Finite Group Databasebeta

← C2×#11740187 #14224404

Subgroups — C2×#11740187 #14224404

Order 1792.

2322 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C22×#6487C211yesmaximal, characteristic
3C2×#90000C212yesmaximal, complemented
4C2×#89979C211yesmaximal, characteristic
5#11740187C2116yesmaximal, direct product factor
6C2×#89822C212yesmaximal, complemented
7C2×#89824C214yesmaximal, complemented
8C2×#89815C212yesmaximal, complemented
9C2×#89743C212yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C23×Dic14C2211yescharacteristic
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8—21no—
15C2×C7×C22⋊Q8—24no—
16C2×C7×C22⋊Q8C2212yes—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×C22⋊Q8C2218yes—
19C22×C7×C4⋊C4—21no—
20C22×C7×C4⋊C4C2211yescharacteristic
21C22×C7×C22⋊C4—21no—
22C22×C22⋊Dic7—21no—
23C22×#1144C2214yescomplemented
24C22×C4⋊Dic7C2211yescharacteristic
25C2×#6487—21no—
26C2×#6487—21no—
27C2×#6487—24no—
28C2×#6487C2212yes—
29C2×#6487C2218yes—
30C2×#6487C2212yes—
31C22×Dic7⋊C4—21no—
32C22×#1136C2212yes—
33#90000C22116yescomplemented
34C2×#6409C2214yescomplemented
35#89979C2218yes—
36C2×C22⋊(C7⋊C8)C2212yes—
37C2×C22⋊(C7⋊C8)C2212yes—
38#89824C22132yescomplemented
39#89822C22116yescomplemented
40#89815C22116yescomplemented
41#89743C22116yescomplemented
42C2×#6384C2214yescomplemented
43C2×#6382C2214yescomplemented
44C2×#3195—71nomaximal, Sylow
45C25×C7D411yescharacteristic
46C22×C7×Q8—21no—
47C22×C7×Q8—22no—
48C22×C7×Q8—44no—
49C22×C7×Q8C2312yes—
50C22×C7×Q8C2314yes—
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7—44no—
54C23×C4×C7D411yescharacteristic
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D414yes—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2314yes—
62C23×Dic7—21no—
63C23×Dic7—21no—
64C22×Dic14—21no—
65C22×Dic14—22no—
66C22×Dic14—44no—
67C22×Dic14C2312yes—
68C22×Dic14C2314yes—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—28no—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8C2318yes—
74C7×C22⋊Q8C2318yes—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—28no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C4⋊C4C2314yes—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—28no—
84C2×C7×C22⋊C4—22no—
85C2×C22⋊Dic7—22no—
86C2×C22⋊Dic7—28no—
87C2×C22⋊Dic7—22no—
88C2×#1144—48no—
89C2×#1144C23116yescomplemented
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—22no—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2314yes—
94#6487—24no—
95#6487—24no—
96#6487—28no—
97#6487—28no—
98#6487C2318yes—
99#6487C2318yes—
100C2×Dic7⋊C4—22no—
101C2×Dic7⋊C4—22no—
102C2×Dic7⋊C4—28no—
103C2×#1136—44no—
104C2×#1136C2318yes—
105C22×C7⋊C8C2314yes—
106#6409C23116yescomplemented
107C22⋊(C7⋊C8)C2318yes—
108C22⋊(C7⋊C8)C2318yes—
109#6384C23116yescomplemented
110#6382C23116yescomplemented
111C22×C22⋊Q8—71no—
112C22×C22⋊Q8D711yescharacteristic
113#3195—716no—
114C2×#483—72no—
115C2×#483—72no—
116C2×#470—74no—
117C2×#451—72no—
118C2×#451—72no—
119C2×#407—71no—
120C24×C7—24no—
121C24×C7—24no—
122C24×C7—42no—
123C24×C7C2×D411yescharacteristic
124C24×C7C2×D414yes—
125C24×C7C2×D411yescharacteristic
126C24×C7C2×D411yescharacteristic
127C2×C7×Q8—24no—
128C2×C7×Q8—22no—
129C2×C7×Q8—28no—
130C2×C7×Q8—48no—
131C2×C7×Q8—44no—
132C2×C7×Q8—48no—
133C2×C7×Q8C2414yes—
134C22×C4×C7—24no—
135C22×C4×C7—24no—
136C22×C4×C7—24no—
137C22×C4×C7—24no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—48no—
142C22×C4×C7—48no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—44no—
147C22×C4×C7—44no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2412yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—24no—
164C22×Dic7—22no—
165C22×Dic7—44no—
166C22×Dic7—44no—
167C2×Dic14—28no—
168C2×Dic14—24no—
169C2×Dic14—22no—
170C2×Dic14—48no—
171C2×Dic14—44no—
172C2×Dic14—48no—
173C2×Dic14C2414yes—
174C7×C4⋊C4—28no—
175C7×C4⋊C4—28no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4C2414yes—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—28no—
181C22⋊Dic7—28no—
182C22⋊Dic7—28no—
183#1144—416no—
184C4⋊Dic7—24no—
185C4⋊Dic7—22no—
186C4⋊Dic7C2414yes—
187Dic7⋊C4—28no—
188Dic7⋊C4—28no—
189#1136—48no—
190C2×C7⋊C8—28no—
191C2×C7⋊C8C2418yes—
192C23×Q8—71no—
193C23×Q8C2×D711yescharacteristic
194C24×C4C2×D711yescharacteristic
195C22×Q16—74no—
196C22×M16—72no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—24no—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—72no—
201C2×C22⋊Q8—78no—
202C2×C22⋊Q8—72no—
203C2×C22⋊Q8—141no—
204C2×C22⋊Q8—144no—
205C2×C22⋊Q8—141no—
206C2×C22⋊Q8C2×D712yes—
207C2×C22⋊Q8C2×D718yes—
208C2×C22⋊Q8C2×D712yes—
209C22×C4⋊C4—21no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—141no—
212C22×C4⋊C4C2×D711yescharacteristic
213C22×C22⋊C4—21no—
214C22×C22⋊C4—141no—
215#483—716no—
216#483—716no—
217#470—732no—
218#451—716no—
219#451—716no—
220C2×#106—74no—
221C2×#103—74no—
222C2×#103—74no—
223#407—78no—
224C2×C22⋊C8—72no—
225C2×C22⋊C8—72no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—28no—
232C23×C7—22no—
233C23×C7—28no—
234C23×C7—22no—
235C23×C7—44no—
236C23×C7—416no—
237C23×C7#13714yes—
238C23×C7#13714yes—
239C23×C7C22×D412yes—
240C23×C7C22×D412yes—
241C23×C7C22×D412yes—
242C23×C7C22×D411yescharacteristic
243C7×Q8—28no—
244C7×Q8—48no—
245C7×Q8—44no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—44no—
258C2×C4×C7—44no—
259C2×C4×C7—44no—
260C2×C4×C7—48no—
261C2×C4×C7—44no—
262C2×C4×C7—48no—
263C2×C4×C7—44no—
264C2×C4×C7C22×D412yes—
265C2×C4×C7C22×D412yes—
266C2×C4×C7C22×D412yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7H(5,𝔽2)14yes—
269C2×C4×C7C2511yescharacteristic
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—48no—
273C2×Dic7—44no—
274C2×Dic7—48no—
275C2×Dic7—44no—
276Dic14—28no—
277Dic14—48no—
278Dic14—44no—
279C7⋊C8—28no—
280C25C22⋊D711yescharacteristic
281C22×Q8—22no—
282C22×Q8—21no—
283C22×Q8—44no—
284C22×Q8—72no—
285C22×Q8—74no—
286C22×Q8—142no—
287C22×Q8—141no—
288C22×Q8—284no—
289C22×Q8C22×D712yes—
290C22×Q8C22×D714yes—
291C23×C4—21no—
292C23×C4—21no—
293C23×C4—44no—
294C23×C4—141no—
295C23×C4—141no—
296C23×C4C22⋊D711yescharacteristic
297C23×C4C22⋊D711yescharacteristic
298C23×C4C22⋊D714yes—
299C23×C4C22⋊D711yescharacteristic
300C23×C4C22×D711yescharacteristic
301C23×C4C22×D711yescharacteristic
302C23×C4C22×D711yescharacteristic
303C23×C4C22×D714yes—
304C2×Q16—716no—
305C2×Q16—288no—
306C2×M16—78no—
307C2×M16—284no—
308C22×C8—74no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—28no—
312C22⋊Q8—28no—
313C22⋊Q8—78no—
314C22⋊Q8—78no—
315C22⋊Q8—144no—
316C22⋊Q8—144no—
317C22⋊Q8—148no—
318C22⋊Q8—148no—
319C22⋊Q8C22×D718yes—
320C22⋊Q8C22×D718yes—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4—74no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—148no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4C22×D712yes—
334C2×C4⋊C4C22×D714yes—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—148no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—142no—
341#106—716no—
342#103—716no—
343#103—716no—
344C22⋊C8—78no—
345C22⋊C8—78no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—28no—
349C22×C7—28no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—416no—
356C22×C7—44no—
357C22×C7#53412yes—
358C22×C7C2×#13714yes—
359C22×C7C2×#13712yes—
360C22×C7C2×#13714yes—
361C22×C7C2×#13712yes—
362C22×C7C23×D411yescharacteristic
363C4×C7—22no—
364C4×C7—22no—
365C4×C7—22no—
366C4×C7—44no—
367C4×C7—44no—
368C4×C7#53412yes—
369C4×C7C2×H(5,𝔽2)12yes—
370Dic7—44no—
371Dic7—44no—
372C24—24no—
373C24—24no—
374C24—42no—
375C24C2×C22⋊D711yescharacteristic
376C24C2×C22⋊D714yes—
377C24C2×C22⋊D711yescharacteristic
378C24C2×C22⋊D711yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—28no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—74no—
386C2×Q8—144no—
387C2×Q8—142no—
388C2×Q8—148no—
389C2×Q8—288no—
390C2×Q8—284no—
391C2×Q8—288no—
392C2×Q8C23×D714yes—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—144no—
408C22×C4—144no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—284no—
412C22×C4—284no—
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D712yes—
415C22×C4C2×C22⋊D712yes—
416C22×C4C2×C22⋊D711yescharacteristic
417C22×C4C2×C22⋊D711yescharacteristic
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D714yes—
420C22×C4C2×C22⋊D714yes—
421C22×C4C2×C22⋊D714yes—
422C22×C4C23×D712yes—
423C22×C4C23×D712yes—
424C22×C4C23×D712yes—
425C22×C4C23×D711yescharacteristic
426Q16—2816no—
427M16—288no—
428C2×C8—78no—
429C2×C8—148no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—74no—
435C4⋊C4—148no—
436C4⋊C4—148no—
437C4⋊C4—144no—
438C4⋊C4—142no—
439C4⋊C4C23×D714yes—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—148no—
443C22⋊C4—148no—
444C2×C7—24no—
445C2×C7—24no—
446C2×C7—42no—
447C2×C7#319514yescomplemented
448C2×C7C2×#53411yescharacteristic
449C2×C7C22×#13711yescharacteristic
450C2×C7C22×#13711yescharacteristic
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—28no—
457C23—22no—
458C23—28no—
459C23—22no—
460C23—44no—
461C23—416no—
462C23#650514yes—
463C23#651414yes—
464C23C22×C22⋊D712yes—
465C23C22×C22⋊D712yes—
466C23C22×C22⋊D712yes—
467C23C22×C22⋊D711yescharacteristic
468Q8—28no—
469Q8—48no—
470Q8—44no—
471Q8—148no—
472Q8—288no—
473Q8—284no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—144no—
493C2×C4—144no—
494C2×C4—288no—
495C2×C4—284no—
496C2×C4—288no—
497C2×C4—284no—
498C2×C4#654814yes—
499C2×C4C22×C22⋊D712yes—
500C2×C4C22×C22⋊D712yes—
501C2×C4C22×C22⋊D712yes—
502C2×C4C22×C22⋊D711yescharacteristic
503C2×C4C24×D711yescharacteristic
504C8—148no—
505C7C2×#319511yescharacteristic, complemented, Sylow
506C22—28no—
507C22—28no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—416no—
516C22—44no—
517C22#9049512yes—
518C22C2×#650514yes—
519C22C2×#650512yes—
520C22C2×#651414yes—
521C22C2×#651412yes—
522C22C23×C22⋊D711yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—44no—
527C4—44no—
528C4—284no—
529C4—284no—
530C4#9049512yes—
531C4C2×#654812yes—
532C2—24no—
533C2—24no—
534C2—42no—
535C2#1174018714yesdirect product factor
536C2C2×#9049511yescharacteristic
537C2C22×#650511yescharacteristic
538C2C22×#651411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C7×C22⋊Q8C2
socleC23×C7C22×D4
prime radicalC25×C7D4
hypercenterC22×C22⋊Q8D7
hypocenterC7C2×#3195
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#90000C2semidirect
5. #11740187C2direct (N × H)
6. C2×#89822C2semidirect
7. C2×#89824C2semidirect
8. C2×#89815C2semidirect
9. C2×#89743C2semidirect
23. C22×#1144C22semidirect
33. #90000C22semidirect
34. C2×#6409C22semidirect
38. #89824C22semidirect
39. #89822C22semidirect
40. #89815C22semidirect
41. #89743C22semidirect
42. C2×#6384C22semidirect
43. C2×#6382C22semidirect
89. C2×#1144C23semidirect
106. #6409C23semidirect
109. #6384C23semidirect
110. #6382C23semidirect
447. C2×C7#3195semidirect
505. C7C2×#3195semidirect
535. C2#11740187direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C22×#64872
C2×#900002
C2×#899792
#117401872
C2×#898222
C2×#898242
C2×#898152
C2×#897432
C2×#31957