Finite Group Databasebeta

← #13332487

Subgroups — #13332487

Order 1792.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×Q82C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic
3#11739887C211yesmaximal, characteristic
4#11739886C211yesmaximal, characteristic
5#11729695C214yesmaximal, complemented
6#11729693C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729375C211yesmaximal, characteristic, complemented
10C7×Q82—22no—
11C7×Q82—22no—
12C7×Q82—22no—
13C7×Q82C2214yes—
14C2×C7×C4⋊Q8—22no—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8—21no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×#6505—21no—
23#90168C2211yescharacteristic
24#89999—21no—
25#89999—22no—
26#89857C2212yes—
27#89855—22no—
28#89853C2212yes—
29#89803C2212yes—
30#89800—22no—
31#89799C2212yes—
32#89799C2212yes—
33C4×#1143—22no—
34#89792C2212yes—
35#89740—21no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yes—
41Q8⋊(C7⋊C8)C2214yes—
42Q8⋊(C7⋊C8)C2214yes—
43#18768—71nomaximal, Sylow
44C22×C7×Q8—22no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8D412yes—
48C2×#1157—21no—
49C7×C4⋊Q8—28no—
50C7×C4⋊Q8—24no—
51C7×C4⋊Q8—24no—
52C7×C4⋊Q8—28no—
53C7×C4⋊Q8—24no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D412yes—
58C7×C4⋊Q8C2312yes—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C4×C7×Q8—24no—
62C4×C7×Q8—24no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8—21no—
67C4×C7×Q8D412yes—
68C4×C7×Q8D412yes—
69C4×C7×Q8C2312yes—
70C4×C7×Q8C2312yes—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C42×C7—22no—
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×#1144—22no—
84#6505—44no—
85C2×#1143—22no—
86C2×#1143—22no—
87C2×#1143—24no—
88#6493—21no—
89#6493—21no—
90C2×#1136—21no—
91#6438—21no—
92#6437C2311yescharacteristic
93C4×D28—21no—
94C4×D28—21no—
95#6432C2311yescharacteristic
96#6409—22no—
97#6409—24no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6383—22no—
102#6383—22no—
103#6382—22no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2312yes—
107C4⋊(C7⋊C8)C2312yes—
108C4×C7⋊C8—21no—
109C4×C7⋊C8C2312yes—
110C2×Q82D711yescharacteristic, complemented
111#3206—71no—
112#3103—71no—
113#3037—71no—
114#1645—74no—
115#1623—74no—
116#1548—71no—
117#1487—71no—
118#1479—71no—
119C2×C7×Q8—24no—
120C2×C7×Q8—28no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—82no—
126C2×C7×Q8—84no—
127C2×C7×Q8SD1614yescomplemented
128C2×C7×Q8C2×D412yes—
129C2×C7×Q8C2×D412yes—
130C2×C7×Q8C2×D414yes—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C2×C22⋊D7—41no—
140#1157—42no—
141#1157—42no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×C4×D7—41no—
146C2×Dic14—21no—
147C7×C4⋊C4—28no—
148C7×C4⋊C4—28no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4C2×D412yes—
160C7×C4⋊C4C2×D412yes—
161C42×C7—28no—
162C42×C7—22no—
163C42×C7—22no—
164C42×C7—22no—
165C42×C7—22no—
166C42×C7C2×D411yescharacteristic
167C42×C7C2×D411yescharacteristic
168C42×C7C2×D411yescharacteristic
169C42×C7C2411yescharacteristic
170#1144—44no—
171#1143—44no—
172#1143—44no—
173#1143—84no—
174(C2×D7)⋊C4—41no—
175(C2×D7)⋊C4—41no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178#1136—42no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×#137—141no—
183Q82—22no—
184Q82—22no—
185Q82—22no—
186Q82C2×D714yes—
187#532—71no—
188C2×C4⋊Q8—22no—
189C2×C4⋊Q8—21no—
190C2×C4⋊Q8—21no—
191C2×C4⋊Q8C2×D711yescharacteristic
192C2×C4×Q8—21no—
193C2×C4×Q8—21no—
194C2×C4×Q8C2×D711yescharacteristic
195C2×C4×Q8C2×D711yescharacteristic
196#492—72no—
197#487—72no—
198#481—141no—
199#475—72no—
200#464—142no—
201#462—72no—
202#461—142no—
203#460—72no—
204#460—72no—
205#448—141no—
206#448—142no—
207C4×SD16—142no—
208#431—71no—
209#431—71no—
210#423—71no—
211#417—141no—
212#333—74no—
213Q8⋊C8—74no—
214Q8⋊C8—74no—
215C23×C7#12011yescharacteristic
216C22×D7—41no—
217C7×Q8—24no—
218C7×Q8—22no—
219C7×Q8—22no—
220C7×Q8—24no—
221C7×Q8—44no—
222C7×Q8—82no—
223C7×Q8—88no—
224C7×Q8C2×SD1614yes—
225C7×Q8#13714yescomplemented
226C2×C4×C7—24no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—41no—
235C2×C4×C7—44no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7C2×SD1612yes—
242C2×C4×C7#13712yes—
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C22⋊D7—82no—
247C2×Dic7—41no—
248D28—42no—
249D28—41no—
250D28—42no—
251D28—41no—
252C4×D7—82no—
253Dic14—41no—
254Dic14—41no—
255C7⋊C8—42no—
256C7⋊C8—42no—
257C7⋊C8—42no—
258C2×#41—141no—
259C22×Q8—22no—
260C22×Q8C22⋊D711yescharacteristic
261C22×Q8C22⋊D711yescharacteristic
262C22×Q8C22⋊D712yes—
263#137—284no—
264C2×Q16—142no—
265C2×SD16—142no—
266C2×SD16—142no—
267C2×SD16—144no—
268C2×M16—141no—
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—28no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—71no—
278C4⋊Q8C22⋊D712yes—
279C4⋊Q8C22×D712yes—
280#127—71no—
281#124—141no—
282#121—141no—
283#121—141no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8C22⋊D712yes—
293C4×Q8C22⋊D712yes—
294C4×Q8C22×D712yes—
295C4×Q8C22×D712yes—
296C4×D4—141no—
297C4×D4—141no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C22⋊D711yescharacteristic
305C2×C4⋊C4C22⋊D711yescharacteristic
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C22×D711yescharacteristic
310#106—142no—
311C4⋊C8—72no—
312C4⋊C8—72no—
313C4⋊C8—141no—
314C4⋊C8—141no—
315#103—142no—
316#103—144no—
317#103—142no—
318#102—142no—
319#102—142no—
320C22⋊C8—141no—
321C22⋊C8—141no—
322C4×C8—72no—
323C4×C8—141no—
324C22×C7—21no—
325C22×C7—21no—
326C22×C7#44812yes—
327C22×C7C2×#12011yescharacteristic
328C2×D7—81no—
329C2×D7—81no—
330C2×D7—81no—
331C4×C7—24no—
332C4×C7—24no—
333C4×C7—21no—
334C4×C7—21no—
335C4×C7—21no—
336C4×C7—44no—
337C4×C7—42no—
338C4×C7—42no—
339C4×C7#44812yes—
340C4×C7C22×SD1611yescharacteristic
341C4×C7C2×#13711yescharacteristic
342C4×C7C2×#13712yes—
343Dic7—81no—
344#41—282no—
345#41—282no—
346C2×Q8—24no—
347C2×Q8—28no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—82no—
353C2×Q8—84no—
354C2×Q8—141no—
355C2×Q8#114314yescomplemented
356C2×Q8C2×C22⋊D712yes—
357C2×Q8C2×C22⋊D712yes—
358C2×Q8C2×C22⋊D714yes—
359C2×D4—141no—
360C2×D4—141no—
361C2×D4—141no—
362C2×D4—281no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—281no—
369C22×C4C2×C22⋊D711yescharacteristic
370C22×C4C2×C22⋊D711yescharacteristic
371C22×C4C2×C22⋊D711yescharacteristic
372Q16—284no—
373SD16—284no—
374SD16—284no—
375SD16—564no—
376M16—282no—
377C2×C8—142no—
378C2×C8—142no—
379C2×C8—142no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—141no—
393C4⋊C4—141no—
394C4⋊C4C2×C22⋊D712yes—
395C4⋊C4C2×C22⋊D712yes—
396C22⋊C4—281no—
397C22⋊C4—281no—
398C42—28no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42C2×C22⋊D711yescharacteristic
404C42C2×C22⋊D711yescharacteristic
405C42C2×C22⋊D711yescharacteristic
406C42C23×D711yescharacteristic
407C2×C7—22no—
408C2×C7C2×#44811yescharacteristic
409C2×C7#300411yescharacteristic
410C2×C7#301911yescharacteristic
411D7—161no—
412C23—281no—
413C23#651611yescharacteristic
414Q8—24no—
415Q8—22no—
416Q8—22no—
417Q8—24no—
418Q8—44no—
419Q8—82no—
420Q8—88no—
421Q8—281no—
422Q8—281no—
423Q8C2×#114314yes—
424Q8#650514yescomplemented
425D4—282no—
426D4—281no—
427D4—282no—
428D4—281no—
429D4—562no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—281no—
443C2×C4—562no—
444C2×C4C2×#114312yes—
445C2×C4#650512yes—
446C2×C4#651611yescharacteristic
447C2×C4#651611yescharacteristic
448C2×C4#651611yescharacteristic
449C2×C4C22×C22⋊D711yescharacteristic
450C2×C4C22×C22⋊D711yescharacteristic
451C2×C4C22×C22⋊D711yescharacteristic
452C8—282no—
453C8—282no—
454C8—282no—
455C7#1876811yescharacteristic, complemented, Sylow
456C22—21no—
457C22—21no—
458C22—561no—
459C22—561no—
460C22—561no—
461C22#8999912yes—
462C22C2×#651611yescharacteristic
463C4—24no—
464C4—24no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—561no—
472C4#8999912yes—
473C4C22×#114311yescharacteristic
474C4C2×#650511yescharacteristic
475C4C2×#650512yes—
476C2—22no—
477C2—1121no—
478C2C2×#8999911yescharacteristic
479C2#1174150411yescharacteristic
480C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×Q82C2
socleC22×C7C2×#120
prime radical#90168C22
hypercenterC2×Q82D7
hypocenterC7#18768
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C7×Q82C2semidirect
5. #11729695C2semidirect
6. #11729693C2semidirect
7. #11729528C2semidirect
8. #11729376C2semidirect
9. #11729375C2semidirect
110. C2×Q82D7semidirect
127. C2×C7×Q8SD16semidirect
225. C7×Q8#137semidirect
355. C2×Q8#1143semidirect
424. Q8#6505semidirect
455. C7#18768semidirect

Maximal subgroups

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

subgroupindex
C2×C7×Q822
#117406582
#117398872
#117398862
#117296952
#117296932
#117295282
#117293762
#117293752
#187687