Finite Group Databasebeta

← C2×#11739178 #14233504

Subgroups — C2×#11739178 #14233504

Order 1792.

1778 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90412C211yesmaximal, characteristic
2C2×#90376C211yesmaximal, characteristic
3C2×C7×#491C211yesmaximal, characteristic
4C2×#89862C211yesmaximal, characteristic
5C2×#89835C211yesmaximal, characteristic
6#11739178C2116yesmaximal, direct product factor
7C2×#89565C211yesmaximal, characteristic
8C2×#89565C211yesmaximal, characteristic
9C2×#89569C211yesmaximal, characteristic
10C2×#89564C211yesmaximal, characteristic
11C2×#89571C211yesmaximal, characteristic
12C2×#89563C211yesmaximal, characteristic
13C2×#89561C211yesmaximal, characteristic
14C2×#89562C211yesmaximal, characteristic
15C2×#89488C211yesmaximal, characteristic
16C2×#89473C211yesmaximal, characteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#125C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6507C2211yescharacteristic
22C2×#6507C2211yescharacteristic
23C22×#1144C2211yescharacteristic
24C22×#1144C2211yescharacteristic
25C22×Dic28C2211yescharacteristic
26#90412—24no—
27#90412C2218yes—
28#90376—24no—
29#90376C2218yes—
30C2×#6453—21no—
31C2×#6453C2211yescharacteristic
32C2×#6452—21no—
33C2×#6452—21no—
34C2×#6452—21no—
35C2×#6452—21no—
36C2×#6451—21no—
37C2×#6451C2211yescharacteristic
38C2×#6450—21no—
39C2×#6450—21no—
40C2×#6450C2211yescharacteristic
41C2×#6450C2211yescharacteristic
42C2×C4×Dic14C2211yescharacteristic
43C2×C4×Dic14C2211yescharacteristic
44C2×C4×Dic14C2211yescharacteristic
45C7×#491C2218yes—
46C2×C7×#103C2211yescharacteristic
47C2×C7×#103C2211yescharacteristic
48C2×C7×#103C2211yescharacteristic
49C2×C7×#103C2211yescharacteristic
50C2×C7×#97C2211yescharacteristic
51C2×#6409C2211yescharacteristic
52C2×#6409C2211yescharacteristic
53C2×#6391C2211yescharacteristic
54C2×#6390C2211yescharacteristic
55C2×#6390C2211yescharacteristic
56C2×#6389C2211yescharacteristic
57C2×Dic7⋊C8C2211yescharacteristic
58C2×Dic7⋊C8C2211yescharacteristic
59C2×C8×Dic7C2211yescharacteristic
60#89862C2218yes—
61#89835C2218yes—
62C2×#6384C2211yescharacteristic
63C2×#6384C2211yescharacteristic
64C2×#6382C2211yescharacteristic
65C2×#6381C2211yescharacteristic
66C2×C4⋊(C7⋊C8)C2211yescharacteristic
67#89571C2218yes—
68#89569C2218yes—
69#89565C2218yes—
70#89565C2218yes—
71#89564C2218yes—
72#89563C2218yes—
73#89562C2218yes—
74#89561C2218yes—
75#89488C2218yes—
76#89473C2218yes—
77C2×#3374—71nomaximal, Sylow
78C22×C7×Q8C2311yescharacteristic
79C22×C7×Q8C2311yescharacteristic
80C22×Dic14—21no—
81C22×Dic14D411yescharacteristic
82C22×Dic14C2311yescharacteristic
83C22×Dic14C2311yescharacteristic
84C22×C7×C8C2311yescharacteristic
85C22×C7×C8C2311yescharacteristic
86C7×C4⋊Q8—22no—
87C7×C4⋊Q8C2314yes—
88C7×#125—22no—
89C7×#125C2314yes—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C42×C7C2311yescharacteristic
97Q8×Dic7—22no—
98Q8×Dic7—22no—
99Q8×Dic7C2314yes—
100Q8×Dic7C2314yes—
101#6507—22no—
102#6507—22no—
103#6507C2314yes—
104#6507C2314yes—
105C2×#1144—24no—
106C2×#1144—24no—
107C2×#1144C2314yes—
108C2×#1144C2314yes—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7—21no—
111C2×C4⋊Dic7—21no—
112C2×C4⋊Dic7D411yescharacteristic
113C2×C4⋊Dic7C2311yescharacteristic
114C2×C4⋊Dic7C2311yescharacteristic
115C2×Dic7⋊C4—21no—
116C2×Dic7⋊C4—21no—
117C2×Dic7⋊C4—21no—
118C2×Dic7⋊C4—21no—
119C2×Dic7⋊C4—21no—
120C2×Dic7⋊C4—21no—
121C2×Dic7⋊C4—21no—
122C2×Dic7⋊C4—21no—
123C2×Dic7⋊C4—21no—
124C2×Dic7⋊C4—21no—
125C2×Dic7⋊C4D411yescharacteristic
126C2×Dic7⋊C4D411yescharacteristic
127C2×C4×Dic7—21no—
128C2×C4×Dic7—21no—
129C2×C4×Dic7—21no—
130C2×C4×Dic7—21no—
131C2×C4×Dic7—21no—
132C2×C4×Dic7—21no—
133C2×C4×Dic7C2311yescharacteristic
134C2×C4×Dic7C2311yescharacteristic
135C22×C7⋊C8C2311yescharacteristic
136C22×C7⋊C8C2311yescharacteristic
137C2×Dic28—24no—
138C2×Dic28C2314yes—
139#6453—28no—
140#6453—22no—
141#6453C2314yes—
142#6452—28no—
143#6452—28no—
144#6452—28no—
145#6452—28no—
146#6451—28no—
147#6451—22no—
148#6451C2314yes—
149#6450—28no—
150#6450—28no—
151#6450—22no—
152#6450—22no—
153#6450C2314yes—
154#6450C2314yes—
155C4×Dic14—22no—
156C4×Dic14—22no—
157C4×Dic14D414yes—
158C4×Dic14C2314yes—
159C4×Dic14C2314yes—
160C4×Dic14C2314yes—
161C7×#103C2314yes—
162C7×#103C2314yes—
163C7×#103C2314yes—
164C7×#103C2314yes—
165C7×#97C2314yes—
166#6409C2314yes—
167#6409C2314yes—
168#6391C2314yes—
169#6390C2314yes—
170#6390C2314yes—
171#6389C2314yes—
172Dic7⋊C8C2314yes—
173Dic7⋊C8C2314yes—
174C8×Dic7C2314yes—
175#6384C2314yes—
176#6384C2314yes—
177#6382C2314yes—
178#6381C2314yes—
179C4⋊(C7⋊C8)C2314yes—
180C2×#557—71no—
181C2×#557—71no—
182#3374—716no—
183C2×#491—71no—
184C2×#491D711yescharacteristic
185C2×#487—71no—
186C2×#479—71no—
187C2×#479—71no—
188C2×#475—71no—
189C2×#462—71no—
190C2×#462—71no—
191C2×#462—71no—
192C2×#446—71no—
193C2×#441—71no—
194C2×#441—71no—
195C2×C4×Q16—71no—
196C2×C7×Q8—21no—
197C2×C7×Q8—21no—
198C2×C7×Q8—22no—
199C2×C7×Q8—22no—
200C2×C7×Q8—22no—
201C2×C7×Q8—22no—
202C2×C7×Q8C2412yes—
203C2×C7×Q8C2412yes—
204C22×C4×C7—21no—
205C22×C4×C7—21no—
206C22×C4×C7—21no—
207C22×C4×C7—21no—
208C22×C4×C7C2×D411yescharacteristic
209C22×C4×C7C2×D411yescharacteristic
210C22×C4×C7C2411yescharacteristic
211C22×Dic7—21no—
212C22×Dic7—21no—
213C22×Dic7—21no—
214C22×Dic7—21no—
215C22×Dic7C2×D411yescharacteristic
216C22×Dic7C2×D411yescharacteristic
217C22×Dic7C2×D411yescharacteristic
218C22×Dic7C2×D411yescharacteristic
219C2×Dic14—21no—
220C2×Dic14—22no—
221C2×Dic14—24no—
222C2×Dic14—22no—
223C2×Dic14—22no—
224C2×Dic14—21no—
225C2×Dic14—24no—
226C2×Dic14—44no—
227C2×Dic14C2×D414yes—
228C2×Dic14#4114yes—
229C2×Dic14C2412yes—
230C2×Dic14C2412yes—
231C2×C7×C8—22no—
232C2×C7×C8—22no—
233C2×C7×C8C2412yes—
234C2×C7×C8C2412yes—
235C7×C4⋊C4—21no—
236C7×C4⋊C4—24no—
237C7×C4⋊C4—24no—
238C7×C4⋊C4—21no—
239C7×C4⋊C4—24no—
240C7×C4⋊C4—24no—
241C7×C4⋊C4C2412yes—
242C7×C4⋊C4C2412yes—
243C42×C7C2×D412yes—
244C42×C7C2412yes—
245#1144—28no—
246#1144—28no—
247C4⋊Dic7—21no—
248C4⋊Dic7—21no—
249C4⋊Dic7—24no—
250C4⋊Dic7—24no—
251C4⋊Dic7—24no—
252C4⋊Dic7C2×D414yes—
253C4⋊Dic7C2412yes—
254C4⋊Dic7C2412yes—
255Dic7⋊C4—24no—
256Dic7⋊C4—24no—
257Dic7⋊C4—24no—
258Dic7⋊C4—24no—
259Dic7⋊C4—24no—
260Dic7⋊C4—24no—
261Dic7⋊C4—24no—
262Dic7⋊C4—24no—
263Dic7⋊C4—24no—
264Dic7⋊C4—24no—
265Dic7⋊C4C2×D414yes—
266Dic7⋊C4C2×D414yes—
267C4×Dic7—24no—
268C4×Dic7—24no—
269C4×Dic7—24no—
270C4×Dic7—24no—
271C4×Dic7—24no—
272C4×Dic7—24no—
273C4×Dic7C2×D412yes—
274C4×Dic7C2×D412yes—
275C4×Dic7C2412yes—
276C4×Dic7C2412yes—
277C2×C7⋊C8—22no—
278C2×C7⋊C8—22no—
279C2×C7⋊C8C2412yes—
280C2×C7⋊C8C2412yes—
281Dic28—28no—
282C22×Q16—71no—
283C22×Q16—71no—
284C22×Q16—71no—
285#557—78no—
286#557—78no—
287#557—144no—
288#557—144no—
289C2×C4⋊Q8—71no—
290C2×C4⋊Q8—71no—
291C2×C4⋊Q8—71no—
292C2×C4⋊Q8—141no—
293C2×C4⋊Q8C2×D711yescharacteristic
294C2×#125—71no—
295C2×#125—141no—
296C2×#125—141no—
297C2×#125—141no—
298C2×#125—141no—
299C2×#125—141no—
300C2×#125C2×D711yescharacteristic
301C2×C4×Q8—71no—
302C2×C4×Q8—71no—
303C2×C4×Q8—71no—
304C2×C4×Q8—71no—
305C2×C4×Q8—71no—
306C2×C4×Q8—71no—
307C2×C4×Q8—71no—
308C2×C4×Q8—141no—
309C2×C4×Q8—141no—
310#491—78no—
311#491C2×D718yes—
312#487—78no—
313#479—78no—
314#479—78no—
315#475—78no—
316#462—78no—
317#462—78no—
318#462—78no—
319#446—78no—
320#441—78no—
321#441—78no—
322C4×Q16—78no—
323C2×#107—71no—
324C2×#106—71no—
325C2×#106—71no—
326C2×C4⋊C8—71no—
327C2×C4⋊C8—71no—
328C2×C4⋊C8—71no—
329C2×#103—71no—
330C2×#103—71no—
331C2×#103—71no—
332C2×#103—71no—
333C2×#103—71no—
334C2×#103—71no—
335C2×#103C2×D711yescharacteristic
336C2×#103C2×D711yescharacteristic
337C2×#103C2×D711yescharacteristic
338C2×#103C2×D711yescharacteristic
339C2×#97—71no—
340C2×#97C2×D711yescharacteristic
341C2×C4×C8—71no—
342C23×C7C22×D411yescharacteristic
343C7×Q8—22no—
344C7×Q8—22no—
345C7×Q8—22no—
346C7×Q8—22no—
347C7×Q8—42no—
348C7×Q8—42no—
349C2×C4×C7—22no—
350C2×C4×C7—22no—
351C2×C4×C7—22no—
352C2×C4×C7—22no—
353C2×C4×C7—22no—
354C2×C4×C7—22no—
355C2×C4×C7—42no—
356C2×C4×C7—42no—
357C2×C4×C7—42no—
358C2×C4×C7—42no—
359C2×C4×C7C22×D412yes—
360C2×C4×C7C22×D412yes—
361C2×C4×C7C22×D411yescharacteristic
362C2×C4×C7C2×#4112yes—
363C2×C4×C72−1+412yes—
364C2×C4×C7C2511yescharacteristic
365C2×Dic7—24no—
366C2×Dic7—22no—
367C2×Dic7—22no—
368C2×Dic7—22no—
369C2×Dic7—22no—
370C2×Dic7—22no—
371C2×Dic7—22no—
372C2×Dic7—22no—
373C2×Dic7—42no—
374C2×Dic7—42no—
375C2×Dic7—42no—
376C2×Dic7#13714yes—
377C2×Dic7C22×D412yes—
378C2×Dic7C22×D412yes—
379C2×Dic7C22×D412yes—
380C2×Dic7C22×D412yes—
381Dic14—24no—
382Dic14—22no—
383Dic14—28no—
384Dic14—22no—
385Dic14—22no—
386Dic14—42no—
387Dic14—48no—
388Dic14C2×#4114yes—
389C7×C8—22no—
390C7×C8—22no—
391C7⋊C8—22no—
392C7⋊C8—22no—
393C22×Q8—71no—
394C22×Q8—71no—
395C22×Q8—71no—
396C22×Q8—141no—
397C22×Q8C22×D711yescharacteristic
398C22×Q8C22×D711yescharacteristic
399C2×Q16—74no—
400C2×Q16—74no—
401C2×Q16—74no—
402C2×Q16—144no—
403C2×Q16—144no—
404C2×Q16—144no—
405C22×C8—71no—
406C22×C8—71no—
407C22×C8C22×D711yescharacteristic
408C22×C8C22×D711yescharacteristic
409C4⋊Q8—22no—
410C4⋊Q8—74no—
411C4⋊Q8—74no—
412C4⋊Q8—74no—
413C4⋊Q8—142no—
414C4⋊Q8—142no—
415C4⋊Q8—142no—
416C4⋊Q8—148no—
417C4⋊Q8C22×D714yes—
418#125—22no—
419#125—74no—
420#125—142no—
421#125—148no—
422#125—148no—
423#125—148no—
424#125—148no—
425#125—148no—
426#125C22×D714yes—
427C4×Q8—74no—
428C4×Q8—74no—
429C4×Q8—74no—
430C4×Q8—74no—
431C4×Q8—74no—
432C4×Q8—74no—
433C4×Q8—74no—
434C4×Q8—74no—
435C4×Q8—142no—
436C4×Q8—142no—
437C4×Q8—142no—
438C4×Q8—142no—
439C4×Q8—148no—
440C4×Q8—148no—
441C4×Q8—142no—
442C4×Q8—142no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—71no—
448C2×C4⋊C4—71no—
449C2×C4⋊C4—71no—
450C2×C4⋊C4—71no—
451C2×C4⋊C4—71no—
452C2×C4⋊C4—141no—
453C2×C4⋊C4—141no—
454C2×C4⋊C4—141no—
455C2×C4⋊C4—141no—
456C2×C4⋊C4—141no—
457C2×C4⋊C4—141no—
458C2×C4⋊C4—141no—
459C2×C4⋊C4—141no—
460C2×C4⋊C4—141no—
461C2×C4⋊C4—141no—
462C2×C4⋊C4—141no—
463C2×C4⋊C4—141no—
464C2×C4⋊C4—141no—
465C2×C4⋊C4C22×D711yescharacteristic
466C2×C4⋊C4C22×D711yescharacteristic
467C2×C42—71no—
468C2×C42—71no—
469C2×C42—141no—
470C2×C42—141no—
471C2×C42—141no—
472C2×C42—141no—
473C2×C42—141no—
474C2×C42—141no—
475C2×C42C22×D711yescharacteristic
476#107—74no—
477#106—74no—
478#106—74no—
479C4⋊C8—74no—
480C4⋊C8—74no—
481C4⋊C8—74no—
482#103—74no—
483#103—74no—
484#103—74no—
485#103—74no—
486#103—74no—
487#103—74no—
488#103C22×D714yes—
489#103C22×D714yes—
490#103C22×D714yes—
491#103C22×D714yes—
492#97—74no—
493#97C22×D714yes—
494C4×C8—74no—
495C22×C7#54812yes—
496C22×C7C2×#13712yes—
497C22×C7#57812yes—
498C22×C7C23×D411yescharacteristic
499C4×C7—22no—
500C4×C7—22no—
501C4×C7—42no—
502C4×C7—42no—
503C4×C7—42no—
504C4×C7—42no—
505C4×C7#54812yes—
506C4×C7C22×#4111yescharacteristic
507C4×C7C2×2−1+411yescharacteristic
508Dic7—24no—
509Dic7—22no—
510Dic7—22no—
511Dic7—22no—
512Dic7—42no—
513Dic7—42no—
514Dic7—42no—
515Dic7C2×#13714yes—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—22no—
519C2×Q8—22no—
520C2×Q8—22no—
521C2×Q8—22no—
522C2×Q8—74no—
523C2×Q8—74no—
524C2×Q8—72no—
525C2×Q8—72no—
526C2×Q8—144no—
527C2×Q8—144no—
528C2×Q8—141no—
529C2×Q8—141no—
530C2×Q8—142no—
531C2×Q8—142no—
532C2×Q8—142no—
533C2×Q8—284no—
534C2×Q8C23×D712yes—
535C2×Q8C23×D712yes—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—71no—
541C22×C4—71no—
542C22×C4—71no—
543C22×C4—71no—
544C22×C4—141no—
545C22×C4—141no—
546C22×C4—141no—
547C22×C4—141no—
548C22×C4D4×D711yescharacteristic
549C22×C4D4×D711yescharacteristic
550C22×C4C23×D711yescharacteristic
551Q16—148no—
552Q16—148no—
553Q16—148no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—72no—
557C2×C8—72no—
558C2×C8—142no—
559C2×C8—142no—
560C2×C8C23×D712yes—
561C2×C8C23×D712yes—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—24no—
565C4⋊C4—24no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—74no—
569C4⋊C4—74no—
570C4⋊C4—74no—
571C4⋊C4—72no—
572C4⋊C4—72no—
573C4⋊C4—144no—
574C4⋊C4—144no—
575C4⋊C4—144no—
576C4⋊C4—144no—
577C4⋊C4—144no—
578C4⋊C4—144no—
579C4⋊C4—144no—
580C4⋊C4—144no—
581C4⋊C4—144no—
582C4⋊C4—144no—
583C4⋊C4—144no—
584C4⋊C4—144no—
585C4⋊C4—144no—
586C4⋊C4—141no—
587C4⋊C4—141no—
588C4⋊C4C23×D712yes—
589C4⋊C4C23×D712yes—
590C42—72no—
591C42—72no—
592C42—72no—
593C42—72no—
594C42—144no—
595C42—144no—
596C42—144no—
597C42—144no—
598C42—144no—
599C42—144no—
600C42D4×D712yes—
601C42C23×D712yes—
602C2×C7#337414yescomplemented
603C2×C7C2×#54811yescharacteristic
604C2×C7C22×#13711yescharacteristic
605C2×C7C2×#57811yescharacteristic
606C23C2×D4×D711yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—22no—
610Q8—22no—
611Q8—42no—
612Q8—42no—
613Q8—74no—
614Q8—148no—
615Q8—144no—
616Q8—142no—
617Q8—142no—
618Q8—142no—
619Q8—288no—
620Q8—282no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—74no—
632C2×C4—72no—
633C2×C4—72no—
634C2×C4—72no—
635C2×C4—72no—
636C2×C4—144no—
637C2×C4—142no—
638C2×C4—142no—
639C2×C4—142no—
640C2×C4—142no—
641C2×C4—142no—
642C2×C4—142no—
643C2×C4—142no—
644C2×C4—282no—
645C2×C4—282no—
646C2×C4—282no—
647C2×C4C2×D4×D712yes—
648C2×C4C2×D4×D712yes—
649C2×C4C2×D4×D711yescharacteristic
650C2×C4D7×#4112yes—
651C2×C4#655412yes—
652C2×C4C24×D711yescharacteristic
653C8—22no—
654C8—22no—
655C8—142no—
656C8—142no—
657C7C2×#337411yescharacteristic, complemented, Sylow
658C22#9036812yes—
659C22D7×#13712yes—
660C22#9047012yes—
661C22C22×D4×D711yescharacteristic
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—74no—
669C4—142no—
670C4—142no—
671C4—142no—
672C4—144no—
673C4—282no—
674C4—282no—
675C4—282no—
676C4#9036812yes—
677C4C2×D7×#4111yescharacteristic
678C4C2×#655411yescharacteristic
679C2#1173917814yesdirect product factor
680C2C2×#9036811yescharacteristic
681C2C2×D7×#13711yescharacteristic
682C2C2×#9047011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#491C2
socle, prime radicalC23×C7C22×D4
hypercenterC2×#491D7
hypocenterC7C2×#3374
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
6. #11739178C2direct (N × H)
602. C2×C7#3374semidirect
657. C7C2×#3374semidirect
679. C2#11739178direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#904122
C2×#903762
C2×C7×#4912
C2×#898622
C2×#898352
#117391782
C2×#895652
C2×#895692
C2×#895642
C2×#895712
C2×#895632
C2×#895612
C2×#895622
C2×#894882
C2×#894732
C2×#33747