Finite Group Databasebeta

← #13319724

Subgroups — #13319724

Order 1792.

1098 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90207C211yesmaximal, characteristic
2#11736759C212yesmaximal, complemented
3C7×#1464C211yesmaximal, characteristic
4#11729326C211yesmaximal, characteristic
5#11728584C212yesmaximal, complemented
6#11725881C212yesmaximal, complemented
7#11725879C212yesmaximal, complemented
8#11725856C212yesmaximal
9#11725854C212yesmaximal
10C23×Dic14C411yescharacteristic
11C2×C7×#117C2211yescharacteristic
12C2×#6489C411yescharacteristic
13#90207—24no—
14#90207—24no—
15C2×#6450C411yescharacteristic
16C2×#6450C411yescharacteristic
17C2×#6450C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×#6440C411yescharacteristic
20C2×#6440C411yescharacteristic
21C2×#6440C2211yescharacteristic
22C2×#6440C2211yescharacteristic
23C2×C4×Dic14C411yescharacteristic
24C2×C4×Dic14C411yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C7×#432C2212yes—
28#89698C2212yes—
29#89600—24no—
30#89491C2214yes—
31#89490C2214yes—
32#89473—24no—
33C7×#324C2212yes—
34C7×#323C2212yes—
35#89323C2212yes—
36#89320C2212yes—
37#89264C2214yes—
38#89264C2214yes—
39#17371—71nomaximal, Sylow
40C23×C4×C7C2×C411yescharacteristic
41C23×Dic7C2×C411yescharacteristic
42C23×Dic7C2×C411yescharacteristic
43C22×Dic14—24no—
44C22×Dic14—22no—
45C22×Dic14—22no—
46C22×Dic14—24no—
47C22×Dic14C2×C411yescharacteristic
48C22×Dic14C2×C411yescharacteristic
49C22×Dic14C2×C411yescharacteristic
50C22×Dic14C2×C411yescharacteristic
51C7×#117—22no—
52C7×#117—22no—
53C2×C7×C4⋊C4C2×C411yescharacteristic
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C22⋊C4C2×C411yescharacteristic
56C2×C7×C22⋊C4C2311yescharacteristic
57C2×C42×C7C2×C411yescharacteristic
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7C2×C411yescharacteristic
60C2×C22⋊Dic7C2×C411yescharacteristic
61#6489—22no—
62#6489—22no—
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×C4⋊Dic7C2×C411yescharacteristic
65C2×Dic7⋊C4C2×C411yescharacteristic
66C2×Dic7⋊C4C2×C411yescharacteristic
67C2×Dic7⋊C4C2×C411yescharacteristic
68C2×Dic7⋊C4C2×C411yescharacteristic
69C2×Dic7⋊C4C2×C411yescharacteristic
70C2×Dic7⋊C4C2×C411yescharacteristic
71C2×Dic7⋊C4C2311yescharacteristic
72C2×Dic7⋊C4C2311yescharacteristic
73C2×C4×Dic7C2×C411yescharacteristic
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7C2×C411yescharacteristic
77C2×C4×Dic7C2×C411yescharacteristic
78C2×C4×Dic7C2×C411yescharacteristic
79C2×C4×Dic7C2×C411yescharacteristic
80C2×C4×Dic7C2×C411yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83#6450—24no—
84#6450—24no—
85#6450—24no—
86#6450—24no—
87#6440—24no—
88#6440—24no—
89#6440—24no—
90#6440—24no—
91C4×Dic14—22no—
92C4×Dic14—22no—
93C4×Dic14—22no—
94C4×Dic14—22no—
95C4×Dic14—22no—
96C4×Dic14—22no—
97C4×Dic14D414yes—
98C4×Dic14D414yes—
99C7×C4⋊C8—22no—
100C7×C22⋊C8C2314yes—
101C7×#97—22no—
102C22⋊(C7⋊C8)C2314yes—
103#6389—24no—
104Dic7⋊C8—24no—
105Dic7⋊C8—24no—
106C8×Dic7—24no—
107C4⋊(C7⋊C8)—22no—
108#6377—22no—
109C2×#519—71no—
110#2983—72no—
111#1467—72no—
112#1467—72no—
113#1464—72no—
114#1464—71no—
115#1464D711yescharacteristic
116#1461—72no—
117#1458—72no—
118C24×C7C22×C411yescharacteristic
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7C22×C411yescharacteristic
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C2411yescharacteristic
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7M1614yes—
135C22×Dic7C22×C411yescharacteristic
136C22×Dic7C22×C411yescharacteristic
137C22×Dic7C22×C411yescharacteristic
138C22×Dic7C22×C411yescharacteristic
139C22×Dic7C22×C411yescharacteristic
140C22×Dic7C22×C411yescharacteristic
141C22×Dic7C22×C411yescharacteristic
142C22×Dic7C22×C411yescharacteristic
143C2×Dic14—216no—
144C2×Dic14—216no—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—22no—
151C2×Dic14—22no—
152C2×Dic14—22no—
153C2×Dic14—22no—
154C2×Dic14—22no—
155C2×Dic14—22no—
156C2×Dic14C22⋊C412yes—
157C2×Dic14C22⋊C412yes—
158C2×Dic14M1614yes—
159C2×C7×C8—24no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C22⋊C4—22no—
163C7×C22⋊C4—22no—
164C42×C7—21no—
165C42×C7—21no—
166C42×C7C2×D412yes—
167C42×C7C2×D412yes—
168C22⋊Dic7—22no—
169C22⋊Dic7—22no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7C22⋊C412yes—
173C4⋊Dic7C22⋊C412yes—
174Dic7⋊C4—22no—
175Dic7⋊C4—22no—
176Dic7⋊C4—22no—
177Dic7⋊C4—22no—
178Dic7⋊C4—22no—
179Dic7⋊C4—22no—
180Dic7⋊C4C22⋊C414yes—
181Dic7⋊C4C22⋊C414yes—
182C4×Dic7—22no—
183C4×Dic7—22no—
184C4×Dic7—22no—
185C4×Dic7—22no—
186C4×Dic7—22no—
187C4×Dic7—22no—
188C4×Dic7—22no—
189C4×Dic7—22no—
190C4×Dic7C2×D414yes—
191C4×Dic7C2×D414yes—
192C2×C7⋊C8—24no—
193C23×Q8—71no—
194#519—144no—
195#519—144no—
196C2×C4×Q8—71no—
197C2×C4×Q8—71no—
198C2×C4×Q8—71no—
199C2×C4×Q8—71no—
200C2×C4×Q8—71no—
201C2×C4×Q8—71no—
202C2×C4×Q8—71no—
203C2×C4×Q8—71no—
204C2×#117—71no—
205C2×#117—71no—
206C2×#117—71no—
207C2×#117—71no—
208C2×#117—71no—
209C2×#117C2×D711yescharacteristic
210#446—144no—
211#446—144no—
212#432—72no—
213#432—74no—
214#432C2×D712yes—
215#431—74no—
216#324—72no—
217#324—74no—
218#324—74no—
219#324C2×D712yes—
220#323—72no—
221#323C2×D712yes—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7C2×M1612yes—
227C23×C7C23×C411yescharacteristic
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7C2×C22⋊C412yes—
241C2×C4×C7C2×C22⋊C411yescharacteristic
242C2×C4×C7C2×C22⋊C411yescharacteristic
243C2×C4×C7C2×M1612yes—
244C2×C4×C7C22×D412yes—
245C2×Dic7—28no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—28no—
253C2×Dic7—21no—
254C2×Dic7—21no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7C2×C22⋊C412yes—
260C2×Dic7C2×C22⋊C412yes—
261C2×Dic7C2×C22⋊C412yes—
262C2×Dic7C2×C22⋊C412yes—
263C2×Dic7C2×M1612yes—
264C2×Dic7C2×M1612yes—
265C2×Dic7C2×M1612yes—
266C2×Dic7C2×M1612yes—
267Dic14—28no—
268Dic14—24no—
269Dic14—24no—
270Dic14—24no—
271Dic14—24no—
272Dic14—28no—
273C7×C8—44no—
274C7⋊C8—44no—
275C22×Q8—71no—
276C22×Q8—71no—
277C22×Q8—71no—
278C22×Q8—71no—
279C22×Q8—144no—
280C22×Q8—142no—
281C22×Q8—142no—
282C22×Q8—144no—
283C23×C4—71no—
284C23×C4—71no—
285C23×C4C4×D711yescharacteristic
286C4×Q8—74no—
287C4×Q8—74no—
288C4×Q8—142no—
289C4×Q8—142no—
290C4×Q8—142no—
291C4×Q8—142no—
292C4×Q8—144no—
293C4×Q8—144no—
294C4×Q8—144no—
295C4×Q8—144no—
296C4×Q8—142no—
297C4×Q8—142no—
298#117—22no—
299#117—22no—
300#117—144no—
301#117—144no—
302#117—144no—
303#117—144no—
304#117—142no—
305#117—142no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—71no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4C4×D711yescharacteristic
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C22⋊C4—71no—
319C2×C22⋊C4—71no—
320C2×C22⋊C4C4×D711yescharacteristic
321C2×C22⋊C4C22×D711yescharacteristic
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—71no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—71no—
332C2×C42C4×D711yescharacteristic
333C2×C42C22×D711yescharacteristic
334C4⋊C8—22no—
335C4⋊C8—144no—
336C4⋊C8—144no—
337C4⋊C8—142no—
338C22⋊C8—74no—
339C22⋊C8C22×D714yes—
340#97—22no—
341#97—144no—
342#97—142no—
343C4×C8—144no—
344C22×C7—21no—
345C22×C7—21no—
346C22×C7—22no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—41no—
350C22×C7—41no—
351C22×C7C22×C22⋊C411yescharacteristic
352C22×C7C22×M1611yescharacteristic
353C22×C7C22×M1611yescharacteristic
354C4×C7—21no—
355C4×C7—21no—
356C4×C7—22no—
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—41no—
360C4×C7—41no—
361Dic7—22no—
362Dic7—22no—
363Dic7—22no—
364Dic7—22no—
365Dic7—22no—
366Dic7—41no—
367Dic7—41no—
368Dic7#40714yes—
369C24C2×C4×D711yescharacteristic
370C2×Q8—72no—
371C2×Q8—72no—
372C2×Q8—74no—
373C2×Q8—142no—
374C2×Q8—142no—
375C2×Q8—142no—
376C2×Q8—142no—
377C2×Q8—142no—
378C2×Q8—142no—
379C2×Q8—142no—
380C2×Q8—142no—
381C2×Q8—1416no—
382C2×Q8—1416no—
383C2×Q8—141no—
384C2×Q8—141no—
385C2×Q8—142no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—71no—
391C22×C4—74no—
392C22×C4—71no—
393C22×C4—71no—
394C22×C4—71no—
395C22×C4—71no—
396C22×C4—71no—
397C22×C4—71no—
398C22×C4—71no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4C2×C4×D711yescharacteristic
405C22×C4C2×C4×D711yescharacteristic
406C22×C4C2×C4×D711yescharacteristic
407C22×C4C2×C4×D711yescharacteristic
408C22×C4C2×C4×D711yescharacteristic
409C22×C4C23×D711yescharacteristic
410C2×C8—24no—
411C2×C8—144no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—74no—
415C4⋊C4—74no—
416C4⋊C4—72no—
417C4⋊C4—72no—
418C4⋊C4—142no—
419C4⋊C4—142no—
420C4⋊C4—142no—
421C4⋊C4—142no—
422C4⋊C4—141no—
423C4⋊C4—141no—
424C4⋊C4—142no—
425C4⋊C4—142no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—142no—
429C22⋊C4—142no—
430C42—21no—
431C42—21no—
432C42—74no—
433C42—74no—
434C42—142no—
435C42—142no—
436C42—142no—
437C42—142no—
438C42—142no—
439C42—142no—
440C42—142no—
441C42—142no—
442C42D4×D712yes—
443C42D4×D712yes—
444C2×C7—21no—
445C2×C7—21no—
446C2×C7—42no—
447C2×C7C2×#40711yescharacteristic
448C2×C7#288411yescharacteristic
449C2×C7#288911yescharacteristic
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23D7×M1612yes—
455C23C22×C4×D711yescharacteristic
456Q8—144no—
457Q8—144no—
458Q8—148no—
459Q8—144no—
460Q8—144no—
461Q8—148no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—72no—
475C2×C4—72no—
476C2×C4—72no—
477C2×C4—72no—
478C2×C4—72no—
479C2×C4—72no—
480C2×C4—72no—
481C2×C4—72no—
482C2×C4—148no—
483C2×C4—148no—
484C2×C4—141no—
485C2×C4—141no—
486C2×C4—141no—
487C2×C4—141no—
488C2×C4—141no—
489C2×C4—141no—
490C2×C4—141no—
491C2×C4—141no—
492C2×C4—281no—
493C2×C4—281no—
494C2×C4—281no—
495C2×C4—281no—
496C2×C4D7×C22⋊C411yescharacteristic
497C2×C4D7×C22⋊C411yescharacteristic
498C2×C4D7×C22⋊C412yes—
499C2×C4D7×M1612yes—
500C2×C4C2×D4×D712yes—
501C8—44no—
502C8—284no—
503C7#1737111yescharacteristic, complemented, Sylow
504C22—21no—
505C22—21no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—41no—
510C22—41no—
511C22C2×D7×C22⋊C411yescharacteristic
512C22C2×D7×M1611yescharacteristic
513C22C2×D7×M1611yescharacteristic
514C4—21no—
515C4—21no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—74no—
522C4—142no—
523C4—142no—
524C4—142no—
525C4—142no—
526C4—142no—
527C4—281no—
528C4—281no—
529C2—21no—
530C2—21no—
531C2—42no—
532C2D7×#40711yescharacteristic
533C2#1173613411yescharacteristic
534C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1464C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1464D7
hypocenterC7#17371
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #11736759C2semidirect
5. #11728584C2semidirect
6. #11725881C2semidirect
7. #11725879C2semidirect
503. C7#17371semidirect

Maximal subgroups

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

subgroupindex
C2×#902072
#117367592
C7×#14642
#117293262
#117285842
#117258812
#117258792
#117258562
#117258542
#173717