Finite Group Databasebeta

← #13321568

Subgroups — #13321568

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
1#11742834C211yesmaximal, characteristic
2C7×#2973C211yesmaximal, characteristic
3#11739781C211yesmaximal, characteristic
4#11728591C212yesmaximal, complemented
5#11728590C212yesmaximal, complemented
6#11727735C212yesmaximal, complemented
7#11727734C212yesmaximal, complemented
8#11727700C212yesmaximal, complemented
9C2×#89262C212yesmaximal, complemented
10C2×C4×C7×D4C2211yescharacteristic
11#90489—21no—
12C2×#6487C2212yes—
13C2×#6487C2212yes—
14#90234C414yescomplemented
15#90230—24no—
16D4×Dic14C414yescomplemented
17#90191—21no—
18#90174—21no—
19#90174—21no—
20#90161C2212yes—
21C7×#435—24no—
22C7×#432C2212yes—
23C7×#407C2212yes—
24C2×C7×C22⋊C8C2212yes—
25#89992—21no—
26#89984C2211yescharacteristic
27#89983C2211yescharacteristic
28C4×C22⋊Dic7C2211yescharacteristic
29C2×C7⋊#95C2211yescharacteristic
30#89753—21no—
31#89723—21no—
32C22⋊C4×Dic7—21no—
33#89709C2211yescharacteristic
34#89704C2211yescharacteristic
35#89262C2214yes—
36#89262C2214yes—
37#89262C2214yes—
38#89262C2214yes—
39#17621—71nomaximal, Sylow
40C22×C7×D4D411yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×Dic7—21no—
44C22×Dic14C2×C412yes—
45C22×Dic14C2×C412yescomplemented
46C2×C7×M16—22no—
47C22×C7×C8—22no—
48C4×C7×D4C2×C414yes—
49C4×C7×D4D414yes—
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C42×C7C2311yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×C22⋊Dic7—22no—
56C2×C22⋊Dic7—21no—
57C2×C22⋊Dic7—21no—
58C2×C22⋊Dic7—21no—
59D4×Dic7—24no—
60D4×Dic7—24no—
61#6489—22no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7C2×C412yes—
64C2×C4⋊Dic7C2×C412yescomplemented
65#6487—22no—
66#6487—22no—
67#6487—22no—
68#6487—22no—
69#6487—22no—
70#6487C2×C414yes—
71#6487C2×C414yescomplemented
72#6487C2314yes—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—21no—
77#6453—24no—
78#6451—24no—
79#6441—24no—
80#6441—24no—
81#6441—24no—
82#6441—24no—
83#6433—22no—
84#6432C2×C412yescomplemented
85#6432C2×C412yescomplemented
86C4×Dic14—22no—
87C4×Dic14C2×C414yescomplemented
88C7×C4⋊C8—22no—
89C7×C22⋊C8—22no—
90C7×C22⋊C8—22no—
91C7×C22⋊C8C2314yes—
92C7×#97—22no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95C2311yescharacteristic
99C7⋊#95C2311yescharacteristic
100C7⋊#95C2311yescharacteristic
101C7⋊#95C2311yescharacteristic
102#3501—71no—
103#2973D711yescharacteristic
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×D4—22no—
114C2×C7×D4—22no—
115C2×C7×D4C22⋊C414yes—
116C22×C4×C7—81no—
117C22×C4×C7—81no—
118C22×C4×C7C22×C412yes—
119C22×C4×C7C22×C412yes—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7—42no—
136C22×Dic7—42no—
137C22×Dic7—81no—
138C2×Dic14—22no—
139C2×Dic14—22no—
140C2×Dic14—22no—
141C2×Dic14—82no—
142C2×Dic14—82no—
143C2×Dic14C22×C414yes—
144C7×M16—82no—
145C2×C7×C8—24no—
146C2×C7×C8—82no—
147C7×C4⋊C4C22⋊C414yes—
148C7×C22⋊C4C22⋊C414yes—
149C7×C22⋊C4C22⋊C414yes—
150C42×C7C22×C412yes—
151C42×C7C2×D412yes—
152C22⋊Dic7—24no—
153C22⋊Dic7—24no—
154C22⋊Dic7—24no—
155C22⋊Dic7—42no—
156C22⋊Dic7—42no—
157C22⋊Dic7—81no—
158C4⋊Dic7—22no—
159C4⋊Dic7—22no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7—22no—
163C4⋊Dic7—81no—
164C4⋊Dic7C22×C414yes—
165Dic7⋊C4—24no—
166Dic7⋊C4—24no—
167Dic7⋊C4—24no—
168Dic7⋊C4—24no—
169C4×Dic7—24no—
170C4×Dic7—42no—
171C4×Dic7—81no—
172#554—74no—
173#554—144no—
174D4×Q8—74no—
175#544—141no—
176#543—141no—
177#543—141no—
178#533—72no—
179C2×C22⋊Q8—72no—
180C2×C22⋊Q8—72no—
181#518—141no—
182C2×C4×D4C2×D711yescharacteristic
183#435—24no—
184#432C2×D712yes—
185#407C2×D712yes—
186C2×C22⋊C8C2×D712yes—
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×D4—28no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—41no—
223C2×C4×C7C2×C22⋊C412yes—
224C2×C4×C7C2×C22⋊C412yes—
225C2×C4×C7C2×C22⋊C412yes—
226C2×C4×C7C2×C22⋊C412yes—
227C2×C4×C7C23×C411yescharacteristic
228C2×C4×C7C22×D411yescharacteristic
229C2×Dic7—22no—
230C2×Dic7—24no—
231C2×Dic7—24no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—82no—
244C2×Dic7—82no—
245C2×Dic7—82no—
246C2×Dic7—82no—
247C2×Dic7—84no—
248Dic14—44no—
249Dic14—82no—
250Dic14—82no—
251Dic14—82no—
252C7×C8—44no—
253C22×Q8—72no—
254C22×Q8—72no—
255C22×D4D2811yescharacteristic
256C23×C4—141no—
257C23×C4C22×D711yescharacteristic
258C23×C4C22×D711yescharacteristic
259C2×M16—22no—
260C22×C8—22no—
261C4⋊Q8—72no—
262C4⋊Q8—72no—
263C4⋊Q8—144no—
264#125—144no—
265#125—142no—
266C22⋊Q8—74no—
267C22⋊Q8—74no—
268C22⋊Q8—74no—
269C22⋊Q8—142no—
270C22⋊Q8—142no—
271C22⋊Q8—142no—
272C22⋊Q8—142no—
273C22⋊Q8—142no—
274C22⋊Q8—144no—
275C22⋊Q8—144no—
276C22⋊Q8—144no—
277C22⋊Q8—144no—
278C4×Q8—74no—
279C4×Q8—142no—
280C4×D4—144no—
281C4×D4—144no—
282C4×D4C4×D714yes—
283C4×D4C22⋊D714yes—
284#117—142no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—72no—
287C2×C4⋊C4—142no—
288C2×C4⋊C4—142no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4D2811yescharacteristic
291C2×C22⋊C4—142no—
292C2×C22⋊C4—142no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C22⋊C4D2811yescharacteristic
297C2×C22⋊C4D2811yescharacteristic
298C2×C42—141no—
299C2×C42—141no—
300C2×C42C22×D711yescharacteristic
301C4⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8C22×D714yes—
305#97—22no—
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—
336Dic7—42no—
337Dic7—44no—
338Dic7—44no—
339Dic7—81no—
340Dic7—81no—
341Dic7—81no—
342C24C2×D2811yescharacteristic
343C24C2×D2811yescharacteristic
344C2×Q8—74no—
345C2×Q8—142no—
346C2×Q8—142no—
347C2×Q8—142no—
348C2×Q8—562no—
349C2×Q8—562no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4(C2×D7)⋊C414yes—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—141no—
356C22×C4—142no—
357C22×C4—142no—
358C22×C4—142no—
359C22×C4—142no—
360C22×C4—141no—
361C22×C4—141no—
362C22×C4—141no—
363C22×C4—282no—
364C22×C4—282no—
365C22×C4—561no—
366C22×C4C2×C4×D712yes—
367C22×C4C2×C4×D712yes—
368C22×C4C2×D2811yescharacteristic
369C22×C4C2×D2811yescharacteristic
370C22×C4C2×D2811yescharacteristic
371C22×C4C2×D2811yescharacteristic
372C22×C4C2×C22⋊D712yes—
373C22×C4C2×C22⋊D712yes—
374C22×C4C23×D711yescharacteristic
375M16—82no—
376C2×C8—24no—
377C2×C8—82no—
378C4⋊C4—74no—
379C4⋊C4—144no—
380C4⋊C4—144no—
381C4⋊C4—144no—
382C4⋊C4—142no—
383C4⋊C4—142no—
384C4⋊C4—142no—
385C4⋊C4—142no—
386C4⋊C4—142no—
387C4⋊C4—144no—
388C4⋊C4—561no—
389C4⋊C4(C2×D7)⋊C414yes—
390C22⋊C4—144no—
391C22⋊C4—144no—
392C22⋊C4—144no—
393C22⋊C4—282no—
394C22⋊C4—282no—
395C22⋊C4—561no—
396C22⋊C4(C2×D7)⋊C414yes—
397C22⋊C4(C2×D7)⋊C414yes—
398C42—144no—
399C42—282no—
400C42—561no—
401C42C2×C4×D712yes—
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#647212yes—
417C23#647212yes—
418C23C2×(C2×D7)⋊C412yes—
419C23C2×(C2×D7)⋊C412yes—
420C23C22×D2811yescharacteristic
421Q8—284no—
422Q8—562no—
423Q8—562no—
424Q8—562no—
425D4—28no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—142no—
437C2×C4—144no—
438C2×C4—144no—
439C2×C4—281no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—281no—
447C2×C4—281no—
448C2×C4—281no—
449C2×C4—281no—
450C2×C4—562no—
451C2×C4—562no—
452C2×C4—562no—
453C2×C4—562no—
454C2×C4—564no—
455C2×C4C2×(C2×D7)⋊C412yes—
456C2×C4C2×(C2×D7)⋊C412yes—
457C2×C4C2×(C2×D7)⋊C412yes—
458C2×C4C2×(C2×D7)⋊C412yes—
459C2×C4C22×C4×D711yescharacteristic
460C2×C4C22×C22⋊D711yescharacteristic
461C8—44no—
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#8989214yes—
475C22C2×#647211yescharacteristic
476C22C2×#647211yescharacteristic
477C22C22×(C2×D7)⋊C411yescharacteristic
478C4—22no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—282no—
485C4—284no—
486C4—284no—
487C4—561no—
488C4—561no—
489C4—561no—
490C2—21no—
491C2—21no—
492C2—24no—
493C2—42no—
494C2#1173999111yescharacteristic
495C2C2×#8989211yescharacteristic
496C2#1174126211yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2973C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4D4
cosocleC22×C4×C7C24
hypercenter#2973D7
hypocenterC7#17621
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #11728591C2semidirect
5. #11728590C2semidirect
6. #11727735C2semidirect
7. #11727734C2semidirect
8. #11727700C2semidirect
9. C2×#89262C2semidirect
14. #90234C4semidirect
16. D4×Dic14C4semidirect
45. C22×Dic14C2×C4semidirect
64. C2×C4⋊Dic7C2×C4semidirect
71. #6487C2×C4semidirect
84. #6432C2×C4semidirect
85. #6432C2×C4semidirect
87. C4×Dic14C2×C4semidirect
462. C7#17621semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#29732
#117397812
#117285912
#117285902
#117277352
#117277342
#117277002
C2×#892622
#176217