Finite Group Databasebeta

← C2×#11737932 #14226674

Subgroups — C2×#11737932 #14226674

Order 1792.

1866 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C2×#90269C211yesmaximal, characteristic
3C2×C7×#462C211yesmaximal, characteristic, complemented
4C2×#89961C212yesmaximal
5C2×#89954C212yesmaximal, complemented
6C2×#89862C211yesmaximal, characteristic, complemented
7C2×#89792C211yesmaximal, characteristic, complemented
8#11737932C2116yesmaximal, direct product factor
9C2×#89623C211yesmaximal, characteristic, complemented
10C2×#89609C211yesmaximal, characteristic
11C2×#89588C212yesmaximal
12C2×#89563C212yesmaximal, complemented
13C22×C7×Q16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C4×C7×Q8C2211yescharacteristic
16C2×#6509C2212yes—
17C2×#6509C2211yescharacteristic
18C2×Q8×Dic7C2212yes—
19C2×#6507C2211yescharacteristic
20C22×#1144C2212yes—
21C22×#1131C2212yes—
22#90419—24no—
23#90419C2218yes—
24#90269—24no—
25#90269C2218yes—
26C2×#6460—21no—
27C2×#6460C2211yescharacteristic
28C2×#6459—21no—
29C2×#6455—21no—
30C2×#6453—21no—
31C2×#6452—21no—
32C2×#6451C2211yescharacteristic
33C2×#6450—21no—
34C2×#6435—21no—
35C2×C42×D7C2211yescharacteristic
36C2×#6432C2211yescharacteristic
37C2×C4×Dic14—21no—
38C2×C4×Dic14C2211yescharacteristic
39C7×#462C2218yescomplemented
40C2×C7×C4⋊C8C2211yescharacteristic
41C2×C7×#103C2212yes—
42C2×#6409C2212yes—
43C2×#6409C2212yes—
44#89961C22116yes—
45#89954C22116yescomplemented
46C2×#6391C2212yes—
47C2×#6390C2212yes—
48C2×Dic7⋊C8C2212yes—
49#89862C2218yescomplemented
50#89792C2218yescomplemented
51C2×#6382C2212yes—
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53#89623C2218yescomplemented
54#89609C2218yes—
55#89588C22116yes—
56#89563C22116yescomplemented
57C2×#3255—71nomaximal, Sylow
58C22×C7×Q8C2312yes—
59C22×C7×Q8C2311yescharacteristic
60C22×C4×D7D411yescharacteristic
61C22×C4×D7D411yescharacteristic
62C22×C4×D7C2311yescharacteristic
63C22×Dic14—21no—
64C22×Dic14—21no—
65C22×Dic14C2311yescharacteristic
66C2×C7×Q16—44no—
67C2×C7×Q16C2318yes—
68C22×C7×C8C2312yes—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8C2314yes—
71C4×C7×Q8—22no—
72C4×C7×Q8C2314yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79#6509—24no—
80#6509—22no—
81#6509C2318yes—
82#6509C2314yes—
83Q8×Dic7—24no—
84Q8×Dic7C2318yes—
85#6507—22no—
86#6507C2314yes—
87C2×#1144—44no—
88C2×#1144C2318yes—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—22no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—22no—
95C2×C4⋊Dic7C2312yes—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—22no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—22no—
105C2×C4×Dic7D411yescharacteristic
106C2×C4×Dic7D411yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C22×C7⋊C8C2312yes—
109C2×#1131—44no—
110C2×#1131C2318yes—
111#6460—28no—
112#6460—22no—
113#6460C2314yes—
114#6459—28no—
115#6455—28no—
116#6453—28no—
117#6452—28no—
118#6451—22no—
119#6451C2314yes—
120#6450—28no—
121#6435—28no—
122C42×D7D414yes—
123C42×D7C2314yes—
124#6432—22no—
125#6432C2314yes—
126C4×Dic14—22no—
127C4×Dic14—28no—
128C4×Dic14C2314yes—
129C7×C4⋊C8C2314yes—
130C7×#103C2318yes—
131#6409C2318yes—
132#6409C2318yes—
133#6391C2318yes—
134#6390C2318yes—
135Dic7⋊C8C2318yes—
136#6382C2318yes—
137C4⋊(C7⋊C8)C2314yes—
138C2×#533—71no—
139C2×#533—71no—
140#3255—716no—
141C2×#475—71no—
142C2×#475—72no—
143C2×#475—71no—
144C2×#470—72no—
145C2×#470—72no—
146C2×#462—72no—
147C2×#462—71no—
148C2×#462D711yescharacteristic, complemented
149C2×#423—71no—
150C23×D7C2×D411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8—22no—
153C2×C7×Q8—24no—
154C2×C7×Q8—22no—
155C2×C7×Q8—24no—
156C2×C7×Q8—22no—
157C2×C7×Q8C2412yes—
158C2×C7×Q8C2414yes—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C2×C4×D7—24no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—24no—
175C2×C4×D7C2×D412yes—
176C2×C4×D7C2×D414yes—
177C2×C4×D7C2×D414yes—
178C2×C4×D7C2412yes—
179C2×Dic14—22no—
180C2×Dic14—22no—
181C2×Dic14—21no—
182C2×Dic14—24no—
183C2×Dic14—24no—
184C2×Dic14—44no—
185C2×Dic14—44no—
186C2×Dic14C2412yes—
187C7×Q16—48no—
188C2×C7×C8—24no—
189C2×C7×C8C2414yes—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4C2412yes—
195C42×C7—24no—
196C42×C7C2×D412yes—
197C42×C7C2412yes—
198#1144—48no—
199(C2×D7)⋊C4—24no—
200(C2×D7)⋊C4—24no—
201(C2×D7)⋊C4—28no—
202C4⋊Dic7—21no—
203C4⋊Dic7—22no—
204C4⋊Dic7—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—28no—
207C4⋊Dic7C2412yes—
208C4⋊Dic7C2414yes—
209Dic7⋊C4—24no—
210Dic7⋊C4—24no—
211Dic7⋊C4—24no—
212Dic7⋊C4—24no—
213Dic7⋊C4—24no—
214Dic7⋊C4—28no—
215C4×Dic7—24no—
216C4×Dic7—28no—
217C4×Dic7C2×D414yes—
218C4×Dic7C2×D412yes—
219C4×Dic7C2×D414yes—
220C4×Dic7C2412yes—
221C2×C7⋊C8—24no—
222C2×C7⋊C8C2414yes—
223#1131—48no—
224C22×Q16—72no—
225C22×Q16C2×D712yes—
226C22×M16—72no—
227#533—78no—
228#533—78no—
229#533—144no—
230#533—144no—
231C2×C4⋊Q8—71no—
232C2×C4⋊Q8—71no—
233C2×C4⋊Q8—71no—
234C2×C4⋊Q8C2×D711yescharacteristic
235C2×#125—141no—
236C2×#125—141no—
237C2×C22⋊Q8—72no—
238C2×C22⋊Q8—71no—
239C2×C22⋊Q8—71no—
240C2×C22⋊Q8—141no—
241C2×C22⋊Q8—141no—
242C2×C4×Q8—72no—
243C2×C4×Q8—71no—
244C2×C4×Q8—141no—
245C2×C4×Q8—141no—
246C2×C4×Q8C2×D711yescharacteristic
247C2×#117—141no—
248C2×#117—141no—
249C22×C42—71no—
250#475—78no—
251#475—78no—
252#475—716no—
253#470—716no—
254#470—716no—
255#462—716no—
256#462—78no—
257#462C2×D718yescomplemented
258C2×#106—72no—
259C2×#106—72no—
260#423—78no—
261C2×C4⋊C8—72no—
262C2×C4⋊C8—71no—
263C2×C4⋊C8C2×D711yescharacteristic
264C2×#103—72no—
265C2×#103—72no—
266C2×#103—72no—
267C2×#103C2×D712yes—
268C23×C7C22×D411yescharacteristic
269C22×D7—22no—
270C22×D7—22no—
271C22×D7—22no—
272C22×D7C22×D412yes—
273C7×Q8—24no—
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8—24no—
277C7×Q8—42no—
278C7×Q8—44no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—44no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7#13714yes—
287C2×C4×C7C22×D412yes—
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D412yes—
290C2×C4×C7H(5,𝔽2)12yes—
291C2×C4×C72−1+412yes—
292C2×C4×C7C2511yescharacteristic
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—24no—
298C2×Dic7—42no—
299C2×Dic7—44no—
300C2×Dic7—42no—
301C2×Dic7#13714yes—
302C2×Dic7C22×D412yes—
303C2×Dic7C22×D412yes—
304C2×Dic7C22×D412yes—
305C4×D7—24no—
306C4×D7—28no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—28no—
310Dic14—22no—
311Dic14—22no—
312Dic14—42no—
313Dic14—48no—
314Dic14—48no—
315C7×C8—24no—
316C7⋊C8—24no—
317C22×Q8—71no—
318C22×Q8—141no—
319C22×Q8—141no—
320C22×Q8C22×D712yes—
321C22×Q8C22×D711yescharacteristic
322C23×C4—71no—
323C23×C4—71no—
324C23×C4—71no—
325C2×Q16—44no—
326C2×Q16—78no—
327C2×Q16—284no—
328C2×Q16C22×D718yes—
329C2×M16—78no—
330C2×M16—284no—
331C22×C8—72no—
332C22×C8C22×D712yes—
333C4⋊Q8—22no—
334C4⋊Q8—74no—
335C4⋊Q8—74no—
336C4⋊Q8—74no—
337C4⋊Q8—142no—
338C4⋊Q8—142no—
339C4⋊Q8—142no—
340C4⋊Q8C22×D714yes—
341#125—148no—
342#125—148no—
343C22⋊Q8—74no—
344C22⋊Q8—78no—
345C22⋊Q8—74no—
346C22⋊Q8—142no—
347C22⋊Q8—148no—
348C22⋊Q8—148no—
349C22⋊Q8—142no—
350C22⋊Q8—144no—
351C4×Q8—22no—
352C4×Q8—78no—
353C4×Q8—74no—
354C4×Q8—148no—
355C4×Q8—148no—
356C4×Q8—144no—
357C4×Q8—142no—
358C4×Q8C22×D714yes—
359#117—148no—
360#117—148no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—72no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—142no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—142no—
379C2×C42—21no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—71no—
383C2×C42—74no—
384C2×C42—74no—
385C2×C42—142no—
386C2×C42—141no—
387C2×C42C22×D711yescharacteristic
388#106—78no—
389#106—78no—
390C4⋊C8—78no—
391C4⋊C8—74no—
392C4⋊C8C22×D714yes—
393#103—78no—
394#103—78no—
395#103—78no—
396#103C22×D718yes—
397C22×C7#53712yes—
398C22×C7C2×#13712yes—
399C22×C7C2×#13712yes—
400C22×C7C23×D411yescharacteristic
401C2×D7—22no—
402C2×D7—22no—
403C2×D7—22no—
404C2×D7—44no—
405C4×C7—22no—
406C4×C7—44no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7#53712yes—
410C4×C7C2×#13714yes—
411C4×C7C2×H(5,𝔽2)11yescharacteristic
412C4×C7C2×2−1+411yescharacteristic
413Dic7—22no—
414Dic7—22no—
415Dic7—42no—
416Dic7—42no—
417Dic7—44no—
418Dic7C2×#13714yes—
419C24—71no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—72no—
427C2×Q8—144no—
428C2×Q8—144no—
429C2×Q8—141no—
430C2×Q8—142no—
431C2×Q8—142no—
432C2×Q8—284no—
433C2×Q8—284no—
434C2×Q8C23×D712yes—
435C2×Q8C23×D714yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—144no—
447C22×C4—144no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4D4×D711yescharacteristic
454C22×C4D4×D711yescharacteristic
455C22×C4C23×D711yescharacteristic
456Q16—48no—
457Q16—288no—
458M16—288no—
459C2×C8—24no—
460C2×C8—74no—
461C2×C8—144no—
462C2×C8C23×D714yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—72no—
468C4⋊C4—74no—
469C4⋊C4—144no—
470C4⋊C4—144no—
471C4⋊C4—144no—
472C4⋊C4—148no—
473C4⋊C4—144no—
474C4⋊C4—148no—
475C4⋊C4—144no—
476C4⋊C4—144no—
477C4⋊C4—141no—
478C4⋊C4—142no—
479C4⋊C4—144no—
480C4⋊C4C23×D712yes—
481C22⋊C4—144no—
482C22⋊C4—144no—
483C22⋊C4—148no—
484C42—24no—
485C42—74no—
486C42—72no—
487C42—74no—
488C42—72no—
489C42—148no—
490C42—144no—
491C42D4×D712yes—
492C42C23×D712yes—
493C2×C7#325514yescomplemented
494C2×C7C2×#53711yescharacteristic
495C2×C7C22×#13711yescharacteristic
496C2×C7C22×#13711yescharacteristic
497D7—42no—
498C23—72no—
499C23—142no—
500C23—142no—
501C23—142no—
502C23C2×D4×D711yescharacteristic
503Q8—22no—
504Q8—24no—
505Q8—22no—
506Q8—24no—
507Q8—42no—
508Q8—44no—
509Q8—142no—
510Q8—142no—
511Q8—288no—
512Q8—288no—
513Q8—282no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—74no—
522C2×C4—72no—
523C2×C4—72no—
524C2×C4—72no—
525C2×C4—148no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—148no—
529C2×C4—144no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—144no—
534C2×C4—142no—
535C2×C4—284no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4#648114yes—
539C2×C4C2×D4×D712yes—
540C2×C4C2×D4×D712yes—
541C2×C4C2×D4×D711yescharacteristic
542C2×C4#654812yes—
543C2×C4#655412yes—
544C2×C4C24×D711yescharacteristic
545C8—24no—
546C8—144no—
547C7C2×#325511yescharacteristic, complemented, Sylow
548C22—142no—
549C22—142no—
550C22—142no—
551C22—284no—
552C22#9029912yes—
553C22C2×#648112yes—
554C22D7×#13712yes—
555C22C22×D4×D711yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—74no—
561C4—142no—
562C4—142no—
563C4—282no—
564C4—282no—
565C4—284no—
566C4#9029912yes—
567C4C2×#648114yes—
568C4C2×#654811yescharacteristic
569C4C2×#655411yescharacteristic
570C2—282no—
571C2#1173793214yesdirect product factor
572C2C2×#9029911yescharacteristic
573C2C22×#648111yescharacteristic
574C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#462C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#462D7
hypocenterC7C2×#3255
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. C2×C7×#462C2semidirect
5. C2×#89954C2semidirect
6. C2×#89862C2semidirect
7. C2×#89792C2semidirect
8. #11737932C2direct (N × H)
9. C2×#89623C2semidirect
12. C2×#89563C2semidirect
39. C7×#462C22semidirect
45. #89954C22semidirect
49. #89862C22semidirect
50. #89792C22semidirect
53. #89623C22semidirect
56. #89563C22semidirect
148. C2×#462D7semidirect
257. #462C2×D7semidirect
493. C2×C7#3255semidirect
547. C7C2×#3255semidirect
571. C2#11737932direct (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×#4622
C2×#899612
C2×#899542
C2×#898622
C2×#897922
#117379322
C2×#896232
C2×#896092
C2×#895882
C2×#895632
C2×#32557