250C7×Q8—22no—
251C7×Q8—42no—
252C7×Q8—84no—
253C7×D4—28no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—28no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—44no—
265C2×C4×C7—88no—
266C2×C4×C7#13714yes—
267C2×C4×C7C22×D414yes—
268C2×C4×C7C22×D412yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D412yes—
271C2×C4×C7C2×#4112yes—
272C2×C4×C72−1+412yes—
273C2×C4×C7C2511yescharacteristic
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—24no—
277C2×Dic7—42no—
278C2×Dic7—44no—
279C2×Dic7—42no—
280Dic14—22no—
281Dic14—22no—
282Dic14—42no—
283Dic14—48no—
284C7⋊C8—24no—
285C7⋊C8—22no—
286C7⋊C8C2×#4114yes—
287C22×Q8—21no—
288C22×Q8—71no—
289C22×Q8—141no—
290C22×Q8C22×D711yescharacteristic
291C22×D4C22⋊D711yescharacteristic
292C23×C4—22no—
293C23×C4C22×D712yes—
294C2×Q16—74no—
295C2×Q16—144no—
296C2×M16—78no—
297C2×M16—148no—
298C22×C8—71no—
299C22×C8—72no—
300C22×C8—71no—
301C4⋊Q8—22no—
302C4⋊Q8—74no—
303C4⋊Q8—142no—
304C4⋊Q8C22×D714yes—
305#126—1416no—
306#125—28no—
307#124—148no—
308C22⋊Q8—24no—
309C22⋊Q8—216no—
310C22⋊Q8—78no—
311C22⋊Q8—144no—
312C22⋊Q8C22×D718yes—
313C4×Q8—22no—
314C4×Q8—74no—
315C4×Q8—148no—
316C4×Q8—142no—
317C4×Q8C22×D714yes—
318C4×D4—28no—
319C4×D4C22⋊D714yes—
320C4×D4C22×D714yes—
321#117—78no—
322#117—144no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—72no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4C22⋊D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D712yes—
336C2×C4⋊C4C22×D718yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4C22⋊D712yes—
341C2×C42—21no—
342C2×C42—142no—
343C2×C42—141no—
344C2×C42C22×D711yescharacteristic
345#107—74no—
346#107—78no—
347#106—78no—
348#106—78no—
349#106—78no—
350C4⋊C8—74no—
351#103—74no—
352#103—78no—
353#103—74no—
354#103—78no—
355C22⋊C8—78no—
356C4×C8—74no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—48no—
361C22×C7#54712yes—
362C22×C7C2×#13712yes—
363C22×C7#57712yes—
364C22×C7C23×D411yescharacteristic
365C4×C7—24no—
366C4×C7—22no—
367C4×C7—42no—
368C4×C7—42no—
369C4×C7—44no—
370C4×C7#54712yes—
371C4×C7C2×#13714yes—
372C4×C7C22×#4111yescharacteristic
373C4×C7C2×2−1+411yescharacteristic
374Dic7—42no—
375Dic7—44no—
376Dic7—42no—
377C24C2×C22⋊D712yes—
378C2×Q8—21no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—72no—
383C2×Q8—82no—
384C2×Q8—141no—
385C2×Q8—144no—
386C2×Q8—142no—
387C2×Q8—142no—
388C2×Q8—284no—
389C2×Q8C23×D712yes—
390C2×D4—24no—
391C2×D4C2×C22⋊D714yes—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—84no—
399C22×C4—141no—
400C22×C4—142no—
401C22×C4—141no—
402C22×C4C2×C22⋊D714yes—
403C22×C4C2×C22⋊D712yes—
404C22×C4C2×C22⋊D711yescharacteristic
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C23×D714yes—
407C22×C4C23×D711yescharacteristic
408Q16—148no—
409M16—1416no—
410C2×C8—74no—
411C2×C8—72no—
412C2×C8—72no—
413C2×C8—74no—
414C2×C8—144no—
415C2×C8—142no—
416C4⋊C4—21no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—74no—
423C4⋊C4—72no—
424C4⋊C4—144no—
425C4⋊C4—148no—
426C4⋊C4—141no—
427C4⋊C4—142no—
428C4⋊C4—144no—
429C4⋊C4C2×C22⋊D714yes—
430C4⋊C4C23×D712yes—
431C4⋊C4C23×D714yes—
432C22⋊C4—28no—
433C22⋊C4—148no—
434C22⋊C4—148no—
435C22⋊C4C2×C22⋊D718yes—
436C42—24no—
437C42—148no—
438C42—144no—
439C42C2×C22⋊D712yes—
440C42C23×D712yes—
441C2×C7—44no—
442C2×C7#335614yescomplemented
443C2×C7C2×#54711yescharacteristic
444C2×C7C22×#13711yescharacteristic
445C2×C7C2×#57711yescharacteristic
446C23—24no—
447C23—24no—
448C23—24no—
449C23C22×C22⋊D714yes—
450C23C22×C22⋊D711yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—42no—
454Q8—84no—
455Q8—142no—
456Q8—142no—
457Q8—288no—
458Q8—282no—
459D4—28no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—88no—
472C2×C4—142no—
473C2×C4—144no—
474C2×C4—142no—
475C2×C4—284no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4#650514yes—
479C2×C4D7×#4112yes—
480C2×C4#655412yes—
481C2×C4C22×C22⋊D714yes—
482C2×C4C22×C22⋊D712yes—
483C2×C4C22×C22⋊D712yes—
484C2×C4C22×C22⋊D711yescharacteristic
485C2×C4C24×D711yescharacteristic
486C8—74no—
487C8—144no—
488C8—142no—
489C7C2×#335611yescharacteristic, complemented, Sylow
490C22—24no—
491C22—24no—
492C22—24no—
493C22—48no—
494C22C2×#650512yes—
495C22#9051512yes—
496C22#9052712yes—
497C22C23×C22⋊D711yescharacteristic
498C4—24no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—44no—
503C4—284no—
504C4—282no—
505C4—282no—
506C4C2×#650514yes—
507C4#9051512yes—
508C4C2×D7×#4111yescharacteristic
509C4C2×#655411yescharacteristic
510C2—44no—
511C2#1174107114yesdirect product factor
512C2C22×#650511yescharacteristic
513C2C2×#9051511yescharacteristic
514C2C2×#9052711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7C2×#3356
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
3. #11741071C2direct (N × H)
4. C2×#89863C2semidirect
5. C2×#89851C2semidirect
6. C2×#89824C2semidirect
7. C2×#89817C2semidirect
8. C2×#89792C2semidirect
9. C2×C4×#1144C2semidirect
11. C2×#89743C2semidirect
12. C2×#89736C2semidirect
37. #89863C22semidirect
38. #89851C22semidirect
39. #89824C22semidirect
40. #89817C22semidirect
41. C4×#1144C22semidirect
42. #89792C22semidirect
44. #89743C22semidirect
47. #89736C22semidirect
442. C2×C7#3356semidirect
489. C7C2×#3356semidirect
511. C2#11741071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#902332
#117410712
C2×#898632
C2×#898512
C2×#898242
C2×#898172
C2×#897922
C2×C4×#11442
C2×#897842
C2×#897432
C2×#897362
C2×#33567