Finite Group Databasebeta

← C7×#18343 #13195559

Subgroups — C7×#18343 #13195559

Order 1792.

1034 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3239C212yesmaximal, complemented
3C7×#2542C211yesmaximal, characteristic, complemented
4C7×#1618C214yesmaximal, complemented
5C7×#1614C212yesmaximal, complemented
6C7×#1613C212yesmaximal, complemented
7C7×#1509C212yesmaximal, complemented
8C7×#1466C211yesmaximal, characteristic, complemented
9C7×#543C2211yescharacteristic
10C7×#539—22no—
11C7×#539—21no—
12C7×#538—22no—
13C7×#538C2212yes—
14C7×#534—21no—
15C7×#534C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic, complemented
17C7×#529—22no—
18C7×#529C2212yes—
19C7×#525—21no—
20C7×#525—22no—
21C2×C7×C22⋊Q8—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C7×#501—22no—
26C7×#489—22no—
27C7×#484C2214yescomplemented
28C7×#481C2214yes—
29C7×#475—22no—
30C7×#474—22no—
31C7×#432C2212yescomplemented
32C7×#399C2212yes—
33C7×#399C2212yescomplemented
34C7×#390—21no—
35C7×#384—21no—
36C4×C7×C4⋊C4C2211yescharacteristic, complemented
37C7×#328C2214yescomplemented
38C7×#327C2214yes—
39C7×#323C2212yes—
40C7×#323C2212yescomplemented
41#18343C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8—21no—
44C22×C7×D4—21no—
45C22×C7×D4D411yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8—22no—
51C7×#126—22no—
52C7×#126—22no—
53C7×#126—22no—
54C7×#125—22no—
55C7×#125D412yes—
56C7×#124—22no—
57C7×#124D412yes—
58C7×#123—22no—
59C7×#123—22no—
60C7×#123—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—44no—
70C7×C22⋊Q8—44no—
71C7×C22⋊Q8C2312yes—
72C7×#121—22no—
73C7×#121—21no—
74C7×#121—22no—
75C7×#121—22no—
76C7×#121—21no—
77C7×#121C2312yes—
78C7×#120—22no—
79C7×#120—22no—
80C7×#120—22no—
81C4×C7×Q8D412yes—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4—22no—
89C4×C7×D4D412yes—
90C7×#117—21no—
91C7×#117—22no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4—21no—
97C2×C7×C4⋊C4D411yescharacteristic
98C2×C7×C4⋊C4C2312yes—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—22no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4C2311yescharacteristic
104C2×C7×C22⋊C4C2311yescharacteristic
105C2×C42×C7—21no—
106C2×C42×C7—21no—
107C2×C42×C7C2311yescharacteristic
108C7×#107—24no—
109C7×#106—24no—
110C7×C4⋊C8—22no—
111C7×#103—24no—
112C7×#102—24no—
113C7×C22⋊C8C2314yes—
114C7×#97—22no—
115C7×#95C2312yes—
116#3459C2×C711yescharacteristic
117#3239C2×C712yescomplemented
118#2542C2×C711yescharacteristic, complemented
119#1618C2×C714yescomplemented
120#1614C2×C712yescomplemented
121#1613C2×C712yescomplemented
122#1509C2×C712yescomplemented
123#1466C2×C711yescharacteristic, complemented
124C24×C7—21no—
125C24×C7C2×D411yescharacteristic
126C24×C7C2×D411yescharacteristic
127C7×#41—82no—
128C2×C7×Q8—21no—
129C2×C7×Q8—42no—
130C2×C7×Q8—42no—
131C2×C7×Q8—41no—
132C2×C7×Q8—41no—
133C2×C7×Q8C2×D412yes—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—21no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—41no—
142C2×C7×D4—42no—
143C2×C7×D4—42no—
144C2×C7×D4C2×D412yes—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—82no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C2×C7×C8—24no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—44no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4C2×D412yes—
180C7×C4⋊C4C2×D412yes—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—44no—
194C7×C22⋊C4—44no—
195C42×C7—22no—
196C42×C7—22no—
197C42×C7—42no—
198C42×C7C2×D412yes—
199C42×C7C2×D412yes—
200#543C22×C711yescharacteristic
201#539—22no—
202#539—21no—
203#538—22no—
204#538C22×C712yes—
205#534—21no—
206#534C22×C711yescharacteristic
207C2×C4⋊Q8C22×C711yescharacteristic, complemented
208#529—22no—
209#529C22×C712yes—
210#525—21no—
211#525—22no—
212C2×C22⋊Q8—21no—
213C2×C22⋊Q8—21no—
214C2×C4×D4—21no—
215C2×C4×D4C22×C711yescharacteristic
216#501—22no—
217#489—22no—
218#484C22×C714yescomplemented
219#481C22×C714yes—
220#475—22no—
221#474—22no—
222#432C22×C712yescomplemented
223#399C22×C712yes—
224#399C22×C712yescomplemented
225#390—21no—
226#384—21no—
227C4×C4⋊C4C22×C711yescharacteristic, complemented
228#328C22×C714yescomplemented
229#327C22×C714yes—
230#323C22×C712yes—
231#323C22×C712yescomplemented
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—22no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—42no—
243C23×C7—81no—
244C23×C7—81no—
245C23×C7#13712yes—
246C23×C7C22×D411yescharacteristic
247C7×Q8—42no—
248C7×Q8—82no—
249C7×Q8—81no—
250C7×Q8—82no—
251C7×D4—44no—
252C7×D4—42no—
253C7×D4—44no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—82no—
257C7×D4—81no—
258C7×D4—82no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—44no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—81no—
281C2×C4×C7—81no—
282C2×C4×C7—81no—
283C2×C4×C7—82no—
284C2×C4×C7—81no—
285C2×C4×C7#12012yes—
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7C22×D412yes—
289C7×C8—44no—
290C2×#41—21no—
291C22×Q8—21no—
292C22×D4—21no—
293C22×D4C7×D411yescharacteristic
294C23×C4—21no—
295C23×C4C7×D411yescharacteristic
296C23×C4C7×D411yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#125—22no—
303#125C7×D412yes—
304#124—22no—
305#124C7×D412yes—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—22no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—21no—
316C22⋊Q8—22no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C22⋊Q8C23×C712yes—
320#121—22no—
321#121—21no—
322#121—22no—
323#121—22no—
324#121—21no—
325#121C23×C712yes—
326#120—22no—
327#120—22no—
328#120—22no—
329C4×Q8C7×D412yes—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4C7×D412yes—
338#117—21no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C7×D411yescharacteristic
346C2×C4⋊C4C23×C712yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C23×C711yescharacteristic
352C2×C22⋊C4C23×C711yescharacteristic
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C23×C711yescharacteristic
356#107—24no—
357#106—24no—
358C4⋊C8—22no—
359#103—24no—
360#102—24no—
361C22⋊C8C23×C714yes—
362#97—22no—
363#95C23×C712yes—
364C22×C7—21no—
365C22×C7—21no—
366C22×C7—22no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—41no—
370C22×C7—41no—
371C22×C7—41no—
372C22×C7—42no—
373C22×C7—41no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—41no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—84no—
380C22×C7C2×#12011yescharacteristic
381C22×C7C2×#13711yescharacteristic
382C22×C7#57511yescharacteristic
383C4×C7—22no—
384C4×C7—21no—
385C4×C7—22no—
386C4×C7—44no—
387C4×C7—42no—
388C4×C7—41no—
389C4×C7—41no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—82no—
393C4×C7#45312yes—
394C24—21no—
395C24C2×C7×D411yescharacteristic
396C24C2×C7×D411yescharacteristic
397#41—82no—
398C2×Q8—21no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8C2×C7×D412yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×C7×D412yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C2×C7×D411yescharacteristic
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C24×C711yescharacteristic
434C2×C8—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4C2×C7×D412yes—
450C4⋊C4C2×C7×D412yes—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42C2×C7×D412yes—
469C42C2×C7×D412yes—
470C2×C7—21no—
471C2×C7—21no—
472C2×C7—42no—
473C2×C7—81no—
474C2×C7—82no—
475C2×C7#300711yescharacteristic
476C2×C7C2×#45311yescharacteristic
477C2×C7#302811yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—81no—
490C23—81no—
491C23C7×#13712yes—
492C23C22×C7×D411yescharacteristic
493Q8—42no—
494Q8—82no—
495Q8—81no—
496Q8—82no—
497D4—44no—
498D4—42no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—82no—
503D4—81no—
504D4—82no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4C7×#12012yes—
532C2×C4C7×#12011yescharacteristic
533C2×C4C7×#12011yescharacteristic
534C2×C4C22×C7×D412yes—
535C8—44no—
536C7#1834311yescharacteristic, direct product factor, Sylow
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—84no—
553C22C2×C7×#12011yescharacteristic
554C22C2×C7×#13711yescharacteristic
555C22C7×#57511yescharacteristic
556C4—22no—
557C4—21no—
558C4—22no—
559C4—44no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4C7×#45312yes—
567C2—21no—
568C2—21no—
569C2—42no—
570C2—81no—
571C2—82no—
572C2C7×#300711yescharacteristic
573C2C2×C7×#45311yescharacteristic
574C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3239C2semidirect
3. C7×#2542C2semidirect
4. C7×#1618C2semidirect
5. C7×#1614C2semidirect
6. C7×#1613C2semidirect
7. C7×#1509C2semidirect
8. C7×#1466C2semidirect
16. C2×C7×C4⋊Q8C22semidirect
27. C7×#484C22semidirect
31. C7×#432C22semidirect
33. C7×#399C22semidirect
36. C4×C7×C4⋊C4C22semidirect
37. C7×#328C22semidirect
40. C7×#323C22semidirect
41. #18343C7direct (N × H)
117. #3239C2×C7semidirect
118. #2542C2×C7semidirect
119. #1618C2×C7semidirect
120. #1614C2×C7semidirect
121. #1613C2×C7semidirect
122. #1509C2×C7semidirect
123. #1466C2×C7semidirect
207. C2×C4⋊Q8C22×C7semidirect
218. #484C22×C7semidirect
222. #432C22×C7semidirect
224. #399C22×C7semidirect
227. C4×C4⋊C4C22×C7semidirect
228. #328C22×C7semidirect
231. #323C22×C7semidirect
536. C7#18343direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#32392
C7×#25422
C7×#16182
C7×#16142
C7×#16132
C7×#15092
C7×#14662
#183437