Finite Group Databasebeta

← C2×#11741104 #14232295

Subgroups — C2×#11741104 #14232295

Order 1792.

1854 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2C2×#90233C211yesmaximal, characteristic
3#11741104C2116yesmaximal, direct product factor
4C2×#89866C211yesmaximal, characteristic, complemented
5C2×#89850C211yesmaximal, characteristic, complemented
6C2×#89823C212yesmaximal, complemented
7C2×#89817C212yesmaximal, complemented
8C2×#89792C211yesmaximal, characteristic, complemented
9C2×C4×#1144C211yesmaximal, characteristic, complemented
10C2×D4×C7⋊C8C211yesmaximal, characteristic
11C2×#89768C212yesmaximal, complemented
12C2×#89770C212yesmaximal, complemented
13C7×#556—24no—
14C7×#556C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#126—22no—
17C2×C7×#124—21no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117C2212yes—
23C22×#1144C2211yescharacteristic
24C2×#6499—21no—
25C2×#6489C2212yes—
26C2×#6487C2212yes—
27C23×C7⋊C8C2212yes—
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#89866C2218yescomplemented
38#89850C2218yescomplemented
39#89823C22116yescomplemented
40#89817C22116yescomplemented
41C4×#1144C2218yescomplemented
42#89792C2218yescomplemented
43D4×C7⋊C8C2218yes—
44#89770C22116yescomplemented
45#89768C22116yescomplemented
46C2×#6384C2212yes—
47C2×#6384C2211yescharacteristic
48C2×#6382C2212yes—
49C2×#6382C2212yes—
50C2×#6381C2211yescharacteristic
51C2×#6381C2212yes—
52C2×#6381C2212yes—
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3342—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4D411yescharacteristic
59C23×C4×C7C2312yes—
60C22×Dic14—21no—
61C22×Dic14C2311yescharacteristic
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8C2314yes—
64C7×#126—216no—
65C7×#124—28no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8C2318yes—
68C4×C7×Q8—28no—
69C4×C7×Q8—22no—
70C4×C7×Q8C2314yes—
71C4×C7×D4D414yes—
72C4×C7×D4C2314yes—
73C7×#117—24no—
74C7×#117C2318yes—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4D412yes—
84C2×C42×C7—22no—
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—
103C22×C7⋊C8—28no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2312yes—
106C22×C7⋊C8C2318yes—
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#6381C2318yes—
122#6381C2314yes—
123#6381C2318yes—
124C4⋊(C7⋊C8)C2314yes—
125C4×C7⋊C8C2314yes—
126C2×#556—71no—
127C2×#556D711yescharacteristic
128#3342—716no—
129C2×#500—71no—
130C2×#485—72no—
131C2×#485—72no—
132C2×#475—71no—
133C2×#475—71no—
134C2×#467—72no—
135C2×C4×Q16—71no—
136C2×C8×D4—71no—
137C2×#427—72no—
138C24×C7C2×D412yes—
139C2×C7×Q8—21no—
140C2×C7×Q8—24no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—44no—
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×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2414yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—21no—
164C2×Dic14—24no—
165C2×Dic14—44no—
166C2×Dic14C2412yes—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4C2×D414yes—
173C7×C4⋊C4C2412yes—
174C7×C4⋊C4C2414yes—
175C7×C22⋊C4—28no—
176C7×C22⋊C4—28no—
177C7×C22⋊C4C2×D418yes—
178C42×C7—28no—
179C42×C7—24no—
180C42×C7C2×D412yes—
181C42×C7C2412yes—
182C22⋊Dic7—28no—
183C22⋊Dic7—28no—
184#1144—28no—
185C4⋊Dic7—21no—
186C4⋊Dic7—22no—
187C4⋊Dic7—24no—
188C4⋊Dic7C2412yes—
189C4⋊Dic7C2414yes—
190Dic7⋊C4—24no—
191Dic7⋊C4—28no—
192C4×Dic7—24no—
193C4×Dic7—28no—
194C2×C7⋊C8—24no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—216no—
197C2×C7⋊C8#4114yes—
198C2×C7⋊C8C2412yes—
199C2×C7⋊C8C2414yes—
200C2×C7⋊C8C2412yes—
201C22×Q16—71no—
202C23×C8—72no—
203#556—24no—
204#556—78no—
205#556—144no—
206#556C2×D718yes—
207C2×C4⋊Q8—71no—
208C2×C4⋊Q8C2×D711yescharacteristic
209C2×#126—22no—
210C2×#126—142no—
211C2×#124—21no—
212C2×#124—141no—
213C2×C22⋊Q8—72no—
214C2×C22⋊Q8C2×D712yes—
215C2×C4×Q8—21no—
216C2×C4×Q8—71no—
217C2×C4×Q8—141no—
218C2×C4×Q8C2×D711yescharacteristic
219C2×C4×D4C2×D711yescharacteristic
220C2×#117—72no—
221C2×#117C2×D712yes—
222#500—78no—
223#485—716no—
224#485—716no—
225#475—78no—
226#475—78no—
227#467—716no—
228C4×Q16—78no—
229C8×D4—78no—
230#427—716no—
231C2×#107—72no—
232C2×#107—72no—
233C2×#107—71no—
234C2×#106—72no—
235C2×#106—72no—
236C2×C4⋊C8—71no—
237C2×#103—72no—
238C2×#103—71no—
239C2×#103—72no—
240C2×#103—71no—
241C2×C22⋊C8—72no—
242C2×C4×C8—71no—
243C23×C7—24no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7C22×D414yes—
247C23×C7C22×D411yescharacteristic
248C7×Q8—22no—
249C7×Q8—22no—
250C7×Q8—42no—
251C7×Q8—48no—
252C7×D4—28no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—28no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—42no—
262C2×C4×C7—44no—
263C2×C4×C7—42no—
264C2×C4×C7#13514yes—
265C2×C4×C7C22×D414yes—
266C2×C4×C7C22×D412yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D412yes—
269C2×C4×C7C2×#4112yes—
270C2×C4×C72−1+412yes—
271C2×C4×C7C2511yescharacteristic
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—24no—
275C2×Dic7—42no—
276C2×Dic7—44no—
277C2×Dic7—42no—
278Dic14—22no—
279Dic14—22no—
280Dic14—42no—
281Dic14—48no—
282C7⋊C8—24no—
283C7⋊C8—22no—
284C7⋊C8C2×#4114yes—
285C22×Q8—21no—
286C22×Q8—71no—
287C22×Q8—141no—
288C22×Q8C22×D711yescharacteristic
289C22×D4C22⋊D711yescharacteristic
290C23×C4C22×D712yes—
291C2×Q16—74no—
292C2×Q16—144no—
293C22×C8—71no—
294C22×C8—72no—
295C22×C8—78no—
296C22×C8—71no—
297C22×C8—148no—
298C4⋊Q8—22no—
299C4⋊Q8—74no—
300C4⋊Q8—142no—
301C4⋊Q8C22×D714yes—
302#126—216no—
303#126—1416no—
304#124—28no—
305#124—148no—
306C22⋊Q8—24no—
307C22⋊Q8—78no—
308C22⋊Q8—144no—
309C22⋊Q8C22×D718yes—
310C4×Q8—28no—
311C4×Q8—22no—
312C4×Q8—74no—
313C4×Q8—148no—
314C4×Q8—142no—
315C4×Q8C22×D714yes—
316C4×D4C22⋊D714yes—
317C4×D4C22×D714yes—
318#117—24no—
319#117—78no—
320#117—144no—
321#117C22×D718yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4C22⋊D711yescharacteristic
331C2×C4⋊C4C22×D711yescharacteristic
332C2×C4⋊C4C22×D712yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—142no—
336C2×C22⋊C4—142no—
337C2×C22⋊C4C22⋊D712yes—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—142no—
341C2×C42—141no—
342C2×C42C22×D711yescharacteristic
343#107—74no—
344#107—78no—
345#107—78no—
346#106—78no—
347#106—78no—
348C4⋊C8—74no—
349#103—74no—
350#103—78no—
351#103—74no—
352#103—78no—
353C22⋊C8—78no—
354C4×C8—74no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—24no—
358C22×C7—48no—
359C22×C7#54712yes—
360C22×C7C2×#13512yes—
361C22×C7#57812yes—
362C22×C7C23×D411yescharacteristic
363C4×C7—24no—
364C4×C7—22no—
365C4×C7—42no—
366C4×C7—44no—
367C4×C7—42no—
368C4×C7#54712yes—
369C4×C7C2×#13514yes—
370C4×C7C22×#4111yescharacteristic
371C4×C7C2×2−1+411yescharacteristic
372Dic7—42no—
373Dic7—44no—
374Dic7—42no—
375C24C2×C22⋊D712yes—
376C2×Q8—21no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—72no—
382C2×Q8—141no—
383C2×Q8—144no—
384C2×Q8—142no—
385C2×Q8—142no—
386C2×Q8—284no—
387C2×Q8C23×D712yes—
388C2×D4—24no—
389C2×D4C2×C22⋊D714yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—141no—
396C22×C4—142no—
397C22×C4—141no—
398C22×C4C2×C22⋊D714yes—
399C22×C4C2×C22⋊D712yes—
400C22×C4C2×C22⋊D711yescharacteristic
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C23×D714yes—
403C22×C4C23×D711yescharacteristic
404Q16—148no—
405C2×C8—74no—
406C2×C8—72no—
407C2×C8—72no—
408C2×C8—74no—
409C2×C8—1416no—
410C2×C8—144no—
411C2×C8—142no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—72no—
418C4⋊C4—74no—
419C4⋊C4—144no—
420C4⋊C4—148no—
421C4⋊C4—141no—
422C4⋊C4—142no—
423C4⋊C4—144no—
424C4⋊C4C2×C22⋊D714yes—
425C4⋊C4C23×D712yes—
426C4⋊C4C23×D714yes—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—148no—
430C22⋊C4—148no—
431C22⋊C4C2×C22⋊D718yes—
432C42—28no—
433C42—24no—
434C42—148no—
435C42—144no—
436C42C2×C22⋊D712yes—
437C42C23×D712yes—
438C2×C7—44no—
439C2×C7#334214yescomplemented
440C2×C7C2×#54711yescharacteristic
441C2×C7C22×#13511yescharacteristic
442C2×C7C2×#57811yescharacteristic
443C23—24no—
444C23—24no—
445C23—24no—
446C23C22×C22⋊D714yes—
447C23C22×C22⋊D711yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—48no—
451Q8—42no—
452Q8—142no—
453Q8—142no—
454Q8—288no—
455Q8—282no—
456D4—28no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—142no—
469C2×C4—144no—
470C2×C4—142no—
471C2×C4—284no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4#651314yes—
475C2×C4D7×#4112yes—
476C2×C4#655412yes—
477C2×C4C22×C22⋊D714yes—
478C2×C4C22×C22⋊D712yes—
479C2×C4C22×C22⋊D712yes—
480C2×C4C22×C22⋊D711yescharacteristic
481C2×C4C24×D711yescharacteristic
482C8—74no—
483C8—144no—
484C8—142no—
485C7C2×#334211yescharacteristic, complemented, Sylow
486C22—24no—
487C22—24no—
488C22—24no—
489C22—48no—
490C22C2×#651312yes—
491C22#9051512yes—
492C22#9052912yes—
493C22C23×C22⋊D711yescharacteristic
494C4—24no—
495C4—22no—
496C4—44no—
497C4—42no—
498C4—42no—
499C4—284no—
500C4—282no—
501C4—282no—
502C4C2×#651314yes—
503C4#9051512yes—
504C4C2×D7×#4111yescharacteristic
505C4C2×#655411yescharacteristic
506C2—44no—
507C2#1174110414yesdirect product factor
508C2C22×#651311yescharacteristic
509C2C2×#9051511yescharacteristic
510C2C2×#9052911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×#556D7
hypocenterC7C2×#3342
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
3. #11741104C2direct (N × H)
4. C2×#89866C2semidirect
5. C2×#89850C2semidirect
6. C2×#89823C2semidirect
7. C2×#89817C2semidirect
8. C2×#89792C2semidirect
9. C2×C4×#1144C2semidirect
11. C2×#89768C2semidirect
12. C2×#89770C2semidirect
37. #89866C22semidirect
38. #89850C22semidirect
39. #89823C22semidirect
40. #89817C22semidirect
41. C4×#1144C22semidirect
42. #89792C22semidirect
44. #89770C22semidirect
45. #89768C22semidirect
439. C2×C7#3342semidirect
485. C7C2×#3342semidirect
507. C2#11741104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#902332
#117411042
C2×#898662
C2×#898502
C2×#898232
C2×#898172
C2×#897922
C2×C4×#11442
C2×D4×C7⋊C82
C2×#897682
C2×#897702
C2×#33427