Finite Group Databasebeta

← #13337370

Subgroups — #13337370

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×#3246C211yesmaximal, characteristic, complemented
2#11740800C211yesmaximal, characteristic, complemented
3#11740775C211yesmaximal, characteristic, complemented
4#11740732C211yesmaximal, characteristic
5#11729558C211yesmaximal, characteristic, complemented
6#11728874C212yesmaximal, complemented
7#11728870C212yesmaximal, complemented
8#11728779C212yesmaximal, complemented
9#11728775C212yesmaximal, complemented
10C2×#89280C211yesmaximal, characteristic, complemented
11#11728174C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C7×#532C2211yescharacteristic
14C2×#6512—21no—
15C2×#6512—21no—
16C2×#6437C2211yescharacteristic
17C7×#476—21no—
18C7×#475C2212yescomplemented
19C7×#465—22no—
20C7×#461—21no—
21C7×#460C2212yescomplemented
22C2×C7×C4⋊C8C2211yescharacteristic
23#89860—21no—
24#89857—21no—
25#89857C2212yes—
26#89845C2212yes—
27#89830—22no—
28#89818—22no—
29#89799C2212yes—
30#89799C2212yes—
31#89796—22no—
32#89796C2212yescomplemented
33Q8×C7⋊C8C2212yescomplemented
34#89771—21no—
35#89769—21no—
36#89697C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C4×#1136C2211yescharacteristic
39#89342C2211yescharacteristic, complemented
40#89342C2211yescharacteristic, complemented
41#89339C2211yescharacteristic, complemented
42#89339C2211yescharacteristic, complemented
43#89280C2212yes—
44#89280C2212yes—
45#89275C2212yescomplemented
46#89274C2212yescomplemented
47#19238—71nomaximal, Sylow
48C2×C7×#41—21no—
49C22×D28—21no—
50C22×D28—21no—
51C22×D28—21no—
52C2×C7×SD16—22no—
53C22×C7×C8—21no—
54C7×C4⋊Q8D411yescharacteristic
55C7×C4⋊Q8D411yescharacteristic
56C7×C4⋊Q8C2311yescharacteristic
57C7×#127C2311yescharacteristic
58C7×#125—21no—
59C7×#124—21no—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8D412yes—
62C7×#121—21no—
63C7×#121—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8C2312yes—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C7×#117—21no—
69C2×C42×C7C2311yescharacteristic
70#6512—44no—
71#6512—44no—
72C2×#1143—22no—
73C2×#1143—22no—
74C2×#1143—22no—
75C2×#1143—22no—
76C2×#1141—22no—
77C2×#1141—22no—
78C2×#1136D411yescharacteristic
79C2×#1136D411yescharacteristic
80C22×C7⋊C8—21no—
81#6437—21no—
82#6437—21no—
83#6437—21no—
84#6437C2312yes—
85C7×#106—22no—
86C7×C4⋊C8—21no—
87C7×C4⋊C8C2312yes—
88C7×#103—22no—
89C7×#102—22no—
90C22⋊(C7⋊C8)—21no—
91#6383—22no—
92#6383—22no—
93C4⋊(C7⋊C8)—21no—
94C4⋊(C7⋊C8)—22no—
95C4⋊(C7⋊C8)—22no—
96C4⋊(C7⋊C8)C2311yescharacteristic
97C4⋊(C7⋊C8)C2311yescharacteristic
98C4⋊(C7⋊C8)C2311yescharacteristic
99C4⋊(C7⋊C8)C2311yescharacteristic
100#6377—22no—
101#6377D412yes—
102C4×C7⋊C8—21no—
103C4×C7⋊C8—22no—
104C4×C7⋊C8C2312yes—
105#3246D711yescharacteristic, complemented
106#3208—71no—
107#3151—71no—
108#2962—71no—
109#1688—72no—
110#1686—72no—
111#1668—72no—
112#1666—72no—
113#1565—71no—
114#1540—71no—
115C2×#331—71no—
116C23×D7—41no—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8C2×D412yes—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—41no—
126C22×C4×C7—41no—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7#4111yescharacteristic
130C2×D28—22no—
131C2×D28—22no—
132C2×D28—22no—
133C2×D28—42no—
134C2×D28—42no—
135C2×D28—41no—
136C2×D28—41no—
137C2×D28—41no—
138C2×D28—81no—
139C2×D28—81no—
140C2×D28—82no—
141C7×SD16—82no—
142C2×C7×C8—22no—
143C2×C7×C8—81no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4C2×D412yes—
152C7×C22⋊C4—22no—
153C7×C22⋊C4—41no—
154C7×C22⋊C4—41no—
155C42×C7—22no—
156C42×C7C2×D411yescharacteristic
157C42×C7C2×D411yescharacteristic
158C42×C7#4111yescharacteristic
159C42×C7C2411yescharacteristic
160#1143—44no—
161#1143—44no—
162#1143—82no—
163#1143—82no—
164#1141—44no—
165#1141—44no—
166#1136—22no—
167#1136SD1614yescomplemented
168C2×C7⋊C8—21no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—21no—
172C2×C7⋊C8—42no—
173C2×C7⋊C8C2×D412yes—
174C2×C7⋊C8C2×D412yes—
175C2×AGL(1,ℤ/8)—141no—
176C2×AGL(1,ℤ/8)—141no—
177#533C2×D711yescharacteristic
178#532C2×D711yescharacteristic
179C2×#127—71no—
180#496—142no—
181#493—72no—
182#492—72no—
183#492—141no—
184#486—141no—
185#476—21no—
186#475C2×D712yescomplemented
187#465—22no—
188#461—21no—
189#460—72no—
190#460—72no—
191#460C2×D712yescomplemented
192#450—142no—
193#446—72no—
194#446—142no—
195C8×Q8—72no—
196#431—71no—
197#424—141no—
198C2×C4⋊C8—71no—
199C2×C4⋊C8C2×D711yescharacteristic
200#405—141no—
201C4×M16—71no—
202#335—72no—
203#334—72no—
204#332—71no—
205#332—71no—
206#331—71no—
207#331—71no—
208#331—72no—
209#331—72no—
210C23×C7—41no—
211C23×C7#12111yescharacteristic
212C22×D7—42no—
213C22×D7—82no—
214C22×D7—82no—
215C22×D7—82no—
216C7×Q8—22no—
217C7×Q8—22no—
218C7×Q8—41no—
219C7×Q8—41no—
220C7×Q8—81no—
221C7×D4—41no—
222C7×D4—41no—
223C7×D4—41no—
224C7×D4—41no—
225C7×D4—81no—
226C7×D4—82no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—82no—
234C2×C4×C7#12111yescharacteristic
235C2×C4×C7#12111yescharacteristic
236C2×C4×C7#12111yescharacteristic
237C2×C4×C7C2×SD1612yes—
238C2×C4×C7AGL(1,ℤ/8)12yes—
239C2×C4×C7C22×D411yescharacteristic
240C2×C4×C7C22×D411yescharacteristic
241C2×C4×C7C2×#4111yescharacteristic
242D28—42no—
243D28—42no—
244D28—42no—
245D28—42no—
246D28—82no—
247D28—82no—
248D28—82no—
249D28—82no—
250C7×C8—42no—
251C7⋊C8—22no—
252C7⋊C8—41no—
253C7⋊C8—41no—
254C7⋊C8—41no—
255C7⋊C8—41no—
256C7⋊C8C2×SD1614yes—
257C2×#41—21no—
258C22×D4—141no—
259C22×D4—141no—
260C22×D4—141no—
261AGL(1,ℤ/8)—284no—
262AGL(1,ℤ/8)—284no—
263C2×SD16—22no—
264C2×SD16—142no—
265C2×SD16—142no—
266C2×SD16—142no—
267C2×SD16—142no—
268C2×D8—142no—
269C2×D8—142no—
270C2×M16—71no—
271C2×M16—71no—
272C22×C8—21no—
273C22×C8—141no—
274C4⋊Q8C22⋊D711yescharacteristic
275C4⋊Q8C22⋊D711yescharacteristic
276C4⋊Q8C22×D711yescharacteristic
277#127—72no—
278#127—141no—
279#127—141no—
280#127—141no—
281#127C22×D711yescharacteristic
282#125—21no—
283#124—21no—
284C22⋊Q8—21no—
285C22⋊Q8C22⋊D712yes—
286#121—21no—
287#121—21no—
288C4×Q8—21no—
289C4×Q8C22×D712yes—
290C4×D4—21no—
291C4×D4—21no—
292#117—21no—
293C2×C42C22×D711yescharacteristic
294#106—22no—
295C4⋊C8—21no—
296C4⋊C8—71no—
297C4⋊C8—71no—
298C4⋊C8—71no—
299C4⋊C8—71no—
300C4⋊C8—142no—
301C4⋊C8—142no—
302C4⋊C8—141no—
303C4⋊C8C22×D712yes—
304#103—22no—
305#102—22no—
306#102—142no—
307#102—142no—
308C22⋊C8—141no—
309#97—72no—
310#97—142no—
311C4×C8—72no—
312C4×C8—142no—
313C4×C8—141no—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7—81no—
319C22×C7#46012yes—
320C22×C7C2×#12111yescharacteristic
321C2×D7—82no—
322C2×D7—82no—
323C2×D7—82no—
324C2×D7—164no—
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—21no—
328C4×C7—42no—
329C4×C7—42no—
330C4×C7—81no—
331C4×C7#46012yes—
332C4×C7C22×SD1611yescharacteristic
333C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
334C4×C7#57611yescharacteristic
335C4×C7#57711yescharacteristic
336C24—281no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8C2×C22⋊D712yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4—142no—
347C2×D4—142no—
348C2×D4—142no—
349C2×D4—282no—
350C2×D4—282no—
351C2×D4—281no—
352C2×D4—281no—
353C2×D4—281no—
354C2×D4—561no—
355C2×D4—561no—
356C2×D4—562no—
357C22×C4—41no—
358C22×C4D4×D711yescharacteristic
359C22×C4#115911yescharacteristic
360C22×C4C2×C22⋊D711yescharacteristic
361SD16—82no—
362SD16—284no—
363SD16—284no—
364SD16—562no—
365SD16—562no—
366D8—284no—
367D8—284no—
368M16—74no—
369M16—142no—
370C2×C8—22no—
371C2×C8—72no—
372C2×C8—72no—
373C2×C8—81no—
374C2×C8—141no—
375C2×C8—141no—
376C2×C8—141no—
377C2×C8—141no—
378C2×C8—282no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4C2×C22⋊D712yes—
387C22⋊C4—22no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C42—22no—
391C42D4×D711yescharacteristic
392C42#115911yescharacteristic
393C42C2×C22⋊D711yescharacteristic
394C42C23×D711yescharacteristic
395C2×C7—22no—
396C2×C7—161no—
397C2×C7C2×#46011yescharacteristic
398C2×C7#306211yescharacteristic
399C2×C7#306811yescharacteristic
400D7—162no—
401C23—41no—
402C23—282no—
403C23—562no—
404C23—562no—
405C23—562no—
406C23#650211yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—41no—
410Q8—41no—
411Q8—81no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—81no—
417D4—82no—
418D4—282no—
419D4—282no—
420D4—282no—
421D4—282no—
422D4—562no—
423D4—562no—
424D4—562no—
425D4—562no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4D7×SD1612yes—
434C2×C4#647712yes—
435C2×C4#650211yescharacteristic
436C2×C4#650211yescharacteristic
437C2×C4#650211yescharacteristic
438C2×C4C2×D4×D711yescharacteristic
439C2×C4C2×#115911yescharacteristic
440C2×C4C22×C22⋊D711yescharacteristic
441C8—42no—
442C8—74no—
443C8—142no—
444C8—281no—
445C8—281no—
446C8—281no—
447C8—281no—
448C7#1923811yescharacteristic, complemented, Sylow
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—562no—
455C22—562no—
456C22—562no—
457C22—1124no—
458C22#8993512yes—
459C22C2×#650211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4#8993512yes—
467C4C2×D7×SD1611yescharacteristic
468C4C2×#647711yescharacteristic
469C4#9052611yescharacteristic
470C4#9052811yescharacteristic
471C2—22no—
472C2—161no—
473C2—1122no—
474C2#1174086711yescharacteristic
475C2C2×#8993511yescharacteristic
476C2#1174144711yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3246C2
socleC22×C7C2×#121
prime radical#11740732C2
hypercenter#3246D7
hypocenterC7#19238
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
1. C7×#3246C2semidirect
2. #11740800C2semidirect
3. #11740775C2semidirect
5. #11729558C2semidirect
6. #11728874C2semidirect
7. #11728870C2semidirect
8. #11728779C2semidirect
9. #11728775C2semidirect
10. C2×#89280C2semidirect
11. #11728174C2semidirect
18. C7×#475C22semidirect
21. C7×#460C22semidirect
32. #89796C22semidirect
33. Q8×C7⋊C8C22semidirect
39. #89342C22semidirect
40. #89342C22semidirect
41. #89339C22semidirect
42. #89339C22semidirect
45. #89275C22semidirect
46. #89274C22semidirect
105. #3246D7semidirect
167. #1136SD16semidirect
186. #475C2×D7semidirect
191. #460C2×D7semidirect
448. C7#19238semidirect

Maximal subgroups

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

subgroupindex
C7×#32462
#117408002
#117407752
#117407322
#117295582
#117288742
#117288702
#117287792
#117287752
C2×#892802
#117281742
#192387