Finite Group Databasebeta

← C2×#11731311 #13512239

Subgroups — C2×#11731311 #13512239

Order 1792.

3854 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C23×C4×Dic7C211yesmaximal, characteristic
3C22×#6450C211yesmaximal, characteristic, complemented
4C2×C7×#380C211yesmaximal, characteristic
5C2×#89746C211yesmaximal, characteristic, complemented
6C2×#89745C211yesmaximal, characteristic, complemented
7C2×C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
8C2×#89718C212yesmaximal
9C2×C22⋊C4×Dic7C212yesmaximal
10#11731311C2116yesmaximal, direct product factor
11C2×#89424C212yesmaximal, complemented
12C2×#89421C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C24×Dic7C2211yescharacteristic
16C23×Dic14C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C22⋊Dic7C2212yes—
21C22×C4⋊Dic7C2211yescharacteristic
22C2×#6487C4116yescomplemented
23C2×#6487C2218yes—
24C22×Dic7⋊C4C2212yes—
25C22×Dic7⋊C4C2212yes—
26C22×Dic7⋊C4C2212yes—
27C22×C4×Dic7—28no—
28C22×C4×Dic7C2211yescharacteristic
29C22×C4×Dic7C2211yescharacteristic
30C22×C4×Dic7C2218yes—
31C22×C4×Dic7C2211yescharacteristic
32C22×C4×Dic7C2212yes—
33C22×C4×Dic7C2212yes—
34C22×C4×Dic7C2211yescharacteristic
35C2×#6450—28no—
36C2×#6450C2218yescomplemented
37C7×#380C2218yes—
38C2×C7×#95C2212yes—
39C2×C7⋊#95C2212yes—
40C2×C7⋊#95C2212yes—
41C2×C7⋊#95C2212yes—
42C2×C7⋊#95C2212yes—
43C4⋊C4×Dic7C2218yescomplemented
44#89746C2218yescomplemented
45#89745C2218yescomplemented
46#89718C22116yes—
47C22⋊C4×Dic7C22116yes—
48#89424C22116yescomplemented
49#89421C22116yescomplemented
50C2×C4×C22⋊Q8—71nomaximal, Sylow
51C25×C7C2311yescharacteristic
52C23×C4×C7—24no—
53C23×C4×C7—24no—
54C23×C4×C7C2×C418yes—
55C23×C4×C7C2312yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—24no—
61C23×Dic7—24no—
62C23×Dic7—24no—
63C23×Dic7—24no—
64C23×Dic7—24no—
65C23×Dic7Q818yescomplemented
66C23×Dic7C2314yes—
67C23×Dic7C2312yes—
68C23×Dic7C2312yes—
69C23×Dic7C2311yescharacteristic
70C23×Dic7C2311yescharacteristic
71C23×Dic7C2314yes—
72C23×Dic7C2311yescharacteristic
73C23×Dic7C2311yescharacteristic
74C22×Dic14—24no—
75C22×Dic14—24no—
76C22×Dic14C2×C418yescomplemented
77C22×Dic14C2314yes—
78C2×C7×C4⋊C4—24no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4C2314yes—
81C2×C7×C4⋊C4C2314yes—
82C2×C7×C22⋊C4—28no—
83C2×C7×C22⋊C4C2318yes—
84C2×C22⋊Dic7C2×C4116yes—
85C2×C22⋊Dic7C2318yes—
86C2×C4⋊Dic7C2×C418yescomplemented
87C2×C4⋊Dic7C2314yes—
88#6487C2×C4164yescomplemented
89C2×Dic7⋊C4—28no—
90C2×Dic7⋊C4—28no—
91C2×Dic7⋊C4C2×C4116yescomplemented
92C2×Dic7⋊C4C2318yes—
93C2×Dic7⋊C4C2318yes—
94C2×Dic7⋊C4C2318yes—
95C2×C4×Dic7—24no—
96C2×C4×Dic7—28no—
97C2×C4×Dic7—24no—
98C2×C4×Dic7—24no—
99C2×C4×Dic7—28no—
100C2×C4×Dic7—232no—
101C2×C4×Dic7D418yes—
102C2×C4×Dic7C2318yes—
103C2×C4×Dic7C2314yes—
104C2×C4×Dic7C2318yes—
105C2×C4×Dic7C2314yes—
106C2×C4×Dic7C2314yes—
107C2×C4×Dic7C2314yes—
108#6450—232no—
109C7×#95C2318yes—
110C7⋊#95C2318yes—
111C7⋊#95C2318yes—
112C7⋊#95C2318yes—
113C7⋊#95C2318yes—
114C22×C22⋊Q8—71no—
115C22×C4×Q8—71no—
116C23×C42—71no—
117C4×C22⋊Q8—716no—
118C2×#391—71no—
119C2×#389—71no—
120C2×#387—72no—
121C2×#385—72no—
122C2×#380D711yescharacteristic
123C2×C4×C4⋊C4—72no—
124C2×C4×C4⋊C4—71no—
125C2×C4×C22⋊C4—72no—
126C24×C7—22no—
127C24×C7—22no—
128C24×C7—22no—
129C24×C7—22no—
130C24×C7C22×C414yes—
131C24×C7C2×Q814yes—
132C24×C7#4114yes—
133C24×C7C2412yes—
134C24×C7C2411yescharacteristic
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—232no—
138C22×C4×C7—28no—
139C22×C4×C7—28no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—24no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—24no—
146C22×C4×C7C22×C414yes—
147C22×C4×C7C22×C4116yes—
148C22×C4×C7C22×C414yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7#4114yes—
151C22×C4×C7C2414yes—
152C22×C4×C7C2414yes—
153C22×C4×C7C2412yes—
154C22×C4×C7C2412yes—
155C22×Dic7—232no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—24no—
159C22×Dic7—24no—
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7—28no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—232no—
168C22×Dic7—28no—
169C22×Dic7—28no—
170C22×Dic7—28no—
171C22×Dic7—28no—
172C22×Dic7C22×C418yes—
173C22×Dic7C22×C418yes—
174C22×Dic7C2×D414yes—
175C22×Dic7C2×D414yes—
176C22×Dic7C2×Q814yes—
177C22×Dic7C2×Q814yes—
178C22×Dic7C2×Q8116yescomplemented
179C22×Dic7#4114yes—
180C22×Dic7#4114yes—
181C22×Dic7C2414yes—
182C22×Dic7C2414yes—
183C22×Dic7C2412yes—
184C22×Dic7C2412yes—
185C22×Dic7C2412yes—
186C22×Dic7C2412yes—
187C2×Dic14—232no—
188C2×Dic14—28no—
189C2×Dic14—28no—
190C2×Dic14C22×C4116yescomplemented
191C7×C4⋊C4—28no—
192C7×C4⋊C4—28no—
193C7×C22⋊C4—216no—
194C22⋊Dic7C22×C4132yes—
195C4⋊Dic7C22×C4116yescomplemented
196Dic7⋊C4—216no—
197Dic7⋊C4—216no—
198Dic7⋊C4C22×C4132yescomplemented
199C4×Dic7—28no—
200C4×Dic7—216no—
201C4×Dic7—28no—
202C4×Dic7—28no—
203C4×Dic7—216no—
204C4×Dic7C2×D4116yes—
205C23×Q8—71no—
206C24×C4—71no—
207C24×C4—71no—
208C24×C4C2×D711yescharacteristic
209C2×C22⋊Q8—716no—
210C2×C22⋊Q8—78no—
211C2×C4×Q8—78no—
212C2×C4×Q8—148no—
213C22×C4⋊C4—72no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—72no—
217C22×C4⋊C4C2×D711yescharacteristic
218C22×C4⋊C4C2×D711yescharacteristic
219C22×C22⋊C4—72no—
220C22×C22⋊C4C2×D712yes—
221C22×C42—71no—
222C22×C42—71no—
223C22×C42—71no—
224C22×C42—71no—
225C22×C42—78no—
226C22×C42—72no—
227C22×C42—72no—
228C22×C42—148no—
229#391—78no—
230#389—78no—
231#387—716no—
232#385—716no—
233#380C2×D718yes—
234C4×C4⋊C4—78no—
235C4×C4⋊C4—716no—
236C4×C22⋊C4—716no—
237C2×#95—72no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95—72no—
241C2×#95C2×D712yes—
242C23×C7—28no—
243C23×C7—28no—
244C23×C7—28no—
245C23×C7—28no—
246C23×C7—28no—
247C23×C7—28no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—22no—
251C23×C7—22no—
252C23×C7C4×Q8116yescomplemented
253C23×C7C23×C414yes—
254C23×C7C23×C412yes—
255C23×C7C22×D412yes—
256C23×C7C22×Q814yes—
257C23×C7C22×Q812yes—
258C23×C7C2×#4114yes—
259C23×C7C2×#4112yes—
260C23×C7C2×#4112yes—
261C23×C7C2×#4112yes—
262C23×C7C2×#4112yes—
263C23×C7C2511yescharacteristic
264C2×C4×C7—28no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—232no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—48no—
275C2×C4×C7—48no—
276C2×C4×C7C4×D4116yes—
277C2×C4×C7C23×C414yes—
278C2×C4×C7C23×C414yes—
279C2×C4×C7C22×D414yes—
280C2×C4×C7C2×#4114yes—
281C2×Dic7—232no—
282C2×Dic7—216no—
283C2×Dic7—216no—
284C2×Dic7—28no—
285C2×Dic7—28no—
286C2×Dic7—28no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—232no—
298C2×Dic7C22⋊Q8116yescomplemented
299C2×Dic7C23×C418yes—
300C2×Dic7C23×C418yes—
301C2×Dic7C22×D414yes—
302C2×Dic7C22×D414yes—
303C2×Dic7C22×Q814yes—
304C2×Dic7C22×Q814yes—
305C2×Dic7C2×#4114yes—
306C2×Dic7C2×#4114yes—
307Dic14—232no—
308C25C22×D711yescharacteristic
309C22×Q8—78no—
310C22×Q8—74no—
311C22×Q8—144no—
312C22×Q8—144no—
313C23×C4—24no—
314C23×C4—24no—
315C23×C4—71no—
316C23×C4—78no—
317C23×C4—74no—
318C23×C4—72no—
319C23×C4—72no—
320C23×C4—71no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—74no—
324C23×C4—144no—
325C23×C4—144no—
326C23×C4—144no—
327C23×C4—144no—
328C23×C4—144no—
329C23×C4C4×D718yes—
330C23×C4C22×D712yes—
331C23×C4C22×D712yes—
332C23×C4C22×D714yes—
333C23×C4C22×D711yescharacteristic
334C23×C4C22×D711yescharacteristic
335C22⋊Q8—764no—
336C4×Q8—1432no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—716no—
340C2×C4⋊C4—78no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—78no—
344C2×C4⋊C4—78no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—148no—
347C2×C4⋊C4C22×D714yes—
348C2×C4⋊C4C22×D714yes—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—716no—
351C2×C22⋊C4—78no—
352C2×C22⋊C4C22×D718yes—
353C2×C42—78no—
354C2×C42—74no—
355C2×C42—78no—
356C2×C42—78no—
357C2×C42—74no—
358C2×C42—74no—
359C2×C42—74no—
360C2×C42—1432no—
361C2×C42—148no—
362C2×C42—148no—
363C2×C42—144no—
364C2×C42—144no—
365C2×C42—144no—
366#95—78no—
367#95—78no—
368#95—78no—
369#95—78no—
370#95C22×D718yes—
371C22×C7—24no—
372C22×C7—24no—
373C22×C7—24no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—24no—
377C22×C7—232no—
378C22×C7C2×C4×D414yes—
379C22×C7C2×C4×Q814yes—
380C22×C7C2×C4×Q818yescomplemented
381C22×C7C4×#4114yes—
382C22×C7C2×C22⋊Q814yes—
383C22×C7#52514yes—
384C22×C7#52914yes—
385C22×C7#53314yes—
386C22×C7C24×C411yescharacteristic
387C22×C7C23×D411yescharacteristic
388C22×C7C23×Q811yescharacteristic
389C22×C7C22×#4111yescharacteristic
390C22×C7C22×#4111yescharacteristic
391C22×C7C22×#4111yescharacteristic
392C22×C7C22×#4111yescharacteristic
393C4×C7—24no—
394C4×C7—44no—
395C4×C7—44no—
396C4×C7C2×C4×D418yes—
397Dic7—28no—
398Dic7—28no—
399Dic7—24no—
400Dic7—24no—
401Dic7—24no—
402Dic7C2×C22⋊Q818yescomplemented
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24C2×C4×D714yes—
408C24#115914yes—
409C24Q8×D714yes—
410C24C23×D712yes—
411C24C23×D711yescharacteristic
412C2×Q8—716no—
413C2×Q8—1432no—
414C2×Q8—148no—
415C2×Q8—148no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—232no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—72no—
431C22×C4—78no—
432C22×C4—78no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—716no—
436C22×C4—74no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—74no—
441C22×C4—74no—
442C22×C4—148no—
443C22×C4—148no—
444C22×C4—1432no—
445C22×C4—148no—
446C22×C4—148no—
447C22×C4—1432no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—144no—
451C22×C4—148no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4C2×C4×D714yes—
460C22×C4C2×C4×D7116yes—
461C22×C4C2×C4×D714yes—
462C22×C4D4×D714yes—
463C22×C4#115914yes—
464C22×C4C23×D714yes—
465C22×C4C23×D714yes—
466C22×C4C23×D712yes—
467C22×C4C23×D712yes—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—732no—
471C4⋊C4—716no—
472C4⋊C4—1416no—
473C4⋊C4—1416no—
474C22⋊C4—216no—
475C22⋊C4—732no—
476C42—716no—
477C42—1416no—
478C42—148no—
479C42—148no—
480C42—1416no—
481C42—148no—
482C2×C7—28no—
483C2×C7C4×C22⋊Q818yescomplemented
484C2×C7C22×C4×D411yescharacteristic
485C2×C7C22×C4×Q811yescharacteristic
486C2×C7C2×C4×#4111yescharacteristic
487C2×C7C22×C22⋊Q811yescharacteristic
488C2×C7C2×#52511yescharacteristic
489C2×C7C2×#52911yescharacteristic
490C2×C7C2×#53311yescharacteristic
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23#6450116yescomplemented
502C23C22×C4×D714yes—
503C23C22×C4×D712yes—
504C23C2×D4×D712yes—
505C23C2×#115914yes—
506C23C2×#115912yes—
507C23C2×#115912yes—
508C23C2×Q8×D714yes—
509C23C2×Q8×D712yes—
510C23D7×#4112yes—
511C23D7×#4112yes—
512C23C24×D711yescharacteristic
513Q8—1432no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—232no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—716no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—74no—
530C2×C4—78no—
531C2×C4—78no—
532C2×C4—74no—
533C2×C4—74no—
534C2×C4—74no—
535C2×C4—1432no—
536C2×C4—1432no—
537C2×C4—1416no—
538C2×C4—1416no—
539C2×C4—148no—
540C2×C4—148no—
541C2×C4—148no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4(C22⋊D7)⋊C4116yes—
553C2×C4C22×C4×D714yes—
554C2×C4C22×C4×D714yes—
555C2×C4C2×D4×D714yes—
556C2×C4C2×#115914yes—
557C7C2×C4×C22⋊Q811yescharacteristic, complemented, Sylow
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—232no—
565C22C2×(C22⋊D7)⋊C414yes—
566C22C2×#645014yes—
567C22C2×#645018yescomplemented
568C22#9021314yes—
569C22#9029114yes—
570C22#9029714yes—
571C22#9031314yes—
572C22D7×C22⋊Q814yes—
573C22C23×C4×D711yescharacteristic
574C22C22×D4×D711yescharacteristic
575C22C22×#115911yescharacteristic
576C22C22×#115911yescharacteristic
577C22C22×Q8×D711yescharacteristic
578C22C2×D7×#4111yescharacteristic
579C22C2×D7×#4111yescharacteristic
580C4—24no—
581C4—44no—
582C4—44no—
583C4—78no—
584C4—144no—
585C4—144no—
586C4—148no—
587C4—144no—
588C4—148no—
589C4C2×(C22⋊D7)⋊C418yes—
590C2—28no—
591C2#1173131118yesdirect product factor
592C2C22×(C22⋊D7)⋊C411yescharacteristic
593C2C22×#645011yescharacteristic
594C2C2×#9021311yescharacteristic
595C2C2×#9029711yescharacteristic
596C2C2×#9029111yescharacteristic
597C2C2×#9031311yescharacteristic
598C2C2×D7×C22⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC2×C7×#380C2
socleC24×C7C24
prime radicalC25×C7C23
cosocleC23×C7C25
hypercenterC2×#380D7
hypocenterC7C2×C4×C22⋊Q8
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. C22×#6450C2semidirect
5. C2×#89746C2semidirect
6. C2×#89745C2semidirect
7. C2×C4⋊C4×Dic7C2semidirect
10. #11731311C2direct (N × H)
11. C2×#89424C2semidirect
12. C2×#89421C2semidirect
22. C2×#6487C4semidirect
36. C2×#6450C22semidirect
43. C4⋊C4×Dic7C22semidirect
44. #89746C22semidirect
45. #89745C22semidirect
48. #89424C22semidirect
49. #89421C22semidirect
65. C23×Dic7Q8semidirect
76. C22×Dic14C2×C4semidirect
86. C2×C4⋊Dic7C2×C4semidirect
88. #6487C2×C4semidirect
91. C2×Dic7⋊C4C2×C4semidirect
178. C22×Dic7C2×Q8semidirect
190. C2×Dic14C22×C4semidirect
195. C4⋊Dic7C22×C4semidirect
198. Dic7⋊C4C22×C4semidirect
252. C23×C7C4×Q8semidirect
298. C2×Dic7C22⋊Q8semidirect
380. C22×C7C2×C4×Q8semidirect
402. Dic7C2×C22⋊Q8semidirect
483. C2×C7C4×C22⋊Q8semidirect
501. C23#6450semidirect
557. C7C2×C4×C22⋊Q8semidirect
567. C22C2×#6450semidirect
591. C2#11731311direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C23×C4×Dic72
C22×#64502
C2×C7×#3802
C2×#897462
C2×#897452
C2×C4⋊C4×Dic72
C2×#897182
C2×C22⋊C4×Dic72
#117313112
C2×#894242
C2×#894212
C2×C4×C22⋊Q87