Finite Group Databasebeta

← C7×#18433 #13195649

Subgroups — C7×#18433 #13195649

Order 1792.

872 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×#3563C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic
3C7×#3209C211yesmaximal, characteristic, complemented
4C7×#2660C211yesmaximal, characteristic, complemented
5C7×#1605C214yesmaximal, complemented
6C7×#1603C214yesmaximal
7C7×#1506C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic
9C7×#1470C211yesmaximal, characteristic, complemented
10C7×#564—21no—
11C7×#563—21no—
12C7×#557—21no—
13C7×#554—21no—
14C7×#552—21no—
15C7×#552—21no—
16C7×#552—22no—
17C7×#544—21no—
18C7×#544C2212yes—
19C7×#541—21no—
20C7×#541—22no—
21C7×#539C2212yes—
22C7×#533C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#497—21no—
27C7×#494—21no—
28C7×#491—21no—
29C7×#487—21no—
30C7×#462—22no—
31C7×#462—21no—
32C7×#461—21no—
33C7×#451C2214yes—
34C7×#451C2212yes—
35C7×#448C2212yes—
36C7×#432C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#398—22no—
39C7×#395C2214yes—
40C7×#384—21no—
41C7×#382—21no—
42C7×#376C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#324C2214yes—
47#18433C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8C2312yes—
49C2×C7×Q16—24no—
50C2×C7×Q16—22no—
51C2×C7×SD16—22no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D411yescharacteristic
60C7×C4⋊Q8D411yescharacteristic
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—21no—
66C7×#125—22no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#125—22no—
70C7×#125—21no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#125—21no—
77C7×#125—21no—
78C7×#125—21no—
79C7×#125—21no—
80C7×#125D411yescharacteristic
81C7×#124D411yescharacteristic
82C7×#123—22no—
83C7×#123—21no—
84C7×#123—22no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8C2311yescharacteristic
93C7×C22⋊Q8C2312yes—
94C7×#121C2311yescharacteristic
95C4×C7×Q8—21no—
96C4×C7×Q8—22no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8D411yescharacteristic
100C4×C7×Q8D412yes—
101C4×C7×D4—21no—
102C4×C7×D4—22no—
103C4×C7×D4D411yescharacteristic
104C7×#117—22no—
105C7×#117—21no—
106C7×#117—21no—
107C7×#117—22no—
108C7×#117—21no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—24no—
113C2×C7×C4⋊C4D411yescharacteristic
114C2×C7×C4⋊C4D411yescharacteristic
115C2×C7×C4⋊C4D412yes—
116C2×C7×C4⋊C4C2312yes—
117C2×C42×C7—21no—
118C2×C42×C7C2311yescharacteristic
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×#103—24no—
122C7×#103—22no—
123C7×#103—22no—
124C7×C22⋊C8C2312yes—
125C7×C22⋊C8C2312yes—
126C7×#97—21no—
127C4×C7×C8—21no—
128C7×#95—21no—
129C7×#95—24no—
130C7×#95C2312yes—
131#3563C2×C711yescharacteristic
132#3218C2×C711yescharacteristic
133#3209C2×C711yescharacteristic, complemented
134#2660C2×C711yescharacteristic, complemented
135#1605C2×C714yescomplemented
136#1603C2×C714yes—
137#1506C2×C711yescharacteristic, complemented
138#1500C2×C711yescharacteristic
139#1470C2×C711yescharacteristic, complemented
140C2×C7×Q8—22no—
141C2×C7×Q8—21no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—82no—
145C2×C7×Q8C2×D412yes—
146C2×C7×Q8C2×D411yescharacteristic
147C2×C7×D4—21no—
148C2×C7×D4C2×D411yescharacteristic
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2411yescharacteristic
158C7×Q16—82no—
159C7×Q16—84no—
160C7×SD16—82no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—41no—
189C7×C4⋊C4—41no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—84no—
193C7×C4⋊C4C2×D411yescharacteristic
194C7×C4⋊C4C2×D411yescharacteristic
195C7×C4⋊C4C2×D412yes—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—21no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C42×C7—24no—
206C42×C7—22no—
207C42×C7—22no—
208C42×C7—22no—
209C42×C7—22no—
210C42×C7—21no—
211C42×C7C2×D411yescharacteristic
212C42×C7C2×D411yescharacteristic
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215#564—21no—
216#563—21no—
217#557—21no—
218#554—21no—
219#552—21no—
220#552—21no—
221#552—22no—
222#544—21no—
223#544C22×C712yes—
224#541—21no—
225#541—22no—
226#539C22×C712yes—
227#533C22×C711yescharacteristic
228C2×C4⋊Q8C22×C711yescharacteristic
229#529C22×C711yescharacteristic
230C2×#125C22×C711yescharacteristic
231#497—21no—
232#494—21no—
233#491—21no—
234#487—21no—
235#462—22no—
236#462—21no—
237#461—21no—
238#451C22×C714yes—
239#451C22×C712yes—
240#448C22×C712yes—
241#432C22×C711yescharacteristic
242#431C22×C711yescharacteristic
243#398—22no—
244#395C22×C714yes—
245#384—21no—
246#382—21no—
247#376C22×C711yescharacteristic
248#328C22×C714yes—
249#328C22×C712yes—
250#327C22×C712yes—
251#324C22×C714yes—
252C23×C7—21no—
253C23×C7C22×D411yescharacteristic
254C7×Q8—41no—
255C7×Q8—42no—
256C7×Q8—42no—
257C7×Q8—42no—
258C7×Q8—44no—
259C7×Q8—42no—
260C7×Q8—82no—
261C7×D4—41no—
262C7×D4—42no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—44no—
280C2×C4×C7—42no—
281C2×C4×C7—82no—
282C2×C4×C7#12011yescharacteristic
283C2×C4×C7#12011yescharacteristic
284C2×C4×C7#12011yescharacteristic
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C7×C8—42no—
289C7×C8—42no—
290C22×Q8C23×C712yes—
291C2×Q16—24no—
292C2×Q16—22no—
293C2×SD16—22no—
294C4⋊Q8—21no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8C7×D411yescharacteristic
302C4⋊Q8C7×D411yescharacteristic
303#126—22no—
304#126—22no—
305#126—22no—
306#126—22no—
307#126—21no—
308#125—22no—
309#125—22no—
310#125—22no—
311#125—22no—
312#125—21no—
313#125—22no—
314#125—22no—
315#125—22no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125—21no—
321#125—21no—
322#125C7×D411yescharacteristic
323#124C7×D411yescharacteristic
324#123—22no—
325#123—21no—
326#123—22no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—21no—
333C22⋊Q8—22no—
334C22⋊Q8C23×C711yescharacteristic
335C22⋊Q8C23×C712yes—
336#121C23×C711yescharacteristic
337C4×Q8—21no—
338C4×Q8—22no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8C7×D411yescharacteristic
342C4×Q8C7×D412yes—
343C4×D4—21no—
344C4×D4—22no—
345C4×D4C7×D411yescharacteristic
346#117—22no—
347#117—21no—
348#117—21no—
349#117—22no—
350#117—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4C7×D411yescharacteristic
356C2×C4⋊C4C7×D412yes—
357C2×C4⋊C4C7×D411yescharacteristic
358C2×C4⋊C4C23×C712yes—
359C2×C42—21no—
360C2×C42C23×C711yescharacteristic
361C4⋊C8—21no—
362C4⋊C8—21no—
363#103—22no—
364#103—22no—
365#103—24no—
366C22⋊C8C23×C712yes—
367C22⋊C8C23×C712yes—
368#97—21no—
369C4×C8—21no—
370#95—21no—
371#95—24no—
372#95C23×C712yes—
373C22×C7—21no—
374C22×C7—21no—
375C22×C7—41no—
376C22×C7—41no—
377C22×C7—41no—
378C22×C7C2×#12011yescharacteristic
379C22×C7#57812yes—
380C4×C7—21no—
381C4×C7—21no—
382C4×C7—21no—
383C4×C7—21no—
384C4×C7—21no—
385C4×C7—44no—
386C4×C7—42no—
387C4×C7—41no—
388C4×C7—41no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—41no—
393C4×C7—82no—
394C4×C7#45612yes—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—82no—
400C2×Q8C2×C7×D412yes—
401C2×Q8C2×C7×D411yescharacteristic
402C2×D4—21no—
403C2×D4C2×C7×D411yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4C2×C7×D412yes—
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D412yes—
412C22×C4C24×C711yescharacteristic
413Q16—82no—
414Q16—84no—
415SD16—82no—
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—84no—
448C4⋊C4C2×C7×D411yescharacteristic
449C4⋊C4C2×C7×D411yescharacteristic
450C4⋊C4C2×C7×D412yes—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—21no—
466C42C2×C7×D411yescharacteristic
467C42C2×C7×D411yescharacteristic
468C42C2×C7×D411yescharacteristic
469C42C2×C7×D411yescharacteristic
470C2×C7—22no—
471C2×C7—81no—
472C2×C7#301111yescharacteristic
473C2×C7C2×#45611yescharacteristic
474C2×C7#302711yescharacteristic
475C23—21no—
476C23C22×C7×D411yescharacteristic
477Q8—41no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—44no—
482Q8—42no—
483Q8—82no—
484D4—41no—
485D4—42no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—82no—
505C2×C4C7×#12011yescharacteristic
506C2×C4C7×#12011yescharacteristic
507C2×C4C7×#12011yescharacteristic
508C2×C4C7×#12011yescharacteristic
509C2×C4C22×C7×D411yescharacteristic
510C2×C4C22×C7×D411yescharacteristic
511C8—42no—
512C8—42no—
513C7#1843311yescharacteristic, direct product factor, Sylow
514C22—21no—
515C22—21no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×C7×#12011yescharacteristic
520C22C7×#57812yes—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—44no—
527C4—42no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—41no—
534C4—82no—
535C4C7×#45612yes—
536C2—22no—
537C2—81no—
538C2C7×#301111yescharacteristic
539C2C2×C7×#45611yescharacteristic
540C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#121C23
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
3. C7×#3209C2semidirect
4. C7×#2660C2semidirect
5. C7×#1605C2semidirect
7. C7×#1506C2semidirect
9. C7×#1470C2semidirect
47. #18433C7direct (N × H)
133. #3209C2×C7semidirect
134. #2660C2×C7semidirect
135. #1605C2×C7semidirect
137. #1506C2×C7semidirect
139. #1470C2×C7semidirect
513. C7#18433direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32182
C7×#32092
C7×#26602
C7×#16052
C7×#16032
C7×#15062
C7×#15002
C7×#14702
#184337