Finite Group Databasebeta

← #13331391

Subgroups — #13331391

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2#11740818C211yesmaximal, characteristic, complemented
3#11740789C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic
5#11729680C212yesmaximal, complemented
6#11729678C212yesmaximal, complemented
7#11729670C212yesmaximal, complemented
8#11729669C212yesmaximal, complemented
9#11729584C211yesmaximal, characteristic, complemented
10#11729580C211yesmaximal, characteristic, complemented
11C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic
12C7×#554—21no—
13C7×#554—22no—
14C7×#554—22no—
15C7×#554C2212yes—
16C7×D4×Q8—21no—
17C7×D4×Q8C2212yes—
18C7×#544—21no—
19C7×#543—22no—
20C7×#533C2211yescharacteristic
21C7×#533C2211yescharacteristic
22C2×C7×C22⋊Q8—22no—
23C7×#518—21no—
24C2×C4×C7×D4C2211yescharacteristic
25#90161C2211yescharacteristic
26#89987—21no—
27#89865C2211yescharacteristic, complemented
28#89864C2211yescharacteristic, complemented
29#89863—21no—
30#89862—21no—
31#89853C2211yescharacteristic, complemented
32#89850C2211yescharacteristic, complemented
33#89823—22no—
34#89823—22no—
35#89803C2212yescomplemented
36C4×#1144—22no—
37#89793C2212yescomplemented
38#89781—22no—
39#89780C2212yes—
40#89779C2212yes—
41#89736—21no—
42#89696C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89342C2212yescomplemented
46#89341C2212yescomplemented
47Q8⋊(C7⋊C8)C2212yescomplemented
48Q8⋊(C7⋊C8)C2212yescomplemented
49D4⋊(C7⋊C8)C2212yes—
50D4⋊(C7⋊C8)C2212yes—
51#18665—71nomaximal, Sylow
52C22×C7×Q8—22no—
53C22×C7×D4D411yescharacteristic
54C23×C4×C7—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8C2311yescharacteristic
60C7×C4⋊Q8C2311yescharacteristic
61C7×#125—22no—
62C7×#125—22no—
63C7×#125—21no—
64C7×#125—21no—
65C7×C22⋊Q8—24no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—21no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8D412yes—
76C7×C22⋊Q8D412yes—
77C4×C7×Q8—21no—
78C4×C7×Q8—21no—
79C4×C7×Q8D412yes—
80C4×C7×Q8C2312yes—
81C4×C7×D4—22no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—21no—
86C4×C7×D4—21no—
87C4×C7×D4D412yes—
88C4×C7×D4C2312yes—
89C7×#117—21no—
90C7×#117—21no—
91C2×C7×C4⋊C4—24no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4D411yescharacteristic
94C2×C7×C22⋊C4—24no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7C2311yescharacteristic
97C2×#1144—22no—
98C2×#1144—22no—
99#6489—21no—
100#6487—21no—
101#6487—21no—
102C2×#1136—21no—
103C22×C7⋊C8—21no—
104C22×C7⋊C8—21no—
105#6433—21no—
106#6432C2311yescharacteristic
107#6432C2311yescharacteristic
108C4×Dic14—21no—
109C4×Dic14—21no—
110#6409—22no—
111#6409—22no—
112#6406—22no—
113#6406—22no—
114#6384—22no—
115#6384—22no—
116#6382—22no—
117#6381—22no—
118#6381—22no—
119#6381—22no—
120C4⋊(C7⋊C8)—21no—
121C4⋊(C7⋊C8)—21no—
122C4⋊(C7⋊C8)C2312yes—
123C4⋊(C7⋊C8)C2312yes—
124C4×C7⋊C8—21no—
125C4×C7⋊C8C2311yescharacteristic
126C4×C7⋊C8C2311yescharacteristic
127#3501D711yescharacteristic
128#3249—71no—
129#3170—71no—
130#3077—71no—
131#1649—72no—
132#1644—72no—
133#1637—72no—
134#1632—72no—
135#1550—71no—
136#1493—71no—
137C2×D4⋊C8—71no—
138C24×C7—21no—
139C2×C7×Q8—24no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—82no—
143C2×C7×Q8—82no—
144C2×C7×Q8C2×D412yes—
145C2×C7×Q8C2×D412yes—
146C2×C7×D4—21no—
147C2×C7×D4—22no—
148C2×C7×D4Q1614yescomplemented
149C2×C7×D4C2×D412yes—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—21no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C2×Dic14—21no—
161C2×Dic14—21no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4C2×D412yes—
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C42×C7—22no—
186C42×C7—22no—
187C42×C7C2×D411yescharacteristic
188C42×C7C2×D411yescharacteristic
189C42×C7C2×D411yescharacteristic
190C42×C7C2411yescharacteristic
191C22⋊Dic7—41no—
192#1144—44no—
193#1144—82no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198Dic7⋊C4—41no—
199Dic7⋊C4—41no—
200C4×Dic7—41no—
201#1136—42no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—42no—
207C2×C7⋊C8—81no—
208#554—21no—
209#554—22no—
210#554—22no—
211#554C2×D712yes—
212D4×Q8—21no—
213D4×Q8C2×D712yes—
214#544—21no—
215#543—22no—
216#533—71no—
217#533C2×D711yescharacteristic
218#533C2×D711yescharacteristic
219C2×C22⋊Q8—22no—
220#518—21no—
221C2×C4×D4C2×D711yescharacteristic
222#500—71no—
223#494—71no—
224#487—71no—
225#487—71no—
226#478—142no—
227#477—72no—
228#477—141no—
229#476—72no—
230#474—72no—
231#467—142no—
232#467—142no—
233#462—72no—
234#462—141no—
235C4×Q16—142no—
236#428—141no—
237#423—71no—
238C2×C4⋊C8—71no—
239#418—141no—
240C2×C4×C8—71no—
241#333—72no—
242#332—72no—
243Q8⋊C8—72no—
244Q8⋊C8—72no—
245D4⋊C8—72no—
246D4⋊C8—72no—
247C23×C7—22no—
248C23×C7—24no—
249C23×C7—81no—
250C23×C7—81no—
251C23×C7#12011yescharacteristic
252C7×Q8—22no—
253C7×Q8—22no—
254C7×Q8—42no—
255C7×Q8—41no—
256C7×Q8—41no—
257C7×Q8—81no—
258C7×Q8—84no—
259C7×D4—22no—
260C7×D4—24no—
261C7×D4C2×Q1614yes—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—82no—
273C2×C4×C7—81no—
274C2×C4×C7—84no—
275C2×C4×C7—84no—
276C2×C4×C7#12011yescharacteristic
277C2×C4×C7#12011yescharacteristic
278C2×C4×C7#12011yescharacteristic
279C2×C4×C7C2×Q1612yes—
280C2×C4×C7#13712yes—
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×Dic7—41no—
285C2×Dic7—41no—
286Dic14—42no—
287Dic14—41no—
288Dic14—41no—
289C7⋊C8—22no—
290C7⋊C8—41no—
291C7⋊C8—42no—
292C7⋊C8—42no—
293C22×Q8—22no—
294C22×D4C22⋊D711yescharacteristic
295C23×C4—21no—
296C2×Q16—142no—
297C2×Q16—142no—
298C2×M16—141no—
299C22×C8—141no—
300C22×C8—141no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—71no—
304C4⋊Q8—71no—
305C4⋊Q8C22⋊D711yescharacteristic
306C4⋊Q8C22⋊D711yescharacteristic
307C4⋊Q8C22×D711yescharacteristic
308C4⋊Q8C22×D711yescharacteristic
309#125—22no—
310#125—22no—
311#125—21no—
312#125—21no—
313#125—141no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—24no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8—141no—
325C22⋊Q8—141no—
326C22⋊Q8C22⋊D712yes—
327C22⋊Q8C22⋊D712yes—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—141no—
331C4×Q8—141no—
332C4×Q8C22⋊D712yes—
333C4×Q8C22×D712yes—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D712yes—
341C4×D4C22×D712yes—
342#117—21no—
343#117—21no—
344#117—141no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4C22⋊D711yescharacteristic
348C2×C22⋊C4—24no—
349C2×C22⋊C4—21no—
350C2×C42C22×D711yescharacteristic
351#107—142no—
352#107—142no—
353#107—142no—
354#106—142no—
355C4⋊C8—72no—
356C4⋊C8—72no—
357C4⋊C8—141no—
358C4⋊C8—141no—
359#103—142no—
360#103—142no—
361#103—142no—
362#103—142no—
363#102—142no—
364#102—142no—
365C4×C8—71no—
366C4×C8—71no—
367C4×C8—141no—
368C22×C7—21no—
369C22×C7—21no—
370C22×C7—24no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—84no—
374C22×C7#45112yes—
375C22×C7C2×#12011yescharacteristic
376C4×C7—21no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—42no—
383C4×C7—42no—
384C4×C7#45112yes—
385C4×C7C22×Q1611yescharacteristic
386C4×C7C2×#13711yescharacteristic
387C4×C7#57611yescharacteristic
388C4×C7#57711yescharacteristic
389Dic7—81no—
390Dic7—81no—
391C24—21no—
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—141no—
398C2×Q8—141no—
399C2×Q8C2×C22⋊D712yes—
400C2×Q8C2×C22⋊D712yes—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4#114414yescomplemented
404C2×D4C2×C22⋊D712yes—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4C2×C22⋊D711yescharacteristic
413C22×C4C2×C22⋊D711yescharacteristic
414C22×C4C2×C22⋊D711yescharacteristic
415Q16—284no—
416Q16—562no—
417M16—282no—
418C2×C8—141no—
419C2×C8—141no—
420C2×C8—142no—
421C2×C8—142no—
422C2×C8—282no—
423C2×C8—561no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—141no—
442C4⋊C4—281no—
443C4⋊C4—281no—
444C4⋊C4C2×C22⋊D712yes—
445C4⋊C4C2×C22⋊D712yes—
446C4⋊C4C2×C22⋊D712yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—281no—
454C42—22no—
455C42—22no—
456C42—281no—
457C42C2×C22⋊D711yescharacteristic
458C42C2×C22⋊D711yescharacteristic
459C42C2×C22⋊D711yescharacteristic
460C42C23×D711yescharacteristic
461C2×C7—22no—
462C2×C7—44no—
463C2×C7C2×#45111yescharacteristic
464C2×C7#301211yescharacteristic
465C2×C7#301811yescharacteristic
466C23—22no—
467C23—24no—
468C23—81no—
469C23—81no—
470C23#651611yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—42no—
474Q8—41no—
475Q8—41no—
476Q8—81no—
477Q8—84no—
478Q8—282no—
479Q8—281no—
480Q8—281no—
481D4—22no—
482D4—24no—
483D4C2×#114414yes—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4#650512yes—
501C2×C4C2×#114412yes—
502C2×C4#651611yescharacteristic
503C2×C4#651611yescharacteristic
504C2×C4#651611yescharacteristic
505C2×C4C22×C22⋊D711yescharacteristic
506C2×C4C22×C22⋊D711yescharacteristic
507C2×C4C22×C22⋊D711yescharacteristic
508C8—142no—
509C8—281no—
510C8—282no—
511C8—282no—
512C7#1866511yescharacteristic, complemented, Sylow
513C22—21no—
514C22—21no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22#9000012yes—
520C22C2×#651611yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—561no—
530C4—561no—
531C4#9000012yes—
532C4C2×#650511yescharacteristic
533C4C22×#114411yescharacteristic
534C4#9052611yescharacteristic
535C4#9052711yescharacteristic
536C2—22no—
537C2—44no—
538C2C2×#9000011yescharacteristic
539C2#1174154711yescharacteristic
540C2#1174156611yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18665
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11740818C2semidirect
3. #11740789C2semidirect
5. #11729680C2semidirect
6. #11729678C2semidirect
7. #11729670C2semidirect
8. #11729669C2semidirect
9. #11729584C2semidirect
10. #11729580C2semidirect
27. #89865C22semidirect
28. #89864C22semidirect
31. #89853C22semidirect
32. #89850C22semidirect
35. #89803C22semidirect
37. #89793C22semidirect
45. #89342C22semidirect
46. #89341C22semidirect
47. Q8⋊(C7⋊C8)C22semidirect
48. Q8⋊(C7⋊C8)C22semidirect
148. C2×C7×D4Q16semidirect
403. C2×D4#1144semidirect
512. C7#18665semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117408182
#117407892
#117397412
#117296802
#117296782
#117296702
#117296692
#117295842
#117295802
C2×D4⋊(C7⋊C8)2
#186657