Finite Group Databasebeta

← #13337362

Subgroups — #13337362

Order 1792.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3151C211yesmaximal, characteristic
2#11740765C211yesmaximal, characteristic, complemented
3#11740614C211yesmaximal, characteristic, complemented
4#11735953C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6C2×#89339C211yesmaximal, characteristic, complemented
7#11729041C212yesmaximal, complemented
8#11729040C212yesmaximal, complemented
9#11729027C212yesmaximal, complemented
10#11729026C212yesmaximal, complemented
11#11727604C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)—21no—
13C2×C7×AGL(1,ℤ/8)—21no—
14C7×#532C2211yescharacteristic
15C2×C7×#127C2211yescharacteristic
16#90161C2211yescharacteristic
17C7×#496—22no—
18C7×#493C2212yes—
19C7×#492C2212yes—
20C4×C7×M16C2211yescharacteristic
21#89990—22no—
22#89894—21no—
23#89891—21no—
24#89866C2212yescomplemented
25#89848—21no—
26#89847C2212yes—
27#89847C2212yes—
28#89847C2212yescomplemented
29#89842—21no—
30#89813—22no—
31#89791—21no—
32#89780—21no—
33#89697C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36#89473—22no—
37#89451C2212yescomplemented
38C8×Dic14C2212yescomplemented
39#89342C2211yescharacteristic, complemented
40#89342C2211yescharacteristic, complemented
41#89339C2212yes—
42#89339C2212yes—
43#89339C2211yescharacteristic, complemented
44#89339C2211yescharacteristic, complemented
45#89253C2212yescomplemented
46#89252C2212yescomplemented
47#19238—71nomaximal, Sylow
48C2×C7×#41—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C7×AGL(1,ℤ/8)—44no—
53C7×AGL(1,ℤ/8)—44no—
54C2×C7×SD16—22no—
55C2×C7×SD16—22no—
56C2×C7×D8—22no—
57C2×C7×D8—22no—
58C2×C7×M16D411yescharacteristic
59C2×C7×M16D411yescharacteristic
60C7×C4⋊Q8C2311yescharacteristic
61C7×#127—21no—
62C7×#127—21no—
63C7×#127—21no—
64C7×#127C2312yes—
65C7×#127C2311yescharacteristic
66C7×#124—21no—
67C7×#121—21no—
68C7×#121—21no—
69C4×C7×D4—21no—
70C4×C7×D4—21no—
71C2×C42×C7C2311yescharacteristic
72C2×#1142—22no—
73C2×#1142—22no—
74C2×#1142—22no—
75#6489—21no—
76#6487—21no—
77#6487D412yes—
78C22×C7⋊C8—21no—
79C22×C7⋊C8—21no—
80#6433—21no—
81#6432D411yescharacteristic
82#6432D411yescharacteristic
83C4×Dic14—21no—
84C4×Dic14C2312yes—
85C7×#97D412yes—
86C4×C7×C8C2312yes—
87#6406—22no—
88#6406—22no—
89#6406—22no—
90C22⋊(C7⋊C8)—21no—
91#6389—22no—
92Dic7⋊C8—22no—
93Dic7⋊C8—22no—
94C8×Dic7—22no—
95#6384—22no—
96#6382—22no—
97C4⋊(C7⋊C8)—21no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)C2312yes—
100C4⋊(C7⋊C8)C2311yescharacteristic
101C4⋊(C7⋊C8)C2311yescharacteristic
102C4⋊(C7⋊C8)C2311yescharacteristic
103C4⋊(C7⋊C8)C2311yescharacteristic
104C4×C7⋊C8—21no—
105#3246—71no—
106#3208—71no—
107#3151D711yescharacteristic
108#2962—71no—
109#1688—72no—
110#1686—72no—
111#1668—72no—
112#1666—72no—
113#1565—71no—
114#1540—71no—
115C2×#331—71no—
116C24×C7—41no—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×D4—21no—
121C2×C7×D4—22no—
122C2×C7×D4—21no—
123C2×C7×D4—22no—
124C2×C7×D4—21no—
125C2×C7×D4—22no—
126C2×C7×D4—41no—
127C2×C7×D4—41no—
128C2×C7×D4—41no—
129C2×C7×D4—41no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—81no—
133C2×C7×D4—81no—
134C2×C7×D4—82no—
135C22×C4×C7—41no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7#4111yescharacteristic
139C2×Dic14—21no—
140C2×Dic14C2×D412yes—
141C7×SD16—44no—
142C7×SD16—44no—
143C7×D8—44no—
144C7×D8—44no—
145C7×M16—22no—
146C7×M16SD1614yescomplemented
147C2×C7×C8C2×D412yes—
148C2×C7×C8C2×D412yes—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—21no—
151C7×C22⋊C4—41no—
152C7×C22⋊C4—41no—
153C42×C7C2×D411yescharacteristic
154C42×C7C2×D411yescharacteristic
155C42×C7#4111yescharacteristic
156C42×C7C2411yescharacteristic
157C22⋊Dic7—22no—
158#1142—82no—
159#1142—82no—
160#1142—82no—
161C4⋊Dic7—21no—
162C4⋊Dic7—21no—
163C4⋊Dic7—21no—
164C4⋊Dic7C2×D412yes—
165Dic7⋊C4—22no—
166Dic7⋊C4—22no—
167C4×Dic7—22no—
168C2×C7⋊C8—21no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—21no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8—42no—
174C2×C7⋊C8—81no—
175C2×AGL(1,ℤ/8)—21no—
176C2×AGL(1,ℤ/8)—21no—
177#533—71no—
178#532C2×D711yescharacteristic
179C2×#127C2×D711yescharacteristic
180#496—22no—
181#493C2×D712yes—
182#492—141no—
183#492C2×D712yes—
184#486—141no—
185#476—141no—
186#475—72no—
187#465—142no—
188#461—141no—
189#460—72no—
190#460—72no—
191#460—72no—
192#450—142no—
193#446—72no—
194#446—142no—
195C8×Q8—72no—
196#431—71no—
197#424—141no—
198C2×C4⋊C8—71no—
199C2×C4⋊C8—71no—
200#405—141no—
201C4×M16C2×D711yescharacteristic
202#335—72no—
203#334—72no—
204#332—71no—
205#332—71no—
206#331—72no—
207#331—72no—
208#331—71no—
209#331—71no—
210C23×C7—41no—
211C23×C7—42no—
212C23×C7—82no—
213C23×C7—82no—
214C23×C7—82no—
215C23×C7#12111yescharacteristic
216C7×Q8—41no—
217C7×Q8—41no—
218C7×D4—41no—
219C7×D4—42no—
220C7×D4—41no—
221C7×D4—42no—
222C7×D4—41no—
223C7×D4—42no—
224C7×D4—42no—
225C7×D4—41no—
226C7×D4—81no—
227C7×D4—82no—
228C7×D4—82no—
229C7×D4—82no—
230C7×D4—82no—
231C7×D4—82no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—82no—
237C2×C4×C7#12111yescharacteristic
238C2×C4×C7#12111yescharacteristic
239C2×C4×C7#12111yescharacteristic
240C2×C4×C7C2×SD1612yes—
241C2×C4×C7AGL(1,ℤ/8)12yes—
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C2×#4111yescharacteristic
245C2×Dic7—22no—
246C2×Dic7—22no—
247Dic14—22no—
248Dic14—22no—
249Dic14—81no—
250C7×C8—22no—
251C7×C8C2×SD1614yes—
252C7⋊C8—41no—
253C7⋊C8—41no—
254C7⋊C8—42no—
255C7⋊C8—41no—
256C7⋊C8—41no—
257C2×#41—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261AGL(1,ℤ/8)—44no—
262AGL(1,ℤ/8)—44no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×SD16—142no—
266C2×SD16—142no—
267C2×SD16—142no—
268C2×D8—22no—
269C2×D8—22no—
270C2×M16C22⋊D711yescharacteristic
271C2×M16C22⋊D711yescharacteristic
272C22×C8—141no—
273C22×C8—141no—
274C4⋊Q8—71no—
275C4⋊Q8—71no—
276C4⋊Q8C22×D711yescharacteristic
277#127—21no—
278#127—21no—
279#127—21no—
280#127C22×D712yes—
281#127C22×D711yescharacteristic
282#125—141no—
283#124—21no—
284C22⋊Q8—72no—
285C22⋊Q8—141no—
286#121—21no—
287#121—21no—
288C4×Q8—72no—
289C4×Q8—141no—
290C4×D4—21no—
291C4×D4—21no—
292#117—141no—
293C2×C42C22×D711yescharacteristic
294#106—142no—
295C4⋊C8—71no—
296C4⋊C8—71no—
297C4⋊C8—72no—
298C4⋊C8—71no—
299C4⋊C8—71no—
300C4⋊C8—142no—
301C4⋊C8—142no—
302C4⋊C8—141no—
303C4⋊C8—141no—
304#103—142no—
305#102—142no—
306#102—142no—
307#102—142no—
308C22⋊C8—141no—
309#97—142no—
310#97C22⋊D712yes—
311C4×C8—142no—
312C4×C8—141no—
313C4×C8C22×D712yes—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—81no—
317C22×C7—82no—
318C22×C7—82no—
319C22×C7—81no—
320C22×C7—82no—
321C22×C7—81no—
322C22×C7—164no—
323C22×C7#46012yes—
324C22×C7C2×#12111yescharacteristic
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—21no—
328C4×C7—81no—
329C4×C7#46012yes—
330C4×C7C22×SD1611yescharacteristic
331C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
332C4×C7#57611yescharacteristic
333C4×C7#57711yescharacteristic
334Dic7—42no—
335Dic7—42no—
336C24—41no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—72no—
341C2×Q8—141no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4—82no—
357C22×C4—41no—
358C22×C4D4×D711yescharacteristic
359C22×C4#115911yescharacteristic
360C22×C4C2×C22⋊D711yescharacteristic
361SD16—44no—
362SD16—44no—
363SD16—562no—
364SD16—562no—
365SD16—562no—
366D8—44no—
367D8—44no—
368M16—22no—
369M16#114214yescomplemented
370C2×C8—141no—
371C2×C8—141no—
372C2×C8—142no—
373C2×C8—141no—
374C2×C8—141no—
375C2×C8—282no—
376C2×C8—561no—
377C2×C8C2×C22⋊D712yes—
378C2×C8C2×C22⋊D712yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—72no—
382C4⋊C4—142no—
383C4⋊C4—141no—
384C4⋊C4—141no—
385C4⋊C4—141no—
386C4⋊C4—142no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—142no—
390C42—142no—
391C42D4×D711yescharacteristic
392C42#115911yescharacteristic
393C42C2×C22⋊D711yescharacteristic
394C42C23×D711yescharacteristic
395C2×C7—22no—
396C2×C7—161no—
397C2×C7—162no—
398C2×C7C2×#46011yescharacteristic
399C2×C7#306211yescharacteristic
400C2×C7#306811yescharacteristic
401C23—41no—
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23#650111yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—142no—
410Q8—142no—
411Q8—561no—
412D4—41no—
413D4—42no—
414D4—42no—
415D4—41no—
416D4—41no—
417D4—42no—
418D4—41no—
419D4—42no—
420D4—81no—
421D4—82no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—142no—
432C2×C4—142no—
433C2×C4#649512yes—
434C2×C4C2×#114212yes—
435C2×C4#650111yescharacteristic
436C2×C4#650111yescharacteristic
437C2×C4#650111yescharacteristic
438C2×C4C2×D4×D711yescharacteristic
439C2×C4C2×#115911yescharacteristic
440C2×C4C22×C22⋊D711yescharacteristic
441C8—22no—
442C8—282no—
443C8—281no—
444C8—281no—
445C8—281no—
446C8—281no—
447C8C2×#114214yes—
448C7#1923811yescharacteristic, complemented, Sylow
449C22—21no—
450C22—21no—
451C22—81no—
452C22—82no—
453C22—82no—
454C22—81no—
455C22—82no—
456C22—81no—
457C22—164no—
458C22#8984712yes—
459C22C2×#650111yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—282no—
465C4—282no—
466C4#8984712yes—
467C4#9046511yescharacteristic
468C4#9046611yescharacteristic
469C4C2×#649511yescharacteristic
470C4C22×#114211yescharacteristic
471C2—22no—
472C2—162no—
473C2—161no—
474C2C2×#8984711yescharacteristic
475C2#1174139611yescharacteristic
476C2#1174140811yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3151C2
socleC22×C7C2×#121
hypercenter#3151D7
hypocenterC7#19238
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
2. #11740765C2semidirect
3. #11740614C2semidirect
4. #11735953C2semidirect
5. #11729558C2semidirect
6. C2×#89339C2semidirect
7. #11729041C2semidirect
8. #11729040C2semidirect
9. #11729027C2semidirect
10. #11729026C2semidirect
11. #11727604C2semidirect
24. #89866C22semidirect
28. #89847C22semidirect
37. #89451C22semidirect
38. C8×Dic14C22semidirect
39. #89342C22semidirect
40. #89342C22semidirect
43. #89339C22semidirect
44. #89339C22semidirect
45. #89253C22semidirect
46. #89252C22semidirect
146. C7×M16SD16semidirect
369. M16#1142semidirect
448. C7#19238semidirect

Maximal subgroups

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

subgroupindex
C7×#31512
#117407652
#117406142
#117359532
#117295582
C2×#893392
#117290412
#117290402
#117290272
#117290262
#117276042
#192387