Finite Group Databasebeta

← #13321564

Subgroups — #13321564

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2#11739781C211yesmaximal, characteristic
3#11739747C211yesmaximal, characteristic
4#11729591C212yesmaximal, complemented
5#11729590C212yesmaximal, complemented
6#11729444C212yesmaximal, complemented
7#11729443C212yesmaximal, complemented
8#11729442C212yesmaximal, complemented
9C2×#89333C212yesmaximal, complemented
10C7×#554—24no—
11C7×#554C414yescomplemented
12C7×D4×Q8C414yescomplemented
13C7×#544—21no—
14C7×#543—21no—
15C7×#543—21no—
16C7×#533C2212yes—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×C22⋊Q8C2212yes—
19C7×#518—21no—
20C2×C4×C7×D4C2211yescharacteristic
21#89992—21no—
22#89984C2211yescharacteristic
23#89983C2211yescharacteristic
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C7⋊#95C2211yescharacteristic
26#89979C2212yes—
27C2×C22⋊(C7⋊C8)C2212yes—
28#89783—24no—
29#89753—21no—
30#89723—21no—
31C22⋊C4×Dic7—21no—
32#89709C2211yescharacteristic
33#89704C2211yescharacteristic
34#89698C2212yes—
35#89333C2214yes—
36#89333C2214yes—
37#89333C2214yes—
38#89333C2214yes—
39#17621—71nomaximal, Sylow
40C22×C7×Q8C2×C412yes—
41C22×C7×Q8C2×C412yescomplemented
42C22×C7×D4D411yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×Dic7—21no—
46C7×C4⋊Q8—24no—
47C7×C4⋊Q8C2×C412yescomplemented
48C7×C4⋊Q8C2×C412yescomplemented
49C7×#125—24no—
50C7×#125—22no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—22no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—24no—
60C7×C22⋊Q8C2×C414yes—
61C7×C22⋊Q8C2×C414yescomplemented
62C7×C22⋊Q8C2314yes—
63C4×C7×Q8—22no—
64C4×C7×Q8C2×C414yescomplemented
65C4×C7×D4—24no—
66C4×C7×D4—24no—
67C4×C7×D4C2×C414yes—
68C4×C7×D4D414yes—
69C7×#117—22no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4C2×C412yes—
73C2×C7×C4⋊C4C2×C412yescomplemented
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×#1136—22no—
87C22×C7⋊C8—22no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95C2311yescharacteristic
94C7⋊#95C2311yescharacteristic
95C7⋊#95C2311yescharacteristic
96C7⋊#95C2311yescharacteristic
97C22⋊(C7⋊C8)—22no—
98C22⋊(C7⋊C8)—22no—
99C22⋊(C7⋊C8)C2314yes—
100C4⋊(C7⋊C8)—22no—
101#6377—22no—
102#3501D711yescharacteristic
103#2973—71no—
104#2353—71no—
105#1520—72no—
106#1515—72no—
107#1511—72no—
108#1508—72no—
109#1506—72no—
110C2×#328—72no—
111C24×C7C2×D411yescharacteristic
112C24×C7C2×D411yescharacteristic
113C2×C7×Q8—22no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—82no—
117C2×C7×Q8—82no—
118C2×C7×Q8C22×C414yes—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4C22⋊C414yes—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—81no—
127C22×C4×C7—81no—
128C22×C4×C7C22×C412yes—
129C22×C4×C7C22×C412yes—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2411yescharacteristic
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—42no—
142C22×Dic7—42no—
143C22×Dic7—81no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—24no—
153C7×C4⋊C4C22⋊C414yes—
154C7×C4⋊C4C22×C414yes—
155C7×C22⋊C4—24no—
156C7×C22⋊C4—24no—
157C7×C22⋊C4—24no—
158C7×C22⋊C4C22⋊C414yes—
159C7×C22⋊C4C22⋊C414yes—
160C42×C7—24no—
161C42×C7C22×C412yes—
162C42×C7C2×D412yes—
163C22⋊Dic7—42no—
164C22⋊Dic7—42no—
165C22⋊Dic7—81no—
166C4⋊Dic7—81no—
167C4×Dic7—42no—
168C4×Dic7—81no—
169#1136—82no—
170C2×C7⋊C8—24no—
171C2×C7⋊C8—82no—
172#554—24no—
173#554Dic714yescomplemented
174D4×Q8Dic714yescomplemented
175#544—21no—
176#543—21no—
177#543—21no—
178#533C2×D712yes—
179C2×C22⋊Q8C2×D712yes—
180C2×C22⋊Q8C2×D712yes—
181#518—21no—
182C2×C4×D4C2×D711yescharacteristic
183#435—144no—
184#432—72no—
185#407—72no—
186C2×C22⋊C8—72no—
187#388—141no—
188#387—141no—
189#386—141no—
190#383—71no—
191#381—71no—
192#380—71no—
193C4×C22⋊C4—71no—
194C4×C22⋊C4—141no—
195#376—71no—
196C2×#95—71no—
197#328—74no—
198#328—74no—
199#328—74no—
200#328—74no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—41no—
204C23×C7—41no—
205C23×C7—41no—
206C23×C7—41no—
207C23×C7C2×C22⋊C412yes—
208C23×C7C2×C22⋊C412yes—
209C23×C7#13712yes—
210C23×C7#13712yes—
211C23×C7C22×D411yescharacteristic
212C7×Q8—44no—
213C7×Q8—82no—
214C7×Q8—82no—
215C7×Q8—82no—
216C7×D4—28no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—24no—
220C2×C4×C7—24no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—82no—
230C2×C4×C7—82no—
231C2×C4×C7—82no—
232C2×C4×C7—82no—
233C2×C4×C7C2×C22⋊C412yes—
234C2×C4×C7C2×C22⋊C412yes—
235C2×C4×C7C2×C22⋊C412yes—
236C2×C4×C7C2×C22⋊C412yes—
237C2×C4×C7C23×C411yescharacteristic
238C2×C4×C7C22×D411yescharacteristic
239C2×Dic7—22no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—41no—
251C2×Dic7—84no—
252C7⋊C8—44no—
253C22×Q8C2×Dic712yes—
254C22×Q8C2×Dic712yescomplemented
255C22×D4C22⋊D711yescharacteristic
256C23×C4—141no—
257C23×C4C22×D711yescharacteristic
258C23×C4C22×D711yescharacteristic
259C2×M16—142no—
260C22×C8—142no—
261C4⋊Q8—24no—
262C4⋊Q8C2×Dic712yescomplemented
263C4⋊Q8C2×Dic712yescomplemented
264#125—24no—
265#125—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8C2×Dic714yes—
276C22⋊Q8C2×Dic714yescomplemented
277C22⋊Q8C22×D714yes—
278C4×Q8—22no—
279C4×Q8C2×Dic714yescomplemented
280C4×D4—24no—
281C4×D4—24no—
282C4×D4C2×Dic714yes—
283C4×D4C22⋊D714yes—
284#117—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4C2×Dic712yes—
289C2×C4⋊C4C2×Dic712yescomplemented
290C2×C4⋊C4C22⋊D711yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C22⋊C4C22⋊D711yescharacteristic
297C2×C22⋊C4C22⋊D711yescharacteristic
298C2×C42—141no—
299C2×C42—141no—
300C2×C42C22×D711yescharacteristic
301C4⋊C8—142no—
302C22⋊C8—74no—
303C22⋊C8—142no—
304C22⋊C8—142no—
305#97—142no—
306#95—71no—
307#95—71no—
308#95—71no—
309#95—71no—
310#95—141no—
311#95—141no—
312#95—141no—
313#95—141no—
314#95—141no—
315C22×C7—21no—
316C22×C7—22no—
317C22×C7—21no—
318C22×C7—22no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7—81no—
325C22×C7—81no—
326C22×C7#41914yes—
327C22×C7C22×C22⋊C411yescharacteristic
328C22×C7C2×#13711yescharacteristic
329C22×C7C2×#13711yescharacteristic
330C4×C7—21no—
331C4×C7—21no—
332C4×C7—22no—
333C4×C7—22no—
334C4×C7—22no—
335C4×C7—22no—
336C4×C7—44no—
337C4×C7—44no—
338Dic7—42no—
339Dic7—81no—
340Dic7—81no—
341Dic7—81no—
342C24C2×C22⋊D711yescharacteristic
343C24C2×C22⋊D711yescharacteristic
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—82no—
348C2×Q8—82no—
349C2×Q8C22×Dic714yes—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4C22⋊Dic714yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—141no—
360C22×C4—141no—
361C22×C4—141no—
362C22×C4—141no—
363C22×C4—282no—
364C22×C4—282no—
365C22×C4—561no—
366C22×C4C22×Dic712yes—
367C22×C4C22×Dic712yes—
368C22×C4C2×C22⋊D711yescharacteristic
369C22×C4C2×C22⋊D711yescharacteristic
370C22×C4C2×C22⋊D711yescharacteristic
371C22×C4C2×C22⋊D711yescharacteristic
372C22×C4C2×C22⋊D712yes—
373C22×C4C2×C22⋊D712yes—
374C22×C4C23×D711yescharacteristic
375M16—562no—
376C2×C8—144no—
377C2×C8—562no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—561no—
388C4⋊C4C22⋊Dic714yes—
389C4⋊C4C22×Dic714yes—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—282no—
394C22⋊C4—282no—
395C22⋊C4—561no—
396C22⋊C4C22⋊Dic714yes—
397C22⋊C4C22⋊Dic714yes—
398C42—24no—
399C42—282no—
400C42—561no—
401C42C22×Dic712yes—
402C42C2×C22⋊D712yes—
403C2×C7—21no—
404C2×C7—21no—
405C2×C7—24no—
406C2×C7—42no—
407C2×C7#288311yescharacteristic
408C2×C7C2×#41911yescharacteristic
409C2×C7#290111yescharacteristic
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#650512yes—
417C23#650512yes—
418C23C2×C22⋊Dic712yes—
419C23C2×C22⋊Dic712yes—
420C23C22×C22⋊D711yescharacteristic
421Q8—44no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425D4—28no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—142no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—281no—
447C2×C4—281no—
448C2×C4—281no—
449C2×C4—281no—
450C2×C4—281no—
451C2×C4—281no—
452C2×C4—281no—
453C2×C4—281no—
454C2×C4—564no—
455C2×C4C2×C22⋊Dic712yes—
456C2×C4C2×C22⋊Dic712yes—
457C2×C4C2×C22⋊Dic712yes—
458C2×C4C2×C22⋊Dic712yes—
459C2×C4C23×Dic711yescharacteristic
460C2×C4C22×C22⋊D711yescharacteristic
461C8—284no—
462C7#1762111yescharacteristic, complemented, Sylow
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—42no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22#8999714yes—
475C22C2×#650511yescharacteristic
476C22C2×#650511yescharacteristic
477C22C22×C22⋊Dic711yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—44no—
485C4—44no—
486C4—282no—
487C4—561no—
488C4—561no—
489C4—561no—
490C2—21no—
491C2—21no—
492C2—24no—
493C2—42no—
494C2C2×#8999711yescharacteristic
495C2#1174155211yescharacteristic
496C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4D4
cosocleC22×C4×C7C24
hypercenter#3501D7
hypocenterC7#17621
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3501

Complements

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

NH ≅ G/Ntype
4. #11729591C2semidirect
5. #11729590C2semidirect
6. #11729444C2semidirect
7. #11729443C2semidirect
8. #11729442C2semidirect
9. C2×#89333C2semidirect
11. C7×#554C4semidirect
12. C7×D4×Q8C4semidirect
41. C22×C7×Q8C2×C4semidirect
47. C7×C4⋊Q8C2×C4semidirect
48. C7×C4⋊Q8C2×C4semidirect
61. C7×C22⋊Q8C2×C4semidirect
64. C4×C7×Q8C2×C4semidirect
73. C2×C7×C4⋊C4C2×C4semidirect
173. #554Dic7semidirect
174. D4×Q8Dic7semidirect
254. C22×Q8C2×Dic7semidirect
262. C4⋊Q8C2×Dic7semidirect
263. C4⋊Q8C2×Dic7semidirect
276. C22⋊Q8C2×Dic7semidirect
279. C4×Q8C2×Dic7semidirect
289. C2×C4⋊C4C2×Dic7semidirect
462. C7#17621semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117397812
#117397472
#117295912
#117295902
#117294442
#117294432
#117294422
C2×#893332
#176217