Finite Group Databasebeta

← C2×#11729665 #13329501

Subgroups — C2×#11729665 #13329501

Order 1792.

1206 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×#89865C211yesmaximal, characteristic, complemented
3C2×#89780C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729665C218yesmaximal, direct product factor
6C2×#89342C211yesmaximal, characteristic, complemented
7C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
9C22×C7×#41—21no—
10C7×#547—22no—
11C7×#547—22no—
12C7×#547—22no—
13C7×#547C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#123—22no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×#121—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C22×C7×C4⋊C4—21no—
21C2×#6432C2211yescharacteristic
22C2×#6406—21no—
23C2×#6406—21no—
24#89865—22no—
25#89865C2214yescomplemented
26#89780—22no—
27#89780—22no—
28#89780C2214yescomplemented
29#89780C2214yescomplemented
30C2×#6384—22no—
31C2×#6382—21no—
32C2×#6382—21no—
33C2×C4⋊(C7⋊C8)C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4×C7⋊C8C2211yescharacteristic
36#89342C2214yescomplemented
37D4⋊(C7⋊C8)C2214yescomplemented
38D4⋊(C7⋊C8)C2214yescomplemented
39C2×#1635—71nomaximal, Sylow
40C2×C7×#41—24no—
41C2×C7×#41—24no—
42C2×C7×#41—44no—
43C2×C7×#41—44no—
44C22×C7×Q8—21no—
45C22×C7×D4—22no—
46C22×C7×D4D411yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—22no—
49C23×C4×C7—21no—
50C23×C4×C7—21no—
51C22×Dic14—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8D412yes—
55C7×C4⋊Q8C2312yes—
56C7×#123—28no—
57C7×#123—28no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—24no—
60C7×#121—24no—
61C7×#121—24no—
62C4×C7×D4—21no—
63C4×C7×D4—21no—
64C4×C7×D4—21no—
65C4×C7×D4—21no—
66C4×C7×D4D412yes—
67C4×C7×D4D412yes—
68C4×C7×D4C2312yes—
69C4×C7×D4C2312yes—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—24no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—24no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D412yes—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85C22×C7⋊C8—21no—
86#6432—21no—
87#6432—21no—
88#6432—21no—
89#6432C2312yes—
90#6406—24no—
91#6406—24no—
92#6384—28no—
93#6382—24no—
94#6382—24no—
95C4⋊(C7⋊C8)—21no—
96C4⋊(C7⋊C8)—21no—
97C4⋊(C7⋊C8)C2312yes—
98C4⋊(C7⋊C8)C2312yes—
99C4×C7⋊C8—21no—
100C4×C7⋊C8C2312yes—
101C2×#547D711yescharacteristic
102C2×#487—71no—
103C2×#476—71no—
104C2×#476—71no—
105#1635—78no—
106C2×#332—71no—
107C2×D4⋊C8—71no—
108C2×D4⋊C8—71no—
109C24×C7—21no—
110C24×C7—21no—
111C7×#41—48no—
112C7×#41—416no—
113C7×#41—48no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—42no—
117C2×C7×Q8—42no—
118C2×C7×D4—28no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—48no—
125C2×C7×D4SD1614yes—
126C2×C7×D4C2×D412yes—
127C2×C7×D4C2×D412yes—
128C22×C4×C7—28no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—48no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×Dic7—41no—
145C2×Dic14—22no—
146C2×Dic14—44no—
147C2×Dic14—41no—
148C7×C4⋊C4—28no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4C2×D412yes—
155C7×C4⋊C4C2×D412yes—
156C7×C4⋊C4C2×D414yes—
157C7×C22⋊C4—28no—
158C7×C22⋊C4—22no—
159C7×C22⋊C4—22no—
160C7×C22⋊C4—22no—
161C7×C22⋊C4—22no—
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C42×C7C2×D411yescharacteristic
165C42×C7C2411yescharacteristic
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4⋊Dic7—41no—
169C4⋊Dic7—41no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—42no—
174C2×C7⋊C8—42no—
175C2×C7⋊C8—42no—
176C22×#41—21no—
177#547—22no—
178#547—22no—
179#547—22no—
180#547C2×D714yes—
181C2×C4⋊Q8—71no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#123—22no—
184C2×C22⋊Q8—21no—
185C2×#121—21no—
186C2×C4×D4C2×D711yescharacteristic
187C2×C4×D4C2×D711yescharacteristic
188C22×C4⋊C4—21no—
189#487—74no—
190#487—142no—
191#476—74no—
192#476—74no—
193#476—142no—
194#476—142no—
195C2×#106—141no—
196C2×#106—141no—
197C2×C4⋊C8—71no—
198C2×C4⋊C8—71no—
199C2×#103—142no—
200C2×#102—141no—
201C2×#102—141no—
202C2×C4×C8—71no—
203#332—74no—
204D4⋊C8—74no—
205D4⋊C8—74no—
206C23×C7—22no—
207C23×C7—24no—
208C23×C7—22no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7#12011yescharacteristic
215C7×Q8—42no—
216C7×Q8—42no—
217C7×Q8—44no—
218C7×D4—22no—
219C7×D4—22no—
220C7×D4—22no—
221C7×D4—24no—
222C7×D4—416no—
223C7×D4—42no—
224C7×D4C2×SD1614yes—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—24no—
229C2×C4×C7—416no—
230C2×C4×C7—42no—
231C2×C4×C7—44no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—44no—
238C2×C4×C7—44no—
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7#12011yescharacteristic
242C2×C4×C7C2×SD1612yes—
243C2×C4×C7#13512yes—
244C2×C4×C7AGL(1,ℤ/8)12yes—
245C2×C4×C7#13712yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C2×Dic7—42no—
250C2×Dic7—82no—
251Dic14—44no—
252Dic14—82no—
253C7⋊C8—42no—
254C7⋊C8—42no—
255C7⋊C8—42no—
256C2×#41—24no—
257C2×#41—24no—
258C2×#41—44no—
259C2×#41—44no—
260C22×Q8—21no—
261C22×Q8—141no—
262C22×D4—22no—
263C22×D4C22⋊D711yescharacteristic
264C22×D4C22⋊D711yescharacteristic
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—21no—
268C22×C8—141no—
269C22×C8—141no—
270C22×C8—141no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—72no—
274C4⋊Q8—141no—
275C4⋊Q8—141no—
276C4⋊Q8—141no—
277C4⋊Q8C22⋊D712yes—
278C4⋊Q8C22×D712yes—
279#123—28no—
280#123—28no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#121—24no—
284#121—24no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C22⋊D712yes—
290C4×D4C22⋊D712yes—
291C4×D4C22×D712yes—
292C4×D4C22×D712yes—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—141no—
299C2×C4⋊C4C22⋊D711yescharacteristic
300C2×C4⋊C4C22⋊D711yescharacteristic
301C2×C4⋊C4C22⋊D712yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C42C22×D711yescharacteristic
306#106—144no—
307#106—144no—
308C4⋊C8—72no—
309C4⋊C8—72no—
310C4⋊C8—141no—
311C4⋊C8—141no—
312#103—148no—
313#102—144no—
314#102—144no—
315C4×C8—72no—
316C4×C8—141no—
317C22×C7—24no—
318C22×C7—48no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—42no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7—84no—
325C22×C7#45012yes—
326C22×C7#45212yes—
327C22×C7#45412yes—
328C22×C7C2×#12011yescharacteristic
329C4×C7—21no—
330C4×C7—21no—
331C4×C7—42no—
332C4×C7—44no—
333C4×C7—42no—
334C4×C7#45012yes—
335C4×C7#45212yes—
336C4×C7C22×SD1611yescharacteristic
337C4×C7C2×#13511yescharacteristic
338C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
339C4×C7C2×#13711yescharacteristic
340Dic7—82no—
341C24—21no—
342C24—21no—
343#41—48no—
344#41—416no—
345#41—48no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—142no—
351C2×Q8—284no—
352C2×Q8—281no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—48no—
360C2×D4#114214yes—
361C2×D4C2×C22⋊D712yes—
362C2×D4C2×C22⋊D712yes—
363C22×C4—28no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—48no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—281no—
377C22×C4C2×C22⋊D711yescharacteristic
378C22×C4C2×C22⋊D711yescharacteristic
379C22×C4C2×C22⋊D711yescharacteristic
380C2×C8—142no—
381C2×C8—142no—
382C2×C8—142no—
383C2×C8—282no—
384C2×C8—282no—
385C2×C8—282no—
386C4⋊C4—28no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—142no—
393C4⋊C4—142no—
394C4⋊C4—281no—
395C4⋊C4—281no—
396C4⋊C4C2×C22⋊D714yes—
397C4⋊C4C2×C22⋊D712yes—
398C4⋊C4C2×C22⋊D712yes—
399C22⋊C4—28no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C42C2×C22⋊D711yescharacteristic
405C42C2×C22⋊D711yescharacteristic
406C42C2×C22⋊D711yescharacteristic
407C42C23×D711yescharacteristic
408C2×C7—44no—
409C2×C7—82no—
410C2×C7#163514yescomplemented
411C2×C7C2×#45011yescharacteristic
412C2×C7C2×#45211yescharacteristic
413C2×C7C2×#45411yescharacteristic
414C23—24no—
415C23—22no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23#651611yescharacteristic
423Q8—42no—
424Q8—42no—
425Q8—44no—
426Q8—284no—
427Q8—562no—
428D4—22no—
429D4—24no—
430D4—22no—
431D4—22no—
432D4—416no—
433D4—42no—
434D4C2×#114214yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—416no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—282no—
450C2×C4—562no—
451C2×C4#649512yes—
452C2×C4C2×#114212yes—
453C2×C4#651312yes—
454C2×C4#651412yes—
455C2×C4#651611yescharacteristic
456C2×C4#651611yescharacteristic
457C2×C4#651611yescharacteristic
458C2×C4C22×C22⋊D711yescharacteristic
459C2×C4C22×C22⋊D711yescharacteristic
460C2×C4C22×C22⋊D711yescharacteristic
461C8—282no—
462C8—282no—
463C8—282no—
464C7C2×#163511yescharacteristic, complemented, Sylow
465C22—24no—
466C22—48no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22#8999012yes—
474C22#9001112yes—
475C22#9001712yes—
476C22C2×#651611yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—562no—
483C4#8999012yes—
484C4#9001112yes—
485C4C2×#649511yescharacteristic
486C4C22×#114211yescharacteristic
487C4C2×#651311yescharacteristic
488C4C2×#651411yescharacteristic
489C2—44no—
490C2—82no—
491C2#1172966514yesdirect product factor
492C2C2×#8999011yescharacteristic
493C2C2×#9001111yescharacteristic
494C2C2×#9001711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC23×C7#120
hypercenterC2×#547D7
hypocenterC7C2×#1635
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#89865C2semidirect
3. C2×#89780C2semidirect
4. C2×#89780C2semidirect
5. #11729665C2direct (N × H)
6. C2×#89342C2semidirect
7. C2×D4⋊(C7⋊C8)C2semidirect
8. C2×D4⋊(C7⋊C8)C2semidirect
25. #89865C22semidirect
28. #89780C22semidirect
29. #89780C22semidirect
36. #89342C22semidirect
37. D4⋊(C7⋊C8)C22semidirect
38. D4⋊(C7⋊C8)C22semidirect
410. C2×C7#1635semidirect
464. C7C2×#1635semidirect
491. C2#11729665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×#898652
C2×#897802
#117296652
C2×#893422
C2×D4⋊(C7⋊C8)2
C2×#16357