Finite Group Databasebeta

← C2×#11728900 #13329499

Subgroups — C2×#11728900 #13329499

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×#90246C211yesmaximal, characteristic
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×#89856C211yesmaximal, characteristic, complemented
4C2×#89853C211yesmaximal, characteristic, complemented
5C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11728900C218yesmaximal, direct product factor
7C2×#89277C211yesmaximal, characteristic, complemented
8C2×#89278C211yesmaximal, characteristic, complemented
9C22×#1159—21no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C2×#6493—21no—
13C22×C4⋊Dic7—21no—
14#90246—22no—
15#90246—22no—
16#90246—22no—
17#90246C2214yes—
18C2×#6460—21no—
19C2×#6449—22no—
20C2×C4×D28C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C7×#476—22no—
23C7×#476C2214yescomplemented
24C2×C7×#106—21no—
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#102—21no—
27C2×#6409—22no—
28#89856—22no—
29#89856C2214yescomplemented
30#89853—22no—
31#89853C2214yescomplemented
32C2×#6383—21no—
33C2×#6382—21no—
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4×C7⋊C8C2211yescharacteristic
36D4⋊(C7⋊C8)C2214yescomplemented
37#89278C2214yescomplemented
38#89277C2214yescomplemented
39C2×#1635—71nomaximal, Sylow
40C22×C7×Q8—21no—
41C22×C7×D4D411yescharacteristic
42C23×C4×C7—21no—
43C22×C22⋊D7—22no—
44C23×Dic7—22no—
45C2×#1159—24no—
46C2×#1159—24no—
47C2×#1159—44no—
48C2×#1159—44no—
49C22×D28D411yescharacteristic
50C22×C4×D7—21no—
51C22×Dic14—21no—
52C22×C7×C8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8C2312yes—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C4×C7×D4D412yes—
60C4×C7×D4C2312yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C42×C7C2311yescharacteristic
66#6493—24no—
67#6493—24no—
68C2×(C2×D7)⋊C4—22no—
69C2×(C2×D7)⋊C4—21no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—24no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—24no—
74C2×C4⋊Dic7D412yes—
75C2×C4⋊Dic7D411yescharacteristic
76C22×C7⋊C8—21no—
77C22×C7⋊C8—21no—
78#6460—24no—
79#6460—24no—
80#6449—28no—
81#6449—28no—
82C4×D28—21no—
83C4×D28—21no—
84C4×D28D412yes—
85C4×D28C2312yes—
86#6432—21no—
87#6432—21no—
88#6432D412yes—
89#6432C2312yes—
90C7×#106—24no—
91C7×C4⋊C8—21no—
92C7×C4⋊C8C2312yes—
93C7×#102—24no—
94#6409—28no—
95#6383—24no—
96#6382—24no—
97C4⋊(C7⋊C8)—21no—
98C4⋊(C7⋊C8)C2312yes—
99C4×C7⋊C8—21no—
100C4×C7⋊C8C2312yes—
101C2×#547—71no—
102C2×#487—71no—
103C2×#476—71no—
104C2×#476D711yescharacteristic, complemented
105#1635—78no—
106C2×#332—71no—
107C2×D4⋊C8—71no—
108C2×D4⋊C8—71no—
109C24×C7—21no—
110C23×D7—21no—
111C2×C7×Q8—22no—
112C2×C7×Q8—44no—
113C2×C7×Q8—41no—
114C2×C7×D4—21no—
115C2×C7×D4—22no—
116C2×C7×D4SD1614yes—
117C2×C7×D4C2×D412yes—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—42no—
122C22×C4×C7—42no—
123C22×C4×C7—41no—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D411yescharacteristic
127C2×C22⋊D7—28no—
128C2×C22⋊D7—48no—
129C22×Dic7—28no—
130C22×Dic7—22no—
131C22×Dic7—21no—
132C22×Dic7—48no—
133#1159—48no—
134#1159—416no—
135#1159—48no—
136C2×D28—22no—
137C2×D28—22no—
138C2×D28—21no—
139C2×D28C2×D412yes—
140C2×C4×D7—22no—
141C2×C4×D7—22no—
142C2×C4×D7—42no—
143C2×C4×D7—42no—
144C2×Dic14—22no—
145C2×Dic14—22no—
146C2×Dic14—42no—
147C2×Dic14—42no—
148C2×C7×C8—22no—
149C2×C7×C8—42no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—41no—
154C7×C4⋊C4—41no—
155C7×C4⋊C4C2×D412yes—
156C7×C22⋊C4—22no—
157C7×C22⋊C4—22no—
158C42×C7C2×D411yescharacteristic
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D411yescharacteristic
161C42×C7C2411yescharacteristic
162(C2×D7)⋊C4—28no—
163(C2×D7)⋊C4—22no—
164(C2×D7)⋊C4—22no—
165C4⋊Dic7—28no—
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4⋊Dic7—21no—
169C4⋊Dic7—22no—
170C4⋊Dic7C2×D414yes—
171C4⋊Dic7C2×D412yes—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—42no—
175C2×C7⋊C8—42no—
176C22×#41—141no—
177#547—74no—
178#547—142no—
179#547—142no—
180#547—142no—
181C2×C4⋊Q8—71no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#123—142no—
184C2×C22⋊Q8—141no—
185C2×#121—141no—
186C2×C4×D4—71no—
187C2×C4×D4C2×D711yescharacteristic
188C22×C4⋊C4—141no—
189#487—74no—
190#487—142no—
191#476—22no—
192#476—74no—
193#476—142no—
194#476C2×D714yescomplemented
195C2×#106—21no—
196C2×#106—141no—
197C2×C4⋊C8—71no—
198C2×C4⋊C8C2×D711yescharacteristic
199C2×#103—142no—
200C2×#102—21no—
201C2×#102—141no—
202C2×C4×C8—71no—
203#332—74no—
204D4⋊C8—74no—
205D4⋊C8—74no—
206C23×C7—22no—
207C23×C7—24no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7#12011yescharacteristic
211C22×D7—22no—
212C22×D7—42no—
213C22×D7—42no—
214C22×D7—42no—
215C7×Q8—44no—
216C7×Q8—82no—
217C7×D4—22no—
218C7×D4—24no—
219C7×D4C2×SD1614yes—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—44no—
226C2×C4×C7—42no—
227C2×C4×C7—82no—
228C2×C4×C7#12011yescharacteristic
229C2×C4×C7#12011yescharacteristic
230C2×C4×C7#12011yescharacteristic
231C2×C4×C7C2×SD1612yes—
232C2×C4×C7#13512yes—
233C2×C4×C7AGL(1,ℤ/8)12yes—
234C2×C4×C7#13712yes—
235C2×C4×C7C22×D411yescharacteristic
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C22×D411yescharacteristic
238C22⋊D7—416no—
239C2×Dic7—24no—
240C2×Dic7—22no—
241C2×Dic7—416no—
242C2×Dic7—44no—
243C2×Dic7—42no—
244D28—22no—
245D28—22no—
246D28—42no—
247C4×D7—42no—
248C4×D7—42no—
249C4×D7—44no—
250Dic14—42no—
251Dic14—42no—
252Dic14—44no—
253C7×C8—42no—
254C7⋊C8—42no—
255C7⋊C8—42no—
256C2×#41—144no—
257C2×#41—144no—
258C2×#41—284no—
259C2×#41—284no—
260C22×Q8—21no—
261C22×Q8—141no—
262C22×D4—71no—
263C22×D4—142no—
264C22×D4C22⋊D711yescharacteristic
265C23×C4—21no—
266C23×C4—142no—
267C23×C4—141no—
268C22×C8—21no—
269C22×C8—141no—
270C22×C8—141no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—72no—
275C4⋊Q8—72no—
276C4⋊Q8—141no—
277C4⋊Q8—141no—
278C4⋊Q8C22×D712yes—
279#123—148no—
280#123—148no—
281C22⋊Q8—144no—
282C22⋊Q8—144no—
283#121—144no—
284#121—144no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—72no—
288C4×D4—72no—
289C4×D4—141no—
290C4×D4—141no—
291C4×D4C22⋊D712yes—
292C4×D4C22×D712yes—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—71no—
296C2×C4⋊C4—72no—
297C2×C4⋊C4—144no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—144no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4C22⋊D711yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—142no—
304C2×C22⋊C4—141no—
305C2×C42C22×D711yescharacteristic
306#106—24no—
307#106—144no—
308C4⋊C8—21no—
309C4⋊C8—72no—
310C4⋊C8—141no—
311C4⋊C8C22×D712yes—
312#103—148no—
313#102—24no—
314#102—144no—
315C4×C8—72no—
316C4×C8—141no—
317C22×C7—24no—
318C22×C7—48no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7#45012yes—
322C22×C7#45212yes—
323C22×C7#45412yes—
324C22×C7C2×#12011yescharacteristic
325C2×D7—42no—
326C2×D7—42no—
327C2×D7—42no—
328C2×D7—84no—
329C4×C7—21no—
330C4×C7—21no—
331C4×C7—42no—
332C4×C7—82no—
333C4×C7#45012yes—
334C4×C7#45212yes—
335C4×C7C22×SD1611yescharacteristic
336C4×C7C2×#13511yescharacteristic
337C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
338C4×C7C2×#13711yescharacteristic
339Dic7—44no—
340Dic7—42no—
341C24—21no—
342C24—141no—
343#41—288no—
344#41—2816no—
345#41—288no—
346C2×Q8—22no—
347C2×Q8—44no—
348C2×Q8—41no—
349C2×Q8—142no—
350C2×Q8—142no—
351C2×Q8—282no—
352C2×Q8—282no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—72no—
356C2×D4—148no—
357C2×D4—142no—
358C2×D4—142no—
359C2×D4—141no—
360C2×D4—288no—
361C2×D4#114314yes—
362C2×D4C2×C22⋊D712yes—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—148no—
370C22×C4—141no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—142no—
374C22×C4—288no—
375C22×C4—282no—
376C22×C4—282no—
377C22×C4D4×D711yescharacteristic
378C22×C4D4×D711yescharacteristic
379C22×C4C2×C22⋊D711yescharacteristic
380C2×C8—22no—
381C2×C8—42no—
382C2×C8—142no—
383C2×C8—142no—
384C2×C8—282no—
385C2×C8—282no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—74no—
392C4⋊C4—72no—
393C4⋊C4—148no—
394C4⋊C4—141no—
395C4⋊C4—142no—
396C4⋊C4—142no—
397C4⋊C4—142no—
398C4⋊C4C2×C22⋊D712yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—148no—
402C22⋊C4—142no—
403C22⋊C4—142no—
404C42D4×D711yescharacteristic
405C42D4×D711yescharacteristic
406C42C2×C22⋊D711yescharacteristic
407C42C23×D711yescharacteristic
408C2×C7—44no—
409C2×C7#163514yescomplemented
410C2×C7C2×#45011yescharacteristic
411C2×C7C2×#45211yescharacteristic
412C2×C7C2×#45411yescharacteristic
413D7—82no—
414C23—22no—
415C23—24no—
416C23—42no—
417C23—42no—
418C23—142no—
419C23—282no—
420C23—282no—
421C23—282no—
422C23#650011yescharacteristic
423Q8—44no—
424Q8—82no—
425Q8—282no—
426Q8—282no—
427Q8—284no—
428D4—22no—
429D4—24no—
430D4—142no—
431D4—142no—
432D4—2816no—
433D4—282no—
434D4C2×#114314yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—142no—
444C2×C4—144no—
445C2×C4—2816no—
446C2×C4—282no—
447C2×C4—284no—
448C2×C4—282no—
449C2×C4—282no—
450C2×C4—284no—
451C2×C4#647812yes—
452C2×C4#647912yes—
453C2×C4#650011yescharacteristic
454C2×C4#650011yescharacteristic
455C2×C4#650011yescharacteristic
456C2×C4C2×#114312yes—
457C2×C4#651212yes—
458C2×C4C2×D4×D711yescharacteristic
459C2×C4C2×D4×D711yescharacteristic
460C2×C4C22×C22⋊D711yescharacteristic
461C8—42no—
462C8—282no—
463C8—282no—
464C7C2×#163511yescharacteristic, complemented, Sylow
465C22—24no—
466C22—48no—
467C22—42no—
468C22—42no—
469C22—282no—
470C22—282no—
471C22—282no—
472C22—564no—
473C22#8981812yes—
474C22#8994512yes—
475C22#8997312yes—
476C22C2×#650011yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—82no—
481C4—282no—
482C4—284no—
483C4#8981812yes—
484C4#8994512yes—
485C4C2×#647811yescharacteristic
486C4C2×#647911yescharacteristic
487C4C22×#114311yescharacteristic
488C4C2×#651211yescharacteristic
489C2—44no—
490C2—562no—
491C2#1172890014yesdirect product factor
492C2C2×#8981811yescharacteristic
493C2C2×#8994511yescharacteristic
494C2C2×#8997311yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#476C2
socleC23×C7#120
prime radicalC2×#90246C2
hypercenterC2×#476D7
hypocenterC7C2×#1635
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. C2×C7×#476C2semidirect
3. C2×#89856C2semidirect
4. C2×#89853C2semidirect
5. C2×D4⋊(C7⋊C8)C2semidirect
6. #11728900C2direct (N × H)
7. C2×#89277C2semidirect
8. C2×#89278C2semidirect
23. C7×#476C22semidirect
29. #89856C22semidirect
31. #89853C22semidirect
36. D4⋊(C7⋊C8)C22semidirect
37. #89278C22semidirect
38. #89277C22semidirect
104. C2×#476D7semidirect
194. #476C2×D7semidirect
409. C2×C7#1635semidirect
464. C7C2×#1635semidirect
491. C2#11728900direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902462
C2×C7×#4762
C2×#898562
C2×#898532
C2×D4⋊(C7⋊C8)2
#117289002
C2×#892772
C2×#892782
C2×#16357