Finite Group Databasebeta

← #13319697

Subgroups — #13319697

Order 1792.

1350 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90177C211yesmaximal, characteristic
2#11736047C212yesmaximal, complemented
3C2×C7×#323C211yesmaximal, characteristic, complemented
4C2×#89320C211yesmaximal, characteristic, complemented
5#11727673C212yesmaximal, complemented
6#11725875C212yesmaximal, complemented
7#11725874C212yesmaximal, complemented
8#11725857C212yesmaximal, complemented
9#11725856C212yesmaximal, complemented
10C23×C22⋊D7C411yescharacteristic
11C22×C7×C22⋊C4C2211yescharacteristic
12C22×C22⋊Dic7C411yescharacteristic
13C2×C4×C22⋊D7C411yescharacteristic
14C2×C4×C22⋊D7C411yescharacteristic
15C2×C4×C22⋊D7C2211yescharacteristic
16C2×C4×C22⋊D7C2211yescharacteristic
17#90177—24no—
18#90177—24no—
19C2×(C22⋊D7)⋊C4C411yescharacteristic
20C2×(C22⋊D7)⋊C4C411yescharacteristic
21C2×(C22⋊D7)⋊C4C2211yescharacteristic
22C2×(C22⋊D7)⋊C4C2211yescharacteristic
23C2×D7×C22⋊C4C411yescharacteristic
24C2×D7×C22⋊C4C2211yescharacteristic
25C2×#6440C411yescharacteristic
26C2×#6440C2211yescharacteristic
27C2×C7×C22⋊C8C2212yescomplemented
28C2×C22⋊(C7⋊C8)C2212yescomplemented
29#89883—24no—
30#89502—24no—
31#89498C2214yescomplemented
32#89491C2214yescomplemented
33C7×#323C2212yes—
34C7×#323C2212yescomplemented
35#89320C2212yes—
36#89320C2212yescomplemented
37#89264C2214yescomplemented
38#89263C2214yescomplemented
39#17367—71nomaximal, Sylow
40C25×C7C2×C411yescharacteristic
41C24×D7C2×C411yescharacteristic
42C23×C4×C7C2×C411yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C22×C22⋊D7—24no—
45C22×C22⋊D7—22no—
46C22×C22⋊D7—22no—
47C22×C22⋊D7—24no—
48C22×C22⋊D7C2×C411yescharacteristic
49C22×C22⋊D7C2×C411yescharacteristic
50C22×C22⋊D7C2×C411yescharacteristic
51C22×C22⋊D7C2×C411yescharacteristic
52C23×Dic7C2×C411yescharacteristic
53C23×Dic7C2×C411yescharacteristic
54C23×Dic7C2×C411yescharacteristic
55C22×C4×D7C2×C411yescharacteristic
56C22×C4×D7C2×C411yescharacteristic
57C22×C4×D7C2×C411yescharacteristic
58C22×C4×D7C2311yescharacteristic
59C22×C7×C8—22no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4C2×C411yescharacteristic
63C2×C7×C22⋊C4C2×C411yescharacteristic
64C2×C7×C22⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4C2311yescharacteristic
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7C2×C411yescharacteristic
69C2×C22⋊Dic7C2×C411yescharacteristic
70C2×C22⋊Dic7C2×C411yescharacteristic
71C2×C22⋊Dic7C2×C411yescharacteristic
72C4×C22⋊D7—22no—
73C4×C22⋊D7—22no—
74C4×C22⋊D7—22no—
75C4×C22⋊D7—22no—
76C4×C22⋊D7—22no—
77C4×C22⋊D7—22no—
78C4×C22⋊D7D414yes—
79C4×C22⋊D7D414yes—
80C2×(C2×D7)⋊C4C2×C411yescharacteristic
81C2×(C2×D7)⋊C4C2×C411yescharacteristic
82C2×(C2×D7)⋊C4C2×C411yescharacteristic
83C2×(C2×D7)⋊C4C2311yescharacteristic
84C2×Dic7⋊C4C2×C411yescharacteristic
85C2×Dic7⋊C4C2×C411yescharacteristic
86C2×Dic7⋊C4C2×C411yescharacteristic
87C2×Dic7⋊C4C2311yescharacteristic
88C2×C4×Dic7C2×C411yescharacteristic
89C2×C4×Dic7C2×C411yescharacteristic
90C2×C4×Dic7C2×C411yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C22×C7⋊C8—22no—
93C2×#1131—24no—
94(C22⋊D7)⋊C4—24no—
95(C22⋊D7)⋊C4—24no—
96(C22⋊D7)⋊C4—24no—
97(C22⋊D7)⋊C4—24no—
98D7×C22⋊C4—24no—
99D7×C22⋊C4—24no—
100#6440—24no—
101#6440—24no—
102C7×C22⋊C8—22no—
103C7×C22⋊C8C2314yes—
104C22⋊(C7⋊C8)—22no—
105C22⋊(C7⋊C8)C2314yes—
106(C2×D7)⋊C8—24no—
107#6389—24no—
108Dic7⋊C8—24no—
109C2×#518—71no—
110#2973—72no—
111#1466—72no—
112#1465—72no—
113#1464—72no—
114#1462—72no—
115#1459—72no—
116C2×#323—71no—
117C2×#323D711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7—21no—
120C24×C7—22no—
121C24×C7—81no—
122C24×C7—81no—
123C24×C7M1614yescomplemented
124C24×C7C22×C411yescharacteristic
125C24×C7C22×C411yescharacteristic
126C24×C7C22×C411yescharacteristic
127C23×D7—22no—
128C23×D7—22no—
129C23×D7—22no—
130C23×D7—81no—
131C23×D7—81no—
132C23×D7C22×C411yescharacteristic
133C23×D7C22×C411yescharacteristic
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—81no—
137C22×C4×C7—81no—
138C22×C4×C7C22×C411yescharacteristic
139C22×C4×C7C22×C411yescharacteristic
140C22×C4×C7C22×C411yescharacteristic
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2411yescharacteristic
144C2×C22⋊D7—216no—
145C2×C22⋊D7—216no—
146C2×C22⋊D7—21no—
147C2×C22⋊D7—21no—
148C2×C22⋊D7—22no—
149C2×C22⋊D7—22no—
150C2×C22⋊D7—22no—
151C2×C22⋊D7—22no—
152C2×C22⋊D7—22no—
153C2×C22⋊D7—22no—
154C2×C22⋊D7—22no—
155C2×C22⋊D7—22no—
156C2×C22⋊D7—22no—
157C2×C22⋊D7C22⋊C412yes—
158C2×C22⋊D7C22⋊C412yes—
159C2×C22⋊D7M1614yescomplemented
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—81no—
166C22×Dic7—81no—
167C22×Dic7C22×C411yescharacteristic
168C22×Dic7C22×C411yescharacteristic
169C22×Dic7C22×C411yescharacteristic
170C22×Dic7C22×C411yescharacteristic
171C22×Dic7C22×C411yescharacteristic
172C22×Dic7C22×C411yescharacteristic
173C2×C4×D7—22no—
174C2×C4×D7—22no—
175C2×C4×D7—22no—
176C2×C4×D7—81no—
177C2×C4×D7—81no—
178C2×C4×D7—81no—
179C2×C4×D7—81no—
180C2×C4×D7C2×D414yes—
181C2×C7×C8—24no—
182C2×C7×C8—82no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C22⋊Dic7—21no—
188C22⋊Dic7—21no—
189C22⋊Dic7—22no—
190C22⋊Dic7—22no—
191C22⋊Dic7C22⋊C412yes—
192C22⋊Dic7C22⋊C412yes—
193(C2×D7)⋊C4—22no—
194(C2×D7)⋊C4—22no—
195(C2×D7)⋊C4—22no—
196(C2×D7)⋊C4C22⋊C414yes—
197Dic7⋊C4—22no—
198Dic7⋊C4—22no—
199Dic7⋊C4—22no—
200Dic7⋊C4C22⋊C414yes—
201C4×Dic7—22no—
202C4×Dic7—22no—
203C4×Dic7—22no—
204C4×Dic7C2×D414yes—
205C2×C7⋊C8—24no—
206C2×C7⋊C8—82no—
207#1131—84no—
208C23×D4—71no—
209#518—144no—
210#518—144no—
211C2×C4×D4—71no—
212C2×C4×D4—71no—
213C2×C4×D4—71no—
214C2×C4×D4—71no—
215C2×C4×D4—71no—
216C2×C4×D4—71no—
217C2×C4×D4—71no—
218C2×C4×D4—71no—
219C2×#117—71no—
220C2×#117—71no—
221C22×C22⋊C4—71no—
222C22×C22⋊C4—71no—
223C22×C22⋊C4—71no—
224C22×C22⋊C4C2×D711yescharacteristic
225#435—144no—
226#435—144no—
227#432—74no—
228#407—74no—
229C2×C22⋊C8—72no—
230C2×C22⋊C8C2×D712yescomplemented
231#324—74no—
232#323—72no—
233#323—72no—
234#323—74no—
235#323C2×D712yes—
236#323C2×D712yescomplemented
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—24no—
242C23×C7—24no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—81no—
250C23×C7—81no—
251C23×C7—81no—
252C23×C7—81no—
253C23×C7—84no—
254C23×C7—84no—
255C23×C7C2×C22⋊C411yescharacteristic
256C23×C7C2×C22⋊C411yescharacteristic
257C23×C7C2×M1612yes—
258C23×C7C2×M1612yes—
259C23×C7C2×M1612yes—
260C23×C7C2×M1612yes—
261C23×C7C23×C411yescharacteristic
262C22×D7—21no—
263C22×D7—21no—
264C22×D7—21no—
265C22×D7—21no—
266C22×D7—28no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—82no—
272C22×D7—82no—
273C22×D7—84no—
274C22×D7—84no—
275C22×D7C2×C22⋊C412yes—
276C22×D7C2×M1612yes—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—81no—
286C2×C4×C7—81no—
287C2×C4×C7C2×C22⋊C412yes—
288C2×C4×C7C22×D412yes—
289C22⋊D7—28no—
290C22⋊D7—28no—
291C22⋊D7—24no—
292C22⋊D7—24no—
293C22⋊D7—24no—
294C22⋊D7—24no—
295C2×Dic7—28no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—82no—
307C2×Dic7—82no—
308C2×Dic7C2×C22⋊C412yes—
309C2×Dic7C2×C22⋊C412yes—
310C2×Dic7C2×C22⋊C412yes—
311C2×Dic7C2×M1612yes—
312C4×D7—44no—
313C4×D7—82no—
314C4×D7—82no—
315C4×D7—82no—
316C7×C8—44no—
317C7⋊C8—44no—
318C25—71no—
319C25C4×D711yescharacteristic
320C22×D4—71no—
321C22×D4—71no—
322C22×D4—71no—
323C22×D4—71no—
324C22×D4—142no—
325C22×D4—142no—
326C22×D4—144no—
327C22×D4—144no—
328C23×C4—71no—
329C23×C4—71no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C23×C4—71no—
334C23×C4—71no—
335C23×C4C4×D711yescharacteristic
336C23×C4C22×D711yescharacteristic
337C2×M16—144no—
338C22×C8—22no—
339C22×C8—142no—
340C4×D4—74no—
341C4×D4—74no—
342C4×D4—144no—
343C4×D4—144no—
344C4×D4—144no—
345C4×D4—144no—
346C4×D4—142no—
347C4×D4—142no—
348C4×D4—142no—
349C4×D4—142no—
350C4×D4—142no—
351C4×D4—142no—
352#117—144no—
353#117—144no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—71no—
361C2×C22⋊C4—71no—
362C2×C22⋊C4—71no—
363C2×C22⋊C4—71no—
364C2×C22⋊C4—71no—
365C2×C22⋊C4—71no—
366C2×C22⋊C4—71no—
367C2×C22⋊C4—71no—
368C2×C22⋊C4—144no—
369C2×C22⋊C4—144no—
370C2×C22⋊C4—142no—
371C2×C22⋊C4—142no—
372C2×C22⋊C4C4×D711yescharacteristic
373C2×C22⋊C4C4×D711yescharacteristic
374C2×C22⋊C4C22×D711yescharacteristic
375C2×C22⋊C4C22×D711yescharacteristic
376C2×C42—71no—
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—71no—
380C4⋊C8—144no—
381C22⋊C8—22no—
382C22⋊C8—74no—
383C22⋊C8—144no—
384C22⋊C8—142no—
385C22⋊C8C22×D714yes—
386#97—144no—
387C22×C7—21no—
388C22×C7—21no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—42no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—84no—
406C22×C7—84no—
407C22×C7#40712yes—
408C22×C7#40712yes—
409C22×C7C22×C22⋊C411yescharacteristic
410C22×C7C22×M1611yescharacteristic
411C22×C7C22×M1611yescharacteristic
412C2×D7—22no—
413C2×D7—22no—
414C2×D7—22no—
415C2×D7—22no—
416C2×D7—42no—
417C2×D7—42no—
418C2×D7—42no—
419C2×D7—41no—
420C2×D7—41no—
421C2×D7—42no—
422C2×D7—88no—
423C4×C7—22no—
424C4×C7—22no—
425C4×C7—41no—
426C4×C7—41no—
427Dic7—22no—
428Dic7—22no—
429Dic7—22no—
430Dic7—22no—
431Dic7—41no—
432Dic7—41no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—71no—
437C24—71no—
438C24—81no—
439C24—81no—
440C24—142no—
441C24—142no—
442C24—142no—
443C24—561no—
444C24—561no—
445C24#113114yescomplemented
446C24C2×C4×D711yescharacteristic
447C24C2×C4×D711yescharacteristic
448C24C2×C4×D711yescharacteristic
449C2×D4—72no—
450C2×D4—72no—
451C2×D4—74no—
452C2×D4—142no—
453C2×D4—142no—
454C2×D4—142no—
455C2×D4—142no—
456C2×D4—1416no—
457C2×D4—1416no—
458C2×D4—141no—
459C2×D4—141no—
460C2×D4—142no—
461C2×D4—142no—
462C2×D4—142no—
463C2×D4—142no—
464C2×D4—142no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—74no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—561no—
485C22×C4—561no—
486C22×C4—561no—
487C22×C4—561no—
488C22×C4—561no—
489C22×C4—561no—
490C22×C4C2×C4×D711yescharacteristic
491C22×C4C2×C4×D711yescharacteristic
492C22×C4C2×C4×D711yescharacteristic
493C22×C4D4×D712yes—
494C22×C4D4×D712yes—
495C22×C4C23×D711yescharacteristic
496M16—564no—
497C2×C8—24no—
498C2×C8—82no—
499C2×C8—144no—
500C2×C8—562no—
501C4⋊C4—74no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—72no—
510C22⋊C4—72no—
511C22⋊C4—74no—
512C22⋊C4—142no—
513C22⋊C4—142no—
514C22⋊C4—142no—
515C22⋊C4—142no—
516C22⋊C4—141no—
517C22⋊C4—141no—
518C22⋊C4—142no—
519C42—74no—
520C42—142no—
521C42—142no—
522C42—142no—
523C2×C7—21no—
524C2×C7—21no—
525C2×C7—22no—
526C2×C7—22no—
527C2×C7—42no—
528C2×C7—42no—
529C2×C7C2×#40711yescharacteristic
530C2×C7#288311yescharacteristic
531C2×C7#288811yescharacteristic
532D7—42no—
533D7—42no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—24no—
539C23—24no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—72no—
547C23—72no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23—84no—
553C23—84no—
554C23—148no—
555C23—141no—
556C23—141no—
557C23—141no—
558C23—141no—
559C23—281no—
560C23—281no—
561C23—281no—
562C23—281no—
563C23—562no—
564C23—562no—
565C23—564no—
566C23—564no—
567C23D7×C22⋊C411yescharacteristic
568C23D7×C22⋊C411yescharacteristic
569C23C2×#113112yes—
570C23C2×#113112yes—
571C23C2×#113112yes—
572C23D7×M1612yes—
573C23C22×C4×D711yescharacteristic
574D4—148no—
575D4—148no—
576D4—144no—
577D4—144no—
578D4—144no—
579D4—144no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—72no—
589C2×C4—72no—
590C2×C4—72no—
591C2×C4—72no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—148no—
595C2×C4—141no—
596C2×C4—141no—
597C2×C4—141no—
598C2×C4—141no—
599C2×C4—141no—
600C2×C4—141no—
601C2×C4—284no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—562no—
607C2×C4—562no—
608C2×C4—562no—
609C2×C4—562no—
610C2×C4—562no—
611C2×C4D7×C22⋊C412yes—
612C2×C4C2×D4×D712yes—
613C8—44no—
614C8—284no—
615C7#1736711yescharacteristic, complemented, Sylow
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—84no—
635C22—84no—
636C22—142no—
637C22—142no—
638C22—142no—
639C22—142no—
640C22—282no—
641C22—282no—
642C22—282no—
643C22—282no—
644C22—281no—
645C22—281no—
646C22—568no—
647C22#8949812yes—
648C22#8949812yes—
649C22C2×D7×C22⋊C411yescharacteristic
650C22C22×#113111yescharacteristic
651C22C2×D7×M1611yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—142no—
657C4—142no—
658C4—142no—
659C4—142no—
660C4—281no—
661C4—281no—
662C2—21no—
663C2—21no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2—42no—
668C2—282no—
669C2—282no—
670C2C2×#8949811yescharacteristic
671C2#1173613311yescharacteristic
672C2#1173614211yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#323C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C22⋊D7C4
cosocleC22×C4×C7C24
hypercenterC2×#323D7
hypocenterC7#17367
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #11736047C2semidirect
3. C2×C7×#323C2semidirect
4. C2×#89320C2semidirect
5. #11727673C2semidirect
6. #11725875C2semidirect
7. #11725874C2semidirect
8. #11725857C2semidirect
9. #11725856C2semidirect
27. C2×C7×C22⋊C8C22semidirect
28. C2×C22⋊(C7⋊C8)C22semidirect
31. #89498C22semidirect
32. #89491C22semidirect
34. C7×#323C22semidirect
36. #89320C22semidirect
37. #89264C22semidirect
38. #89263C22semidirect
117. C2×#323D7semidirect
123. C24×C7M16semidirect
159. C2×C22⋊D7M16semidirect
230. C2×C22⋊C8C2×D7semidirect
236. #323C2×D7semidirect
445. C24#1131semidirect
615. C7#17367semidirect

Maximal subgroups

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

subgroupindex
C2×#901772
#117360472
C2×C7×#3232
C2×#893202
#117276732
#117258752
#117258742
#117258572
#117258562
#173677