Finite Group Databasebeta

← C2×#11738462 #14225947

Subgroups — C2×#11738462 #14225947

Order 1792.

1758 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C2×#90269C211yesmaximal, characteristic
3C2×C7×#475C211yesmaximal, characteristic, complemented
4C2×#89866C211yesmaximal, characteristic, complemented
5C2×#89793C211yesmaximal, characteristic, complemented
6#11738462C2116yesmaximal, direct product factor
7C2×#89639C212yesmaximal
8C2×#89628C212yesmaximal, complemented
9C2×#89622C211yesmaximal, characteristic, complemented
10C2×#89607C211yesmaximal, characteristic
11C2×#89569C212yesmaximal, complemented
12C2×#89586C212yesmaximal
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6509—21no—
16C2×#6509C2211yescharacteristic
17C2×Q8×Dic7—21no—
18C2×#6507C2211yescharacteristic
19#90419—24no—
20#90419C2218yes—
21#90269—24no—
22#90269C2218yes—
23C2×#6460—21no—
24C2×#6460C2211yescharacteristic
25C2×#6459—21no—
26C2×#6455C2212yes—
27C2×#6453C2212yes—
28C2×#6452—21no—
29C2×#6451C2211yescharacteristic
30C2×#6450—21no—
31C2×#6435—21no—
32C2×C42×D7C2211yescharacteristic
33C2×#6432C2211yescharacteristic
34C2×C4×Dic14—21no—
35C2×C4×Dic14C2211yescharacteristic
36C7×#475C2218yescomplemented
37C2×C7×#106C2212yes—
38C2×C7×C4⋊C8C2211yescharacteristic
39C2×C7×#103C2212yes—
40C2×#6409C2212yes—
41C2×(C2×D7)⋊C8C2212yes—
42C2×#6392C2212yes—
43C2×#6390C2212yes—
44C2×#6389C2212yes—
45#89866C2218yescomplemented
46#89793C2218yescomplemented
47C2×#6384C2212yes—
48C2×#6382C2212yes—
49C2×#6381C2212yes—
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89639C22116yes—
52#89628C22116yescomplemented
53#89622C2218yescomplemented
54#89607C2218yes—
55#89586C22116yes—
56#89569C22116yescomplemented
57C2×#3237—71nomaximal, Sylow
58C22×C7×Q8—21no—
59C22×C7×Q8C2311yescharacteristic
60C22×C4×D7D411yescharacteristic
61C22×C4×D7D411yescharacteristic
62C22×C4×D7C2311yescharacteristic
63C22×Dic14—21no—
64C22×Dic14—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2312yes—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8C2314yes—
69C4×C7×Q8—22no—
70C4×C7×Q8C2314yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2312yes—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77#6509—28no—
78#6509—22no—
79#6509C2314yes—
80Q8×Dic7—28no—
81#6507—22no—
82#6507C2314yes—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7C2312yes—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—22no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—22no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8C2312yes—
103#6460—28no—
104#6460—22no—
105#6460C2314yes—
106#6459—28no—
107#6455—24no—
108#6455C2318yes—
109#6453—24no—
110#6453C2318yes—
111#6452—28no—
112#6451—22no—
113#6451C2314yes—
114#6450—28no—
115#6435—28no—
116C42×D7D414yes—
117C42×D7C2314yes—
118#6432—22no—
119#6432C2314yes—
120C4×Dic14—22no—
121C4×Dic14—28no—
122C4×Dic14C2314yes—
123C7×#106C2318yes—
124C7×C4⋊C8C2314yes—
125C7×#103C2318yes—
126#6409C2318yes—
127(C2×D7)⋊C8C2318yes—
128#6392C2318yes—
129#6390C2318yes—
130#6389C2318yes—
131#6384C2318yes—
132#6382C2318yes—
133#6381C2318yes—
134C4⋊(C7⋊C8)C2314yes—
135C2×#533—71no—
136C2×#533—71no—
137#3237—716no—
138C2×#501—72no—
139C2×#491—72no—
140C2×#485—72no—
141C2×#485—72no—
142C2×#477—71no—
143C2×#477—71no—
144C2×#475—71no—
145C2×#475D711yescharacteristic, complemented
146C2×#432—71no—
147C23×D7C2×D411yescharacteristic
148C2×C7×Q8—21no—
149C2×C7×Q8—24no—
150C2×C7×Q8—22no—
151C2×C7×Q8—22no—
152C2×C7×Q8—44no—
153C2×C7×Q8C2412yes—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C2×C4×D7—24no—
167C2×C4×D7—22no—
168C2×C4×D7—22no—
169C2×C4×D7—42no—
170C2×C4×D7C2×D412yes—
171C2×C4×D7C2×D414yes—
172C2×C4×D7C2×D414yes—
173C2×C4×D7C2412yes—
174C2×Dic14—22no—
175C2×Dic14—22no—
176C2×Dic14—21no—
177C2×Dic14—24no—
178C2×Dic14—24no—
179C2×Dic14—44no—
180C2×Dic14—44no—
181C2×Dic14C2412yes—
182C2×C7×C8—24no—
183C2×C7×C8C2414yes—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4C2414yes—
189C7×C4⋊C4C2412yes—
190C42×C7—24no—
191C42×C7C2×D412yes—
192C42×C7C2412yes—
193(C2×D7)⋊C4—24no—
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—28no—
196C4⋊Dic7—21no—
197C4⋊Dic7—22no—
198C4⋊Dic7—24no—
199C4⋊Dic7—24no—
200C4⋊Dic7—28no—
201C4⋊Dic7C2414yes—
202C4⋊Dic7C2412yes—
203Dic7⋊C4—24no—
204Dic7⋊C4—24no—
205Dic7⋊C4—24no—
206Dic7⋊C4—24no—
207Dic7⋊C4—24no—
208Dic7⋊C4—28no—
209C4×Dic7—24no—
210C4×Dic7—28no—
211C4×Dic7C2×D414yes—
212C4×Dic7C2×D412yes—
213C4×Dic7C2×D414yes—
214C4×Dic7C2412yes—
215C2×C7⋊C8—24no—
216C2×C7⋊C8C2414yes—
217#533—78no—
218#533—78no—
219#533—144no—
220#533—144no—
221C2×C4⋊Q8—71no—
222C2×C4⋊Q8—71no—
223C2×C4⋊Q8—71no—
224C2×C4⋊Q8C2×D711yescharacteristic
225C2×#125—72no—
226C2×#125—141no—
227C2×C22⋊Q8—71no—
228C2×C22⋊Q8—71no—
229C2×C22⋊Q8—141no—
230C2×C22⋊Q8—141no—
231C2×C22⋊Q8—141no—
232C2×C4×Q8—71no—
233C2×C4×Q8—141no—
234C2×C4×Q8—141no—
235C2×C4×Q8—141no—
236C2×C4×Q8C2×D711yescharacteristic
237C2×#117—72no—
238C2×#117—141no—
239C22×C42—71no—
240#501—716no—
241#491—716no—
242#485—716no—
243#485—716no—
244#477—78no—
245#477—78no—
246#475—78no—
247#475C2×D718yescomplemented
248#432—78no—
249C2×#107—72no—
250C2×#107—72no—
251C2×#106—72no—
252C2×#106C2×D712yes—
253C2×C4⋊C8—71no—
254C2×C4⋊C8C2×D711yescharacteristic
255C2×#103—72no—
256C2×#103—72no—
257C2×#103—72no—
258C2×#103C2×D712yes—
259C2×C22⋊C8—72no—
260C2×#97—72no—
261C23×C7C22×D411yescharacteristic
262C22×D7—22no—
263C22×D7—22no—
264C22×D7—22no—
265C22×D7C22×D412yes—
266C7×Q8—22no—
267C7×Q8—22no—
268C7×Q8—42no—
269C7×Q8—48no—
270C2×C4×C7—22no—
271C2×C4×C7—24no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—42no—
275C2×C4×C7—44no—
276C2×C4×C7—42no—
277C2×C4×C7#13714yes—
278C2×C4×C7C22×D412yes—
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D412yes—
281C2×C4×C72−1+412yes—
282C2×C4×C72−1+412yes—
283C2×C4×C7C2511yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—24no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—44no—
293C2×Dic7C22×D412yes—
294C2×Dic7C22×D412yes—
295C2×Dic7C22×D412yes—
296C4×D7—24no—
297C4×D7—28no—
298C4×D7—22no—
299C4×D7—22no—
300C4×D7—44no—
301Dic14—22no—
302Dic14—22no—
303Dic14—48no—
304Dic14—48no—
305Dic14—42no—
306C7×C8—24no—
307C7⋊C8—24no—
308C22×Q8—21no—
309C22×Q8—71no—
310C22×Q8—141no—
311C22×Q8—141no—
312C22×Q8C22×D711yescharacteristic
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—71no—
316C22×C8—72no—
317C22×C8C22×D712yes—
318C4⋊Q8—22no—
319C4⋊Q8—74no—
320C4⋊Q8—74no—
321C4⋊Q8—74no—
322C4⋊Q8—142no—
323C4⋊Q8—142no—
324C4⋊Q8—142no—
325C4⋊Q8C22×D714yes—
326#125—78no—
327#125—144no—
328#125—148no—
329C22⋊Q8—74no—
330C22⋊Q8—74no—
331C22⋊Q8—142no—
332C22⋊Q8—148no—
333C22⋊Q8—148no—
334C22⋊Q8—142no—
335C22⋊Q8—148no—
336C4×Q8—22no—
337C4×Q8—74no—
338C4×Q8—148no—
339C4×Q8—148no—
340C4×Q8—148no—
341C4×Q8—142no—
342C4×Q8C22×D714yes—
343#117—78no—
344#117—148no—
345#117—144no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—72no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4C22×D712yes—
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C22⋊C4—142no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C42—21no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—74no—
369C2×C42—74no—
370C2×C42—142no—
371C2×C42—141no—
372C2×C42C22×D711yescharacteristic
373#107—78no—
374#107—78no—
375#106—78no—
376#106C22×D718yes—
377C4⋊C8—74no—
378C4⋊C8C22×D714yes—
379#103—78no—
380#103—78no—
381#103—78no—
382#103C22×D718yes—
383C22⋊C8—78no—
384#97—78no—
385C22×C7#53612yes—
386C22×C7C2×#13712yes—
387C22×C7#57512yes—
388C22×C7C23×D411yescharacteristic
389C2×D7—22no—
390C2×D7—22no—
391C2×D7—22no—
392C2×D7—44no—
393C4×C7—22no—
394C4×C7—42no—
395C4×C7—44no—
396C4×C7—42no—
397C4×C7#53612yes—
398C4×C7C2×#13714yes—
399C4×C7C2×2−1+411yescharacteristic
400C4×C7C2×2−1+411yescharacteristic
401Dic7—22no—
402Dic7—22no—
403Dic7—22no—
404Dic7—42no—
405Dic7—42no—
406Dic7—44no—
407C24—71no—
408C2×Q8—21no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—72no—
414C2×Q8—144no—
415C2×Q8—144no—
416C2×Q8—141no—
417C2×Q8—142no—
418C2×Q8—142no—
419C2×Q8—284no—
420C2×Q8—284no—
421C2×Q8C23×D712yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—72no—
429C22×C4—74no—
430C22×C4—72no—
431C22×C4—74no—
432C22×C4—144no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—282no—
439C22×C4D4×D711yescharacteristic
440C22×C4D4×D711yescharacteristic
441C22×C4C23×D711yescharacteristic
442C2×C8—24no—
443C2×C8—74no—
444C2×C8—144no—
445C2×C8C23×D714yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—74no—
451C4⋊C4—72no—
452C4⋊C4—144no—
453C4⋊C4—144no—
454C4⋊C4—144no—
455C4⋊C4—148no—
456C4⋊C4—144no—
457C4⋊C4—144no—
458C4⋊C4—148no—
459C4⋊C4—144no—
460C4⋊C4—142no—
461C4⋊C4—141no—
462C4⋊C4—144no—
463C4⋊C4C23×D714yes—
464C4⋊C4C23×D712yes—
465C22⋊C4—148no—
466C22⋊C4—144no—
467C22⋊C4—144no—
468C42—24no—
469C42—74no—
470C42—74no—
471C42—72no—
472C42—72no—
473C42—148no—
474C42—144no—
475C42D4×D712yes—
476C42C23×D712yes—
477C2×C7#323714yescomplemented
478C2×C7C2×#53611yescharacteristic
479C2×C7C22×#13711yescharacteristic
480C2×C7C2×#57511yescharacteristic
481D7—42no—
482C23—72no—
483C23—142no—
484C23—142no—
485C23—142no—
486C23C2×D4×D711yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—48no—
490Q8—42no—
491Q8—142no—
492Q8—142no—
493Q8—288no—
494Q8—288no—
495Q8—282no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—72no—
504C2×C4—72no—
505C2×C4—72no—
506C2×C4—142no—
507C2×C4—142no—
508C2×C4—142no—
509C2×C4—148no—
510C2×C4—144no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—144no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—284no—
517C2×C4—284no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4#647814yes—
521C2×C4C2×D4×D712yes—
522C2×C4C2×D4×D712yes—
523C2×C4C2×D4×D711yescharacteristic
524C2×C4#655112yes—
525C2×C4#655412yes—
526C2×C4C24×D711yescharacteristic
527C8—24no—
528C8—144no—
529C7C2×#323711yescharacteristic, complemented, Sylow
530C22—142no—
531C22—142no—
532C22—142no—
533C22—284no—
534C22#9031912yes—
535C22C2×#647812yes—
536C22#9046812yes—
537C22C22×D4×D711yescharacteristic
538C4—22no—
539C4—42no—
540C4—42no—
541C4—44no—
542C4—142no—
543C4—142no—
544C4—142no—
545C4—284no—
546C4—282no—
547C4—282no—
548C4#9031912yes—
549C4C2×#647814yes—
550C4C2×#655111yescharacteristic
551C4C2×#655411yescharacteristic
552C2—282no—
553C2#1173846214yesdirect product factor
554C2C2×#9031911yescharacteristic
555C2C22×#647811yescharacteristic
556C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#475C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#475D7
hypocenterC7C2×#3237
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. C2×C7×#475C2semidirect
4. C2×#89866C2semidirect
5. C2×#89793C2semidirect
6. #11738462C2direct (N × H)
8. C2×#89628C2semidirect
9. C2×#89622C2semidirect
11. C2×#89569C2semidirect
36. C7×#475C22semidirect
45. #89866C22semidirect
46. #89793C22semidirect
52. #89628C22semidirect
53. #89622C22semidirect
56. #89569C22semidirect
145. C2×#475D7semidirect
247. #475C2×D7semidirect
477. C2×C7#3237semidirect
529. C7C2×#3237semidirect
553. C2#11738462direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904192
C2×#902692
C2×C7×#4752
C2×#898662
C2×#897932
#117384622
C2×#896392
C2×#896282
C2×#896222
C2×#896072
C2×#895692
C2×#895862
C2×#32377