Finite Group Databasebeta

← #13516700

Subgroups — #13516700

Order 1792.

2098 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C7×#2365C211yesmaximal, characteristic, complemented
3#11740674C211yesmaximal, characteristic, complemented
4#11739933C212yesmaximal
5Q8×Dic7⋊C4C212yesmaximal, complemented
6#11734056C212yesmaximal
7#11733934C212yesmaximal, complemented
8#11733642C212yesmaximal
9#11733426C212yesmaximal, complemented
10#11732423C214yesmaximal, complemented
11#11732389C214yesmaximal
12#11731207C212yesmaximal
13#11731198C211yesmaximal, characteristic
14#11731198C211yesmaximal, characteristic
15#11731096C211yesmaximal, characteristic, complemented
16#11731095C211yesmaximal, characteristic, complemented
17#11731086C212yesmaximal, complemented
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C4×C7×Q8C2212yes—
20C2×#6509C2212yes—
21C2×Q8×Dic7C2212yes—
22C2×Q8×Dic7C2212yes—
23#90419C4116yescomplemented
24C2×#6460C2212yes—
25C2×#6455C2212yes—
26C2×#6453C2212yes—
27C2×#6450C2212yes—
28C2×#6450C2212yes—
29C2×#6435C2212yes—
30C2×#6435C2212yes—
31C2×C42×D7C2211yescharacteristic
32C2×#6432C2211yescharacteristic
33C2×C4×Dic14C2212yes—
34C7×#391C2212yes—
35C7×#389C2212yes—
36C7×#387C2214yes—
37C4×C7×C4⋊C4C2212yes—
38C4×C7×C4⋊C4C2211yescharacteristic
39C4×C7×C4⋊C4C2211yescharacteristic
40#90002C2212yes—
41#89762C2214yes—
42#89759C2211yescharacteristic
43#89759C2211yescharacteristic
44#89759C2211yescharacteristic
45#89759C2211yescharacteristic
46#89757C2212yes—
47#89753C2214yes—
48#89748C2211yescharacteristic
49#89748C2214yes—
50#89748C2211yescharacteristic
51#89748C2211yescharacteristic
52#89748C2211yescharacteristic
53C4⋊C4×Dic7C2212yes—
54C4⋊C4×Dic7C2212yes—
55#89746C2212yes—
56#89745C2212yes—
57#89745C2214yes—
58#89744C2214yes—
59#89712C2214yes—
60C4×C4⋊Dic7C2211yescharacteristic
61C4×C4⋊Dic7C2211yescharacteristic
62C4×C4⋊Dic7C2212yes—
63#89705C2214yes—
64C4×Dic7⋊C4C2214yes—
65C4×Dic7⋊C4C2212yes—
66C4×Dic7⋊C4C2212yes—
67#89701C2212yes—
68#89700C2214yes—
69#89700C2212yes—
70#89442C2214yes—
71#89441C2214yes—
72#89439C2212yes—
73#89439C2212yes—
74#89426C2214yes—
75#89426C2214yes—
76#89425C2214yes—
77#89425C2214yes—
78#89424C2214yes—
79#89422C2214yes—
80#89422C2212yes—
81#89422C2212yes—
82#89421C2214yes—
83#89419C2214yes—
84#34374—71nomaximal, Sylow
85C22×C7×Q8C2312yes—
86C22×C4×D7C2311yescharacteristic
87C22×C4×D7C2311yescharacteristic
88C22×C4×D7C2311yescharacteristic
89C22×Dic14C2312yes—
90C7×C4⋊Q8C2×C418yes—
91C4×C7×Q8—44no—
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C4⋊C4C2314yes—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C4⋊C4C2312yes—
99C2×C42×C7C2312yes—
100C2×C42×C7C2312yes—
101C2×C42×C7C2314yes—
102C2×C42×C7C2311yescharacteristic
103#6509C2×C4116yes—
104Q8×Dic7—44no—
105Q8×Dic7C2×C4116yes—
106C2×(C2×D7)⋊C4C2312yes—
107C2×(C2×D7)⋊C4C2312yes—
108C2×(C2×D7)⋊C4C2312yes—
109C2×(C2×D7)⋊C4C2312yes—
110C2×(C2×D7)⋊C4C2314yes—
111C2×C4⋊Dic7C2311yescharacteristic
112C2×C4⋊Dic7C2311yescharacteristic
113C2×C4⋊Dic7C2311yescharacteristic
114C2×C4⋊Dic7C2311yescharacteristic
115C2×C4⋊Dic7C2314yes—
116C2×C4⋊Dic7C2312yes—
117C2×C4⋊Dic7C2312yes—
118C2×C4⋊Dic7C2314yes—
119C2×Dic7⋊C4C2312yes—
120C2×Dic7⋊C4C2312yes—
121C2×Dic7⋊C4C2312yes—
122C2×Dic7⋊C4C2312yes—
123C2×Dic7⋊C4C2314yes—
124C2×Dic7⋊C4C2312yes—
125C2×Dic7⋊C4C2312yes—
126C2×Dic7⋊C4C2312yes—
127C2×Dic7⋊C4C2312yes—
128C2×Dic7⋊C4C2314yes—
129C2×Dic7⋊C4C2314yes—
130C2×C4×Dic7C2312yes—
131C2×C4×Dic7C2312yes—
132C2×C4×Dic7C2314yes—
133C2×C4×Dic7C2314yes—
134C2×C4×Dic7C2314yes—
135C2×C4×Dic7C2311yescharacteristic
136C2×C4×Dic7C2311yescharacteristic
137C2×C4×Dic7C2311yescharacteristic
138C2×C4×Dic7C2314yes—
139#6460C2×C4116yes—
140#6455C2×C4116yes—
141#6453C2×C4116yes—
142#6450—44no—
143#6450C2×C4116yes—
144#6435—44no—
145#6435—44no—
146C42×D7C2×C418yes—
147#6432C2×C418yes—
148C4×Dic14—44no—
149C7×#95C2314yes—
150C7×#95C2314yes—
151C7×#95C2312yes—
152C7×#95C2312yes—
153C7⋊#95C2312yes—
154C7⋊#95C2314yes—
155C7⋊#95C2314yes—
156C7⋊#95C2312yes—
157C7⋊#95C2312yes—
158C7⋊#95C2314yes—
159C7⋊#95C2312yes—
160C7⋊#95C2312yes—
161C7⋊#95C2314yes—
162C7⋊#95C2312yes—
163C7⋊#95C2314yes—
164C7⋊#95C2314yes—
165C2×#533—71no—
166#2382—74no—
167#2365—71no—
168#2365—71no—
169#2365—71no—
170#2365D711yescharacteristic, complemented
171#2355—72no—
172#2355—72no—
173Q8×C4⋊C4—72no—
174Q8×C4⋊C4—72no—
175#2323—72no—
176#2323—72no—
177#2323—72no—
178#2323—72no—
179#2314—74no—
180#2285—71no—
181#2285—71no—
182C23×D7C2411yescharacteristic
183C2×C7×Q8—24no—
184C2×C7×Q8—24no—
185C2×C7×Q8C22×C418yes—
186C22×C4×C7C2412yes—
187C22×C4×C7C2412yes—
188C22×C4×C7C2412yes—
189C22×C4×C7C2412yes—
190C22×C4×C7C2411yescharacteristic
191C22×C4×C7C2414yes—
192C22×C4×C7C2411yescharacteristic
193C22×C4×C7C2411yescharacteristic
194C22×Dic7C2412yes—
195C22×Dic7C2412yes—
196C22×Dic7C2412yes—
197C22×Dic7C2412yes—
198C22×Dic7C2414yes—
199C22×Dic7C2411yescharacteristic
200C22×Dic7C2411yescharacteristic
201C22×Dic7C2411yescharacteristic
202C2×C4×D7—22no—
203C2×C4×D7—22no—
204C2×C4×D7—22no—
205C2×C4×D7—22no—
206C2×C4×D7—22no—
207C2×C4×D7—22no—
208C2×C4×D7C22×C414yes—
209C2×C4×D7C22×C414yes—
210C2×C4×D7C22×C414yes—
211C2×Dic14—24no—
212C2×Dic14—24no—
213C2×Dic14C22×C418yes—
214C7×C4⋊C4—22no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—22no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—44no—
219C7×C4⋊C4C22×C418yes—
220C7×C4⋊C4C22×C418yes—
221C42×C7—24no—
222C42×C7—24no—
223C42×C7—44no—
224C42×C7C22×C414yes—
225(C2×D7)⋊C4—42no—
226(C2×D7)⋊C4—42no—
227(C2×D7)⋊C4—42no—
228(C2×D7)⋊C4—42no—
229(C2×D7)⋊C4C22×C4116yes—
230C4⋊Dic7—22no—
231C4⋊Dic7—22no—
232C4⋊Dic7—22no—
233C4⋊Dic7—22no—
234C4⋊Dic7—44no—
235C4⋊Dic7C22×C418yes—
236C4⋊Dic7C22×C418yes—
237C4⋊Dic7C22×C4116yes—
238Dic7⋊C4—24no—
239Dic7⋊C4—24no—
240Dic7⋊C4—24no—
241Dic7⋊C4—24no—
242Dic7⋊C4—44no—
243Dic7⋊C4—42no—
244Dic7⋊C4—42no—
245Dic7⋊C4—42no—
246Dic7⋊C4—42no—
247Dic7⋊C4—44no—
248Dic7⋊C4C22×C4116yes—
249C4×Dic7—24no—
250C4×Dic7—24no—
251C4×Dic7—44no—
252C4×Dic7—44no—
253C4×Dic7—44no—
254C4×Dic7C22×C414yes—
255C4×Dic7C22×C414yes—
256C4×Dic7C22×C414yes—
257C4×Dic7C22×C4116yes—
258#533—716no—
259C2×C4⋊Q8—71no—
260C2×C4⋊Q8C2×D711yescharacteristic
261C2×#125—72no—
262C2×C22⋊Q8—72no—
263C2×C22⋊Q8—72no—
264C2×C4×Q8—72no—
265C2×C4×Q8—72no—
266C2×C4×Q8—72no—
267C2×C4×Q8—72no—
268C2×C4×Q8—72no—
269C2×C4×Q8C2×D712yes—
270C2×#117—72no—
271C2×#117—72no—
272C2×#117—72no—
273C22×C42—71no—
274#391—72no—
275#391—72no—
276#391—72no—
277#391C2×D712yes—
278#389—72no—
279#389—72no—
280#389—74no—
281#389—74no—
282#389—72no—
283#389—74no—
284#389—74no—
285#389C2×D712yes—
286#388—74no—
287#388—74no—
288#388—74no—
289#388—74no—
290#387—74no—
291#387—74no—
292#387—74no—
293#387—74no—
294#387—74no—
295#387—74no—
296#387—74no—
297#387C2×D714yes—
298#384—74no—
299#384—71no—
300#384—71no—
301#384—71no—
302#384—71no—
303#382—74no—
304#382—72no—
305#382—72no—
306#380—71no—
307#380—71no—
308#380—71no—
309#380—71no—
310#380—72no—
311#380—72no—
312C4×C4⋊C4—72no—
313C4×C4⋊C4—72no—
314C4×C4⋊C4—74no—
315C4×C4⋊C4—72no—
316C4×C4⋊C4—72no—
317C4×C4⋊C4—72no—
318C4×C4⋊C4—71no—
319C4×C4⋊C4—71no—
320C4×C4⋊C4—74no—
321C4×C4⋊C4C2×D712yes—
322C4×C4⋊C4C2×D711yescharacteristic
323C4×C4⋊C4C2×D711yescharacteristic
324C23×C7C2511yescharacteristic
325C22×D7—21no—
326C22×D7—21no—
327C22×D7—21no—
328C22×D7—21no—
329C22×D7—21no—
330C22×D7—21no—
331C22×D7C23×C412yes—
332C7×Q8—216no—
333C2×C4×C7—24no—
334C2×C4×C7—21no—
335C2×C4×C7—21no—
336C2×C4×C7—24no—
337C2×C4×C7—22no—
338C2×C4×C7—22no—
339C2×C4×C7—22no—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—22no—
344C2×C4×C7—22no—
345C2×C4×C7—22no—
346C2×C4×C7—22no—
347C2×C4×C7—22no—
348C2×C4×C7—22no—
349C2×C4×C7C23×C418yes—
350C2×C4×C7C23×C412yes—
351C2×C4×C7C23×C412yes—
352C2×C4×C7C23×C412yes—
353C2×C4×C7H(5,𝔽2)12yes—
354C2×C4×C7H(5,𝔽2)12yes—
355C2×C4×C72−1+412yes—
356C2×C4×C72−1+412yes—
357C2×Dic7—24no—
358C2×Dic7—21no—
359C2×Dic7—21no—
360C2×Dic7—24no—
361C2×Dic7—21no—
362C2×Dic7—21no—
363C2×Dic7—22no—
364C2×Dic7—22no—
365C2×Dic7—22no—
366C2×Dic7—22no—
367C2×Dic7—22no—
368C2×Dic7—22no—
369C2×Dic7—22no—
370C2×Dic7—22no—
371C2×Dic7—22no—
372C2×Dic7—22no—
373C2×Dic7—22no—
374C2×Dic7—22no—
375C2×Dic7C23×C412yes—
376C2×Dic7C23×C412yes—
377C2×Dic7C23×C418yes—
378C2×Dic7C23×C412yes—
379C2×Dic72−1+412yes—
380C2×Dic72−1+412yes—
381C4×D7—28no—
382C4×D7—28no—
383C4×D7—28no—
384Dic14—216no—
385C22×Q8—72no—
386C22×Q8C22×D712yes—
387C23×C4—71no—
388C23×C4—71no—
389C23×C4—71no—
390C4⋊Q8—78no—
391C4⋊Q8C4×D718yes—
392#125—716no—
393C22⋊Q8—716no—
394C22⋊Q8—716no—
395C4×Q8—44no—
396C4×Q8—716no—
397C4×Q8—716no—
398C4×Q8—284no—
399C4×Q8—284no—
400C4×Q8—284no—
401#117—716no—
402#117—284no—
403#117—284no—
404C2×C4⋊C4—72no—
405C2×C4⋊C4—74no—
406C2×C4⋊C4—74no—
407C2×C4⋊C4—72no—
408C2×C4⋊C4—71no—
409C2×C4⋊C4—71no—
410C2×C4⋊C4—72no—
411C2×C4⋊C4—72no—
412C2×C4⋊C4—72no—
413C2×C4⋊C4—72no—
414C2×C4⋊C4—71no—
415C2×C4⋊C4—71no—
416C2×C4⋊C4—72no—
417C2×C4⋊C4—72no—
418C2×C4⋊C4—72no—
419C2×C4⋊C4—72no—
420C2×C4⋊C4—74no—
421C2×C4⋊C4—74no—
422C2×C4⋊C4—74no—
423C2×C4⋊C4C22×D711yescharacteristic
424C2×C4⋊C4C22×D711yescharacteristic
425C2×C4⋊C4C22×D711yescharacteristic
426C2×C4⋊C4C22×D711yescharacteristic
427C2×C4⋊C4C22×D714yes—
428C2×C4⋊C4C22×D712yes—
429C2×C4⋊C4C22×D712yes—
430C2×C22⋊C4—74no—
431C2×C22⋊C4—72no—
432C2×C22⋊C4—72no—
433C2×C22⋊C4—72no—
434C2×C22⋊C4—72no—
435C2×C42—74no—
436C2×C42—71no—
437C2×C42—71no—
438C2×C42—71no—
439C2×C42—78no—
440C2×C42—72no—
441C2×C42—72no—
442C2×C42—74no—
443C2×C42—74no—
444C2×C42—74no—
445C2×C42C22×D712yes—
446C2×C42C22×D712yes—
447C2×C42C22×D714yes—
448C2×C42C22×D711yescharacteristic
449#95—72no—
450#95—72no—
451#95—72no—
452#95—72no—
453#95—72no—
454#95—72no—
455#95—74no—
456#95—74no—
457#95—74no—
458#95—74no—
459#95—74no—
460#95—74no—
461#95C22×D714yes—
462#95C22×D712yes—
463#95C22×D712yes—
464#95C22×D714yes—
465C22×C7C24×C411yescharacteristic
466C22×C7C2×H(5,𝔽2)11yescharacteristic
467C22×C7C2×H(5,𝔽2)11yescharacteristic
468C22×C7C2×2−1+411yescharacteristic
469C22×C7C2×2−1+411yescharacteristic
470C22×C7C2×2−1+411yescharacteristic
471C22×C7C2×2−1+411yescharacteristic
472C2×D7—22no—
473C2×D7—22no—
474C2×D7—22no—
475C2×D7—41no—
476C2×D7—41no—
477C2×D7—41no—
478C2×D7—41no—
479C4×C7—22no—
480C4×C7—28no—
481C4×C7—42no—
482C4×C7—42no—
483C4×C7—42no—
484C4×C7—42no—
485C4×C7#52014yes—
486C4×C7#52014yes—
487Dic7—28no—
488Dic7—22no—
489Dic7—22no—
490Dic7—42no—
491Dic7—42no—
492Dic7—42no—
493Dic7—42no—
494Dic7#51914yes—
495C24—71no—
496C2×Q8—24no—
497C2×Q8—24no—
498C2×Q8—78no—
499C2×Q8—144no—
500C2×Q8—144no—
501C2×Q8C2×C4×D718yes—
502C22×C4—71no—
503C22×C4—74no—
504C22×C4—71no—
505C22×C4—71no—
506C22×C4—74no—
507C22×C4—74no—
508C22×C4—74no—
509C22×C4—72no—
510C22×C4—72no—
511C22×C4—72no—
512C22×C4—72no—
513C22×C4—142no—
514C22×C4—142no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4C23×D712yes—
520C22×C4C23×D712yes—
521C22×C4C23×D712yes—
522C22×C4C23×D712yes—
523C22×C4C23×D711yescharacteristic
524C22×C4C23×D714yes—
525C22×C4C23×D711yescharacteristic
526C22×C4C23×D711yescharacteristic
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—716no—
533C4⋊C4—78no—
534C4⋊C4—716no—
535C4⋊C4—78no—
536C4⋊C4—144no—
537C4⋊C4—144no—
538C4⋊C4—144no—
539C4⋊C4—144no—
540C4⋊C4—142no—
541C4⋊C4—142no—
542C4⋊C4—142no—
543C4⋊C4—142no—
544C4⋊C4—282no—
545C4⋊C4—282no—
546C4⋊C4—282no—
547C4⋊C4—282no—
548C4⋊C4—284no—
549C4⋊C4—284no—
550C4⋊C4—284no—
551C4⋊C4C2×C4×D718yes—
552C4⋊C4C2×C4×D718yes—
553C22⋊C4—716no—
554C22⋊C4—282no—
555C22⋊C4—282no—
556C22⋊C4—282no—
557C22⋊C4—282no—
558C42—24no—
559C42—24no—
560C42—44no—
561C42—74no—
562C42—74no—
563C42—74no—
564C42—716no—
565C42—144no—
566C42—144no—
567C42—284no—
568C42—284no—
569C42—284no—
570C42C2×C4×D714yes—
571C2×C7C2×#51911yescharacteristic
572C2×C7C2×#52011yescharacteristic
573C2×C7C2×#52011yescharacteristic
574C2×C7#354811yescharacteristic
575C2×C7#355011yescharacteristic
576C2×C7#355011yescharacteristic
577C2×C7#355611yescharacteristic
578D7—42no—
579C23—72no—
580C23—141no—
581C23—141no—
582C23—141no—
583C23—141no—
584C23—141no—
585C23—141no—
586C23C24×D711yescharacteristic
587Q8—216no—
588Q8—1416no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—72no—
606C2×C4—72no—
607C2×C4—72no—
608C2×C4—72no—
609C2×C4—72no—
610C2×C4—78no—
611C2×C4—148no—
612C2×C4—141no—
613C2×C4—141no—
614C2×C4—144no—
615C2×C4—144no—
616C2×C4—141no—
617C2×C4—141no—
618C2×C4—148no—
619C2×C4—148no—
620C2×C4—142no—
621C2×C4—142no—
622C2×C4—142no—
623C2×C4—142no—
624C2×C4—142no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—142no—
631C2×C4—142no—
632C2×C4C22×C4×D712yes—
633C2×C4C22×C4×D718yes—
634C2×C4C22×C4×D712yes—
635C2×C4C22×C4×D712yes—
636C2×C4#654812yes—
637C2×C4#655112yes—
638C2×C4#655312yes—
639C2×C4#655412yes—
640C7#3437411yescharacteristic, complemented, Sylow
641C22—142no—
642C22—142no—
643C22—142no—
644C22—281no—
645C22—281no—
646C22—281no—
647C22—281no—
648C22C23×C4×D711yescharacteristic
649C22C2×#654811yescharacteristic
650C22C2×#655111yescharacteristic
651C22C2×#655311yescharacteristic
652C22C2×#655411yescharacteristic
653C22D7×2−1+411yescharacteristic
654C22D7×2−1+411yescharacteristic
655C4—28no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—74no—
662C4—142no—
663C4—142no—
664C4—148no—
665C4—282no—
666C4—282no—
667C4—282no—
668C4—282no—
669C4#9023714yes—
670C4#9026514yes—
671C2—282no—
672C2C2×#9023711yescharacteristic
673C2C2×#9026511yescharacteristic
674C2D7×#51911yescharacteristic
675C2#1174376111yescharacteristic
676C2#1174382311yescharacteristic
677C2#1174406311yescharacteristic
678C2#1174409511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2365C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2365D7
hypocenterC7#34374
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. C7×#2365C2semidirect
3. #11740674C2semidirect
5. Q8×Dic7⋊C4C2semidirect
7. #11733934C2semidirect
9. #11733426C2semidirect
10. #11732423C2semidirect
15. #11731096C2semidirect
16. #11731095C2semidirect
17. #11731086C2semidirect
23. #90419C4semidirect
170. #2365D7semidirect
640. C7#34374semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
C7×#23652
#117406742
#117399332
Q8×Dic7⋊C42
#117340562
#117339342
#117336422
#117334262
#117324232
#117323892
#117312072
#117311982
#117310962
#117310952
#117310862
#343747