Finite Group Databasebeta

← C2×#11734569 #13520154

Subgroups — C2×#11734569 #13520154

Order 1792.

3718 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×#6501C211yesmaximal, characteristic
2C22×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#6446C212yesmaximal, complemented
4C2×C7×#394C211yesmaximal, characteristic, complemented
5C2×#89994C211yesmaximal, characteristic, complemented
6C2×#89755C211yesmaximal, characteristic, complemented
7C2×#89728C212yesmaximal, complemented
8C2×#89720C212yesmaximal, complemented
9C2×#89720C212yesmaximal, complemented
10#11734569C2116yesmaximal, direct product factor
11C2×#89439C211yesmaximal, characteristic, complemented
12C2×#89433C211yesmaximal, characteristic, complemented
13C23×C7×D4C2211yescharacteristic
14C23×C22⋊D7C2212yes—
15C23×C4×D7C2211yescharacteristic
16C23×C4×D7C2211yescharacteristic
17C23×C4×D7C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C22⋊Dic7C2212yes—
22C22×C22⋊Dic7C2212yes—
23C22×C22⋊Dic7C2212yes—
24C2×#6501—28no—
25C2×#6501C2218yes—
26C22×(C2×D7)⋊C4C2212yes—
27C22×C4⋊Dic7C2212yes—
28C22×C4⋊Dic7C2211yescharacteristic
29C22×C4⋊Dic7C2211yescharacteristic
30C22×Dic7⋊C4C2212yes—
31C22×Dic7⋊C4C2211yescharacteristic
32C22×Dic7⋊C4C2211yescharacteristic
33C22×Dic7⋊C4C2212yes—
34C2×D7×C4⋊C4—28no—
35C2×D7×C4⋊C4C2218yescomplemented
36C2×#6446—216no—
37C2×#6446C22116yescomplemented
38C7×#394C2218yescomplemented
39C2×C7×#95C2211yescharacteristic
40#89994C2218yescomplemented
41C2×C7⋊#95C2211yescharacteristic
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44#89755C2218yescomplemented
45#89728C22116yescomplemented
46#89720C22116yescomplemented
47#89720C22116yescomplemented
48#89439C2218yescomplemented
49#89433C2218yescomplemented
50C2×#2618—71nomaximal, Sylow
51C25×C7C2312yes—
52C24×D7C2311yescharacteristic
53C22×C7×D4—24no—
54C22×C7×D4—44no—
55C22×C7×D4C2314yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—28no—
61C22×C22⋊D7—28no—
62C22×C22⋊D7—28no—
63C22×C22⋊D7C2318yes—
64C23×Dic7C2312yes—
65C23×Dic7C2312yes—
66C23×Dic7C2311yescharacteristic
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C22×C4×D7—24no—
70C22×C4×D7—24no—
71C22×C4×D7—24no—
72C22×C4×D7—44no—
73C22×C4×D7—44no—
74C22×C4×D7D418yes—
75C22×C4×D7D418yescomplemented
76C22×C4×D7C2314yes—
77C22×C4×D7C2314yes—
78C22×C4×D7C2314yes—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4C2314yes—
81C2×C7×C22⋊C4—28no—
82C2×C7×C22⋊C4—44no—
83C2×C7×C22⋊C4C2318yes—
84C2×C7×C22⋊C4C2318yes—
85C2×C22⋊Dic7—28no—
86C2×C22⋊Dic7—28no—
87C2×C22⋊Dic7—44no—
88C2×C22⋊Dic7C2318yes—
89C2×C22⋊Dic7C2318yes—
90C2×C22⋊Dic7C2318yes—
91#6501—232no—
92C2×(C2×D7)⋊C4—28no—
93C2×(C2×D7)⋊C4C2318yes—
94C2×C4⋊Dic7—24no—
95C2×C4⋊Dic7—28no—
96C2×C4⋊Dic7—24no—
97C2×C4⋊Dic7C2318yes—
98C2×C4⋊Dic7C2314yes—
99C2×C4⋊Dic7C2314yes—
100C2×Dic7⋊C4—28no—
101C2×Dic7⋊C4—24no—
102C2×Dic7⋊C4—24no—
103C2×Dic7⋊C4—44no—
104C2×Dic7⋊C4C2318yes—
105C2×Dic7⋊C4C2314yes—
106C2×Dic7⋊C4C2314yes—
107C2×Dic7⋊C4C2318yes—
108D7×C4⋊C4—232no—
109#6446—264no—
110C7×#95C2314yes—
111C7⋊#95C2318yes—
112C7⋊#95C2314yes—
113C7⋊#95C2318yes—
114C22×#123—72no—
115C22×#121—71no—
116C23×C4⋊C4—71no—
117#2618—716no—
118C2×#398—71no—
119C2×#398—71no—
120C2×#396—72no—
121C2×#396—72no—
122C2×#394—71no—
123C2×#394D711yescharacteristic, complemented
124C2×#386—72no—
125C2×#380—71no—
126C24×C7—24no—
127C24×C7—24no—
128C24×C7—24no—
129C24×C7—24no—
130C24×C7—24no—
131C24×C7—24no—
132C24×C7—24no—
133C24×C7C2414yes—
134C24×C7C2411yescharacteristic
135C23×D7—24no—
136C23×D7—24no—
137C23×D7C2×D414yes—
138C23×D7C2×D414yes—
139C23×D7#4114yes—
140C23×D7C2412yes—
141C2×C7×D4—28no—
142C2×C7×D4—48no—
143C2×C7×D4—416no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—22no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2414yes—
156C22×C4×C7C2412yes—
157C22×C4×C7C2412yes—
158C2×C22⋊D7—216no—
159C2×C22⋊D7—216no—
160C2×C22⋊D7—216no—
161C2×C22⋊D7—432no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—24no—
165C22×Dic7—22no—
166C22×Dic7—24no—
167C22×Dic7—24no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7—24no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7C2×D414yes—
174C22×Dic7C2×D414yes—
175C22×Dic7C2414yes—
176C22×Dic7C2414yes—
177C22×Dic7C2412yes—
178C22×Dic7C2412yes—
179C22×Dic7C2412yes—
180C2×C4×D7—232no—
181C2×C4×D7—28no—
182C2×C4×D7—28no—
183C2×C4×D7—28no—
184C2×C4×D7—416no—
185C2×C4×D7—48no—
186C2×C4×D7—48no—
187C2×C4×D7—416no—
188C2×C4×D7C2×D4116yes—
189C2×C4×D7C2×D4116yescomplemented
190C7×C4⋊C4—28no—
191C7×C22⋊C4—216no—
192C7×C22⋊C4—48no—
193C22⋊Dic7—216no—
194C22⋊Dic7—216no—
195C22⋊Dic7—48no—
196(C2×D7)⋊C4—216no—
197C4⋊Dic7—28no—
198C4⋊Dic7—216no—
199C4⋊Dic7—28no—
200Dic7⋊C4—216no—
201Dic7⋊C4—28no—
202Dic7⋊C4—28no—
203Dic7⋊C4—48no—
204C23×D4—72no—
205C23×D4C2×D711yescharacteristic
206C24×C4—71no—
207C24×C4—71no—
208C24×C4—71no—
209C2×#123—716no—
210C2×#123—1416no—
211C2×#121—78no—
212C2×#121—148no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—71no—
216C22×C4⋊C4—72no—
217C22×C4⋊C4—72no—
218C22×C4⋊C4—71no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—78no—
221C22×C4⋊C4—148no—
222C22×C4⋊C4C2×D711yescharacteristic
223C22×C22⋊C4—72no—
224C22×C22⋊C4—72no—
225C22×C22⋊C4—72no—
226C22×C22⋊C4—72no—
227C22×C22⋊C4C2×D712yes—
228C22×C22⋊C4C2×D712yes—
229#398—78no—
230#398—78no—
231#396—716no—
232#396—716no—
233#394—78no—
234#394C2×D718yescomplemented
235#386—716no—
236#380—78no—
237C2×#95—71no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95C2×D711yescharacteristic
241C23×C7—24no—
242C23×C7—24no—
243C23×C7—24no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—48no—
249C23×C7—48no—
250C23×C7—48no—
251C23×C7—48no—
252C23×C7—48no—
253C23×C7—48no—
254C23×C7—48no—
255C23×C7C22×D412yes—
256C23×C7C22×D412yes—
257C23×C7C2×#4112yes—
258C23×C7H(5,𝔽2)14yes—
259C23×C72−1+414yes—
260C23×C7C2511yescharacteristic
261C22×D7—28no—
262C22×D7—28no—
263C22×D7—216no—
264C22×D7—216no—
265C22×D7—24no—
266C22×D7—24no—
267C22×D7#121116yes—
268C22×D7C22×D414yes—
269C22×D7C22×D414yes—
270C22×D7C2×#4114yes—
271C7×D4—416no—
272C2×C4×C7—28no—
273C2×C4×C7—28no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—44no—
283C2×C4×C7—48no—
284C2×C4×C7C22×D414yes—
285C2×C4×C7C22×D414yes—
286C22⋊D7—432no—
287C2×Dic7—28no—
288C2×Dic7—28no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—24no—
294C2×Dic7—24no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—48no—
301C2×Dic7—44no—
302C2×Dic7—48no—
303C2×Dic7C22×D414yes—
304C2×Dic7C22×D414yes—
305C4×D7—232no—
306C4×D7—416no—
307C4×D7—416no—
308C25—71no—
309C25C22×D712yes—
310C22×D4—24no—
311C22×D4—44no—
312C22×D4—78no—
313C22×D4—148no—
314C22×D4—148no—
315C22×D4—148no—
316C22×D4C22×D714yes—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—78no—
320C23×C4—74no—
321C23×C4—78no—
322C23×C4—74no—
323C23×C4—72no—
324C23×C4—71no—
325C23×C4—72no—
326C23×C4—74no—
327C23×C4—144no—
328C23×C4—144no—
329C23×C4—144no—
330C23×C4—284no—
331C23×C4—284no—
332C23×C4C22×D712yes—
333C23×C4C22×D711yescharacteristic
334C23×C4C22×D711yescharacteristic
335C23×C4C22×D711yescharacteristic
336#123—1464no—
337#121—1432no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—78no—
344C2×C4⋊C4—78no—
345C2×C4⋊C4—78no—
346C2×C4⋊C4—1432no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—148no—
350C2×C4⋊C4—144no—
351C2×C4⋊C4—144no—
352C2×C4⋊C4—148no—
353C2×C4⋊C4—284no—
354C2×C4⋊C4C22×D714yes—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—78no—
358C2×C22⋊C4—78no—
359C2×C22⋊C4—78no—
360C2×C22⋊C4—78no—
361C2×C22⋊C4—148no—
362C2×C22⋊C4—148no—
363C2×C22⋊C4—148no—
364C2×C22⋊C4—284no—
365C2×C22⋊C4C22×D718yes—
366C2×C22⋊C4C22×D718yes—
367#95—74no—
368#95—78no—
369#95—78no—
370#95C22×D714yes—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—44no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—44no—
378C22×C7—816no—
379C22×C7C2×#12114yes—
380C22×C7#53414yes—
381C22×C7#53614yes—
382C22×C7#53718yes—
383C22×C7#53918yes—
384C22×C7C23×D411yescharacteristic
385C22×C7C23×D411yescharacteristic
386C22×C7C22×#4111yescharacteristic
387C22×C7C2×H(5,𝔽2)12yes—
388C22×C7C2×2−1+412yes—
389C2×D7—232no—
390C2×D7—24no—
391C2×D7—24no—
392C2×D7—28no—
393C2×D7—28no—
394C2×D7C2×#12118yes—
395C4×C7—24no—
396C4×C7—24no—
397C4×C7—42no—
398C4×C7—44no—
399Dic7—24no—
400Dic7—24no—
401Dic7—44no—
402Dic7—42no—
403Dic7—44no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—74no—
412C24—74no—
413C24—74no—
414C24—72no—
415C24—144no—
416C24—144no—
417C24C23×D714yes—
418C24C23×D711yescharacteristic
419C2×D4—28no—
420C2×D4—48no—
421C2×D4—416no—
422C2×D4—1416no—
423C2×D4—1416no—
424C2×D4—1416no—
425C2×D4—2832no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—74no—
435C22×C4—74no—
436C22×C4—72no—
437C22×C4—72no—
438C22×C4—716no—
439C22×C4—716no—
440C22×C4—74no—
441C22×C4—72no—
442C22×C4—74no—
443C22×C4—148no—
444C22×C4—1432no—
445C22×C4—148no—
446C22×C4—148no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—142no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—142no—
453C22×C4—144no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—144no—
458C22×C4—2816no—
459C22×C4—288no—
460C22×C4—288no—
461C22×C4—2816no—
462C22×C4D4×D714yes—
463C22×C4D4×D714yes—
464C22×C4C23×D712yes—
465C22×C4C23×D714yes—
466C22×C4C23×D712yes—
467C22×C4C23×D712yes—
468C4⋊C4—28no—
469C4⋊C4—148no—
470C4⋊C4—148no—
471C4⋊C4—1416no—
472C4⋊C4—148no—
473C4⋊C4—1416no—
474C4⋊C4—148no—
475C4⋊C4—288no—
476C22⋊C4—216no—
477C22⋊C4—48no—
478C22⋊C4—1416no—
479C22⋊C4—1416no—
480C22⋊C4—1416no—
481C22⋊C4—288no—
482C2×C7—84no—
483C2×C7#261818yescomplemented
484C2×C7C22×#12111yescharacteristic
485C2×C7C2×#53411yescharacteristic
486C2×C7C2×#53611yescharacteristic
487C2×C7C2×#53712yes—
488C2×C7C2×#53912yes—
489D7—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—716no—
505C23—74no—
506C23—74no—
507C23—74no—
508C23—148no—
509C23—1416no—
510C23—1416no—
511C23—144no—
512C23—144no—
513C23—148no—
514C23C2×D4×D712yes—
515C23C2×D4×D712yes—
516C23#654814yes—
517C23D7×#4112yes—
518C23#655414yes—
519C23C24×D711yescharacteristic
520D4—416no—
521D4—2832no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—74no—
535C2×C4—74no—
536C2×C4—1432no—
537C2×C4—148no—
538C2×C4—148no—
539C2×C4—142no—
540C2×C4—144no—
541C2×C4—142no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—144no—
546C2×C4—144no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—2816no—
551C2×C4—2816no—
552C2×C4—284no—
553C2×C4—288no—
554C2×C4—288no—
555C2×C4C2×D4×D714yes—
556C2×C4C2×D4×D714yes—
557C7C2×#261811yescharacteristic, complemented, Sylow
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—78no—
566C22—816no—
567C22—144no—
568C22—148no—
569C22—1432no—
570C22—148no—
571C22—144no—
572C22#9028014yes—
573C22D7×#12114yes—
574C22#9029918yes—
575C22#9035118yes—
576C22#9036614yes—
577C22C22×D4×D711yescharacteristic
578C22C22×D4×D711yescharacteristic
579C22C2×#654812yes—
580C22C2×D7×#4111yescharacteristic
581C22C2×#655412yes—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—44no—
586C4—144no—
587C4—144no—
588C4—282no—
589C4—284no—
590C4—284no—
591C2—84no—
592C2—148no—
593C2#1173456918yesdirect product factor
594C2C2×#9028011yescharacteristic
595C2C2×D7×#12111yescharacteristic
596C2C2×#9029912yes—
597C2C2×#9035112yes—
598C2C2×#9036611yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#394C2
socleC24×C7C24
prime radicalC22×#6501C2
hypercenterC2×#394D7
hypocenterC7C2×#2618
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×D7×C4⋊C4C2semidirect
3. C22×#6446C2semidirect
4. C2×C7×#394C2semidirect
5. C2×#89994C2semidirect
6. C2×#89755C2semidirect
7. C2×#89728C2semidirect
8. C2×#89720C2semidirect
9. C2×#89720C2semidirect
10. #11734569C2direct (N × H)
11. C2×#89439C2semidirect
12. C2×#89433C2semidirect
35. C2×D7×C4⋊C4C22semidirect
37. C2×#6446C22semidirect
38. C7×#394C22semidirect
40. #89994C22semidirect
44. #89755C22semidirect
45. #89728C22semidirect
46. #89720C22semidirect
47. #89720C22semidirect
48. #89439C22semidirect
49. #89433C22semidirect
75. C22×C4×D7D4semidirect
123. C2×#394D7semidirect
189. C2×C4×D7C2×D4semidirect
234. #394C2×D7semidirect
483. C2×C7#2618semidirect
557. C7C2×#2618semidirect
593. C2#11734569direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#65012
C22×D7×C4⋊C42
C22×#64462
C2×C7×#3942
C2×#899942
C2×#897552
C2×#897282
C2×#897202
#117345692
C2×#894392
C2×#894332
C2×#26187