Finite Group Databasebeta

← C2×#11738866 #14225921

Subgroups — C2×#11738866 #14225921

Order 1792.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90315C211yesmaximal, characteristic
2C22×#6451C211yesmaximal, characteristic, complemented
3C2×C7×#483C211yesmaximal, characteristic, complemented
4C2×#89815C211yesmaximal, characteristic, complemented
5C2×#89739C211yesmaximal, characteristic
6#11738866C2116yesmaximal, direct product factor
7C2×#89642C212yesmaximal, complemented
8C2×#89628C212yesmaximal, complemented
9C2×#89567C212yesmaximal, complemented
10C2×#89566C212yesmaximal, complemented
11C2×#89508C211yesmaximal, characteristic
12C2×#89491C211yesmaximal, characteristic, complemented
13C23×Dic14—21no—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6510C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×#6493C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4—21no—
21C22×C4×Dic7C2211yescharacteristic
22#90315—24no—
23#90315C2218yes—
24C2×#6461—21no—
25C2×#6456C2211yescharacteristic
26C2×#6455—21no—
27C2×#6453C2211yescharacteristic
28C2×#6451—24no—
29C2×#6451—21no—
30C2×#6451—21no—
31C2×#6451C2212yes—
32C2×#6451C2212yes—
33C2×#6451C2218yescomplemented
34C2×#6449—21no—
35C2×(C22⋊D7)⋊C4—21no—
36C7×#483C2218yescomplemented
37C2×C7×#106C2212yes—
38C2×C7×#103C2212yes—
39C2×C7×C22⋊C8C2211yescharacteristic
40C2×#6409C2212yes—
41C2×C22⋊(C7⋊C8)C2211yescharacteristic
42C2×#6395C2212yes—
43C2×#6392C2212yes—
44C2×#6389C2212yes—
45C2×Dic7⋊C8C2212yes—
46#89815C2218yescomplemented
47#89739C2218yes—
48C2×#6383C2212yes—
49C2×#6382C2212yes—
50C2×#6381C2212yes—
51#89642C22116yescomplemented
52#89628C22116yescomplemented
53#89567C22116yescomplemented
54#89566C22116yescomplemented
55#89508C2218yes—
56#89491C2218yescomplemented
57C2×#3238—71nomaximal, Sylow
58C22×C7×Q8C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—21no—
62C23×Dic7—21no—
63C23×Dic7D411yescharacteristic
64C23×Dic7D411yescharacteristic
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C22×Dic14—22no—
68C22×Dic14—22no—
69C22×Dic14—28no—
70C22×Dic14—48no—
71C22×C7×C8C2312yes—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8C2314yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4C2314yes—
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81#6510—22no—
82#6510C2314yes—
83Q8×Dic7—22no—
84Q8×Dic7C2314yes—
85#6493—22no—
86#6493C2314yes—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2314yes—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—22no—
100C2×Dic7⋊C4—22no—
101C2×Dic7⋊C4—28no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7D411yescharacteristic
104C2×C4×Dic7D411yescharacteristic
105C2×C4×Dic7D414yes—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2314yes—
109C22×C7⋊C8C2312yes—
110#6461—28no—
111#6456—22no—
112#6456C2314yes—
113#6455—28no—
114#6453—22no—
115#6453C2314yes—
116#6451—24no—
117#6451—24no—
118#6451—28no—
119#6451—28no—
120#6451C2318yes—
121#6451C2318yes—
122#6449—28no—
123(C22⋊D7)⋊C4—28no—
124C7×#106C2318yes—
125C7×#103C2318yes—
126C7×C22⋊C8C2314yes—
127#6409C2318yes—
128C22⋊(C7⋊C8)C2314yes—
129#6395C2318yes—
130#6392C2318yes—
131#6389C2318yes—
132Dic7⋊C8C2318yes—
133#6383C2318yes—
134#6382C2318yes—
135#6381C2318yes—
136C22×C4⋊Q8—71no—
137C2×#529—71no—
138#3238—716no—
139C2×#501—72no—
140C2×#489—72no—
141C2×#483—71no—
142C2×#483D711yescharacteristic, complemented
143C2×#480—71no—
144C2×#480—71no—
145C2×#477—72no—
146C2×#476—72no—
147C2×#432—71no—
148C24×C7C2×D411yescharacteristic
149C23×D7—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—24no—
152C2×C7×Q8C2412yes—
153C22×C4×C7—24no—
154C22×C4×C7—21no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—21no—
158C22×C4×C7—44no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2412yes—
162C22×C4×C7C2411yescharacteristic
163C2×C22⋊D7—24no—
164C2×C22⋊D7—44no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—21no—
168C22×Dic7—24no—
169C22×Dic7—22no—
170C22×Dic7—44no—
171C22×Dic7C2×D414yes—
172C22×Dic7C2×D414yes—
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C2×D28—24no—
178C2×D28—21no—
179C2×D28C2412yes—
180C2×C4×D7—24no—
181C2×C4×D7—44no—
182C2×Dic14—28no—
183C2×Dic14—28no—
184C2×Dic14—48no—
185C2×Dic14—48no—
186C2×Dic14—432no—
187C2×C7×C8—24no—
188C2×C7×C8C2414yes—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4C2414yes—
194C7×C4⋊C4C2412yes—
195C7×C22⋊C4—24no—
196(C2×D7)⋊C4—24no—
197(C2×D7)⋊C4—24no—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—24no—
200C4⋊Dic7—24no—
201C4⋊Dic7—22no—
202C4⋊Dic7—21no—
203C4⋊Dic7—24no—
204C4⋊Dic7—24no—
205C4⋊Dic7C2414yes—
206C4⋊Dic7C2412yes—
207Dic7⋊C4—24no—
208Dic7⋊C4—28no—
209Dic7⋊C4—28no—
210C4×Dic7—24no—
211C4×Dic7C2×D412yes—
212C4×Dic7C2×D414yes—
213C4×Dic7C2×D412yes—
214C4×Dic7C2×D414yes—
215C4×Dic7C2412yes—
216C4×Dic7C2412yes—
217C2×C7⋊C8—24no—
218C2×C7⋊C8C2414yes—
219C23×Q8—141no—
220C2×C4⋊Q8—72no—
221C2×C4⋊Q8—72no—
222C2×C4⋊Q8—78no—
223C2×C4⋊Q8—141no—
224C2×C4⋊Q8—141no—
225C2×C4⋊Q8—144no—
226#529—78no—
227#529—144no—
228C2×#126—141no—
229C2×#125—71no—
230C2×#124—71no—
231C2×#123—141no—
232C2×C22⋊Q8C2×D711yescharacteristic
233C2×#121—71no—
234C2×C4×Q8—71no—
235C2×C4×D4—71no—
236C2×C4×D4—141no—
237C2×#117—141no—
238C22×C4⋊C4—71no—
239C22×C4⋊C4—141no—
240C22×C4⋊C4C2×D711yescharacteristic
241C22×C42—71no—
242#501—716no—
243#489—716no—
244#483—78no—
245#483C2×D718yescomplemented
246#480—78no—
247#480—78no—
248#477—716no—
249#476—716no—
250#432—78no—
251C2×#107—72no—
252C2×#107—72no—
253C2×#106—72no—
254C2×#106C2×D712yes—
255C2×C4⋊C8—72no—
256C2×#103—72no—
257C2×#103C2×D712yes—
258C2×#102—72no—
259C2×#102—72no—
260C2×C22⋊C8—71no—
261C2×C22⋊C8C2×D711yescharacteristic
262C2×#97—72no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7AGL(1,ℤ/8)14yes—
266C23×C7C22×D412yes—
267C23×C7C22×D411yescharacteristic
268C22×D7—22no—
269C22×D7—42no—
270C22×D7—42no—
271C22×D7—42no—
272C7×Q8—24no—
273C7×Q8—42no—
274C2×C4×C7—22no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—42no—
281C2×C4×C7—48no—
282C2×C4×C7—44no—
283C2×C4×C7—42no—
284C2×C4×C7C22×D412yes—
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C72−1+414yes—
287C2×C4×C7C2511yescharacteristic
288C22⋊D7—48no—
289C2×Dic7—28no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—28no—
294C2×Dic7—22no—
295C2×Dic7—24no—
296C2×Dic7—48no—
297C2×Dic7—44no—
298C2×Dic7—42no—
299C2×Dic7#13714yes—
300C2×Dic7C22×D412yes—
301C2×Dic7C22×D412yes—
302C2×Dic7C22×D412yes—
303C2×Dic7C22×D412yes—
304D28—24no—
305D28—42no—
306C4×D7—48no—
307Dic14—416no—
308Dic14—416no—
309C7×C8—24no—
310C7⋊C8—24no—
311C22×Q8—148no—
312C22×Q8—142no—
313C22×Q8—142no—
314C22×Q8—288no—
315C22×Q8C22×D711yescharacteristic
316C22×D4—71no—
317C22×D4—141no—
318C23×C4—21no—
319C23×C4—71no—
320C23×C4—71no—
321C23×C4—141no—
322C23×C4—141no—
323C23×C4C22×D711yescharacteristic
324C22×C8—72no—
325C22×C8C22×D712yes—
326C4⋊Q8—78no—
327C4⋊Q8—78no—
328C4⋊Q8—148no—
329C4⋊Q8—144no—
330C4⋊Q8—144no—
331C4⋊Q8—148no—
332#126—148no—
333#125—74no—
334#125—142no—
335#124—74no—
336#124—142no—
337#123—148no—
338C22⋊Q8—22no—
339C22⋊Q8C22×D714yes—
340#121—74no—
341#121—142no—
342C4×Q8—74no—
343C4×Q8—142no—
344C4×D4—74no—
345C4×D4—142no—
346C4×D4—148no—
347#117—148no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—72no—
352C2×C4⋊C4—74no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—148no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—142no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4C22×D712yes—
363C2×C4⋊C4C22×D714yes—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—74no—
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—74no—
376C2×C42—141no—
377#107—78no—
378#107—78no—
379#106—78no—
380#106C22×D718yes—
381C4⋊C8—78no—
382#103—78no—
383#103C22×D718yes—
384#102—78no—
385#102—78no—
386C22⋊C8—74no—
387C22⋊C8C22×D714yes—
388#97—78no—
389C22×C7—22no—
390C22×C7—28no—
391C22×C7—22no—
392C22×C7#53612yes—
393C22×C7C2×AGL(1,ℤ/8)14yes—
394C22×C7C2×AGL(1,ℤ/8)12yes—
395C22×C7C2×#13712yes—
396C22×C7C23×D411yescharacteristic
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—84no—
401C4×C7—22no—
402C4×C7—42no—
403C4×C7—44no—
404C4×C7—42no—
405C4×C7#53612yes—
406C4×C7C2×2−1+412yes—
407Dic7—22no—
408Dic7—22no—
409Dic7—22no—
410Dic7—44no—
411Dic7—42no—
412Dic7C2×#13714yes—
413C24—141no—
414C24D4×D711yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8—148no—
418C2×Q8—148no—
419C2×Q8—2832no—
420C2×Q8—288no—
421C2×Q8—288no—
422C2×Q8C23×D712yes—
423C2×D4—72no—
424C2×D4—144no—
425C2×D4—141no—
426C2×D4—144no—
427C2×D4—284no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—71no—
435C22×C4—74no—
436C22×C4—74no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—144no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—144no—
444C22×C4—141no—
445C22×C4—142no—
446C22×C4—284no—
447C22×C4—284no—
448C22×C4D4×D712yes—
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D712yes—
451C22×C4C23×D711yescharacteristic
452C2×C8—24no—
453C2×C8—74no—
454C2×C8—144no—
455C2×C8C23×D714yes—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—74no—
461C4⋊C4—72no—
462C4⋊C4—148no—
463C4⋊C4—148no—
464C4⋊C4—144no—
465C4⋊C4—144no—
466C4⋊C4—144no—
467C4⋊C4—144no—
468C4⋊C4—142no—
469C4⋊C4—141no—
470C4⋊C4C23×D714yes—
471C4⋊C4C23×D712yes—
472C22⋊C4—24no—
473C22⋊C4—144no—
474C22⋊C4—144no—
475C22⋊C4—144no—
476C22⋊C4—144no—
477C42—74no—
478C42—72no—
479C42—72no—
480C42—74no—
481C42—72no—
482C42—72no—
483C42—144no—
484C2×C7—24no—
485C2×C7#323814yescomplemented
486C2×C7C2×#53611yescharacteristic
487C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
488C2×C7C22×#13711yescharacteristic
489D7—82no—
490C23—22no—
491C23—22no—
492C23—142no—
493C23—282no—
494C23—282no—
495C23—282no—
496C23#647714yes—
497C23C2×D4×D712yes—
498C23C2×D4×D711yescharacteristic
499Q8—24no—
500Q8—42no—
501Q8—2816no—
502Q8—2816no—
503D4—144no—
504D4—288no—
505D4—282no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—74no—
517C2×C4—72no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—72no—
521C2×C4—148no—
522C2×C4—148no—
523C2×C4—142no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—144no—
528C2×C4—288no—
529C2×C4—288no—
530C2×C4—284no—
531C2×C4—282no—
532C2×C4C2×D4×D712yes—
533C2×C4C2×D4×D711yescharacteristic
534C2×C4#655414yes—
535C2×C4C24×D711yescharacteristic
536C8—24no—
537C8—144no—
538C7C2×#323811yescharacteristic, complemented, Sylow
539C22—22no—
540C22—22no—
541C22—28no—
542C22—282no—
543C22—282no—
544C22—282no—
545C22—564no—
546C22#9033912yes—
547C22C2×#647714yes—
548C22C2×#647712yes—
549C22D7×#13712yes—
550C22C22×D4×D711yescharacteristic
551C4—22no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—74no—
556C4—142no—
557C4—142no—
558C4—142no—
559C4—284no—
560C4—282no—
561C4#9033912yes—
562C4C2×#655412yes—
563C2—24no—
564C2—562no—
565C2#1173886614yesdirect product factor
566C2C2×#9033911yescharacteristic
567C2C22×#647711yescharacteristic
568C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#483C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC2×#483D7
hypocenterC7C2×#3238
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
2. C22×#6451C2semidirect
3. C2×C7×#483C2semidirect
4. C2×#89815C2semidirect
6. #11738866C2direct (N × H)
7. C2×#89642C2semidirect
8. C2×#89628C2semidirect
9. C2×#89567C2semidirect
10. C2×#89566C2semidirect
12. C2×#89491C2semidirect
33. C2×#6451C22semidirect
36. C7×#483C22semidirect
46. #89815C22semidirect
51. #89642C22semidirect
52. #89628C22semidirect
53. #89567C22semidirect
54. #89566C22semidirect
56. #89491C22semidirect
142. C2×#483D7semidirect
245. #483C2×D7semidirect
485. C2×C7#3238semidirect
538. C7C2×#3238semidirect
565. C2#11738866direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#903152
C22×#64512
C2×C7×#4832
C2×#898152
C2×#897392
#117388662
C2×#896422
C2×#896282
C2×#895672
C2×#895662
C2×#895082
C2×#894912
C2×#32387