Finite Group Databasebeta

← #13321534

Subgroups — #13321534

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742856C211yesmaximal, characteristic
2C7×#2357C211yesmaximal, characteristic, complemented
3#11739746C211yesmaximal, characteristic, complemented
4#11727300C211yesmaximal, characteristic, complemented
5#11727299C211yesmaximal, characteristic, complemented
6#11727286C214yesmaximal, complemented
7#11727225C211yesmaximal, characteristic, complemented
8#11727223C211yesmaximal, characteristic, complemented
9#11727197C214yesmaximal, complemented
10C2×C4×C7×D4C2211yescharacteristic
11#90496—21no—
12#90482C2214yes—
13#90254C412yescomplemented
14#90250—22no—
15#90249—22no—
16#90244C412yescomplemented
17#90242C412yescomplemented
18#90233C412yescomplemented
19#90195—21no—
20#90185—22no—
21#90168C2211yescharacteristic
22#90161C2211yescharacteristic
23C7×#390—21no—
24C7×#388—22no—
25C7×#382C2211yescharacteristic
26C7×#381C2212yes—
27C4×C7×C4⋊C4—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4C2212yes—
30C2×C22⋊(C7⋊C8)C2214yes—
31D4×C7⋊C8—22no—
32D4×C7⋊C8—22no—
33#89697C2211yescharacteristic
34#89697C2211yescharacteristic
35#89247C2212yes—
36#89247C2214yes—
37#89247C2212yes—
38#89246C2212yes—
39#89246C2214yes—
40#89246C2212yes—
41#17617—71nomaximal, Sylow
42C22×C7×D4D411yescharacteristic
43C23×C4×C7C2312yes—
44C2×#1157—21no—
45C2×#1157C2×C414yes—
46C4×C7×D4C2×C412yes—
47C4×C7×D4C2×C412yes—
48C4×C7×D4D412yes—
49C4×C7×D4D412yes—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4D412yes—
56C2×C42×C7—21no—
57C2×C42×C7—22no—
58C2×C42×C7C2311yescharacteristic
59#6516—24no—
60#6503—22no—
61#6501—22no—
62#6501—22no—
63#6499—22no—
64#6493—21no—
65#6493C2×C414yes—
66#6493C2312yes—
67#6492—24no—
68C4×C22⋊D7—24no—
69C4×C22⋊D7—24no—
70#6489—21no—
71#6489C2×C414yes—
72#6487—21no—
73#6487C2×C414yes—
74#6487C2312yes—
75C22×C7⋊C8—24no—
76#6456—22no—
77#6455—22no—
78D7×C4⋊C4—22no—
79#6450—22no—
80#6448—24no—
81#6447—24no—
82#6446—24no—
83#6442—24no—
84#6438—21no—
85#6437C2×C411yescharacteristic
86C4×D28—21no—
87C4×D28C2×C412yes—
88#6433—21no—
89#6432C2×C411yescharacteristic
90#6432C2×C411yescharacteristic
91#6432C2×C411yescharacteristic
92C4×Dic14—21no—
93C4×Dic14C2×C412yes—
94C7×#95—22no—
95C7×#95—21no—
96C7×#95C2311yescharacteristic
97C7×#95C2312yes—
98C7×#95C2311yescharacteristic
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)C2312yes—
101C22⋊(C7⋊C8)C2312yes—
102C4⋊(C7⋊C8)—21no—
103C4⋊(C7⋊C8)—21no—
104C4×C7⋊C8—21no—
105C4×C7⋊C8—21no—
106#3506—71no—
107#2972—71no—
108#2357D711yescharacteristic, complemented
109#1518—71no—
110#1517—71no—
111#1516—74no—
112#1503—71no—
113#1502—71no—
114#1501—74no—
115C24×C7C2×D412yes—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4C22⋊C412yes—
119C2×C7×D4C22⋊C412yes—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—82no—
124C22×C4×C7C22×C414yes—
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D414yes—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2411yescharacteristic
130C2×C22⋊D7—22no—
131C2×C22⋊D7—24no—
132C2×C22⋊D7—24no—
133#1157—42no—
134#1157—42no—
135#1157—84no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28C22×C412yes—
139C2×C4×D7—22no—
140C2×C4×D7—24no—
141C2×Dic14—21no—
142C2×Dic14—21no—
143C2×Dic14C22×C412yes—
144C7×C4⋊C4—42no—
145C7×C4⋊C4—42no—
146C7×C4⋊C4C22⋊C412yes—
147C7×C4⋊C4C22⋊C412yes—
148C7×C22⋊C4—44no—
149C7×C22⋊C4—82no—
150C7×C22⋊C4C22⋊C414yes—
151C7×C22⋊C4C22⋊C414yes—
152C42×C7—42no—
153C42×C7—82no—
154C42×C7C22×C411yescharacteristic
155C42×C7C22×C411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7C2×D411yescharacteristic
158C22⋊Dic7—24no—
159C22⋊Dic7—24no—
160C22⋊Dic7—22no—
161(C2×D7)⋊C4—22no—
162(C2×D7)⋊C4—24no—
163(C2×D7)⋊C4—22no—
164C4⋊Dic7—21no—
165C4⋊Dic7—21no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7C22×C412yes—
169C4⋊Dic7C22×C412yes—
170Dic7⋊C4—22no—
171Dic7⋊C4—24no—
172Dic7⋊C4—22no—
173C4×Dic7—22no—
174C4×Dic7—24no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—84no—
178#559—142no—
179#556—72no—
180#555—72no—
181#553—142no—
182#550—72no—
183#547—72no—
184#538—142no—
185#535—141no—
186#533—71no—
187#532—71no—
188#525—74no—
189#520—141no—
190C2×C4×D4C2×D711yescharacteristic
191C8×D4—142no—
192C8×D4—142no—
193#431—71no—
194#431—71no—
195C2×C22⋊C8—74no—
196#390—21no—
197#388—22no—
198#382C2×D711yescharacteristic
199#381C2×D712yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2×D711yescharacteristic
202C4×C22⋊C4C2×D712yes—
203#328—72no—
204#328—74no—
205#328—72no—
206#327—72no—
207#327—74no—
208#327—72no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7C2×C22⋊C414yes—
214C23×C7C22×D411yescharacteristic
215C22×D7—22no—
216C7×D4—24no—
217C7×D4—24no—
218C2×C4×C7—22no—
219C2×C4×C7—44no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—44no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7C2×C22⋊C414yes—
233C2×C4×C7C2×C22⋊C411yescharacteristic
234C2×C4×C7C2×C22⋊C411yescharacteristic
235C2×C4×C7C2×C22⋊C411yescharacteristic
236C2×C4×C7C2×C22⋊C411yescharacteristic
237C2×C4×C7#13512yes—
238C2×C4×C7#13512yes—
239C2×C4×C7C23×C411yescharacteristic
240C2×C4×C7C22×D411yescharacteristic
241C22⋊D7—42no—
242C22⋊D7—42no—
243C22⋊D7—84no—
244C22⋊D7—84no—
245C2×Dic7—22no—
246C2×Dic7—22no—
247C2×Dic7—22no—
248D28—42no—
249D28—42no—
250D28—81no—
251C4×D7—44no—
252C4×D7—84no—
253Dic14—42no—
254Dic14—42no—
255Dic14—81no—
256C7⋊C8—42no—
257C7⋊C8—42no—
258C2×#41—74no—
259C2×#41—141no—
260C22×D4C22⋊D711yescharacteristic
261C23×C4C22×D712yes—
262C22×C8—144no—
263C4⋊Q8—71no—
264C4⋊Q8—71no—
265C4⋊Q8—71no—
266#127—71no—
267#127—142no—
268#126—144no—
269#125—141no—
270#124—142no—
271#124—144no—
272#124—141no—
273#123—144no—
274#123—144no—
275C22⋊Q8—74no—
276C22⋊Q8—72no—
277C22⋊Q8—141no—
278#121—74no—
279#121—72no—
280#121—142no—
281#121—142no—
282#121—141no—
283#121—144no—
284#120—144no—
285C4×Q8—72no—
286C4×Q8—142no—
287C4×Q8—141no—
288C4×D4—72no—
289C4×D4—144no—
290C4×D4—144no—
291C4×D4—142no—
292C4×D4—141no—
293C4×D4C4×D712yes—
294C4×D4C4×D712yes—
295C4×D4D2812yes—
296C4×D4D2812yes—
297#117—74no—
298#117—141no—
299#117—142no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4C22⋊D711yescharacteristic
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4C22⋊D712yes—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42C22×D711yescharacteristic
310C4⋊C8—141no—
311C4⋊C8—141no—
312C22⋊C8—72no—
313C22⋊C8—72no—
314C22⋊C8—144no—
315C4×C8—141no—
316C4×C8—141no—
317#95—22no—
318#95—21no—
319#95C22×D711yescharacteristic
320#95C22×D712yes—
321#95C22×D711yescharacteristic
322C22×C7—21no—
323C22×C7—21no—
324C22×C7—24no—
325C22×C7—24no—
326C22×C7—24no—
327C22×C7—82no—
328C22×C7—82no—
329C22×C7C22×C22⋊C411yescharacteristic
330C22×C7C2×#13511yescharacteristic
331C22×C7C2×#13511yescharacteristic
332C2×D7—42no—
333C2×D7—42no—
334C2×D7—42no—
335C4×C7—21no—
336C4×C7—21no—
337C4×C7—24no—
338C4×C7—21no—
339C4×C7—21no—
340C4×C7—42no—
341C4×C7—81no—
342C4×C7—82no—
343C4×C7#41612yes—
344C4×C7#41612yes—
345Dic7—42no—
346Dic7—42no—
347Dic7—42no—
348C24C2×C22⋊D712yes—
349#41—282no—
350#41—282no—
351#41—564no—
352C2×Q8—72no—
353C2×Q8—141no—
354C2×Q8—141no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—72no—
358C2×D4—142no—
359C2×D4—144no—
360C2×D4—144no—
361C2×D4—141no—
362C2×D4—141no—
363C2×D4(C2×D7)⋊C412yes—
364C2×D4(C2×D7)⋊C412yes—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—82no—
369C22×C4—142no—
370C22×C4—144no—
371C22×C4C2×C4×D714yes—
372C22×C4C2×D2814yes—
373C22×C4C2×C22⋊D712yes—
374C22×C4C2×C22⋊D711yescharacteristic
375C22×C4C2×C22⋊D711yescharacteristic
376C22×C4C23×D711yescharacteristic
377C2×C8—142no—
378C2×C8—142no—
379C2×C8—564no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—72no—
383C4⋊C4—72no—
384C4⋊C4—144no—
385C4⋊C4—142no—
386C4⋊C4—141no—
387C4⋊C4—141no—
388C4⋊C4—141no—
389C4⋊C4—141no—
390C4⋊C4—142no—
391C4⋊C4(C2×D7)⋊C412yes—
392C4⋊C4(C2×D7)⋊C412yes—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—144no—
396C22⋊C4—144no—
397C22⋊C4—142no—
398C22⋊C4—144no—
399C22⋊C4—142no—
400C22⋊C4—142no—
401C22⋊C4(C2×D7)⋊C414yes—
402C22⋊C4(C2×D7)⋊C414yes—
403C42—42no—
404C42—82no—
405C42—144no—
406C42—142no—
407C42C2×C4×D711yescharacteristic
408C42C2×C4×D711yescharacteristic
409C42C2×D2811yescharacteristic
410C42C2×D2811yescharacteristic
411C2×C7—21no—
412C2×C7—21no—
413C2×C7—44no—
414C2×C7#288311yescharacteristic
415C2×C7C2×#41611yescharacteristic
416C2×C7#290111yescharacteristic
417D7—82no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—142no—
423C23C2×(C2×D7)⋊C414yes—
424C23C22×C22⋊D711yescharacteristic
425Q8—282no—
426Q8—282no—
427Q8—561no—
428D4—24no—
429D4—24no—
430D4—282no—
431D4—282no—
432D4—282no—
433D4—282no—
434D4—561no—
435D4—564no—
436D4—564no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—142no—
452C2×C4—142no—
453C2×C4—142no—
454C2×C4—284no—
455C2×C4—564no—
456C2×C4C2×(C2×D7)⋊C414yes—
457C2×C4C2×(C2×D7)⋊C411yescharacteristic
458C2×C4C2×(C2×D7)⋊C411yescharacteristic
459C2×C4C2×(C2×D7)⋊C411yescharacteristic
460C2×C4C2×(C2×D7)⋊C411yescharacteristic
461C2×C4#651312yes—
462C2×C4#651312yes—
463C2×C4C22×C4×D711yescharacteristic
464C2×C4C22×D2811yescharacteristic
465C8—282no—
466C8—282no—
467C7#1761711yescharacteristic, complemented, Sylow
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—82no—
474C22—82no—
475C22—282no—
476C22—282no—
477C22—282no—
478C22C22×(C2×D7)⋊C411yescharacteristic
479C22C2×#651311yescharacteristic
480C22C2×#651311yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—42no—
487C4—81no—
488C4—82no—
489C4—282no—
490C4—282no—
491C4—282no—
492C4#8977412yes—
493C4#8977412yes—
494C2—21no—
495C2—21no—
496C2—44no—
497C2—562no—
498C2C2×#8977411yescharacteristic
499C2#1173998111yescharacteristic
500C2#1173999111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2357C2
socleC22×C7C22×C22⋊C4
prime radical#11742856C2
cosocleC22×C4×C7C24
hypercenter#2357D7
hypocenterC7#17617
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. C7×#2357C2semidirect
3. #11739746C2semidirect
4. #11727300C2semidirect
5. #11727299C2semidirect
6. #11727286C2semidirect
7. #11727225C2semidirect
8. #11727223C2semidirect
9. #11727197C2semidirect
13. #90254C4semidirect
16. #90244C4semidirect
17. #90242C4semidirect
18. #90233C4semidirect
108. #2357D7semidirect
467. C7#17617semidirect

Maximal subgroups

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

subgroupindex
#117428562
C7×#23572
#117397462
#117273002
#117272992
#117272862
#117272252
#117272232
#117271972
#176177