Finite Group Databasebeta

← C2×#11735482 #13525572

Subgroups — C2×#11735482 #13525572

Order 1792.

4006 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6459C212yesmaximal, complemented
2C22×#6451C211yesmaximal, characteristic, complemented
3C22×#6441C212yesmaximal, complemented
4C22×(C22⋊D7)⋊C4C211yesmaximal, characteristic
5C2×C7×#399C211yesmaximal, characteristic, complemented
6C2×#89764C211yesmaximal, characteristic, complemented
7C2×#89748C212yesmaximal, complemented
8C2×#89721C211yesmaximal, characteristic, complemented
9C2×#89729C212yesmaximal
10#11735482C2116yesmaximal, direct product factor
11C2×#89449C211yesmaximal, characteristic, complemented
12C2×#89421C211yesmaximal, characteristic, complemented
13C23×C22⋊D7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C23×C4×D7C2211yescharacteristic
16C23×Dic14C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2212yes—
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2212yes—
21C22×C22⋊Dic7C2212yes—
22C22×(C2×D7)⋊C4C2211yescharacteristic
23C22×(C2×D7)⋊C4C2212yes—
24C22×(C2×D7)⋊C4C2212yes—
25C22×C4⋊Dic7C2212yes—
26C22×C4⋊Dic7C2211yescharacteristic
27C22×Dic7⋊C4C2211yescharacteristic
28C22×Dic7⋊C4C2212yes—
29C22×Dic7⋊C4C2212yes—
30C22×Dic7⋊C4C2212yes—
31C22×Dic7⋊C4C2212yes—
32C22×C4×Dic7C2212yes—
33C22×C4×Dic7C2211yescharacteristic
34C2×#6459—216no—
35C2×#6459C22116yescomplemented
36C2×#6451—28no—
37C2×#6451C2218yescomplemented
38C2×(C22⋊D7)⋊C4—28no—
39C2×(C22⋊D7)⋊C4C2218yes—
40C2×#6441—216no—
41C2×#6441C22116yescomplemented
42C7×#399C2218yescomplemented
43C2×C7×#95C2211yescharacteristic
44C2×C7⋊#95C2211yescharacteristic
45#89764C2218yescomplemented
46#89748C22116yescomplemented
47#89729C22116yes—
48#89721C2218yescomplemented
49#89449C2218yescomplemented
50#89421C2218yescomplemented
51C2×#2542—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C22×C22⋊D7—24no—
59C22×C22⋊D7—24no—
60C22×C22⋊D7Q818yescomplemented
61C22×C22⋊D7C2314yes—
62C23×Dic7—28no—
63C23×Dic7—24no—
64C23×Dic7C2314yes—
65C23×Dic7C2311yescharacteristic
66C23×Dic7C2312yes—
67C23×Dic7C2312yes—
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2311yescharacteristic
70C22×C4×D7—24no—
71C22×C4×D7—28no—
72C22×C4×D7C2314yes—
73C22×Dic14—28no—
74C22×Dic14—28no—
75C22×Dic14—28no—
76C22×Dic14C2318yes—
77C2×C7×C4⋊C4—24no—
78C2×C7×C4⋊C4—28no—
79C2×C7×C4⋊C4C2314yes—
80C2×C7×C4⋊C4C2318yes—
81C2×C7×C22⋊C4—28no—
82C2×C7×C22⋊C4—24no—
83C2×C7×C22⋊C4C2318yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C22⋊Dic7—28no—
86C2×C22⋊Dic7C2318yes—
87C2×(C2×D7)⋊C4—24no—
88C2×(C2×D7)⋊C4—28no—
89C2×(C2×D7)⋊C4—28no—
90C2×(C2×D7)⋊C4C2314yes—
91C2×(C2×D7)⋊C4C2318yes—
92C2×(C2×D7)⋊C4C2318yes—
93C2×C4⋊Dic7—28no—
94C2×C4⋊Dic7—24no—
95C2×C4⋊Dic7C2318yes—
96C2×C4⋊Dic7C2314yes—
97C2×Dic7⋊C4—28no—
98C2×Dic7⋊C4—24no—
99C2×Dic7⋊C4—28no—
100C2×Dic7⋊C4—28no—
101C2×Dic7⋊C4D4116yescomplemented
102C2×Dic7⋊C4C2318yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2318yes—
105C2×Dic7⋊C4C2318yes—
106C2×Dic7⋊C4C2318yes—
107C2×C4×Dic7—28no—
108C2×C4×Dic7D418yes—
109C2×C4×Dic7C2318yes—
110C2×C4×Dic7C2314yes—
111#6459—264no—
112#6451—232no—
113(C22⋊D7)⋊C4—232no—
114#6441—264no—
115C7×#95C2314yes—
116C7⋊#95C2314yes—
117C22×C4⋊Q8—71no—
118C22×C22⋊Q8—72no—
119C22×C22⋊Q8—72no—
120C22×C4×D4—71no—
121#2542—716no—
122C2×#399—71no—
123C2×#399—71no—
124C2×#399—71no—
125C2×#399D711yescharacteristic, complemented
126C2×#390—72no—
127C2×#384—72no—
128C2×C4×C4⋊C4—71no—
129C24×C7—24no—
130C24×C7—22no—
131C24×C7—22no—
132C24×C7—24no—
133C24×C7C2×Q814yes—
134C24×C7C2412yes—
135C24×C7C2411yescharacteristic
136C23×D7—24no—
137C23×D7—22no—
138C23×D7—22no—
139C23×D7—24no—
140C23×D7C2×Q814yes—
141C23×D7C2412yes—
142C22×C4×C7—24no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D418yes—
152C22×C4×C7C2412yes—
153C22×C4×C7C2414yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2414yes—
156C2×C22⋊D7—232no—
157C2×C22⋊D7—28no—
158C2×C22⋊D7—28no—
159C2×C22⋊D7C2×Q8116yescomplemented
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7—24no—
163C22×Dic7—24no—
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—24no—
167C22×Dic7—216no—
168C22×Dic7—28no—
169C22×Dic7—416no—
170C22×Dic7C2×D418yes—
171C22×Dic7C2×D414yes—
172C22×Dic7C2×D414yes—
173C22×Dic7C2×D418yes—
174C22×Dic7C2×Q814yes—
175C22×Dic7#4114yes—
176C22×Dic7C2412yes—
177C22×Dic7C2414yes—
178C22×Dic7C2414yes—
179C22×Dic7C2412yes—
180C22×Dic7C2412yes—
181C2×C4×D7—28no—
182C2×C4×D7—216no—
183C2×C4×D7—416no—
184C2×Dic14—216no—
185C2×Dic14—216no—
186C2×Dic14—216no—
187C2×Dic14—432no—
188C7×C4⋊C4—28no—
189C7×C4⋊C4—216no—
190C7×C22⋊C4—216no—
191C7×C22⋊C4—28no—
192C22⋊Dic7—216no—
193(C2×D7)⋊C4—28no—
194(C2×D7)⋊C4—216no—
195(C2×D7)⋊C4—216no—
196C4⋊Dic7—216no—
197C4⋊Dic7—28no—
198Dic7⋊C4—216no—
199Dic7⋊C4—28no—
200Dic7⋊C4—216no—
201Dic7⋊C4—216no—
202Dic7⋊C4C2×D4132yescomplemented
203C4×Dic7—216no—
204C4×Dic7C2×D4116yes—
205C23×Q8—72no—
206C23×D4—71no—
207C24×C4—71no—
208C24×C4—71no—
209C2×C4⋊Q8—78no—
210C2×C4⋊Q8—148no—
211C2×C22⋊Q8—716no—
212C2×C22⋊Q8—716no—
213C2×C22⋊Q8—1416no—
214C2×C22⋊Q8—1416no—
215C2×C4×D4—78no—
216C2×C4×D4—148no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—71no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—72no—
221C22×C4⋊C4—72no—
222C22×C4⋊C4—71no—
223C22×C4⋊C4—72no—
224C22×C4⋊C4C2×D711yescharacteristic
225C22×C4⋊C4C2×D712yes—
226C22×C22⋊C4—72no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4—72no—
229C22×C22⋊C4—72no—
230C22×C22⋊C4C2×D711yescharacteristic
231C22×C22⋊C4C2×D712yes—
232C22×C42—71no—
233C22×C42—72no—
234#399—78no—
235#399—78no—
236#399—78no—
237#399C2×D718yescomplemented
238#390—716no—
239#384—716no—
240C4×C4⋊C4—78no—
241C2×#95—71no—
242C2×#95C2×D711yescharacteristic
243C23×C7—28no—
244C23×C7—216no—
245C23×C7—24no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—24no—
249C23×C7—44no—
250C23×C7—44no—
251C23×C7—48no—
252C23×C7C22×D412yes—
253C23×C7C22×D414yes—
254C23×C7C22×Q814yes—
255C23×C7C22×Q812yes—
256C23×C7C2×#4112yes—
257C23×C72−1+414yes—
258C23×C7C2511yescharacteristic
259C22×D7—28no—
260C22×D7—216no—
261C22×D7—24no—
262C22×D7—22no—
263C22×D7—22no—
264C22×D7—24no—
265C22×D7—44no—
266C22×D7—44no—
267C22×D7—48no—
268C22×D7C22×Q814yes—
269C2×C4×C7—28no—
270C2×C4×C7—216no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—48no—
280C2×C4×C7—44no—
281C2×C4×C7C22×D414yes—
282C2×C4×C7C22×D418yes—
283C22⋊D7—232no—
284C2×Dic7—216no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—48no—
293C2×Dic7—44no—
294C2×Dic7—416no—
295C2×Dic7C22⋊Q8116yescomplemented
296C2×Dic7#127116yes—
297C2×Dic7C22×D418yes—
298C2×Dic7C22×D414yes—
299C2×Dic7C22×D414yes—
300C2×Dic7C22×D418yes—
301C2×Dic7C22×Q814yes—
302C2×Dic7C2×#4114yes—
303C4×D7—416no—
304Dic14—432no—
305C25—71no—
306C25C22×D711yescharacteristic
307C22×Q8—78no—
308C22×Q8—148no—
309C22×Q8—148no—
310C22×Q8—148no—
311C22×D4—78no—
312C22×D4—74no—
313C22×D4—144no—
314C22×D4—144no—
315C23×C4—71no—
316C23×C4—71no—
317C23×C4—72no—
318C23×C4—74no—
319C23×C4—72no—
320C23×C4—71no—
321C23×C4—74no—
322C23×C4—144no—
323C23×C4—148no—
324C23×C4—144no—
325C23×C4—148no—
326C23×C4C22×D712yes—
327C23×C4C22×D711yescharacteristic
328C23×C4C22×D712yes—
329C23×C4C22×D711yescharacteristic
330C4⋊Q8—1432no—
331C22⋊Q8—1464no—
332C22⋊Q8—1464no—
333C4×D4—1432no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—716no—
337C2×C4⋊C4—78no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—78no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—78no—
343C2×C4⋊C4—74no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—148no—
347C2×C4⋊C4—148no—
348C2×C4⋊C4—148no—
349C2×C4⋊C4—144no—
350C2×C4⋊C4C22×D718yes—
351C2×C4⋊C4C22×D714yes—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—78no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—78no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—148no—
359C2×C22⋊C4—148no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—148no—
362C2×C22⋊C4C22×D714yes—
363C2×C22⋊C4C22×D718yes—
364C2×C42—78no—
365C2×C42—74no—
366C2×C42—78no—
367C2×C42—148no—
368#95—74no—
369#95C22×D714yes—
370C22×C7—24no—
371C22×C7—28no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—416no—
376C22×C7C2×C22⋊Q814yes—
377C22×C7C2×#12714yes—
378C22×C7#53614yes—
379C22×C7#54718yes—
380C22×C7D4×Q818yes—
381C22×C7C23×D411yescharacteristic
382C22×C7C23×D412yes—
383C22×C7C23×Q811yescharacteristic
384C22×C7C22×#4111yescharacteristic
385C22×C7C2×2−1+412yes—
386C2×D7—24no—
387C2×D7—28no—
388C2×D7—42no—
389C2×D7—42no—
390C2×D7—44no—
391C2×D7—416no—
392C4×C7—24no—
393C4×C7—28no—
394C4×C7—44no—
395C4×C7—42no—
396Dic7—28no—
397Dic7—44no—
398Dic7—42no—
399Dic7C2×C22⋊Q818yescomplemented
400Dic7C2×#12718yes—
401C24—24no—
402C24—22no—
403C24—22no—
404C24—24no—
405C24—74no—
406C24—72no—
407C24—144no—
408C24—142no—
409C24—142no—
410C24—144no—
411C24Q8×D714yes—
412C24C23×D712yes—
413C24C23×D711yescharacteristic
414C2×Q8—1416no—
415C2×Q8—1416no—
416C2×Q8—1416no—
417C2×Q8—2832no—
418C2×D4—716no—
419C2×D4—148no—
420C2×D4—1432no—
421C2×D4—148no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—78no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—78no—
438C22×C4—74no—
439C22×C4—74no—
440C22×C4—72no—
441C22×C4—148no—
442C22×C4—1416no—
443C22×C4—148no—
444C22×C4—1416no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—2816no—
453C22×C4—2816no—
454C22×C4D4×D714yes—
455C22×C4D4×D718yes—
456C22×C4C23×D714yes—
457C22×C4C23×D712yes—
458C22×C4C23×D712yes—
459C22×C4C23×D714yes—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—732no—
463C4⋊C4—148no—
464C4⋊C4—1416no—
465C4⋊C4—1416no—
466C4⋊C4—1416no—
467C4⋊C4—1416no—
468C4⋊C4—148no—
469C22⋊C4—28no—
470C22⋊C4—216no—
471C22⋊C4—1416no—
472C22⋊C4—1416no—
473C22⋊C4—148no—
474C22⋊C4—1416no—
475C42—716no—
476C42—1416no—
477C2×C7—44no—
478C2×C7#254218yescomplemented
479C2×C7C22×C22⋊Q811yescharacteristic
480C2×C7C22×#12711yescharacteristic
481C2×C7C2×#53611yescharacteristic
482C2×C7C2×#54712yes—
483C2×C7C2×D4×Q812yes—
484D7—44no—
485C23—28no—
486C23—216no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—74no—
495C23—148no—
496C23—1416no—
497C23—144no—
498C23—142no—
499C23—142no—
500C23—144no—
501C23—284no—
502C23—284no—
503C23—288no—
504C23C2×D4×D712yes—
505C23C2×D4×D714yes—
506C23C2×Q8×D714yes—
507C23C2×Q8×D712yes—
508C23D7×#4112yes—
509C23#655414yes—
510C23C24×D711yescharacteristic
511Q8—2832no—
512D4—1432no—
513C2×C4—28no—
514C2×C4—216no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—716no—
526C2×C4—716no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—78no—
530C2×C4—74no—
531C2×C4—74no—
532C2×C4—78no—
533C2×C4—1416no—
534C2×C4—144no—
535C2×C4—144no—
536C2×C4—142no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—144no—
541C2×C4—2816no—
542C2×C4—2816no—
543C2×C4—284no—
544C2×C4—288no—
545C2×C4C2×D4×D714yes—
546C2×C4C2×D4×D718yes—
547C7C2×#254211yescharacteristic, complemented, Sylow
548C22—28no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—416no—
554C22—148no—
555C22—144no—
556C22—282no—
557C22—282no—
558C22—284no—
559C22—2816no—
560C22D7×C22⋊Q814yes—
561C22#9032318yes—
562C22#9033914yes—
563C22#9034618yes—
564C22D7×#12714yes—
565C22C22×D4×D712yes—
566C22C22×D4×D711yescharacteristic
567C22C22×Q8×D711yescharacteristic
568C22C2×D7×#4111yescharacteristic
569C22C2×#655412yes—
570C4—24no—
571C4—28no—
572C4—42no—
573C4—44no—
574C4—78no—
575C4—78no—
576C4—148no—
577C4—282no—
578C4—284no—
579C2—44no—
580C2—284no—
581C2#1173548218yesdirect product factor
582C2C2×#9032312yes—
583C2C2×D7×C22⋊Q811yescharacteristic
584C2C2×#9033911yescharacteristic
585C2C2×#9034612yes—
586C2C2×D7×#12711yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#399C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC2×#399D7
hypocenterC7C2×#2542
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#6459C2semidirect
2. C22×#6451C2semidirect
3. C22×#6441C2semidirect
5. C2×C7×#399C2semidirect
6. C2×#89764C2semidirect
7. C2×#89748C2semidirect
8. C2×#89721C2semidirect
10. #11735482C2direct (N × H)
11. C2×#89449C2semidirect
12. C2×#89421C2semidirect
35. C2×#6459C22semidirect
37. C2×#6451C22semidirect
41. C2×#6441C22semidirect
42. C7×#399C22semidirect
45. #89764C22semidirect
46. #89748C22semidirect
48. #89721C22semidirect
49. #89449C22semidirect
50. #89421C22semidirect
60. C22×C22⋊D7Q8semidirect
101. C2×Dic7⋊C4D4semidirect
125. C2×#399D7semidirect
159. C2×C22⋊D7C2×Q8semidirect
202. Dic7⋊C4C2×D4semidirect
237. #399C2×D7semidirect
295. C2×Dic7C22⋊Q8semidirect
399. Dic7C2×C22⋊Q8semidirect
478. C2×C7#2542semidirect
547. C7C2×#2542semidirect
581. C2#11735482direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64592
C22×#64512
C22×#64412
C22×(C22⋊D7)⋊C42
C2×C7×#3992
C2×#897642
C2×#897482
C2×#897212
C2×#897292
#117354822
C2×#894492
C2×#894212
C2×#25427