Finite Group Databasebeta

← C7×#19649 #13196865

Subgroups — C7×#19649 #13196865

Order 1792.

720 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3231C211yesmaximal, characteristic, complemented
2C7×#3207C211yesmaximal, characteristic
3C7×#3182C211yesmaximal, characteristic, complemented
4C7×#3182C211yesmaximal, characteristic, complemented
5C7×#3167C211yesmaximal, characteristic, complemented
6C7×#3131C211yesmaximal, characteristic, complemented
7C7×#1718C211yesmaximal, characteristic, complemented
8C7×#1717C211yesmaximal, characteristic, complemented
9C7×#1714C211yesmaximal, characteristic, complemented
10C7×#1714C211yesmaximal, characteristic, complemented
11C7×#1711C211yesmaximal, characteristic, complemented
12C7×#1710C211yesmaximal, characteristic, complemented
13C7×#1708C211yesmaximal, characteristic, complemented
14C7×#1706C211yesmaximal, characteristic, complemented
15C7×C8×M16C211yesmaximal, characteristic, complemented
16C7×#533C2211yescharacteristic
17C7×#533C2211yescharacteristic
18C7×#532C2211yescharacteristic
19C7×#532C2211yescharacteristic
20C7×#500C2211yescharacteristic
21C7×#500C2211yescharacteristic
22C7×#500C2211yescharacteristic
23C7×#498C2211yescharacteristic
24C7×#498C2211yescharacteristic
25C7×#493C2211yescharacteristic
26C7×#492C2211yescharacteristic
27C7×#492C2211yescharacteristic
28C7×#492C2211yescharacteristic
29C7×#492C2211yescharacteristic
30C7×#488—22no—
31C7×#487C2211yescharacteristic
32C7×#487C2211yescharacteristic
33C7×#487C2211yescharacteristic
34C7×#487C2211yescharacteristic
35C7×#487C2211yescharacteristic
36C7×#487C2211yescharacteristic
37C7×#487C2211yescharacteristic
38C7×#487C2211yescharacteristic
39C7×#486C2211yescharacteristic
40C7×#486C2211yescharacteristic
41C7×#486C2211yescharacteristic
42C7×#486C2211yescharacteristic
43C7×#486C2211yescharacteristic
44C7×#486C2211yescharacteristic
45C7×#485—22no—
46C7×#479—22no—
47C7×#468—22no—
48C7×#468—22no—
49C7×#465—22no—
50C7×#463—22no—
51C7×#449—22no—
52C7×#440—22no—
53C7×#440—22no—
54C4×C7×SD16—22no—
55C7×#431C2211yescharacteristic
56C7×#416—22no—
57C4×C7×M16C2211yescharacteristic
58C2×C4×C7×C8C2211yescharacteristic
59C7×#322C2211yescharacteristic
60C7×#322C2211yescharacteristic
61C7×C82C2211yescharacteristic
62C7×C82C2211yescharacteristic
63#19649C711yesmaximal, characteristic, direct product factor, Sylow
64C2×C7×#41—21no—
65C2×C7×#41—21no—
66C2×C7×SD16—22no—
67C2×C7×SD16—22no—
68C2×C7×SD16—22no—
69C2×C7×SD16—22no—
70C2×C7×D8—22no—
71C2×C7×M16D412yes—
72C22×C7×C8D412yes—
73C7×C4⋊Q8C2311yescharacteristic
74C7×C4⋊Q8C2311yescharacteristic
75C7×C4⋊Q8C2311yescharacteristic
76C7×C4⋊Q8C2311yescharacteristic
77C7×C4⋊Q8C2311yescharacteristic
78C7×C4⋊Q8C2311yescharacteristic
79C7×#127C2311yescharacteristic
80C7×#127C2311yescharacteristic
81C7×#125—21no—
82C7×#125—21no—
83C7×#124—21no—
84C7×#124—21no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—21no—
89C7×#121—21no—
90C7×#121—21no—
91C7×#121—21no—
92C7×#121—21no—
93C4×C7×Q8—21no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C7×#117—21no—
102C7×#117—21no—
103C2×C42×C7C2311yescharacteristic
104C7×#107—22no—
105C7×#107—22no—
106C7×#107—22no—
107C7×#106—22no—
108C7×#106—22no—
109C7×C4⋊C8D412yes—
110C7×#103—22no—
111C7×#103—22no—
112C7×#103—22no—
113C7×#103—22no—
114C7×#103—22no—
115C7×#103—22no—
116C7×#103—22no—
117C7×#103—22no—
118C7×#102—22no—
119C7×#102—22no—
120C7×#102—22no—
121C7×#102—22no—
122C7×#102—22no—
123C7×#102—22no—
124C7×C22⋊C8D412yes—
125C7×#97D412yes—
126C4×C7×C8D412yes—
127C4×C7×C8C2311yescharacteristic
128C4×C7×C8C2311yescharacteristic
129C4×C7×C8C2311yescharacteristic
130C4×C7×C8C2311yescharacteristic
131C4×C7×C8C2311yescharacteristic
132C4×C7×C8C2311yescharacteristic
133#3231C2×C711yescharacteristic, complemented
134#3207C2×C711yescharacteristic
135#3182C2×C711yescharacteristic, complemented
136#3182C2×C711yescharacteristic, complemented
137#3167C2×C711yescharacteristic, complemented
138#3131C2×C711yescharacteristic, complemented
139#1718C2×C711yescharacteristic, complemented
140#1717C2×C711yescharacteristic, complemented
141#1714C2×C711yescharacteristic, complemented
142#1714C2×C711yescharacteristic, complemented
143#1711C2×C711yescharacteristic, complemented
144#1710C2×C711yescharacteristic, complemented
145#1708C2×C711yescharacteristic, complemented
146#1706C2×C711yescharacteristic, complemented
147C8×M16C2×C711yescharacteristic, complemented
148C7×#41—44no—
149C7×#41—82no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—41no—
163C2×C7×D4—41no—
164C22×C4×C7—41no—
165C22×C4×C7—41no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C7×SD16—44no—
170C7×SD16—44no—
171C7×SD16—44no—
172C7×SD16—82no—
173C7×D8—82no—
174C7×M16—24no—
175C2×C7×C8—42no—
176C2×C7×C8C2×D412yes—
177C2×C7×C8C2×D412yes—
178C2×C7×C8C2×D412yes—
179C2×C7×C8C2×D412yes—
180C2×C7×C8C2×D412yes—
181C2×C7×C8C2×D412yes—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—41no—
195C7×C4⋊C4—41no—
196C7×C4⋊C4—41no—
197C7×C4⋊C4—41no—
198C7×C22⋊C4—41no—
199C7×C22⋊C4—41no—
200C7×C22⋊C4—41no—
201C7×C22⋊C4—41no—
202C7×C22⋊C4—41no—
203C7×C22⋊C4—41no—
204C42×C7—41no—
205C42×C7—41no—
206C42×C7C2×D411yescharacteristic
207C42×C7C2×D411yescharacteristic
208C42×C7C2×D411yescharacteristic
209C42×C7C2411yescharacteristic
210#533C22×C711yescharacteristic
211#533C22×C711yescharacteristic
212#532C22×C711yescharacteristic
213#532C22×C711yescharacteristic
214#500C22×C711yescharacteristic
215#500C22×C711yescharacteristic
216#500C22×C711yescharacteristic
217#498C22×C711yescharacteristic
218#498C22×C711yescharacteristic
219#493C22×C711yescharacteristic
220#492C22×C711yescharacteristic
221#492C22×C711yescharacteristic
222#492C22×C711yescharacteristic
223#492C22×C711yescharacteristic
224#488—22no—
225#487C22×C711yescharacteristic
226#487C22×C711yescharacteristic
227#487C22×C711yescharacteristic
228#487C22×C711yescharacteristic
229#487C22×C711yescharacteristic
230#487C22×C711yescharacteristic
231#487C22×C711yescharacteristic
232#487C22×C711yescharacteristic
233#486C22×C711yescharacteristic
234#486C22×C711yescharacteristic
235#486C22×C711yescharacteristic
236#486C22×C711yescharacteristic
237#486C22×C711yescharacteristic
238#486C22×C711yescharacteristic
239#485—22no—
240#479—22no—
241#468—22no—
242#468—22no—
243#465—22no—
244#463—22no—
245#449—22no—
246#440—22no—
247#440—22no—
248C4×SD16—22no—
249#431C22×C711yescharacteristic
250#416—22no—
251C4×M16C22×C711yescharacteristic
252C2×C4×C8C22×C711yescharacteristic
253#322C22×C711yescharacteristic
254#322C22×C711yescharacteristic
255C82C22×C711yescharacteristic
256C82C22×C711yescharacteristic
257C23×C7—41no—
258C23×C7—41no—
259C23×C7#12711yescharacteristic
260C7×Q8—42no—
261C7×Q8—42no—
262C7×Q8—42no—
263C7×Q8—42no—
264C7×Q8—42no—
265C7×Q8—42no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—42no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—82no—
273C7×D4—82no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—82no—
284C2×C4×C7—82no—
285C2×C4×C7#12711yescharacteristic
286C2×C4×C7#12711yescharacteristic
287C2×C4×C7#12711yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C7×C8—22no—
292C7×C8—22no—
293C7×C8—22no—
294C7×C8—22no—
295C7×C8—22no—
296C7×C8#13514yes—
297C2×#41—21no—
298C2×#41—21no—
299C2×SD16—22no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—22no—
303C2×D8—22no—
304C2×M16C7×D412yes—
305C22×C8C7×D412yes—
306C4⋊Q8C23×C711yescharacteristic
307C4⋊Q8C23×C711yescharacteristic
308C4⋊Q8C23×C711yescharacteristic
309C4⋊Q8C23×C711yescharacteristic
310C4⋊Q8C23×C711yescharacteristic
311C4⋊Q8C23×C711yescharacteristic
312#127C23×C711yescharacteristic
313#127C23×C711yescharacteristic
314#125—21no—
315#125—21no—
316#124—21no—
317#124—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324#121—21no—
325#121—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334#117—21no—
335#117—21no—
336C2×C42C23×C711yescharacteristic
337#107—22no—
338#107—22no—
339#107—22no—
340#106—22no—
341#106—22no—
342C4⋊C8C7×D412yes—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—22no—
347#103—22no—
348#103—22no—
349#103—22no—
350#103—22no—
351#102—22no—
352#102—22no—
353#102—22no—
354#102—22no—
355#102—22no—
356#102—22no—
357C22⋊C8C7×D412yes—
358#97C7×D412yes—
359C4×C8C7×D412yes—
360C4×C8C23×C711yescharacteristic
361C4×C8C23×C711yescharacteristic
362C4×C8C23×C711yescharacteristic
363C4×C8C23×C711yescharacteristic
364C4×C8C23×C711yescharacteristic
365C4×C8C23×C711yescharacteristic
366C22×C7—21no—
367C22×C7—21no—
368C22×C7—21no—
369C22×C7—81no—
370C22×C7—81no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7C2×#12711yescharacteristic
376C4×C7—21no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—81no—
380C4×C7—81no—
381C4×C7—81no—
382C4×C7—81no—
383C4×C7—81no—
384C4×C7—81no—
385C4×C7C2×#13512yes—
386C4×C7#57512yes—
387C4×C7#57712yes—
388#41—44no—
389#41—82no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409SD16—44no—
410SD16—44no—
411SD16—44no—
412SD16—82no—
413D8—82no—
414M16—24no—
415C2×C8—42no—
416C2×C8C2×C7×D412yes—
417C2×C8C2×C7×D412yes—
418C2×C8C2×C7×D412yes—
419C2×C8C2×C7×D412yes—
420C2×C8C2×C7×D412yes—
421C2×C8C2×C7×D412yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—41no—
445C42—41no—
446C42C2×C7×D411yescharacteristic
447C42C2×C7×D411yescharacteristic
448C42C2×C7×D411yescharacteristic
449C42C24×C711yescharacteristic
450C2×C7—41no—
451C2×C7—161no—
452C2×C7—161no—
453C2×C7#314711yescharacteristic
454C2×C7#315311yescharacteristic
455C2×C7#315411yescharacteristic
456C23—41no—
457C23—41no—
458C23C7×#12711yescharacteristic
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—82no—
472D4—82no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4C7×#12711yescharacteristic
485C2×C4C7×#12711yescharacteristic
486C2×C4C7×#12711yescharacteristic
487C2×C4C22×C7×D411yescharacteristic
488C2×C4C22×C7×D411yescharacteristic
489C2×C4C22×C7×D411yescharacteristic
490C8—22no—
491C8—22no—
492C8—22no—
493C8—22no—
494C8—22no—
495C8C7×#13514yes—
496C7#1964911yescharacteristic, direct product factor, Sylow
497C22—21no—
498C22—21no—
499C22—21no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22C2×C7×#12711yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4C2×C7×#13512yes—
517C4C7×#57512yes—
518C4C7×#57712yes—
519C2—41no—
520C2—161no—
521C2—161no—
522C2C7×#314711yescharacteristic
523C2C7×#315311yescharacteristic
524C2C7×#315411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#127
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3207C2
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×#3231C2semidirect
3. C7×#3182C2semidirect
4. C7×#3182C2semidirect
5. C7×#3167C2semidirect
6. C7×#3131C2semidirect
7. C7×#1718C2semidirect
8. C7×#1717C2semidirect
9. C7×#1714C2semidirect
10. C7×#1714C2semidirect
11. C7×#1711C2semidirect
12. C7×#1710C2semidirect
13. C7×#1708C2semidirect
14. C7×#1706C2semidirect
15. C7×C8×M16C2semidirect
63. #19649C7direct (N × H)
133. #3231C2×C7semidirect
135. #3182C2×C7semidirect
136. #3182C2×C7semidirect
137. #3167C2×C7semidirect
138. #3131C2×C7semidirect
139. #1718C2×C7semidirect
140. #1717C2×C7semidirect
141. #1714C2×C7semidirect
142. #1714C2×C7semidirect
143. #1711C2×C7semidirect
144. #1710C2×C7semidirect
145. #1708C2×C7semidirect
146. #1706C2×C7semidirect
147. C8×M16C2×C7semidirect
496. C7#19649direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32312
C7×#32072
C7×#31822
C7×#31672
C7×#31312
C7×#17182
C7×#17172
C7×#17142
C7×#17112
C7×#17102
C7×#17082
C7×#17062
C7×C8×M162
#196497