Finite Group Databasebeta

← #13319710

Subgroups — #13319710

Order 1792.

1210 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90265C211yesmaximal, characteristic
2#11736780C212yesmaximal, complemented
3C7×#1467C211yesmaximal, characteristic, complemented
4#11729379C211yesmaximal, characteristic, complemented
5#11728580C212yesmaximal, complemented
6#11725899C212yesmaximal, complemented
7#11725898C212yesmaximal, complemented
8#11725897C212yesmaximal, complemented
9D7×#324C212yesmaximal, complemented
10C22×#1161C411yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×Q8×Dic7C411yescharacteristic
13#90265—24no—
14#90265—24no—
15C2×#6456C411yescharacteristic
16C2×#6456C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×D7×C4⋊C4C411yescharacteristic
19C2×D7×C4⋊C4C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×C4×D28C411yescharacteristic
22C2×C4×D28C411yescharacteristic
23C2×C4×D28C2211yescharacteristic
24C2×#6435C411yescharacteristic
25C2×#6435C411yescharacteristic
26C2×#6435C2211yescharacteristic
27C7×#432C2212yes—
28#89698C2212yes—
29#89608—24no—
30#89498C2214yes—
31D7×C22⋊C8C2214yes—
32#89478—24no—
33C7×#324C2212yes—
34C7×#324C2212yes—
35#89323C2212yes—
36#89323C2212yes—
37#89264C2214yes—
38#89263C2214yes—
39#17368—71nomaximal, Sylow
40C22×C7×Q8C2×C411yescharacteristic
41C2×#1161—24no—
42C2×#1161—24no—
43C2×#1161—24no—
44C22×D28C2×C411yescharacteristic
45C22×D28C2×C411yescharacteristic
46C22×D28C2×C411yescharacteristic
47C22×C4×D7C2×C411yescharacteristic
48C22×C4×D7C2×C411yescharacteristic
49C22×C4×D7C2×C411yescharacteristic
50C22×C4×D7C2×C411yescharacteristic
51C22×C4×D7C2×C411yescharacteristic
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7C2×C411yescharacteristic
54C22×C4×D7C2311yescharacteristic
55C22×C4×D7C2311yescharacteristic
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C2×C7×C4⋊C4C2×C411yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C42×C7C2×C411yescharacteristic
62C2×C42×C7C2×C411yescharacteristic
63C2×C42×C7C2311yescharacteristic
64Q8×Dic7—24no—
65C2×(C2×D7)⋊C4C2×C411yescharacteristic
66C2×(C2×D7)⋊C4C2×C411yescharacteristic
67C2×(C2×D7)⋊C4C2×C411yescharacteristic
68C2×(C2×D7)⋊C4C2×C411yescharacteristic
69C2×(C2×D7)⋊C4C2×C411yescharacteristic
70C2×(C2×D7)⋊C4C2311yescharacteristic
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×C4⋊Dic7C2×C411yescharacteristic
73C2×C4⋊Dic7C2×C411yescharacteristic
74C2×Dic7⋊C4C2×C411yescharacteristic
75C2×Dic7⋊C4C2×C411yescharacteristic
76C2×Dic7⋊C4C2×C411yescharacteristic
77C2×Dic7⋊C4C2311yescharacteristic
78C2×C4×Dic7C2×C411yescharacteristic
79C2×C4×Dic7C2×C411yescharacteristic
80C2×C4×Dic7C2×C411yescharacteristic
81C2×#1131—24no—
82C2×C8×D7—24no—
83#6456—24no—
84#6456—24no—
85#6456—24no—
86D7×C4⋊C4—24no—
87D7×C4⋊C4—24no—
88D7×C4⋊C4—24no—
89C4×D28—22no—
90C4×D28—22no—
91C4×D28—22no—
92C4×D28—22no—
93C4×D28D414yes—
94C4×D28D414yes—
95#6435—22no—
96#6435—22no—
97#6435—22no—
98#6435—22no—
99#6435—22no—
100#6435—22no—
101C7×C4⋊C8—22no—
102C7×C22⋊C8C2314yes—
103C7×#97—22no—
104C22⋊(C7⋊C8)C2314yes—
105(C2×D7)⋊C8—24no—
106(C2×D7)⋊C8—24no—
107C4⋊(C7⋊C8)—22no—
108#6377—22no—
109C2×#520—71no—
110#2973—72no—
111#1467—71no—
112#1467D711yescharacteristic, complemented
113#1465—72no—
114#1464—72no—
115#1463—72no—
116#1459—72no—
117C2×#324—72no—
118C23×D7C22×C411yescharacteristic
119C23×D7C22×C411yescharacteristic
120C23×D7C22×C411yescharacteristic
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C22×C411yescharacteristic
130C22×C4×C7C2411yescharacteristic
131C22×Dic7C22×C411yescharacteristic
132C22×Dic7C22×C411yescharacteristic
133C22×Dic7C22×C411yescharacteristic
134C22×Dic7C22×C411yescharacteristic
135C22×Dic7C22×C411yescharacteristic
136#1161—232no—
137C2×D28—22no—
138C2×D28—22no—
139C2×D28—22no—
140C2×D28—22no—
141C2×D28—22no—
142C2×D28—22no—
143C2×D28—22no—
144C2×D28C22⋊C414yes—
145C2×D28M1614yescomplemented
146C2×C4×D7—22no—
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—22no—
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—81no—
159C2×C4×D7—81no—
160C2×C4×D7—81no—
161C2×C4×D7—81no—
162C2×C4×D7—81no—
163C2×C4×D7—81no—
164C2×C4×D7M1614yescomplemented
165C2×C4×D7C2×D414yes—
166C2×C4×D7C2×D414yes—
167C2×C7×C8—24no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C42×C7—21no—
172C42×C7—21no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7C2×D412yes—
176C42×C7C2×D412yes—
177(C2×D7)⋊C4—22no—
178(C2×D7)⋊C4—22no—
179(C2×D7)⋊C4—21no—
180(C2×D7)⋊C4—21no—
181(C2×D7)⋊C4—22no—
182(C2×D7)⋊C4C22⋊C412yes—
183(C2×D7)⋊C4C22⋊C412yes—
184(C2×D7)⋊C4C22⋊C412yes—
185(C2×D7)⋊C4C22⋊C412yes—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7C22⋊C414yes—
189Dic7⋊C4—21no—
190Dic7⋊C4—21no—
191Dic7⋊C4—22no—
192Dic7⋊C4—22no—
193Dic7⋊C4—22no—
194C4×Dic7—22no—
195C4×Dic7—22no—
196C4×Dic7—22no—
197C2×C7⋊C8—24no—
198#1131—84no—
199C8×D7—84no—
200C22×#41—71no—
201#520—144no—
202#520—144no—
203C2×C4×Q8—71no—
204C2×C4×Q8C2×D711yescharacteristic
205C2×C4×D4—71no—
206C2×C4×D4—71no—
207C2×C4×D4—71no—
208C2×C4×D4—71no—
209C2×C4×D4—71no—
210C2×C4×D4—71no—
211C2×#117—71no—
212C2×#117—71no—
213C2×#117—71no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—71no—
216C22×C4⋊C4—71no—
217#435—144no—
218#435—144no—
219#432—72no—
220#432C2×D712yes—
221#407—74no—
222C2×C22⋊C8—74no—
223#324—72no—
224#324—72no—
225#324—74no—
226#324C2×D712yes—
227#324C2×D712yes—
228#323—74no—
229C23×C7C23×C411yescharacteristic
230C22×D7—21no—
231C22×D7—21no—
232C22×D7—21no—
233C22×D7—21no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—41no—
239C22×D7—41no—
240C22×D7C2×C22⋊C412yes—
241C22×D7C2×C22⋊C412yes—
242C22×D7C2×M1612yes—
243C22×D7C2×M1612yes—
244C22×D7C2×M1612yes—
245C7×Q8—28no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7C2×C22⋊C412yes—
259C2×C4×C7C2×C22⋊C411yescharacteristic
260C2×C4×C7C2×C22⋊C411yescharacteristic
261C2×C4×C7C2×M1612yes—
262C2×C4×C7C2×M1612yes—
263C2×C4×C7C22×D412yes—
264C2×Dic7—21no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7C2×C22⋊C412yes—
273C2×Dic7C2×C22⋊C412yes—
274C2×Dic7C2×M1612yes—
275D28—28no—
276D28—28no—
277D28—28no—
278C4×D7—28no—
279C4×D7—28no—
280C4×D7—28no—
281C4×D7—44no—
282C4×D7—44no—
283C4×D7—82no—
284C4×D7—82no—
285C4×D7—82no—
286C4×D7—82no—
287C7×C8—44no—
288C7⋊C8—44no—
289C2×#41—144no—
290C2×#41—144no—
291C2×#41—144no—
292C22×Q8C4×D711yescharacteristic
293C22×D4—71no—
294C22×D4—71no—
295C22×D4—71no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—71no—
299C23×C4—71no—
300C23×C4—71no—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4—71no—
304C23×C4—71no—
305C2×M16—144no—
306C22×C8—144no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—144no—
310C4×D4—74no—
311C4×D4—74no—
312C4×D4—142no—
313C4×D4—142no—
314C4×D4—142no—
315C4×D4—142no—
316C4×D4—144no—
317C4×D4—144no—
318C4×D4—144no—
319#117—142no—
320#117—142no—
321#117—142no—
322#117—142no—
323#117—142no—
324#117—142no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—144no—
333C2×C4⋊C4—144no—
334C2×C4⋊C4—144no—
335C2×C4⋊C4C4×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C22⋊C4—71no—
339C2×C22⋊C4—71no—
340C2×C22⋊C4—71no—
341C2×C22⋊C4—71no—
342C2×C22⋊C4—71no—
343C2×C22⋊C4—71no—
344C2×C42—71no—
345C2×C42—71no—
346C2×C42—71no—
347C2×C42C4×D711yescharacteristic
348C2×C42C4×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C4⋊C8—22no—
351C4⋊C8—142no—
352C22⋊C8—74no—
353C22⋊C8—144no—
354C22⋊C8—144no—
355C22⋊C8C22×D714yes—
356#97—22no—
357#97—142no—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7C22×C22⋊C411yescharacteristic
361C22×C7C22×M1611yescharacteristic
362C22×C7C22×M1611yescharacteristic
363C2×D7—22no—
364C2×D7—22no—
365C2×D7—22no—
366C2×D7—22no—
367C2×D7—22no—
368C2×D7—22no—
369C2×D7—42no—
370C2×D7—42no—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—41no—
375C2×D7—41no—
376C2×D7—42no—
377C2×D7#40714yes—
378C4×C7—22no—
379C4×C7—22no—
380C4×C7—22no—
381C4×C7—22no—
382C4×C7—21no—
383C4×C7—21no—
384C4×C7—41no—
385C4×C7—41no—
386Dic7—22no—
387Dic7—22no—
388Dic7—22no—
389Dic7—41no—
390Dic7—41no—
391C24—71no—
392C24—71no—
393C24—71no—
394#41—1432no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×D4—74no—
399C2×D4—74no—
400C2×D4—142no—
401C2×D4—142no—
402C2×D4—142no—
403C2×D4—142no—
404C2×D4—142no—
405C2×D4—142no—
406C2×D4—142no—
407C22×C4—71no—
408C22×C4—71no—
409C22×C4—71no—
410C22×C4—74no—
411C22×C4—74no—
412C22×C4—74no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—561no—
428C22×C4—561no—
429C22×C4—561no—
430C22×C4—561no—
431C22×C4—561no—
432C22×C4—561no—
433C22×C4C2×C4×D711yescharacteristic
434C22×C4C2×C4×D711yescharacteristic
435C22×C4C2×C4×D711yescharacteristic
436C22×C4C2×C4×D711yescharacteristic
437C22×C4C2×C4×D711yescharacteristic
438C22×C4C2×C4×D711yescharacteristic
439C22×C4C23×D711yescharacteristic
440M16—564no—
441C2×C8—24no—
442C2×C8—144no—
443C2×C8—564no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—74no—
448C4⋊C4—142no—
449C4⋊C4—142no—
450C4⋊C4—142no—
451C4⋊C4—142no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—142no—
455C22⋊C4—72no—
456C22⋊C4—72no—
457C22⋊C4—72no—
458C22⋊C4—72no—
459C22⋊C4—142no—
460C22⋊C4—142no—
461C22⋊C4—141no—
462C22⋊C4—141no—
463C22⋊C4—142no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—142no—
469C42—142no—
470C42—142no—
471C42D4×D712yes—
472C42D4×D712yes—
473C2×C7—21no—
474C2×C7—21no—
475C2×C7C2×#40711yescharacteristic
476C2×C7#288311yescharacteristic
477C2×C7#288911yescharacteristic
478D7—24no—
479D7—42no—
480D7—42no—
481C23—72no—
482C23—72no—
483C23—72no—
484C23—72no—
485C23—72no—
486C23—141no—
487C23—141no—
488C23—141no—
489C23—141no—
490C23—281no—
491C23—281no—
492C23—281no—
493C23—281no—
494C23—281no—
495C23—281no—
496C23C22×C4×D711yescharacteristic
497Q8—28no—
498D4—148no—
499D4—148no—
500D4—148no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—72no—
514C2×C4—72no—
515C2×C4—72no—
516C2×C4—148no—
517C2×C4—148no—
518C2×C4—148no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—284no—
524C2×C4—284no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—562no—
530C2×C4—562no—
531C2×C4—562no—
532C2×C4—562no—
533C2×C4D7×C22⋊C411yescharacteristic
534C2×C4D7×C22⋊C411yescharacteristic
535C2×C4D7×C22⋊C412yes—
536C2×C4D7×M1612yes—
537C2×C4D7×M1612yes—
538C2×C4C2×D4×D712yes—
539C8—44no—
540C8—284no—
541C7#1736811yescharacteristic, complemented, Sylow
542C22—21no—
543C22—21no—
544C22—74no—
545C22—142no—
546C22—142no—
547C22—142no—
548C22—142no—
549C22—142no—
550C22—142no—
551C22—282no—
552C22—282no—
553C22—282no—
554C22—282no—
555C22—282no—
556C22—282no—
557C22—281no—
558C22—281no—
559C22C2×D7×C22⋊C411yescharacteristic
560C22C2×D7×M1611yescharacteristic
561C22C2×D7×M1611yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—142no—
571C4—142no—
572C4—142no—
573C4—281no—
574C4—281no—
575C2—21no—
576C2—21no—
577C2—144no—
578C2—282no—
579C2—282no—
580C2D7×#40711yescharacteristic
581C2#1173613911yescharacteristic
582C2#1173615111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1467C2
socleC22×C7C22×C22⋊C4
prime radicalC22×#1161C4
cosocleC22×C4×C7C24
hypercenter#1467D7
hypocenterC7#17368
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #11736780C2semidirect
3. C7×#1467C2semidirect
4. #11729379C2semidirect
5. #11728580C2semidirect
6. #11725899C2semidirect
7. #11725898C2semidirect
8. #11725897C2semidirect
9. D7×#324C2semidirect
112. #1467D7semidirect
145. C2×D28M16semidirect
164. C2×C4×D7M16semidirect
541. C7#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#902652
#117367802
C7×#14672
#117293792
#117285802
#117258992
#117258982
#117258972
D7×#3242
#173687