Finite Group Databasebeta

← #13322293

Subgroups — #13322293

Order 1792.

940 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743293C211yesmaximal, characteristic
2#11736485C211yesmaximal, characteristic, complemented
3#11736458C211yesmaximal, characteristic, complemented
4C7×#1547C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6#11728261C211yesmaximal, characteristic, complemented
7C2×#89276C211yesmaximal, characteristic, complemented
8#11726181C212yesmaximal, complemented
9#11726172C212yesmaximal, complemented
10#11726139C212yesmaximal, complemented
11#11726136C212yesmaximal, complemented
12C7×#532C2211yescharacteristic
13#90524—21no—
14#90518—21no—
15#90419C2212yes—
16#90407C2212yes—
17#90373—22no—
18#90309—22no—
19#90289C414yes—
20#90254—21no—
21#90246C412yes—
22#90234—21no—
23#90233C412yes—
24C4×#1157C2211yescharacteristic
25C2×#6432C2211yescharacteristic
26C7×#423C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28#89901—21no—
29#89894—21no—
30C8×C22⋊D7—22no—
31#89697C2211yescharacteristic
32C2×C4⋊(C7⋊C8)C2211yescharacteristic
33#89612—22no—
34#89607C2212yes—
35#89606—21no—
36#89605C2212yes—
37#89605C2212yes—
38D7×C4⋊C8C2212yes—
39#89602—21no—
40C7×#332C2212yes—
41C7×#331C2212yes—
42#89342C2211yescharacteristic
43#89342C2211yescharacteristic
44#89339C2211yescharacteristic
45#89339C2211yescharacteristic
46#89276C2212yes—
47#89276C2212yes—
48#89276C2212yes—
49#89276C2212yes—
50#17726—71nomaximal, Sylow
51C2×C7×#41—21no—
52C2×#1159C2×C414yes—
53C2×#1157D411yescharacteristic
54C22×Dic14C2×C412yes—
55C2×C7×M16—21no—
56C22×C7×C8—21no—
57C7×C4⋊Q8C2311yescharacteristic
58C7×#127C2311yescharacteristic
59C7×#124—21no—
60C7×#121—21no—
61C7×#121C2×C412yes—
62C4×C7×D4—21no—
63C4×C7×D4C2×C412yes—
64C2×C42×C7C2311yescharacteristic
65#6509—22no—
66Q8×Dic7—22no—
67#6502—24no—
68#6501—22no—
69#6501—22no—
70#6499C2×C414yes—
71D4×Dic7—24no—
72D4×Dic7—22no—
73D4×Dic7—22no—
74#6493—22no—
75C4×C22⋊D7C2×C412yes—
76C4×C22⋊D7D412yes—
77#6489—22no—
78#6489D411yescharacteristic
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—22no—
82#6487—22no—
83C22×C7⋊C8—21no—
84C2×C8×D7—22no—
85#6460C2×C414yes—
86#6455—22no—
87#6453—22no—
88#6450C2×C414yes—
89#6449—24no—
90#6448—24no—
91#6442—24no—
92#6441—24no—
93C4×D28C2×C411yescharacteristic
94C4×D28D411yescharacteristic
95#6435D412yes—
96C42×D7C2312yes—
97#6432—21no—
98#6432—21no—
99#6432C2×C412yes—
100#6432C2312yes—
101C4×Dic14C2×C411yescharacteristic
102C4×Dic14D411yescharacteristic
103C7×C4⋊C8—21no—
104C7×C4⋊C8—21no—
105C7×C4⋊C8C2312yes—
106C7×C4⋊C8C2312yes—
107C22⋊(C7⋊C8)—21no—
108(C2×D7)⋊C8—22no—
109(C2×D7)⋊C8—22no—
110(C2×D7)⋊C8—22no—
111#6389—22no—
112Dic7⋊C8—22no—
113C8×Dic7—22no—
114C8×Dic7—22no—
115C4⋊(C7⋊C8)—21no—
116C4⋊(C7⋊C8)C2311yescharacteristic
117C4⋊(C7⋊C8)C2311yescharacteristic
118C4⋊(C7⋊C8)C2311yescharacteristic
119C4⋊(C7⋊C8)C2311yescharacteristic
120C4×C7⋊C8—21no—
121#3511—71no—
122#2988—71no—
123#2925—71no—
124#1554—72no—
125#1551—72no—
126#1548—71no—
127#1547D711yescharacteristic, complemented
128#1543—72no—
129#1542—72no—
130C2×#333—71no—
131#1540—71no—
132C7×#41—42no—
133C7×#41—42no—
134C2×C7×Q8—21no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4C22×C412yes—
139C22×C4×C7—22no—
140C22×C4×C7C22×C411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C2×C22⋊D7—24no—
144C2×C22⋊D7C22⋊C412yes—
145C22×Dic7—22no—
146C22×Dic7—24no—
147#1159—84no—
148#1157—22no—
149#1157—22no—
150C2×D28C22⋊C411yescharacteristic
151C2×C4×D7C22×C412yes—
152C2×C4×D7C2×D412yes—
153C2×C4×D7C2×D412yes—
154C2×Dic14—22no—
155C2×Dic14—82no—
156C2×Dic14C22⋊C411yescharacteristic
157C2×Dic14C22×C414yes—
158C7×M16—42no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—81no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4C22×C412yes—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C42×C7C22×C411yescharacteristic
167C42×C7C2×D411yescharacteristic
168C42×C7C2×D411yescharacteristic
169C42×C7C2411yescharacteristic
170C22⋊Dic7—24no—
171C22⋊Dic7—24no—
172C22⋊Dic7C22⋊C412yes—
173(C2×D7)⋊C4—24no—
174(C2×D7)⋊C4C22⋊C412yes—
175(C2×D7)⋊C4C22⋊C412yes—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—24no—
181C4⋊Dic7C22⋊C411yescharacteristic
182C4⋊Dic7C22⋊C411yescharacteristic
183Dic7⋊C4—24no—
184Dic7⋊C4C22⋊C412yes—
185Dic7⋊C4C22⋊C412yes—
186C4×Dic7—24no—
187C4×Dic7C22×C412yes—
188C4×Dic7C2×D412yes—
189C4×Dic7C2×D412yes—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—42no—
195C8×D7—82no—
196#556—72no—
197#555—141no—
198#554—141no—
199#553—142no—
200#548—74no—
201#547—72no—
202#542—142no—
203#537—141no—
204#533—72no—
205#532—72no—
206#532C2×D711yescharacteristic
207C2×C4⋊Q8—71no—
208#520—141no—
209C4×#41—71no—
210C8×Q8—141no—
211C8×D4—141no—
212C8×D4—142no—
213#433—142no—
214#432—72no—
215#431—71no—
216#431—72no—
217#431—72no—
218#423C2×D711yescharacteristic
219C2×C4⋊C8—71no—
220C2×C4⋊C8—72no—
221C2×C4⋊C8C2×D711yescharacteristic
222#408—141no—
223#405—141no—
224#333—72no—
225#333—72no—
226#333—72no—
227#333—72no—
228#332—71no—
229#332—71no—
230#332C2×D712yes—
231#331—71no—
232#331—71no—
233#331C2×D712yes—
234C23×C7—22no—
235C23×C7C2×C22⋊C411yescharacteristic
236C22×D7C2×C22⋊C412yes—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—42no—
240C7×D4—44no—
241C7×D4—41no—
242C7×D4—41no—
243C7×D4—41no—
244C7×D4—41no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—41no—
249C2×C4×C7—44no—
250C2×C4×C7C2×C22⋊C411yescharacteristic
251C2×C4×C7C2×C22⋊C411yescharacteristic
252C2×C4×C7C2×C22⋊C411yescharacteristic
253C2×C4×C7#13512yes—
254C2×C4×C7#13512yes—
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C22⋊D7—24no—
259C22⋊D7—84no—
260C2×Dic7—24no—
261C2×Dic7—84no—
262C2×Dic7—82no—
263C2×Dic7C2×C22⋊C412yes—
264C2×Dic7C2×C22⋊C412yes—
265C2×Dic7C2×C22⋊C412yes—
266D28—22no—
267C4×D7—22no—
268C4×D7—22no—
269C4×D7—22no—
270C4×D7—22no—
271C4×D7—42no—
272Dic14—22no—
273Dic14—44no—
274Dic14—82no—
275C7×C8—42no—
276C7×C8—42no—
277C7⋊C8—41no—
278C7⋊C8—41no—
279C7⋊C8—41no—
280C7⋊C8—41no—
281C2×#41—21no—
282C2×#41—71no—
283C2×#41—74no—
284C22×Q8—72no—
285C2×M16—21no—
286C22×C8—21no—
287C22×C8—142no—
288C22×C8—141no—
289C4⋊Q8—72no—
290C4⋊Q8—72no—
291C4⋊Q8—141no—
292C4⋊Q8—141no—
293C4⋊Q8C22×D711yescharacteristic
294#127C22×D711yescharacteristic
295#126—144no—
296#125—142no—
297#124—21no—
298#124—74no—
299#124—144no—
300#123—144no—
301C22⋊Q8—74no—
302C22⋊Q8—144no—
303C22⋊Q8—142no—
304C22⋊Q8—142no—
305#121—21no—
306#121—142no—
307#121—142no—
308#121—142no—
309#121—144no—
310#121C4×D712yes—
311C4×Q8—71no—
312C4×Q8—71no—
313C4×Q8—74no—
314C4×Q8—142no—
315C4×D4—21no—
316C4×D4—72no—
317C4×D4—72no—
318C4×D4—71no—
319C4×D4—71no—
320C4×D4—142no—
321C4×D4—142no—
322C4×D4—144no—
323C4×D4C4×D712yes—
324#117—71no—
325#117—72no—
326#117—142no—
327#117—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—141no—
331C2×C42—72no—
332C2×C42C22×D711yescharacteristic
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—71no—
336C4⋊C8—71no—
337C4⋊C8—71no—
338C4⋊C8—71no—
339C4⋊C8—142no—
340C4⋊C8—141no—
341C4⋊C8C22×D712yes—
342C4⋊C8C22×D712yes—
343C22⋊C8—142no—
344C22⋊C8—142no—
345C22⋊C8—142no—
346C22⋊C8—141no—
347#97—142no—
348C4×C8—142no—
349C4×C8—142no—
350C4×C8—141no—
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7#41612yes—
357C22×C7C22×C22⋊C411yescharacteristic
358C2×D7—22no—
359C2×D7—22no—
360C2×D7—22no—
361C4×C7—21no—
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—42no—
365C4×C7#41612yes—
366C4×C7C2×#13511yescharacteristic
367C4×C7C2×#13511yescharacteristic
368C4×C7#57711yescharacteristic
369C4×C7#57811yescharacteristic
370Dic7—22no—
371Dic7—22no—
372Dic7—22no—
373Dic7—44no—
374#41—42no—
375#41—42no—
376#41—142no—
377#41—142no—
378#41—564no—
379C2×Q8—21no—
380C2×Q8—71no—
381C2×Q8—74no—
382C2×Q8—142no—
383C2×Q8—562no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—72no—
388C2×D4—71no—
389C2×D4—144no—
390C2×D4C2×C4×D712yes—
391C22×C4—22no—
392C22×C4—72no—
393C22×C4—72no—
394C22×C4—72no—
395C22×C4—144no—
396C22×C4—142no—
397C22×C4C2×C4×D711yescharacteristic
398C22×C4D4×D711yescharacteristic
399C22×C4D4×D711yescharacteristic
400M16—42no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—81no—
404C2×C8—141no—
405C2×C8—141no—
406C2×C8—141no—
407C2×C8—141no—
408C2×C8—282no—
409C2×C8—562no—
410C4⋊C4—21no—
411C4⋊C4—72no—
412C4⋊C4—72no—
413C4⋊C4—71no—
414C4⋊C4—71no—
415C4⋊C4—144no—
416C4⋊C4—144no—
417C4⋊C4—142no—
418C4⋊C4—142no—
419C4⋊C4—142no—
420C4⋊C4—142no—
421C4⋊C4C2×C4×D712yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—72no—
425C22⋊C4—72no—
426C22⋊C4—72no—
427C22⋊C4—144no—
428C22⋊C4—144no—
429C22⋊C4—144no—
430C42—72no—
431C42—72no—
432C42—72no—
433C42—144no—
434C42C2×C4×D711yescharacteristic
435C42D4×D711yescharacteristic
436C42D4×D711yescharacteristic
437C42C23×D711yescharacteristic
438C2×C7—22no—
439C2×C7—82no—
440C2×C7#288811yescharacteristic
441C2×C7C2×#41611yescharacteristic
442C2×C7#289811yescharacteristic
443D7—42no—
444C23—22no—
445C23—72no—
446C23D7×C22⋊C411yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—142no—
450Q8—284no—
451Q8—562no—
452D4—42no—
453D4—44no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—144no—
459D4—142no—
460D4—564no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—72no—
467C2×C4—72no—
468C2×C4—72no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—142no—
472C2×C4—142no—
473C2×C4—144no—
474C2×C4—282no—
475C2×C4—564no—
476C2×C4—562no—
477C2×C4D7×C22⋊C411yescharacteristic
478C2×C4D7×C22⋊C411yescharacteristic
479C2×C4D7×C22⋊C411yescharacteristic
480C2×C4#647512yes—
481C2×C4#647912yes—
482C2×C4C22×C4×D711yescharacteristic
483C2×C4C2×D4×D711yescharacteristic
484C2×C4C2×D4×D711yescharacteristic
485C8—42no—
486C8—42no—
487C8—281no—
488C8—281no—
489C8—281no—
490C8—281no—
491C7#1772611yescharacteristic, complemented, Sylow
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—142no—
498C22—142no—
499C22—142no—
500C22#8954412yes—
501C22C2×D7×C22⋊C411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—142no—
507C4—142no—
508C4—142no—
509C4—284no—
510C4#8954412yes—
511C4C2×#647511yescharacteristic
512C4C2×#647911yescharacteristic
513C4#9046611yescharacteristic
514C4#9047011yescharacteristic
515C2—22no—
516C2—82no—
517C2—282no—
518C2#1173615911yescharacteristic
519C2C2×#8954411yescharacteristic
520C2#1173632811yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1547C2
socleC22×C7C22×C22⋊C4
prime radical#11743293C2
cosocleC42×C7C24
hypercenter#1547D7
hypocenterC7#17726
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #11736485C2semidirect
3. #11736458C2semidirect
4. C7×#1547C2semidirect
5. #11729558C2semidirect
6. #11728261C2semidirect
7. C2×#89276C2semidirect
8. #11726181C2semidirect
9. #11726172C2semidirect
10. #11726139C2semidirect
11. #11726136C2semidirect
127. #1547D7semidirect
491. C7#17726semidirect

Maximal subgroups

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

subgroupindex
#117432932
#117364852
#117364582
C7×#15472
#117295582
#117282612
C2×#892762
#117261812
#117261722
#117261392
#117261362
#177267