Finite Group Databasebeta

← #13332328

Subgroups — #13332328

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 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#11739876C211yesmaximal, characteristic, complemented
4#11739759C211yesmaximal, characteristic
5#11729679C214yesmaximal, complemented
6#11729678C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, 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—22no—
24#89997—21no—
25#89990—21no—
26#89864C2212yescomplemented
27#89862—22no—
28#89853C2212yescomplemented
29#89794—22no—
30#89793C2212yescomplemented
31#89792C2212yescomplemented
32#89791C2212yes—
33C4×#1142—22no—
34#89780C2212yes—
35#89743—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18753—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—24no—
83C2×#1142—22no—
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#6409—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
100#6382—22no—
101#6382—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#3101—71no—
112#3078—71no—
113#1643—74no—
114#1632—74no—
115#1548—71no—
116#1486—71no—
117#1479—71no—
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—82no—
124C2×C7×Q8C2×D412yes—
125C2×C7×Q8C2×D414yes—
126C2×C7×D4—24no—
127C2×C7×D4—21no—
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—42no—
137C22×C4×C7—42no—
138C22×C4×C7—84no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C2×Dic14—21no—
143C2×Dic14—21no—
144C7×C4⋊C4—28no—
145C7×C4⋊C4—28no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4C2×D412yes—
155C7×C4⋊C4C2×D412yes—
156C7×C22⋊C4—28no—
157C7×C22⋊C4—22no—
158C7×C22⋊C4—22no—
159C42×C7—22no—
160C42×C7—22no—
161C42×C7C2×D411yescharacteristic
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C42×C7C2411yescharacteristic
165C22⋊Dic7—41no—
166#1144—84no—
167#1142—44no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172Dic7⋊C4—41no—
173Dic7⋊C4—41no—
174C4×Dic7—41no—
175#1136—42no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179#554—22no—
180#554—22no—
181#554—22no—
182#554C2×D714yes—
183#533—71no—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×#125—21no—
186C2×C22⋊Q8—21no—
187C2×C22⋊Q8—22no—
188C2×C4×Q8C2×D711yescharacteristic
189C2×C4×D4—21no—
190C2×C4×D4C2×D711yescharacteristic
191C22×C4⋊C4—21no—
192#494—72no—
193#487—72no—
194#483—141no—
195#479—142no—
196#477—72no—
197#476—72no—
198#475—72no—
199#462—142no—
200#461—72no—
201#451—142no—
202#450—141no—
203C4×SD16—142no—
204#431—71no—
205#431—71no—
206#428—141no—
207#423—71no—
208#419—141no—
209#333—74no—
210Q8⋊C8—74no—
211D4⋊C8—74no—
212C23×C7—24no—
213C23×C7—22no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7#12011yescharacteristic
217C7×Q8—22no—
218C7×Q8—22no—
219C7×Q8—44no—
220C7×Q8—42no—
221C7×Q8—82no—
222C7×D4—24no—
223C7×D4—22no—
224C7×D4#13714yes—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—44no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—44no—
238C2×C4×C7—88no—
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7#12011yescharacteristic
242C2×C4×C7#13512yes—
243C2×C4×C7#13712yes—
244C2×C4×C7C22×D411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7C22×D411yescharacteristic
247C2×Dic7—41no—
248C2×Dic7—41no—
249Dic14—42no—
250Dic14—42no—
251C7⋊C8—42no—
252C7⋊C8—42no—
253C7⋊C8—42no—
254C22×Q8C22⋊D711yescharacteristic
255C22×Q8C22⋊D712yes—
256C22×D4C22⋊D711yescharacteristic
257C23×C4—22no—
258C23×C4—21no—
259C2×Q16—144no—
260C2×SD16—142no—
261C2×M16—141no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—71no—
265C4⋊Q8—71no—
266C4⋊Q8C22⋊D712yes—
267C4⋊Q8C22×D712yes—
268#125—24no—
269#125—24no—
270#125—141no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—141no—
276C22⋊Q8—141no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—141no—
280C4×Q8—141no—
281C4×Q8C22⋊D712yes—
282C4×Q8C22×D712yes—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4C22⋊D712yes—
288C4×D4C22×D712yes—
289#117—141no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4C22⋊D711yescharacteristic
298C2×C4⋊C4C22⋊D711yescharacteristic
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42C22×D711yescharacteristic
303#107—142no—
304#106—142no—
305#106—142no—
306C4⋊C8—72no—
307C4⋊C8—72no—
308C4⋊C8—141no—
309C4⋊C8—141no—
310#103—142no—
311#103—142no—
312#103—144no—
313#103—142no—
314#102—142no—
315C22⋊C8—141no—
316C22⋊C8—141no—
317C4×C8—72no—
318C4×C8—141no—
319C22×C7—21no—
320C22×C7—21no—
321C22×C7—24no—
322C22×C7—48no—
323C22×C7—42no—
324C22×C7—42no—
325C22×C7#45212yes—
326C22×C7C2×#12011yescharacteristic
327C4×C7—21no—
328C4×C7—21no—
329C4×C7—21no—
330C4×C7—42no—
331C4×C7—42no—
332C4×C7—42no—
333C4×C7—44no—
334C4×C7#45212yes—
335C4×C7C2×#13511yescharacteristic
336C4×C7C2×#13712yes—
337C4×C7C2×#13711yescharacteristic
338Dic7—81no—
339Dic7—81no—
340C24—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—82no—
346C2×Q8—141no—
347C2×Q8—141no—
348C2×Q8C2×C22⋊D712yes—
349C2×Q8C2×C22⋊D714yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4C2×C22⋊D712yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—84no—
363C22×C4C2×C22⋊D711yescharacteristic
364C22×C4C2×C22⋊D711yescharacteristic
365C22×C4C2×C22⋊D711yescharacteristic
366Q16—564no—
367SD16—284no—
368M16—282no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—142no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—141no—
383C4⋊C4—141no—
384C4⋊C4—141no—
385C4⋊C4—141no—
386C4⋊C4—281no—
387C4⋊C4—281no—
388C4⋊C4C2×C22⋊D712yes—
389C4⋊C4C2×C22⋊D712yes—
390C22⋊C4—28no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—281no—
394C42—22no—
395C42—22no—
396C42—281no—
397C42C2×C22⋊D711yescharacteristic
398C42C2×C22⋊D711yescharacteristic
399C42C2×C22⋊D711yescharacteristic
400C42C23×D711yescharacteristic
401C2×C7—22no—
402C2×C7—44no—
403C2×C7C2×#45211yescharacteristic
404C2×C7#300411yescharacteristic
405C2×C7#301911yescharacteristic
406C23—24no—
407C23—22no—
408C23—42no—
409C23—42no—
410C23#651611yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—44no—
414Q8—42no—
415Q8—82no—
416Q8—282no—
417Q8—282no—
418D4—24no—
419D4—22no—
420D4#650514yes—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—88no—
435C2×C4—281no—
436C2×C4—281no—
437C2×C4#651312yes—
438C2×C4#651412yes—
439C2×C4#651611yescharacteristic
440C2×C4#651611yescharacteristic
441C2×C4#651611yescharacteristic
442C2×C4C22×C22⋊D711yescharacteristic
443C2×C4C22×C22⋊D711yescharacteristic
444C2×C4C22×C22⋊D711yescharacteristic
445C8—282no—
446C8—282no—
447C8—282no—
448C7#1875311yescharacteristic, complemented, Sylow
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22#9001112yes—
456C22C2×#651611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—44no—
464C4—561no—
465C4—561no—
466C4#9001112yes—
467C4C2×#650512yes—
468C4C2×#651311yescharacteristic
469C4C2×#651411yescharacteristic
470C2—22no—
471C2—44no—
472C2#1174150411yescharacteristic
473C2C2×#9001111yescharacteristic
474C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7#18753
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. #11739876C2semidirect
5. #11729679C2semidirect
6. #11729678C2semidirect
7. #11729528C2semidirect
8. #11729375C2semidirect
26. #89864C22semidirect
28. #89853C22semidirect
30. #89793C22semidirect
31. #89792C22semidirect
40. #89341C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
448. C7#18753semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117406672
#117398762
#117397592
#117296792
#117296782
#117295282
#117293752
#117293522
#187537