Finite Group Databasebeta

← #13338995

Subgroups — #13338995

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3110C211yesmaximal, characteristic, complemented
2#11740801C211yesmaximal, characteristic, complemented
3#11739753C211yesmaximal, characteristic
4#11739746C211yesmaximal, characteristic, complemented
5#11729583C211yesmaximal, characteristic, complemented
6#11728855C212yesmaximal, complemented
7#11728854C212yesmaximal, complemented
8#11728793C212yesmaximal, complemented
9#11728792C212yesmaximal, complemented
10#11728571C211yesmaximal, characteristic, complemented
11#11728550C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14#90168C2211yescharacteristic
15C7×#484—21no—
16C7×#474C2212yes—
17C7×#461C2212yes—
18C7×#450—21no—
19C7×#440—22no—
20C7×#432C2211yescharacteristic
21#89989—21no—
22C2×C22⋊(C7⋊C8)—22no—
23#89865C2211yescharacteristic, complemented
24#89864C2211yescharacteristic, complemented
25#89860C2211yescharacteristic, complemented
26#89857C2211yescharacteristic, complemented
27#89819—22no—
28#89800—22no—
29#89787C2212yes—
30C4×#1141—22no—
31D4×C7⋊C8—22no—
32D4×C7⋊C8C2212yes—
33D4×C7⋊C8C2212yes—
34#89779C2212yes—
35#89739—21no—
36#89697C2211yescharacteristic
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89342C2212yescomplemented
40#89342C2212yescomplemented
41#89281C2212yescomplemented
42#89280C2212yescomplemented
43#89275C2212yescomplemented
44#89274C2212yescomplemented
45#19388—71nomaximal, Sylow
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—21no—
48C2×#1157—21no—
49C2×C7×SD16—22no—
50C7×C4⋊Q8C2311yescharacteristic
51C7×C4⋊Q8C2311yescharacteristic
52C7×#125—21no—
53C7×C22⋊Q8—21no—
54C7×C22⋊Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C4×C7×D4D412yes—
60C4×C7×D4C2312yes—
61C7×#117—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C22⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65C2×#1144—22no—
66C2×#1143—22no—
67C2×#1141—22no—
68#6493—21no—
69#6493—21no—
70C22×C7⋊C8—22no—
71C22×C7⋊C8—22no—
72#6438—21no—
73#6437C2311yescharacteristic
74C4×D28—21no—
75C4×D28—21no—
76#6432C2311yescharacteristic
77C7×#107—22no—
78C7×C4⋊C8C2312yes—
79C7×#103—22no—
80C7×#102—22no—
81C7×C22⋊C8—21no—
82C7×#97—21no—
83#6406—22no—
84C22⋊(C7⋊C8)—21no—
85C22⋊(C7⋊C8)—21no—
86C22⋊(C7⋊C8)—22no—
87C22⋊(C7⋊C8)—22no—
88C22⋊(C7⋊C8)D412yes—
89#6384—22no—
90#6383—22no—
91#6383—22no—
92#6381—22no—
93C4⋊(C7⋊C8)D412yes—
94C4⋊(C7⋊C8)C2312yes—
95C4⋊(C7⋊C8)C2312yes—
96C4×C7⋊C8—21no—
97C4×C7⋊C8—21no—
98C4×C7⋊C8C2311yescharacteristic
99C4×C7⋊C8C2311yescharacteristic
100#3210—71no—
101#3110D711yescharacteristic, complemented
102#3100—71no—
103#2972—71no—
104#1681—72no—
105#1679—72no—
106#1669—72no—
107#1665—72no—
108#1572—71no—
109#1553—71no—
110#1544—71no—
111C24×C7—21no—
112C2×C7×Q8—21no—
113C2×C7×Q8—21no—
114C2×C7×D4—24no—
115C2×C7×D4—21no—
116C2×C7×D4C2×D412yes—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—82no—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7#4111yescharacteristic
124C2×C22⋊D7—41no—
125#1157—82no—
126C2×D28—21no—
127C2×D28—21no—
128C2×D28—21no—
129C2×C4×D7—41no—
130C2×Dic14—21no—
131C7×SD16—44no—
132C2×C7×C8—22no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4C2×D412yes—
141C7×C22⋊C4—22no—
142C7×C22⋊C4—22no—
143C7×C22⋊C4—41no—
144C42×C7—41no—
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7#4111yescharacteristic
148C42×C7C2411yescharacteristic
149#1144—82no—
150#1143—82no—
151#1141—44no—
152(C2×D7)⋊C4—41no—
153(C2×D7)⋊C4—41no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C2×C7⋊C8—22no—
157C2×C7⋊C8—22no—
158C2×C7⋊C8—82no—
159C2×C7⋊C8—82no—
160C2×C7⋊C8C2×D412yes—
161C2×C7⋊C8C2×D412yes—
162#533C2×D711yescharacteristic
163#532—71no—
164C2×C4×D4C2×D711yescharacteristic
165#494—71no—
166#492—71no—
167#487—71no—
168#486—71no—
169#484—21no—
170#480—141no—
171#474—72no—
172#474C2×D712yes—
173#464—142no—
174#461C2×D712yes—
175#459—72no—
176#452—142no—
177#450—21no—
178#447—141no—
179#440—22no—
180C4×D8—142no—
181C8×D4—72no—
182C8×D4—72no—
183C8×D4—142no—
184#432C2×D711yescharacteristic
185#431—71no—
186#431—71no—
187#431—71no—
188C2×C22⋊C8—142no—
189#335—72no—
190#334—72no—
191#332—72no—
192#332—72no—
193#332—72no—
194#331—72no—
195C23×C7—24no—
196C23×C7—22no—
197C23×C7—81no—
198C23×C7—81no—
199C23×C7#12111yescharacteristic
200C22×D7—41no—
201C7×Q8—42no—
202C7×Q8—42no—
203C7×D4—24no—
204C7×D4—22no—
205C7×D4#13514yes—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—82no—
215C2×C4×C7—81no—
216C2×C4×C7#12111yescharacteristic
217C2×C4×C7#12111yescharacteristic
218C2×C4×C7#12111yescharacteristic
219C2×C4×C7C22×D411yescharacteristic
220C2×C4×C7C22×D411yescharacteristic
221C2×C4×C7C2×#4111yescharacteristic
222C22⋊D7—82no—
223C2×Dic7—41no—
224D28—42no—
225D28—42no—
226D28—81no—
227C4×D7—82no—
228Dic14—42no—
229C7×C8—42no—
230C7⋊C8—22no—
231C7⋊C8—22no—
232C7⋊C8—42no—
233C7⋊C8—42no—
234C2×#41—141no—
235C22×D4C22⋊D711yescharacteristic
236C23×C4—21no—
237C2×Q16—142no—
238C2×SD16—22no—
239C2×SD16—142no—
240C2×D8—142no—
241C22×C8—142no—
242C22×C8—142no—
243C4⋊Q8—71no—
244C4⋊Q8C22×D711yescharacteristic
245C4⋊Q8C22×D711yescharacteristic
246#127—71no—
247#125—21no—
248#124—141no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251#121—141no—
252#121—141no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—141no—
258C4×D4—141no—
259C4×D4C22⋊D712yes—
260C4×D4C22×D712yes—
261#117—21no—
262C2×C4⋊C4C22⋊D711yescharacteristic
263C2×C22⋊C4—21no—
264C2×C42C22×D711yescharacteristic
265#107—22no—
266#107—142no—
267C4⋊C8—72no—
268C4⋊C8—72no—
269C4⋊C8—72no—
270C4⋊C8C22×D712yes—
271#103—22no—
272#103—142no—
273#102—22no—
274#102—142no—
275#102—142no—
276#102—142no—
277C22⋊C8—21no—
278C22⋊C8—72no—
279C22⋊C8—142no—
280C22⋊C8—142no—
281C22⋊C8—141no—
282C22⋊C8—141no—
283#97—21no—
284C4×C8—71no—
285C4×C8—71no—
286C4×C8—141no—
287C4×C8—141no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—24no—
291C22×C7—21no—
292C22×C7—42no—
293C22×C7—42no—
294C22×C7—84no—
295C22×C7C2×#12111yescharacteristic
296C2×D7—81no—
297C2×D7—81no—
298C2×D7—81no—
299C4×C7—21no—
300C4×C7—21no—
301C4×C7—21no—
302C4×C7—21no—
303C4×C7—42no—
304C4×C7—81no—
305C4×C7—81no—
306C4×C7C2×#13512yes—
307C4×C7#57612yes—
308Dic7—81no—
309C24—21no—
310#41—562no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—141no—
314C2×D4—24no—
315C2×D4—21no—
316C2×D4—141no—
317C2×D4—141no—
318C2×D4—141no—
319C2×D4—281no—
320C2×D4C2×C22⋊D712yes—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—82no—
325C22×C4—281no—
326C22×C4D4×D711yescharacteristic
327C22×C4#115911yescharacteristic
328C22×C4C2×C22⋊D711yescharacteristic
329Q16—562no—
330SD16—44no—
331SD16—562no—
332D8—284no—
333C2×C8—22no—
334C2×C8—72no—
335C2×C8—72no—
336C2×C8—142no—
337C2×C8—142no—
338C2×C8—562no—
339C2×C8—562no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—141no—
348C4⋊C4—141no—
349C4⋊C4C2×C22⋊D712yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—281no—
354C22⋊C4—281no—
355C42—41no—
356C42D4×D711yescharacteristic
357C42#115911yescharacteristic
358C42C2×C22⋊D711yescharacteristic
359C42C23×D711yescharacteristic
360C2×C7—41no—
361C2×C7—44no—
362C2×C7#304111yescharacteristic
363C2×C7#305011yescharacteristic
364C2×C7#306911yescharacteristic
365D7—161no—
366C23—24no—
367C23—22no—
368C23—81no—
369C23—81no—
370C23—281no—
371C23#650211yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—282no—
375D4—24no—
376D4—22no—
377D4—282no—
378D4—282no—
379D4—561no—
380D4—562no—
381D4#651314yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—281no—
393C2×C4—562no—
394C2×C4#650211yescharacteristic
395C2×C4#650211yescharacteristic
396C2×C4#650211yescharacteristic
397C2×C4C2×D4×D711yescharacteristic
398C2×C4C2×#115911yescharacteristic
399C2×C4C22×C22⋊D711yescharacteristic
400C8—42no—
401C8—142no—
402C8—142no—
403C8—282no—
404C8—282no—
405C7#1938811yescharacteristic, complemented, Sylow
406C22—21no—
407C22—21no—
408C22—24no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—561no—
414C22—561no—
415C22—561no—
416C22C2×#650211yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—561no—
425C4#9046512yes—
426C4C2×#651312yes—
427C2—41no—
428C2—44no—
429C2—1121no—
430C2#1174036711yescharacteristic
431C2#1174137911yescharacteristic
432C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3110C2
socleC22×C7C2×#121
prime radical#11739753C2
hypercenter#3110D7
hypocenterC7#19388
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. C7×#3110C2semidirect
2. #11740801C2semidirect
4. #11739746C2semidirect
5. #11729583C2semidirect
6. #11728855C2semidirect
7. #11728854C2semidirect
8. #11728793C2semidirect
9. #11728792C2semidirect
10. #11728571C2semidirect
11. #11728550C2semidirect
23. #89865C22semidirect
24. #89864C22semidirect
25. #89860C22semidirect
26. #89857C22semidirect
39. #89342C22semidirect
40. #89342C22semidirect
41. #89281C22semidirect
42. #89280C22semidirect
43. #89275C22semidirect
44. #89274C22semidirect
101. #3110D7semidirect
405. C7#19388semidirect

Maximal subgroups

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

subgroupindex
C7×#31102
#117408012
#117397532
#117397462
#117295832
#117288552
#117288542
#117287932
#117287922
#117285712
#117285502
#193887