Finite Group Databasebeta

← C7×#18650 #13195866

Subgroups — C7×#18650 #13195866

Order 1792.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3499C211yesmaximal, characteristic
2C7×#3226C211yesmaximal, characteristic, complemented
3C2×C7×#500C211yesmaximal, characteristic, complemented
4C7×#3039C211yesmaximal, characteristic, complemented
5C7×#1649C214yesmaximal, complemented
6C7×#1647C214yesmaximal, complemented
7C7×#1546C211yesmaximal, characteristic, complemented
8C7×#1492C211yesmaximal, characteristic, complemented
9C7×#1476C211yesmaximal, characteristic, complemented
10C2×C7×2−1+4—21no—
11C2×C7×#137—21no—
12C7×D4×Q8—21no—
13C7×D4×Q8C2212yes—
14C7×#547—22no—
15C7×#547—21no—
16C7×#547—22no—
17C7×#547C2212yes—
18C7×#536—22no—
19C7×#536—21no—
20C7×#532—21no—
21C7×#532C2211yescharacteristic
22C2×C7×C4⋊Q8—21no—
23C2×C7×C4⋊Q8C2211yescharacteristic
24C2×C7×C4⋊Q8C2211yescharacteristic
25C4×C7×#41C2211yescharacteristic
26C7×#500—22no—
27C7×#500—21no—
28C7×#500C2214yes—
29C7×#486—21no—
30C7×#480—22no—
31C7×#474C2214yes—
32C7×#462—21no—
33C7×#462C2212yes—
34C7×#461—21no—
35C7×#461C2212yes—
36C7×#431C2211yescharacteristic
37C2×C7×#107—22no—
38C7×#423C2211yescharacteristic
39C7×#419—21no—
40C2×C4×C7×C8C2211yescharacteristic
41C7×#332C2214yes—
42C7×Q8⋊C8C2212yes—
43C7×D4⋊C8C2212yes—
44C7×D4⋊C8C2214yes—
45#18650C711yesmaximal, characteristic, direct product factor, Sylow
46C7×2−1+4—44no—
47C7×2−1+4—44no—
48C2×C7×#41—22no—
49C2×C7×#41—24no—
50C2×C7×#41—22no—
51C2×C7×#41—21no—
52C2×C7×#41D411yescharacteristic
53C22×C7×Q8—22no—
54C22×C7×Q8—22no—
55C22×C7×Q8—21no—
56C22×C7×Q8—21no—
57C7×#137—44no—
58C2×C7×Q16—22no—
59C2×C7×SD16—22no—
60C2×C7×M16—21no—
61C22×C7×C8—21no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8D411yescharacteristic
71C7×C4⋊Q8D412yes—
72C7×C4⋊Q8C2312yes—
73C7×C4⋊Q8C2312yes—
74C7×#127D411yescharacteristic
75C7×#124—22no—
76C7×#124—21no—
77C7×#123—24no—
78C7×#123—24no—
79C7×#123—24no—
80C7×C22⋊Q8—24no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—24no—
83C7×C22⋊Q8—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—21no—
87C7×#121D412yes—
88C4×C7×Q8D411yescharacteristic
89C4×C7×Q8C2311yescharacteristic
90C4×C7×D4—22no—
91C4×C7×D4—22no—
92C4×C7×D4—21no—
93C4×C7×D4—21no—
94C4×C7×D4—21no—
95C4×C7×D4D411yescharacteristic
96C4×C7×D4C2312yes—
97C4×C7×D4C2311yescharacteristic
98C7×#117—21no—
99C7×#117D411yescharacteristic
100C2×C7×C4⋊C4—22no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4—22no—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4D412yes—
106C2×C42×C7—21no—
107C2×C42×C7C2311yescharacteristic
108C7×#107—24no—
109C7×#107—28no—
110C7×C4⋊C8—21no—
111C7×C4⋊C8C2312yes—
112C7×C4⋊C8C2312yes—
113C7×#103—22no—
114C7×#103—22no—
115C7×#102—24no—
116C7×C22⋊C8—21no—
117C4×C7×C8—21no—
118C4×C7×C8—21no—
119C4×C7×C8C2312yes—
120#3499C2×C711yescharacteristic
121#3226C2×C711yescharacteristic, complemented
122C2×#500C2×C711yescharacteristic, complemented
123#3039C2×C711yescharacteristic, complemented
124#1649C2×C714yescomplemented
125#1647C2×C714yescomplemented
126#1546C2×C711yescharacteristic, complemented
127#1492C2×C711yescharacteristic, complemented
128#1476C2×C711yescharacteristic, complemented
129C7×#41—22no—
130C7×#41—48no—
131C7×#41—416no—
132C7×#41—44no—
133C7×#41—44no—
134C7×#41—42no—
135C7×#41—42no—
136C7×#41D814yes—
137C2×C7×Q8—24no—
138C2×C7×Q8—22no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—48no—
143C2×C7×Q8—44no—
144C2×C7×Q8—44no—
145C2×C7×Q8—41no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—42no—
149C2×C7×Q8—42no—
150C2×C7×Q8C2×D411yescharacteristic
151C2×C7×D4—21no—
152C2×C7×D4—24no—
153C2×C7×D4—22no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4C2×D412yes—
157C2×C7×D4C2×D411yescharacteristic
158C22×C4×C7—21no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—22no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—22no—
165C22×C4×C7—41no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C7×Q16—44no—
170C7×SD16—44no—
171C7×M16—42no—
172C2×C7×C8—24no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4C2×D411yescharacteristic
189C7×C4⋊C4C2×D411yescharacteristic
190C7×C4⋊C4C2×D412yes—
191C7×C4⋊C4C2×D414yes—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—21no—
195C7×C22⋊C4—21no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—21no—
198C7×C22⋊C4—22no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7C2×D411yescharacteristic
203C42×C7C2×D411yescharacteristic
204C42×C7C2×D411yescharacteristic
205C42×C7C2411yescharacteristic
206C2×2−1+4—21no—
207C2×#137—21no—
208D4×Q8—21no—
209D4×Q8C22×C712yes—
210#547—22no—
211#547—21no—
212#547—22no—
213#547C22×C712yes—
214#536—22no—
215#536—21no—
216#532—21no—
217#532C22×C711yescharacteristic
218C2×C4⋊Q8—21no—
219C2×C4⋊Q8C22×C711yescharacteristic
220C2×C4⋊Q8C22×C711yescharacteristic
221C4×#41C22×C711yescharacteristic
222#500—22no—
223#500—21no—
224#500C22×C714yes—
225#486—21no—
226#480—22no—
227#474C22×C714yes—
228#462—21no—
229#462C22×C712yes—
230#461—21no—
231#461C22×C712yes—
232#431C22×C711yescharacteristic
233C2×#107—22no—
234#423C22×C711yescharacteristic
235#419—21no—
236C2×C4×C8C22×C711yescharacteristic
237#332C22×C714yes—
238Q8⋊C8C22×C712yes—
239D4⋊C8C22×C712yes—
240D4⋊C8C22×C714yes—
241C23×C7—21no—
242C23×C7—22no—
243C23×C7#12011yescharacteristic
244C7×Q8—21no—
245C7×Q8—48no—
246C7×Q8—42no—
247C7×Q8—42no—
248C7×Q8—42no—
249C7×Q8—44no—
250C7×Q8—42no—
251C7×Q8—42no—
252C7×Q8—82no—
253C7×Q8C2×D812yes—
254C7×D4—24no—
255C7×D4—21no—
256C7×D4—24no—
257C7×D4—48no—
258C7×D4—44no—
259C7×D4—42no—
260C7×D4—42no—
261C7×D4C2×D812yes—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—24no—
269C2×C4×C7—22no—
270C2×C4×C7—48no—
271C2×C4×C7—48no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—82no—
279C2×C4×C7#12011yescharacteristic
280C2×C4×C7#12011yescharacteristic
281C2×C4×C7#12011yescharacteristic
282C2×C4×C7C2×D812yes—
283C2×C4×C7AGL(1,ℤ/8)12yes—
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C7×C8—24no—
288C7×C8—42no—
289C7×C8—42no—
2902−1+4—44no—
2912−1+4—44no—
292C2×#41—22no—
293C2×#41—24no—
294C2×#41—22no—
295C2×#41—21no—
296C2×#41C7×D411yescharacteristic
297C22×Q8—22no—
298C22×Q8—22no—
299C22×Q8—21no—
300C22×Q8—21no—
301#137—44no—
302C2×Q16—22no—
303C2×SD16—22no—
304C2×M16—21no—
305C22×C8—21no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8C7×D411yescharacteristic
315C4⋊Q8C7×D412yes—
316C4⋊Q8C23×C712yes—
317C4⋊Q8C23×C712yes—
318#127C7×D411yescharacteristic
319#124—22no—
320#124—21no—
321#123—24no—
322#123—24no—
323#123—24no—
324C22⋊Q8—24no—
325C22⋊Q8—22no—
326C22⋊Q8—24no—
327C22⋊Q8—22no—
328#121—22no—
329#121—22no—
330#121—21no—
331#121C7×D412yes—
332C4×Q8C7×D411yescharacteristic
333C4×Q8C23×C711yescharacteristic
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4C7×D411yescharacteristic
340C4×D4C23×C711yescharacteristic
341C4×D4C23×C712yes—
342#117—21no—
343#117C7×D411yescharacteristic
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C7×D412yes—
350C2×C42—21no—
351C2×C42C23×C711yescharacteristic
352#107—28no—
353#107—24no—
354C4⋊C8—21no—
355C4⋊C8C23×C712yes—
356C4⋊C8C23×C712yes—
357#103—22no—
358#103—22no—
359#102—24no—
360C22⋊C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8C23×C712yes—
364C22×C7—21no—
365C22×C7—21no—
366C22×C7—22no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7#44712yes—
373C22×C7C2×#12011yescharacteristic
374C4×C7—22no—
375C4×C7—21no—
376C4×C7—21no—
377C4×C7—44no—
378C4×C7—42no—
379C4×C7—41no—
380C4×C7—41no—
381C4×C7—82no—
382C4×C7#44712yes—
383C4×C7#45612yes—
384C4×C7C22×D811yescharacteristic
385C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
386C4×C7#57812yes—
387#41—22no—
388#41—48no—
389#41—416no—
390#41—44no—
391#41—44no—
392#41—42no—
393#41—42no—
394#41C7×D814yes—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×C7×D411yescharacteristic
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C7×D412yes—
415C2×D4C2×C7×D411yescharacteristic
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427Q16—44no—
428SD16—44no—
429M16—42no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4C2×C7×D411yescharacteristic
447C4⋊C4C2×C7×D411yescharacteristic
448C4⋊C4C2×C7×D412yes—
449C4⋊C4C2×C7×D414yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42C2×C7×D411yescharacteristic
461C42C2×C7×D411yescharacteristic
462C42C2×C7×D411yescharacteristic
463C42C24×C711yescharacteristic
464C2×C7—22no—
465C2×C7—42no—
466C2×C7—82no—
467C2×C7C2×#44711yescharacteristic
468C2×C7#301111yescharacteristic
469C2×C7C2×#45611yescharacteristic
470C23—21no—
471C23—22no—
472C23C7×#12011yescharacteristic
473Q8—21no—
474Q8—48no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—44no—
479Q8—42no—
480Q8—42no—
481Q8—82no—
482Q8C2×C7×D812yes—
483D4—24no—
484D4—21no—
485D4—24no—
486D4—48no—
487D4—44no—
488D4—42no—
489D4—42no—
490D4C2×C7×D812yes—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4C7×#12011yescharacteristic
509C2×C4C7×#12011yescharacteristic
510C2×C4C7×#12011yescharacteristic
511C2×C4C2×C7×D812yes—
512C2×C4C7×AGL(1,ℤ/8)12yes—
513C2×C4C22×C7×D411yescharacteristic
514C2×C4C22×C7×D411yescharacteristic
515C2×C4C22×C7×D411yescharacteristic
516C8—24no—
517C8—42no—
518C8—42no—
519C7#1865011yescharacteristic, direct product factor, Sylow
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22C7×#44712yes—
529C22C2×C7×#12011yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—44no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—82no—
538C4C7×#44712yes—
539C4C7×#45612yes—
540C4C22×C7×D811yescharacteristic
541C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
542C4C7×#57812yes—
543C2—22no—
544C2—42no—
545C2—82no—
546C2C2×C7×#44711yescharacteristic
547C2C7×#301111yescharacteristic
548C2C2×C7×#45611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3499C2
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×#3226C2semidirect
3. C2×C7×#500C2semidirect
4. C7×#3039C2semidirect
5. C7×#1649C2semidirect
6. C7×#1647C2semidirect
7. C7×#1546C2semidirect
8. C7×#1492C2semidirect
9. C7×#1476C2semidirect
45. #18650C7direct (N × H)
121. #3226C2×C7semidirect
122. C2×#500C2×C7semidirect
123. #3039C2×C7semidirect
124. #1649C2×C7semidirect
125. #1647C2×C7semidirect
126. #1546C2×C7semidirect
127. #1492C2×C7semidirect
128. #1476C2×C7semidirect
519. C7#18650direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34992
C7×#32262
C2×C7×#5002
C7×#30392
C7×#16492
C7×#16472
C7×#15462
C7×#14922
C7×#14762
#186507