Finite Group Databasebeta

← #13332403

Subgroups — #13332403

Order 1792.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic
3#11739887C211yesmaximal, characteristic
4#11739886C211yesmaximal, characteristic
5#11729691C214yesmaximal, complemented
6#11729689C214yesmaximal, complemented
7#11729527C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729375C211yesmaximal, characteristic, complemented
10C7×#557—22no—
11C7×#557—22no—
12C7×#557—22no—
13C7×#557C2214yes—
14C2×C7×C4⋊Q8—21no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—22no—
17C2×C7×#125—21no—
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#89860C2212yes—
26#89859—22no—
27#89850C2212yes—
28#89803C2212yes—
29#89800—22no—
30#89799C2212yes—
31#89799C2212yes—
32C4×#1143—22no—
33#89792C2212yes—
34#89740—21no—
35#89739—22no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yes—
41Q8⋊(C7⋊C8)C2214yes—
42Q8⋊(C7⋊C8)C2214yes—
43#18760—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×Q8D411yescharacteristic
47C2×#1157—21no—
48C7×C4⋊Q8—24no—
49C7×C4⋊Q8—24no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8D412yes—
53C7×C4⋊Q8C2312yes—
54C7×#125—28no—
55C7×#125—24no—
56C7×#125—28no—
57C7×#125—24no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C4×C7×Q8—24no—
62C4×C7×Q8—21no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8D412yes—
67C4×C7×Q8D412yes—
68C4×C7×Q8C2312yes—
69C4×C7×Q8C2312yes—
70C2×C7×C4⋊C4—22no—
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⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D412yes—
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—
87#6493—21no—
88#6493—21no—
89C2×#1136—21no—
90#6438—21no—
91#6437C2311yescharacteristic
92C4×D28—21no—
93C4×D28—21no—
94#6432C2311yescharacteristic
95#6409—22no—
96C22⋊(C7⋊C8)—21no—
97C22⋊(C7⋊C8)—21no—
98#6384—22no—
99#6383—24no—
100#6383—22no—
101#6383—22no—
102#6382—22no—
103#6381—24no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2312yes—
107C4⋊(C7⋊C8)C2312yes—
108C4×C7⋊C8—21no—
109C4×C7⋊C8C2312yes—
110C2×#557D711yescharacteristic, complemented
111#3226—71no—
112#3103—71no—
113#3037—71no—
114#1648—74no—
115#1626—74no—
116#1546—71no—
117#1487—71no—
118#1479—71no—
119C2×C7×Q8—24no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×Q8C2×D412yes—
129C2×C7×Q8C2×D412yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C2×C22⋊D7—41no—
139#1157—42no—
140#1157—42no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×C4×D7—41no—
145C2×Dic14—21no—
146C7×C4⋊C4—28no—
147C7×C4⋊C4—28no—
148C7×C4⋊C4—28no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4C2×D412yes—
159C7×C4⋊C4C2×D412yes—
160C7×C4⋊C4C2×D414yes—
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(C2×D7)⋊C4—41no—
174(C2×D7)⋊C4—41no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177#1136—42no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×#137—141no—
182#557—22no—
183#557—22no—
184#557—22no—
185#557C2×D714yes—
186#532—71no—
187C2×C4⋊Q8—21no—
188C2×C4⋊Q8C2×D711yescharacteristic
189C2×#125—22no—
190C2×#125—21no—
191C2×C4×Q8—21no—
192C2×C4×Q8—21no—
193C2×C4×Q8C2×D711yescharacteristic
194C2×C4×Q8C2×D711yescharacteristic
195#500—72no—
196#486—72no—
197#481—141no—
198#480—142no—
199#475—72no—
200#474—142no—
201#464—142no—
202#462—72no—
203#460—72no—
204#460—72no—
205#448—141no—
206C4×SD16—142no—
207#431—71no—
208#431—71no—
209#423—71no—
210#417—141no—
211#332—74no—
212Q8⋊C8—74no—
213Q8⋊C8—74no—
214C23×C7#12011yescharacteristic
215C22×D7—41no—
216C7×Q8—24no—
217C7×Q8—22no—
218C7×Q8—22no—
219C7×Q8—22no—
220C7×Q8—42no—
221C7×Q8—42no—
222C7×Q8—42no—
223C7×Q8—44no—
224C7×Q8#13714yescomplemented
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—44no—
237C2×C4×C7#12011yescharacteristic
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#13512yes—
241C2×C4×C7AGL(1,ℤ/8)12yes—
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C22⋊D7—82no—
246C2×Dic7—41no—
247D28—42no—
248D28—41no—
249D28—41no—
250D28—42no—
251C4×D7—82no—
252Dic14—41no—
253Dic14—41no—
254C7⋊C8—42no—
255C7⋊C8—42no—
256C7⋊C8—42no—
257C2×#41—141no—
258C22×Q8—21no—
259C22×Q8C22⋊D711yescharacteristic
260C22×Q8C22⋊D711yescharacteristic
261#137—284no—
262C2×Q16—142no—
263C2×SD16—142no—
264C2×SD16—142no—
265C2×M16—141no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—71no—
271C4⋊Q8C22⋊D712yes—
272C4⋊Q8C22×D712yes—
273#127—71no—
274#125—28no—
275#125—24no—
276#125—28no—
277#125—24no—
278#124—141no—
279#121—141no—
280#121—141no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8C22⋊D712yes—
290C4×Q8C22⋊D712yes—
291C4×Q8C22×D712yes—
292C4×Q8C22×D712yes—
293C4×D4—141no—
294C4×D4—141no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C22⋊D711yescharacteristic
302C2×C4⋊C4C22⋊D711yescharacteristic
303C2×C4⋊C4C22⋊D712yes—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C22×D711yescharacteristic
308#107—144no—
309#106—142no—
310C4⋊C8—72no—
311C4⋊C8—72no—
312C4⋊C8—141no—
313C4⋊C8—141no—
314#103—142no—
315#103—142no—
316#102—142no—
317#102—142no—
318#102—144no—
319C22⋊C8—141no—
320C22⋊C8—141no—
321C4×C8—72no—
322C4×C8—141no—
323C22×C7—21no—
324C22×C7—21no—
325C22×C7#44912yes—
326C22×C7C2×#12011yescharacteristic
327C2×D7—81no—
328C2×D7—81no—
329C2×D7—81no—
330C4×C7—24no—
331C4×C7—21no—
332C4×C7—21no—
333C4×C7—21no—
334C4×C7—42no—
335C4×C7—44no—
336C4×C7—42no—
337C4×C7—42no—
338C4×C7#44912yes—
339C4×C7C2×#13511yescharacteristic
340C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
341C4×C7C2×#13712yes—
342Dic7—81no—
343#41—282no—
344#41—282no—
345C2×Q8—24no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—141no—
355C2×Q8C2×C22⋊D712yes—
356C2×Q8C2×C22⋊D712yes—
357C2×D4—141no—
358C2×D4—141no—
359C2×D4—141no—
360C2×D4—281no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—281no—
367C22×C4C2×C22⋊D711yescharacteristic
368C22×C4C2×C22⋊D711yescharacteristic
369C22×C4C2×C22⋊D711yescharacteristic
370Q16—284no—
371SD16—284no—
372SD16—284no—
373M16—282no—
374C2×C8—142no—
375C2×C8—142no—
376C2×C8—142no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—141no—
390C4⋊C4—141no—
391C4⋊C4C2×C22⋊D712yes—
392C4⋊C4C2×C22⋊D712yes—
393C4⋊C4C2×C22⋊D714yes—
394C22⋊C4—281no—
395C22⋊C4—281no—
396C42—28no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42C2×C22⋊D711yescharacteristic
402C42C2×C22⋊D711yescharacteristic
403C42C2×C22⋊D711yescharacteristic
404C42C23×D711yescharacteristic
405C2×C7—22no—
406C2×C7C2×#44911yescharacteristic
407C2×C7#300411yescharacteristic
408C2×C7#302011yescharacteristic
409D7—161no—
410C23—281no—
411C23#651611yescharacteristic
412Q8—24no—
413Q8—22no—
414Q8—22no—
415Q8—22no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419Q8—44no—
420Q8—281no—
421Q8—281no—
422Q8#650514yescomplemented
423D4—282no—
424D4—282no—
425D4—281no—
426D4—281no—
427D4—562no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—281no—
441C2×C4—562no—
442C2×C4#651212yes—
443C2×C4#651312yes—
444C2×C4#651611yescharacteristic
445C2×C4#651611yescharacteristic
446C2×C4#651611yescharacteristic
447C2×C4C22×C22⋊D711yescharacteristic
448C2×C4C22×C22⋊D711yescharacteristic
449C2×C4C22×C22⋊D711yescharacteristic
450C8—282no—
451C8—282no—
452C8—282no—
453C7#1876011yescharacteristic, complemented, Sylow
454C22—21no—
455C22—21no—
456C22—561no—
457C22—561no—
458C22—561no—
459C22#9001012yes—
460C22C2×#651611yescharacteristic
461C4—24no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—561no—
470C4#9001012yes—
471C4C2×#650512yes—
472C4C2×#651211yescharacteristic
473C4C2×#651311yescharacteristic
474C2—22no—
475C2—1121no—
476C2#1174150411yescharacteristic
477C2C2×#9001011yescharacteristic
478C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#557C2
socleC22×C7C2×#120
prime radical#90168C22
hypercenterC2×#557D7
hypocenterC7#18760
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
5. #11729691C2semidirect
6. #11729689C2semidirect
7. #11729527C2semidirect
8. #11729376C2semidirect
9. #11729375C2semidirect
110. C2×#557D7semidirect
224. C7×Q8#137semidirect
422. Q8#6505semidirect
453. C7#18760semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
#117406602
#117398872
#117398862
#117296912
#117296892
#117295272
#117293762
#117293752
#187607