Finite Group Databasebeta

← C7×#19017 #13196233

Subgroups — C7×#19017 #13196233

Order 1792.

944 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic
2C7×#3238C211yesmaximal, characteristic, complemented
3C2×C7×#501C211yesmaximal, characteristic, complemented
4C7×#3152C211yesmaximal, characteristic, complemented
5C7×#1661C214yesmaximal, complemented
6C7×#1660C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C2×C7×#137—22no—
11C7×#563C2212yes—
12C7×#562C2212yes—
13C7×#554—21no—
14C7×#553—21no—
15C7×#553—21no—
16C7×#547—22no—
17C7×#547—21no—
18C7×#544—21no—
19C7×#540—21no—
20C7×#539—22no—
21C7×#539—21no—
22C7×#537—21no—
23C7×#532C2211yescharacteristic
24C2×C7×C4⋊Q8C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C7×#501—22no—
28C7×#501C2214yes—
29C7×#501C2212yes—
30C7×#497C2214yes—
31C7×#494—21no—
32C7×#492—21no—
33C7×#489C2212yes—
34C7×#483—21no—
35C7×#480—21no—
36C7×#477—21no—
37C7×#476—21no—
38C7×#432C2211yescharacteristic
39C2×C7×#107—21no—
40C2×C7×#106—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yes—
44C7×#330C2214yes—
45C7×#329C2212yes—
46C7×#329C2214yes—
47#19017C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41D412yes—
50C22×C7×Q8—21no—
51C7×#137—48no—
52C2×C7×Q16—24no—
53C2×C7×SD16—24no—
54C2×C7×M16—21no—
55C22×C7×C8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8D411yescharacteristic
62C7×C4⋊Q8C2312yes—
63C7×#127D411yescharacteristic
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—21no—
67C7×#125—22no—
68C7×#125—21no—
69C7×#125—21no—
70C7×#125—21no—
71C7×#125D412yes—
72C7×#125C2312yes—
73C7×#125C2311yescharacteristic
74C7×#124C2311yescharacteristic
75C7×#124C2312yes—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—21no—
81C7×#123—22no—
82C7×#123—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8D411yescharacteristic
90C7×#121—22no—
91C7×#121—22no—
92C7×#121—21no—
93C7×#121—21no—
94C7×#121—22no—
95C7×#121—22no—
96C7×#121—21no—
97C7×#121—21no—
98C7×#121D411yescharacteristic
99C4×C7×Q8D411yescharacteristic
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4—21no—
105C4×C7×D4—21no—
106C4×C7×D4—21no—
107C4×C7×D4D411yescharacteristic
108C7×#117—21no—
109C7×#117—21no—
110C2×C7×C4⋊C4—22no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4D411yescharacteristic
117C2×C7×C4⋊C4D411yescharacteristic
118C2×C42×C7C2311yescharacteristic
119C7×#107—22no—
120C7×#107—24no—
121C7×#106—22no—
122C7×#106—24no—
123C7×C4⋊C8—21no—
124C7×#103—22no—
125C7×#102—22no—
126C7×C22⋊C8—21no—
127C7×#97—21no—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C7×#97C2312yes—
131C4×C7×C8—21no—
132#3559C2×C711yescharacteristic
133#3238C2×C711yescharacteristic, complemented
134C2×#501C2×C711yescharacteristic, complemented
135#3152C2×C711yescharacteristic, complemented
136#1661C2×C714yescomplemented
137#1660C2×C714yescomplemented
138#1530C2×C711yescharacteristic, complemented
139#1528C2×C711yescharacteristic, complemented
140#1524C2×C711yescharacteristic, complemented
141C7×#41—44no—
142C7×#41—84no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8—44no—
146C2×C7×Q8—41no—
147C2×C7×Q8C2×D411yescharacteristic
148C2×C7×Q8C2×D412yes—
149C2×C7×D4—22no—
150C2×C7×D4—21no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4C2×D411yescharacteristic
156C2×C7×D4C2×D412yes—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—22no—
164C22×C4×C7—41no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C7×Q16—48no—
169C7×SD16—48no—
170C7×M16—24no—
171C2×C7×C8—24no—
172C2×C7×C8—22no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4—41no—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D411yescharacteristic
197C7×C4⋊C4C2×D412yes—
198C7×C4⋊C4C2×D411yescharacteristic
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—21no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—21no—
211C42×C7—22no—
212C42×C7—21no—
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215C42×C7C2×D411yescharacteristic
216C42×C7C2411yescharacteristic
217C2×#137—22no—
218#563C22×C712yes—
219#562C22×C712yes—
220#554—21no—
221#553—21no—
222#553—21no—
223#547—22no—
224#547—21no—
225#544—21no—
226#540—21no—
227#539—22no—
228#539—21no—
229#537—21no—
230#532C22×C711yescharacteristic
231C2×C4⋊Q8C22×C711yescharacteristic
232#529C22×C711yescharacteristic
233C2×#125C22×C711yescharacteristic
234#501—22no—
235#501C22×C714yes—
236#501C22×C712yes—
237#497C22×C714yes—
238#494—21no—
239#492—21no—
240#489C22×C712yes—
241#483—21no—
242#480—21no—
243#477—21no—
244#476—21no—
245#432C22×C711yescharacteristic
246C2×#107—21no—
247C2×#106—21no—
248C4×M16C22×C711yescharacteristic
249C2×#97C22×C711yescharacteristic
250#330C22×C712yes—
251#330C22×C714yes—
252#329C22×C714yes—
253#329C22×C712yes—
254C23×C7—22no—
255C23×C7—21no—
256C23×C7#12011yescharacteristic
257C7×Q8—42no—
258C7×Q8—41no—
259C7×Q8—44no—
260C7×Q8—82no—
261C7×Q8—82no—
262C7×D4—42no—
263C7×D4—41no—
264C7×D4—44no—
265C7×D4—42no—
266C7×D4—42no—
267C7×D4—81no—
268C7×D4—82no—
269C7×D4—82no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—82no—
283C2×C4×C7—81no—
284C2×C4×C7—82no—
285C2×C4×C7—82no—
286C2×C4×C7—82no—
287C2×C4×C7—82no—
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7#12011yescharacteristic
290C2×C4×C7#12011yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C7×C8—24no—
295C7×C8—24no—
296C7×C8—42no—
297C2×#41—22no—
298C2×#41C7×D412yes—
299C22×Q8—21no—
300#137—48no—
301C2×Q16—24no—
302C2×SD16—24no—
303C2×M16—21no—
304C22×C8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—22no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C7×D411yescharacteristic
311C4⋊Q8C23×C712yes—
312#127C7×D411yescharacteristic
313#126—22no—
314#126—22no—
315#126—21no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125C7×D412yes—
321#125C23×C712yes—
322#125C23×C711yescharacteristic
323#124C23×C711yescharacteristic
324#124C23×C712yes—
325#123—22no—
326#123—22no—
327#123—22no—
328#123—22no—
329#123—21no—
330#123—22no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8C7×D411yescharacteristic
339#121—22no—
340#121—22no—
341#121—21no—
342#121—21no—
343#121—22no—
344#121—22no—
345#121—21no—
346#121—21no—
347#121C7×D411yescharacteristic
348C4×Q8C7×D411yescharacteristic
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4C7×D411yescharacteristic
357#117—21no—
358#117—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C7×D411yescharacteristic
366C2×C4⋊C4C7×D411yescharacteristic
367C2×C42C23×C711yescharacteristic
368#107—24no—
369#107—22no—
370#106—24no—
371#106—22no—
372C4⋊C8—21no—
373#103—22no—
374#102—22no—
375C22⋊C8—21no—
376#97—21no—
377#97C23×C712yes—
378#97C23×C712yes—
379#97C23×C712yes—
380C4×C8—21no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7—42no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—41no—
389C22×C7#45312yes—
390C22×C7C2×#12011yescharacteristic
391C4×C7—21no—
392C4×C7—21no—
393C4×C7—21no—
394C4×C7—21no—
395C4×C7—21no—
396C4×C7—42no—
397C4×C7—42no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—42no—
401C4×C7—41no—
402C4×C7—82no—
403C4×C7#45612yes—
404#41—44no—
405#41—84no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—41no—
410C2×Q8C2×C7×D411yescharacteristic
411C2×Q8C2×C7×D412yes—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C7×D411yescharacteristic
419C2×D4C2×C7×D412yes—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C2×C7×D411yescharacteristic
431Q16—48no—
432SD16—48no—
433M16—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4C2×C7×D412yes—
459C4⋊C4C2×C7×D411yescharacteristic
460C4⋊C4C2×C7×D412yes—
461C4⋊C4C2×C7×D411yescharacteristic
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C42—22no—
475C42—21no—
476C42C2×C7×D411yescharacteristic
477C42C2×C7×D411yescharacteristic
478C42C2×C7×D411yescharacteristic
479C42C24×C711yescharacteristic
480C2×C7—22no—
481C2×C7—82no—
482C2×C7—81no—
483C2×C7C2×#45311yescharacteristic
484C2×C7C2×#45611yescharacteristic
485C2×C7#302211yescharacteristic
486C23—22no—
487C23—21no—
488C23C7×#12011yescharacteristic
489Q8—42no—
490Q8—41no—
491Q8—44no—
492Q8—82no—
493Q8—82no—
494D4—42no—
495D4—41no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—81no—
500D4—82no—
501D4—82no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4C7×#12011yescharacteristic
521C2×C4C7×#12011yescharacteristic
522C2×C4C7×#12011yescharacteristic
523C2×C4C22×C7×D411yescharacteristic
524C2×C4C22×C7×D411yescharacteristic
525C2×C4C22×C7×D411yescharacteristic
526C8—24no—
527C8—24no—
528C8—42no—
529C7#1901711yescharacteristic, direct product factor, Sylow
530C22—21no—
531C22—21no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22C7×#45312yes—
539C22C2×C7×#12011yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—82no—
552C4C7×#45612yes—
553C2—22no—
554C2—82no—
555C2—81no—
556C2C2×C7×#45311yescharacteristic
557C2C2×C7×#45611yescharacteristic
558C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3559C2
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
2. C7×#3238C2semidirect
3. C2×C7×#501C2semidirect
4. C7×#3152C2semidirect
5. C7×#1661C2semidirect
6. C7×#1660C2semidirect
7. C7×#1530C2semidirect
8. C7×#1528C2semidirect
9. C7×#1524C2semidirect
47. #19017C7direct (N × H)
133. #3238C2×C7semidirect
134. C2×#501C2×C7semidirect
135. #3152C2×C7semidirect
136. #1661C2×C7semidirect
137. #1660C2×C7semidirect
138. #1530C2×C7semidirect
139. #1528C2×C7semidirect
140. #1524C2×C7semidirect
529. C7#19017direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#32382
C2×C7×#5012
C7×#31522
C7×#16612
C7×#16602
C7×#15302
C7×#15282
C7×#15242
#190177