Finite Group Databasebeta

← C7×#19028 #13196244

Subgroups — C7×#19028 #13196244

Order 1792.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic
3C7×#3138C211yesmaximal, characteristic, complemented
4C2×C7×#490C211yesmaximal, characteristic, complemented
5C7×#1658C214yesmaximal, complemented
6C7×#1657C214yesmaximal, complemented
7C7×#1531C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C7×#564C2212yescomplemented
11C7×#563C2212yescomplemented
12C7×#557—21no—
13C7×#554—21no—
14C7×#552—21no—
15C7×#552—22no—
16C7×#552—21no—
17C7×#544—21no—
18C7×#544—21no—
19C7×#541—21no—
20C7×#541—22no—
21C7×#539—21no—
22C7×#533C2211yescharacteristic
23C2×C7×#127C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#496C2212yes—
27C7×#490—22no—
28C7×#490C2212yes—
29C7×#490C2214yes—
30C7×#490C2214yes—
31C7×#486—21no—
32C7×#486—21no—
33C7×#460—21no—
34C7×#459—21no—
35C7×#450—21no—
36C7×#447—21no—
37C7×#432C2211yescharacteristic
38C7×#418—22no—
39C2×C7×#102—21no—
40C2×C7×#102—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yescomplemented
44C7×#330C2214yescomplemented
45C7×#330C2214yescomplemented
46C7×#329C2212yescomplemented
47#19028C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C2×C7×SD16—22no—
52C2×C7×D8—22no—
53C2×C7×M16—21no—
54C22×C7×C8—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8D411yescharacteristic
60C7×#127—21no—
61C7×#127—21no—
62C7×#127—21no—
63C7×#127C2312yes—
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#126—22no—
68C7×#126—21no—
69C7×#125—22no—
70C7×#125—22no—
71C7×#125—21no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—21no—
79C7×#125—21no—
80C7×#125D412yes—
81C7×#125C2312yes—
82C7×#125C2311yescharacteristic
83C7×#125C2312yes—
84C7×#124C2311yescharacteristic
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—21no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8D411yescharacteristic
97C7×#121D411yescharacteristic
98C4×C7×Q8—21no—
99C4×C7×Q8—22no—
100C4×C7×Q8—21no—
101C4×C7×Q8—21no—
102C4×C7×Q8—21no—
103C4×C7×Q8D411yescharacteristic
104C4×C7×D4—22no—
105C4×C7×D4—21no—
106C4×C7×D4D411yescharacteristic
107C7×#117—22no—
108C7×#117—21no—
109C7×#117—22no—
110C7×#117—21no—
111C7×#117D412yes—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4D411yescharacteristic
116C2×C7×C4⋊C4D411yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×C4⋊C8—21no—
119C7×#102—24no—
120C7×#102—24no—
121C7×#102—24no—
122C7×#102—24no—
123C7×#102—22no—
124C7×#102—22no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C4×C7×C8—21no—
131#3563C2×C711yescharacteristic, complemented
132#3217C2×C711yescharacteristic
133#3138C2×C711yescharacteristic, complemented
134C2×#490C2×C711yescharacteristic, complemented
135#1658C2×C714yescomplemented
136#1657C2×C714yescomplemented
137#1531C2×C711yescharacteristic, complemented
138#1528C2×C711yescharacteristic, complemented
139#1524C2×C711yescharacteristic, complemented
140C24×C7—41no—
141C2×C7×Q8—22no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8C2×D411yescharacteristic
145C2×C7×D4—21no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—22no—
149C2×C7×D4—44no—
150C2×C7×D4—44no—
151C2×C7×D4—41no—
152C2×C7×D4—41no—
153C2×C7×D4—41no—
154C2×C7×D4—82no—
155C2×C7×D4C2×D411yescharacteristic
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C7×SD16—82no—
164C7×D8—82no—
165C7×M16—42no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—42no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D412yes—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D411yescharacteristic
197C7×C4⋊C4C2×D411yescharacteristic
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—21no—
207C42×C7—22no—
208C42×C7—22no—
209C42×C7—22no—
210C42×C7—22no—
211C42×C7—21no—
212C42×C7C2×D411yescharacteristic
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215C42×C7C2411yescharacteristic
216#564C22×C712yescomplemented
217#563C22×C712yescomplemented
218#557—21no—
219#554—21no—
220#552—21no—
221#552—22no—
222#552—21no—
223#544—21no—
224#544—21no—
225#541—21no—
226#541—22no—
227#539—21no—
228#533C22×C711yescharacteristic
229C2×#127C22×C711yescharacteristic
230#529C22×C711yescharacteristic
231C2×#125C22×C711yescharacteristic
232#496C22×C712yes—
233#490—22no—
234#490C22×C714yes—
235#490C22×C714yes—
236#490C22×C712yes—
237#486—21no—
238#486—21no—
239#460—21no—
240#459—21no—
241#450—21no—
242#447—21no—
243#432C22×C711yescharacteristic
244#418—22no—
245C2×#102—21no—
246C2×#102—21no—
247C4×M16C22×C711yescharacteristic
248C2×#97C22×C711yescharacteristic
249#330C22×C714yescomplemented
250#330C22×C714yescomplemented
251#330C22×C712yescomplemented
252#329C22×C712yescomplemented
253C23×C7—21no—
254C23×C7—42no—
255C23×C7—82no—
256C23×C7—82no—
257C23×C7—82no—
258C23×C7#12011yescharacteristic
259C7×Q8—41no—
260C7×Q8—42no—
261C7×Q8—81no—
262C7×Q8—82no—
263C7×D4—41no—
264C7×D4—42no—
265C7×D4—42no—
266C7×D4—44no—
267C7×D4—44no—
268C7×D4—42no—
269C7×D4—82no—
270C7×D4—82no—
271C7×D4—82no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—81no—
285C2×C4×C7—82no—
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C7×C8—42no—
293C7×C8—42no—
294C7×C8—42no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×SD16—22no—
299C2×D8—22no—
300C2×M16—21no—
301C22×C8—21no—
302C4⋊Q8—22no—
303C4⋊Q8—21no—
304C4⋊Q8—22no—
305C4⋊Q8C7×D411yescharacteristic
306C4⋊Q8C7×D411yescharacteristic
307#127—21no—
308#127—21no—
309#127—21no—
310#127C23×C712yes—
311#126—22no—
312#126—22no—
313#126—22no—
314#126—22no—
315#126—21no—
316#125—22no—
317#125—22no—
318#125—21no—
319#125—22no—
320#125—22no—
321#125—22no—
322#125—22no—
323#125—22no—
324#125—22no—
325#125—21no—
326#125—21no—
327#125C7×D412yes—
328#125C23×C712yes—
329#125C23×C711yescharacteristic
330#125C23×C712yes—
331#124C23×C711yescharacteristic
332#123—22no—
333#123—22no—
334#123—21no—
335C22⋊Q8—22no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8C7×D411yescharacteristic
344#121C7×D411yescharacteristic
345C4×Q8—21no—
346C4×Q8—22no—
347C4×Q8—21no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8C7×D411yescharacteristic
351C4×D4—22no—
352C4×D4—21no—
353C4×D4C7×D411yescharacteristic
354#117—22no—
355#117—21no—
356#117—22no—
357#117—21no—
358#117C7×D412yes—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C7×D411yescharacteristic
363C2×C4⋊C4C7×D411yescharacteristic
364C2×C42C23×C711yescharacteristic
365C4⋊C8—21no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—24no—
370#102—22no—
371#102—22no—
372C22⋊C8—21no—
373#97—21no—
374#97C23×C712yes—
375#97C23×C712yes—
376#97C23×C712yes—
377C4×C8—21no—
378C22×C7—21no—
379C22×C7—21no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—41no—
383C22×C7—82no—
384C22×C7—82no—
385C22×C7—82no—
386C22×C7—164no—
387C22×C7#45412yes—
388C22×C7C2×#12011yescharacteristic
389C4×C7—21no—
390C4×C7—21no—
391C4×C7—21no—
392C4×C7—21no—
393C4×C7—21no—
394C4×C7—42no—
395C4×C7—41no—
396C4×C7—41no—
397C4×C7—42no—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—41no—
401C4×C7#45412yes—
402C24—41no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8C2×C7×D411yescharacteristic
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—82no—
417C2×D4C2×C7×D411yescharacteristic
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425SD16—82no—
426D8—82no—
427M16—42no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—42no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4C2×C7×D412yes—
455C4⋊C4C2×C7×D412yes—
456C4⋊C4C2×C7×D412yes—
457C4⋊C4C2×C7×D412yes—
458C4⋊C4C2×C7×D411yescharacteristic
459C4⋊C4C2×C7×D411yescharacteristic
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—21no—
474C42C2×C7×D411yescharacteristic
475C42C2×C7×D411yescharacteristic
476C42C2×C7×D411yescharacteristic
477C42C24×C711yescharacteristic
478C2×C7—22no—
479C2×C7—81no—
480C2×C7—162no—
481C2×C7C2×#45411yescharacteristic
482C2×C7C2×#45411yescharacteristic
483C2×C7#302111yescharacteristic
484C23—21no—
485C23—42no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23C7×#12011yescharacteristic
490Q8—41no—
491Q8—42no—
492Q8—81no—
493Q8—82no—
494D4—41no—
495D4—42no—
496D4—42no—
497D4—44no—
498D4—44no—
499D4—42no—
500D4—82no—
501D4—82no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4C7×#12011yescharacteristic
518C2×C4C7×#12011yescharacteristic
519C2×C4C7×#12011yescharacteristic
520C2×C4C22×C7×D411yescharacteristic
521C2×C4C22×C7×D411yescharacteristic
522C2×C4C22×C7×D411yescharacteristic
523C8—42no—
524C8—42no—
525C8—42no—
526C7#1902811yescharacteristic, direct product factor, Sylow
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—164no—
536C22C7×#45412yes—
537C22C2×C7×#12011yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4C7×#45412yes—
551C2—22no—
552C2—81no—
553C2—162no—
554C2C2×C7×#45411yescharacteristic
555C2C2×C7×#45411yescharacteristic
556C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3563C2semidirect
3. C7×#3138C2semidirect
4. C2×C7×#490C2semidirect
5. C7×#1658C2semidirect
6. C7×#1657C2semidirect
7. C7×#1531C2semidirect
8. C7×#1528C2semidirect
9. C7×#1524C2semidirect
10. C7×#564C22semidirect
11. C7×#563C22semidirect
43. C7×#330C22semidirect
44. C7×#330C22semidirect
45. C7×#330C22semidirect
46. C7×#329C22semidirect
47. #19028C7direct (N × H)
131. #3563C2×C7semidirect
133. #3138C2×C7semidirect
134. C2×#490C2×C7semidirect
135. #1658C2×C7semidirect
136. #1657C2×C7semidirect
137. #1531C2×C7semidirect
138. #1528C2×C7semidirect
139. #1524C2×C7semidirect
216. #564C22×C7semidirect
217. #563C22×C7semidirect
249. #330C22×C7semidirect
250. #330C22×C7semidirect
251. #330C22×C7semidirect
252. #329C22×C7semidirect
526. C7#19028direct (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×#32172
C7×#31382
C2×C7×#4902
C7×#16582
C7×#16572
C7×#15312
C7×#15282
C7×#15242
#190287