Finite Group Databasebeta

← #13332292

Subgroups — #13332292

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2#11740667C211yesmaximal, characteristic, complemented
3#11739892C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic
5#11729680C214yesmaximal, complemented
6#11729678C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic
10C7×#554—22no—
11C7×#554—22no—
12C7×#554—22no—
13C7×#554C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22#90161C2211yescharacteristic
23#90000—21no—
24#90000—22no—
25#89987—21no—
26#89864C2212yescomplemented
27#89862—22no—
28#89853C2212yescomplemented
29#89803C2212yescomplemented
30C4×#1144—22no—
31#89793C2212yescomplemented
32#89781—22no—
33#89780C2212yes—
34#89779C2212yes—
35#89742—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18749—71nomaximal, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D412yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8D412yes—
52C7×C4⋊Q8C2312yes—
53C7×#125—24no—
54C7×#125—24no—
55C7×C22⋊Q8—24no—
56C7×C22⋊Q8—28no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—28no—
59C4×C7×Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×Q8D412yes—
62C4×C7×Q8C2312yes—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C4×C7×D4—21no—
66C4×C7×D4—21no—
67C4×C7×D4D412yes—
68C4×C7×D4C2312yes—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×#1144—22no—
83C2×#1144—24no—
84#6489—21no—
85#6487—21no—
86#6487—21no—
87C2×#1136—21no—
88#6433—21no—
89#6432C2311yescharacteristic
90#6432C2311yescharacteristic
91C4×Dic14—21no—
92C4×Dic14—21no—
93#6409—24no—
94#6409—22no—
95#6406—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
100#6382—22no—
101#6381—22no—
102#6381—22no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)C2312yes—
106C4⋊(C7⋊C8)C2312yes—
107C4×C7⋊C8—21no—
108C4×C7⋊C8C2312yes—
109C2×#554D711yescharacteristic
110#3209—71no—
111#3104—71no—
112#3077—71no—
113#1644—74no—
114#1632—74no—
115#1548—71no—
116#1487—71no—
117#1478—71no—
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—42no—
122C2×C7×Q8—82no—
123C2×C7×Q8C2×D412yes—
124C2×C7×Q8C2×D414yes—
125C2×C7×D4—21no—
126C2×C7×D4—22no—
127C2×C7×D4Q1614yescomplemented
128C2×C7×D4C2×D412yes—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—28no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—44no—
137C22×C4×C7—84no—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C2×Dic14—21no—
142C2×Dic14—21no—
143C7×C4⋊C4—28no—
144C7×C4⋊C4—28no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4C2×D412yes—
154C7×C4⋊C4C2×D412yes—
155C7×C22⋊C4—28no—
156C7×C22⋊C4—22no—
157C7×C22⋊C4—22no—
158C42×C7—22no—
159C42×C7—22no—
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2×D411yescharacteristic
163C42×C7C2411yescharacteristic
164C22⋊Dic7—41no—
165#1144—44no—
166#1144—84no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171Dic7⋊C4—41no—
172Dic7⋊C4—41no—
173C4×Dic7—41no—
174#1136—42no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178#554—22no—
179#554—22no—
180#554—22no—
181#554C2×D714yes—
182#533—71no—
183C2×C4⋊Q8C2×D711yescharacteristic
184C2×#125—21no—
185C2×C22⋊Q8—21no—
186C2×C22⋊Q8—22no—
187C2×C4×Q8C2×D711yescharacteristic
188C2×C4×D4—21no—
189C2×C4×D4C2×D711yescharacteristic
190C22×C4⋊C4—21no—
191#494—72no—
192#487—72no—
193#484—141no—
194#478—142no—
195#477—72no—
196#476—72no—
197#474—72no—
198#462—72no—
199#462—142no—
200#451—141no—
201#451—142no—
202C4×Q16—142no—
203#431—71no—
204#431—71no—
205#428—141no—
206#423—71no—
207#418—141no—
208#333—74no—
209Q8⋊C8—74no—
210D4⋊C8—74no—
211C23×C7—22no—
212C23×C7—24no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7#12011yescharacteristic
216C7×Q8—22no—
217C7×Q8—22no—
218C7×Q8—44no—
219C7×Q8—42no—
220C7×Q8—82no—
221C7×D4—22no—
222C7×D4—24no—
223C7×D4C2×Q1614yes—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—24no—
229C2×C4×C7—41no—
230C2×C4×C7—44no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—44no—
234C2×C4×C7—42no—
235C2×C4×C7—44no—
236C2×C4×C7—88no—
237C2×C4×C7#12011yescharacteristic
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7C2×Q1612yes—
241C2×C4×C7#13712yes—
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C2×Dic7—41no—
246C2×Dic7—41no—
247Dic14—42no—
248Dic14—42no—
249C7⋊C8—42no—
250C7⋊C8—42no—
251C7⋊C8—42no—
252C22×Q8C22⋊D711yescharacteristic
253C22×Q8C22⋊D712yes—
254C22×D4C22⋊D711yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C2×Q16—142no—
258C2×Q16—144no—
259C2×M16—141no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—71no—
263C4⋊Q8—71no—
264C4⋊Q8C22⋊D712yes—
265C4⋊Q8C22×D712yes—
266#125—24no—
267#125—24no—
268#125—141no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—141no—
274C22⋊Q8—141no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—141no—
278C4×Q8—141no—
279C4×Q8C22⋊D712yes—
280C4×Q8C22×D712yes—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4C22⋊D712yes—
286C4×D4C22×D712yes—
287#117—141no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4C22⋊D711yescharacteristic
296C2×C4⋊C4C22⋊D711yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C42—21no—
300C2×C42C22×D711yescharacteristic
301#107—142no—
302#107—142no—
303#106—142no—
304C4⋊C8—72no—
305C4⋊C8—72no—
306C4⋊C8—141no—
307C4⋊C8—141no—
308#103—142no—
309#103—144no—
310#103—142no—
311#102—142no—
312#102—142no—
313C22⋊C8—141no—
314C22⋊C8—141no—
315C4×C8—72no—
316C4×C8—141no—
317C22×C7—21no—
318C22×C7—21no—
319C22×C7—24no—
320C22×C7—48no—
321C22×C7—42no—
322C22×C7—42no—
323C22×C7#45112yes—
324C22×C7C2×#12011yescharacteristic
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—21no—
328C4×C7—42no—
329C4×C7—42no—
330C4×C7—42no—
331C4×C7—44no—
332C4×C7#45112yes—
333C4×C7C22×Q1611yescharacteristic
334C4×C7C2×#13711yescharacteristic
335C4×C7#57512yes—
336Dic7—81no—
337Dic7—81no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—82no—
343C2×Q8—141no—
344C2×Q8—141no—
345C2×Q8C2×C22⋊D712yes—
346C2×Q8C2×C22⋊D714yes—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4#114414yescomplemented
350C2×D4C2×C22⋊D712yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4C2×C22⋊D711yescharacteristic
361C22×C4C2×C22⋊D711yescharacteristic
362C22×C4C2×C22⋊D711yescharacteristic
363Q16—284no—
364Q16—564no—
365M16—282no—
366C2×C8—142no—
367C2×C8—142no—
368C2×C8—142no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4—141no—
382C4⋊C4—141no—
383C4⋊C4—281no—
384C4⋊C4—281no—
385C4⋊C4C2×C22⋊D712yes—
386C4⋊C4C2×C22⋊D712yes—
387C22⋊C4—28no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—281no—
391C42—22no—
392C42—22no—
393C42—281no—
394C42C2×C22⋊D711yescharacteristic
395C42C2×C22⋊D711yescharacteristic
396C42C2×C22⋊D711yescharacteristic
397C42C23×D711yescharacteristic
398C2×C7—22no—
399C2×C7—44no—
400C2×C7C2×#45111yescharacteristic
401C2×C7#300511yescharacteristic
402C2×C7#301911yescharacteristic
403C23—24no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23#651611yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—42no—
412Q8—82no—
413Q8—282no—
414Q8—282no—
415D4—22no—
416D4—24no—
417D4C2×#114414yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—88no—
431C2×C4—281no—
432C2×C4—281no—
433C2×C4#650512yes—
434C2×C4C2×#114412yes—
435C2×C4#651611yescharacteristic
436C2×C4#651611yescharacteristic
437C2×C4#651611yescharacteristic
438C2×C4C22×C22⋊D711yescharacteristic
439C2×C4C22×C22⋊D711yescharacteristic
440C2×C4C22×C22⋊D711yescharacteristic
441C8—282no—
442C8—282no—
443C8—282no—
444C7#1874911yescharacteristic, complemented, Sylow
445C22—21no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—42no—
450C22—42no—
451C22#9000012yes—
452C22C2×#651611yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—561no—
461C4—561no—
462C4#9000012yes—
463C4C2×#650511yescharacteristic
464C4C22×#114411yescharacteristic
465C4#9051312yes—
466C2—22no—
467C2—44no—
468C2C2×#9000011yescharacteristic
469C2#1174152611yescharacteristic
470C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7#18749
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #11740667C2semidirect
3. #11739892C2semidirect
5. #11729680C2semidirect
6. #11729678C2semidirect
7. #11729528C2semidirect
8. #11729376C2semidirect
26. #89864C22semidirect
28. #89853C22semidirect
29. #89803C22semidirect
31. #89793C22semidirect
40. #89341C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
127. C2×C7×D4Q16semidirect
349. C2×D4#1144semidirect
444. C7#18749semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117406672
#117398922
#117397412
#117296802
#117296782
#117295282
#117293762
#117293512
#187497