Finite Group Databasebeta

← #13339199

Subgroups — #13339199

Order 1792.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2988C211yesmaximal, characteristic, complemented
2#11739971C211yesmaximal, characteristic
3#11739952C211yesmaximal, characteristic, complemented
4#11736750C211yesmaximal, characteristic, complemented
5#11728694C212yesmaximal, complemented
6#11728692C212yesmaximal, complemented
7#11728602C212yesmaximal, complemented
8#11728600C212yesmaximal, complemented
9#11728503C211yesmaximal, characteristic, complemented
10#11728192C211yesmaximal, characteristic, complemented
11C2×#89280C211yesmaximal, characteristic, complemented
12C4×C7×#41C2211yescharacteristic
13C2×#6512—21no—
14C2×#6437C2211yescharacteristic
15#90161C2211yescharacteristic
16C7×C8×Q8C2212yescomplemented
17C7×C8×D4—22no—
18C7×C8×D4C2212yescomplemented
19C7×#431—21no—
20C7×#431C2211yescharacteristic
21C2×C7×C4⋊C8—21no—
22C2×C7×C4⋊C8C2211yescharacteristic
23#90005—21no—
24#89799—21no—
25#89799C2212yes—
26#89794—21no—
27#89793C2211yescharacteristic, complemented
28#89792C2211yescharacteristic, complemented
29#89787—21no—
30#89787C2212yes—
31#89781—21no—
32#89780C2211yescharacteristic, complemented
33#89779C2211yescharacteristic, complemented
34#89773—21no—
35#89770—21no—
36#89696C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38#89620—21no—
39#89620—21no—
40#89503—22no—
41#89466C2212yes—
42#89465C2212yes—
43#89280C2212yes—
44#89280C2212yes—
45#89277C2212yescomplemented
46#89276C2212yescomplemented
47#89252C2212yescomplemented
48#89252C2212yescomplemented
49#19409—71nomaximal, Sylow
50C2×C7×#41D411yescharacteristic
51C22×D28—21no—
52C22×D28—21no—
53C22×D28—21no—
54C22×C7×C8—21no—
55C22×C7×C8—22no—
56C4×C7×Q8D411yescharacteristic
57C4×C7×Q8C2311yescharacteristic
58C4×C7×D4—21no—
59C4×C7×D4—21no—
60C4×C7×D4D411yescharacteristic
61C4×C7×D4C2311yescharacteristic
62C7×#117—21no—
63C7×#117D411yescharacteristic
64C2×C42×C7—21no—
65C2×C42×C7C2311yescharacteristic
66#6512—44no—
67C2×#1143—22no—
68C2×#1141—22no—
69#6489—21no—
70#6487—21no—
71#6487—21no—
72C2×#1136—21no—
73C22×C7⋊C8—21no—
74C2×#1132—22no—
75C2×#1132—22no—
76#6437—21no—
77#6437—21no—
78#6437—21no—
79#6437C2312yes—
80#6433—21no—
81#6432C2311yescharacteristic
82#6432C2311yescharacteristic
83C4×Dic14—21no—
84C4×Dic14—21no—
85C7×C4⋊C8—22no—
86C7×C4⋊C8—21no—
87C7×C4⋊C8D411yescharacteristic
88C7×C4⋊C8D411yescharacteristic
89C7×C4⋊C8C2312yes—
90C7×C22⋊C8—22no—
91C7×C22⋊C8D412yes—
92C4×C7×C8—22no—
93C4×C7×C8C2312yes—
94#6409—21no—
95#6409—21no—
96#6406—21no—
97#6406—21no—
98#6395—22no—
99#6395—22no—
100#6383—22no—
101#6383—22no—
102#6382—21no—
103#6382—21no—
104#6381—21no—
105#6381—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)—21no—
108C4⋊(C7⋊C8)C2311yescharacteristic
109C4⋊(C7⋊C8)C2311yescharacteristic
110C4⋊(C7⋊C8)C2312yes—
111#3208—71no—
112#3076—71no—
113#3038—71no—
114#2988D711yescharacteristic, complemented
115#1678—72no—
116#1677—72no—
117#1666—72no—
118#1665—72no—
119#1570—71no—
120#1550—71no—
121C2×#331—71no—
122C23×D7—41no—
123C7×#41—22no—
124C7×#41SD1614yescomplemented
125C2×C7×Q8C2×D411yescharacteristic
126C2×C7×D4—21no—
127C2×C7×D4C2×D411yescharacteristic
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7#4111yescharacteristic
134C2×D28—22no—
135C2×D28—22no—
136C2×D28—22no—
137C2×D28—42no—
138C2×D28—42no—
139C2×D28—41no—
140C2×D28—41no—
141C2×D28—41no—
142C2×D28—81no—
143C2×D28—81no—
144C2×D28—82no—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×C7×C8—22no—
148C2×C7×C8—82no—
149C2×C7×C8—81no—
150C2×C7×C8C2×D412yes—
151C2×C7×C8C2×D412yes—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4C2×D411yescharacteristic
155C7×C4⋊C4C2×D411yescharacteristic
156C7×C22⋊C4—21no—
157C7×C22⋊C4—21no—
158C7×C22⋊C4—21no—
159C42×C7—21no—
160C42×C7—21no—
161C42×C7—21no—
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C42×C7#4111yescharacteristic
165C42×C7C2411yescharacteristic
166C22⋊Dic7—41no—
167#1143—44no—
168#1141—44no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173Dic7⋊C4—41no—
174Dic7⋊C4—41no—
175C4×Dic7—41no—
176#1136—42no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—42no—
181#1132—82no—
182#1132—82no—
183C2×AGL(1,ℤ/8)—141no—
184#533—71no—
185C2×#127—71no—
186C4×#41C2×D711yescharacteristic
187#492—72no—
188#486—72no—
189#479—141no—
190#478—141no—
191#477—71no—
192#476—71no—
193#475—71no—
194#474—71no—
195#460—72no—
196#460—141no—
197#460—141no—
198#460—141no—
199#459—72no—
200#459—141no—
201#450—142no—
202C8×Q8C2×D712yescomplemented
203C8×D4—22no—
204C8×D4C2×D712yescomplemented
205#431—21no—
206#431C2×D711yescharacteristic
207#427—141no—
208#423—71no—
209C2×C4⋊C8—21no—
210C2×C4⋊C8—71no—
211C2×C4⋊C8C2×D711yescharacteristic
212#418—141no—
213#416—141no—
214#334—72no—
215#334—72no—
216#333—72no—
217#332—72no—
218#331—72no—
219#331—72no—
220C23×C7—21no—
221C23×C7#12111yescharacteristic
222C22×D7—42no—
223C22×D7—82no—
224C22×D7—82no—
225C22×D7—82no—
226C7×Q8—21no—
227C7×Q8C2×SD1612yes—
228C7×D4—24no—
229C7×D4—21no—
230C7×D4C2×SD1612yes—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—24no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—81no—
241C2×C4×C7#12111yescharacteristic
242C2×C4×C7#12111yescharacteristic
243C2×C4×C7#12111yescharacteristic
244C2×C4×C7C2×SD1612yes—
245C2×C4×C7#13512yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C2×#4111yescharacteristic
249C2×Dic7—41no—
250C2×Dic7—41no—
251D28—42no—
252D28—42no—
253D28—42no—
254D28—42no—
255D28—82no—
256D28—82no—
257D28—82no—
258D28—82no—
259Dic14—41no—
260Dic14—41no—
261Dic14—81no—
262C7×C8—22no—
263C7×C8—22no—
264C7×C8—42no—
265C7⋊C8—42no—
266C7⋊C8—41no—
267C7⋊C8—41no—
268C2×#41D2811yescharacteristic
269C22×D4—141no—
270C22×D4—141no—
271C22×D4—141no—
272AGL(1,ℤ/8)—284no—
273C2×SD16—142no—
274C2×SD16—142no—
275C2×SD16—142no—
276C2×D8—142no—
277C2×M16—141no—
278C22×C8—21no—
279C22×C8—22no—
280C22×C8—141no—
281C4⋊Q8—71no—
282C4⋊Q8—71no—
283#127—72no—
284#127—141no—
285#127—141no—
286#127—141no—
287#125—141no—
288C22⋊Q8—141no—
289C22⋊Q8—141no—
290C4×Q8—141no—
291C4×Q8—141no—
292C4×Q8D2811yescharacteristic
293C4×Q8C22×D711yescharacteristic
294C4×D4—21no—
295C4×D4—21no—
296C4×D4D2811yescharacteristic
297C4×D4C22×D711yescharacteristic
298#117—21no—
299#117—141no—
300#117D2811yescharacteristic
301C2×C42—21no—
302C2×C42C22×D711yescharacteristic
303#107—141no—
304#107—141no—
305#106—141no—
306#106—141no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8—71no—
310C4⋊C8—71no—
311C4⋊C8—72no—
312C4⋊C8—141no—
313C4⋊C8—141no—
314C4⋊C8C22⋊D711yescharacteristic
315C4⋊C8C22⋊D711yescharacteristic
316C4⋊C8C22×D712yes—
317#103—141no—
318#103—141no—
319#102—141no—
320#102—141no—
321#102—142no—
322#102—142no—
323#102—142no—
324#102—142no—
325C22⋊C8—22no—
326C22⋊C8C22⋊D712yes—
327C4×C8—22no—
328C4×C8C22×D712yes—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—22no—
332C22×C7—41no—
333C22×C7—41no—
334C22×C7#46512yes—
335C22×C7C2×#12111yescharacteristic
336C2×D7—82no—
337C2×D7—82no—
338C2×D7—82no—
339C2×D7—164no—
340C4×C7—21no—
341C4×C7—22no—
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—41no—
345C4×C7—41no—
346C4×C7#46512yes—
347C4×C7C22×SD1611yescharacteristic
348C4×C7C2×#13511yescharacteristic
349C4×C7#57611yescharacteristic
350C4×C7#57711yescharacteristic
351Dic7—81no—
352Dic7—81no—
353C24—281no—
354#41—22no—
355#41#113214yescomplemented
356C2×Q8—141no—
357C2×Q8—141no—
358C2×Q8C2×D2811yescharacteristic
359C2×D4—21no—
360C2×D4—142no—
361C2×D4—142no—
362C2×D4—142no—
363C2×D4—282no—
364C2×D4—282no—
365C2×D4—281no—
366C2×D4—281no—
367C2×D4—281no—
368C2×D4—561no—
369C2×D4—561no—
370C2×D4—562no—
371C2×D4C2×D2811yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C2×D2811yescharacteristic
376C22×C4#115711yescharacteristic
377C22×C4C2×C22⋊D711yescharacteristic
378SD16—284no—
379SD16—562no—
380SD16—562no—
381D8—284no—
382M16—282no—
383C2×C8—22no—
384C2×C8—82no—
385C2×C8—81no—
386C2×C8—142no—
387C2×C8—141no—
388C2×C8—141no—
389C2×C8—282no—
390C2×C8C2×C22⋊D712yes—
391C2×C8C2×C22⋊D712yes—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—141no—
395C4⋊C4—141no—
396C4⋊C4—141no—
397C4⋊C4—141no—
398C4⋊C4—281no—
399C4⋊C4—281no—
400C4⋊C4C2×D2811yescharacteristic
401C4⋊C4C2×D2811yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—281no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—281no—
410C42C2×D2811yescharacteristic
411C42#115711yescharacteristic
412C42C2×C22⋊D711yescharacteristic
413C42C23×D711yescharacteristic
414C2×C7—22no—
415C2×C7—42no—
416C2×C7#304511yescharacteristic
417C2×C7C2×#46511yescharacteristic
418C2×C7#306811yescharacteristic
419D7—162no—
420C23—21no—
421C23—282no—
422C23—562no—
423C23—562no—
424C23—562no—
425C23#649311yescharacteristic
426Q8—21no—
427Q8—281no—
428Q8—281no—
429Q8—561no—
430Q8C2×#113212yes—
431D4—24no—
432D4—21no—
433D4—282no—
434D4—282no—
435D4—282no—
436D4—282no—
437D4—562no—
438D4—562no—
439D4—562no—
440D4—562no—
441D4C2×#113212yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—281no—
453C2×C4—281no—
454C2×C4C2×#113212yes—
455C2×C4#646712yes—
456C2×C4#649311yescharacteristic
457C2×C4#649311yescharacteristic
458C2×C4#649311yescharacteristic
459C2×C4C22×D2811yescharacteristic
460C2×C4C2×#115711yescharacteristic
461C2×C4C22×C22⋊D711yescharacteristic
462C8—22no—
463C8—22no—
464C8—42no—
465C8—282no—
466C8—281no—
467C8—281no—
468C7#1940911yescharacteristic, complemented, Sylow
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—562no—
475C22—562no—
476C22—562no—
477C22—1124no—
478C22#8988612yes—
479C22C2×#649311yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—561no—
487C4—561no—
488C4#8988612yes—
489C4C22×#113211yescharacteristic
490C4C2×#646711yescharacteristic
491C4#9052611yescharacteristic
492C4#9052811yescharacteristic
493C2—22no—
494C2—42no—
495C2—1122no—
496C2#1174001011yescharacteristic
497C2C2×#8988611yescharacteristic
498C2#1174125911yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2988C2
socleC22×C7C2×#121
prime radical#11739971C2
hypercenter#2988D7
hypocenterC7#19409
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. C7×#2988C2semidirect
3. #11739952C2semidirect
4. #11736750C2semidirect
5. #11728694C2semidirect
6. #11728692C2semidirect
7. #11728602C2semidirect
8. #11728600C2semidirect
9. #11728503C2semidirect
10. #11728192C2semidirect
11. C2×#89280C2semidirect
16. C7×C8×Q8C22semidirect
18. C7×C8×D4C22semidirect
27. #89793C22semidirect
28. #89792C22semidirect
32. #89780C22semidirect
33. #89779C22semidirect
45. #89277C22semidirect
46. #89276C22semidirect
47. #89252C22semidirect
48. #89252C22semidirect
114. #2988D7semidirect
124. C7×#41SD16semidirect
202. C8×Q8C2×D7semidirect
204. C8×D4C2×D7semidirect
355. #41#1132semidirect
468. C7#19409semidirect

Maximal subgroups

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

subgroupindex
C7×#29882
#117399712
#117399522
#117367502
#117286942
#117286922
#117286022
#117286002
#117285032
#117281922
C2×#892802
#194097