Finite Group Databasebeta

← C7×#18337 #13195553

Subgroups — C7×#18337 #13195553

Order 1792.

1016 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3461C211yesmaximal, characteristic, complemented
2C7×#3217C212yesmaximal, complemented
3C7×#2548C211yesmaximal, characteristic, complemented
4C7×#1599C212yesmaximal, complemented
5C7×#1598C212yesmaximal, complemented
6C7×#1597C214yesmaximal, complemented
7C7×#1509C212yesmaximal, complemented
8C7×#1467C211yesmaximal, characteristic, complemented
9C7×#544C2211yescharacteristic
10C7×#542—22no—
11C7×#541—22no—
12C7×#539—21no—
13C7×#539C2212yes—
14C7×#537C2211yescharacteristic
15C7×#536—21no—
16C7×#533—22no—
17C2×C7×#127C2211yescharacteristic
18C7×#529—22no—
19C7×#529—21no—
20C7×#529C2212yes—
21C2×C7×#124—21no—
22C4×C7×#41—21no—
23C2×C4×C7×Q8C2211yescharacteristic
24C7×#496—22no—
25C7×#490—22no—
26C7×#460—22no—
27C7×#459—22no—
28C7×#450C2214yes—
29C7×#447C2214yes—
30C7×#432C2212yes—
31C7×#394C2212yes—
32C7×#394C2212yes—
33C7×#390—21no—
34C7×#390—21no—
35C7×#390—21no—
36C4×C7×C4⋊C4C2211yescharacteristic
37C7×#328C2214yes—
38C7×#327C2214yes—
39C7×#324C2212yes—
40C7×#324C2212yes—
41#18337C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8D411yescharacteristic
44C22×C7×D4—21no—
45C22×C7×D4—21no—
46C22×C7×D4C2312yes—
47C2×C7×SD16—24no—
48C2×C7×D8—24no—
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8—22no—
51C7×#127—22no—
52C7×#127—22no—
53C7×#126—22no—
54C7×#126—22no—
55C7×#126—22no—
56C7×#126—22no—
57C7×#125—21no—
58C7×#125—22no—
59C7×#125—22no—
60C7×#125—22no—
61C7×#125D412yes—
62C7×#124—22no—
63C7×#124—21no—
64C7×#124—44no—
65C7×#124D412yes—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8C2312yes—
77C7×#121—22no—
78C7×#121—21no—
79C7×#121C2312yes—
80C4×C7×Q8—21no—
81C4×C7×Q8—21no—
82C4×C7×Q8—21no—
83C4×C7×Q8—22no—
84C4×C7×Q8—22no—
85C4×C7×Q8D412yes—
86C4×C7×D4—21no—
87C4×C7×D4—21no—
88C4×C7×D4—21no—
89C4×C7×D4—22no—
90C4×C7×D4—22no—
91C4×C7×D4D412yes—
92C7×#117—22no—
93C7×#117—22no—
94C7×#117—21no—
95C7×#117—22no—
96C2×C7×C4⋊C4—21no—
97C2×C7×C4⋊C4D411yescharacteristic
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C4⋊C4C2311yescharacteristic
100C2×C7×C22⋊C4—22no—
101C2×C7×C22⋊C4—22no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4—22no—
104C2×C42×C7—22no—
105C2×C42×C7—21no—
106C2×C42×C7D411yescharacteristic
107C2×C42×C7D411yescharacteristic
108C2×C42×C7C2311yescharacteristic
109C7×C4⋊C8—22no—
110C7×#102—24no—
111C7×#102—24no—
112C7×C22⋊C8C2314yes—
113C7×#97—22no—
114C7×#95C2312yes—
115#3461C2×C711yescharacteristic, complemented
116#3217C2×C712yescomplemented
117#2548C2×C711yescharacteristic, complemented
118#1599C2×C712yescomplemented
119#1598C2×C712yescomplemented
120#1597C2×C714yescomplemented
121#1509C2×C712yescomplemented
122#1467C2×C711yescharacteristic, complemented
123C24×C7—22no—
124C7×#41—42no—
125C7×#41—42no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—41no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4—24no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4—41no—
141C2×C7×D4—42no—
142C2×C7×D4—41no—
143C2×C7×D4—82no—
144C2×C7×D4C2×D412yes—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C7×SD16—84no—
157C7×D8—84no—
158C2×C7×C8—24no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4C2×D412yes—
176C7×C4⋊C4C2×D412yes—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—44no—
188C7×C22⋊C4—82no—
189C7×C22⋊C4—82no—
190C42×C7—22no—
191C42×C7—22no—
192C42×C7—21no—
193C42×C7—22no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—42no—
198C42×C7C2×D412yes—
199C42×C7C2×D412yes—
200#544C22×C711yescharacteristic
201#542—22no—
202#541—22no—
203#539—21no—
204#539C22×C712yes—
205#537C22×C711yescharacteristic
206#536—21no—
207#533—22no—
208C2×#127C22×C711yescharacteristic
209#529—22no—
210#529—21no—
211#529C22×C712yes—
212C2×#124—21no—
213C4×#41—21no—
214C2×C4×Q8C22×C711yescharacteristic
215#496—22no—
216#490—22no—
217#460—22no—
218#459—22no—
219#450C22×C714yes—
220#447C22×C714yes—
221#432C22×C712yes—
222#394C22×C712yes—
223#394C22×C712yes—
224#390—21no—
225#390—21no—
226#390—21no—
227C4×C4⋊C4C22×C711yescharacteristic
228#328C22×C714yes—
229#327C22×C714yes—
230#324C22×C712yes—
231#324C22×C712yes—
232C23×C7—22no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7C22×D411yescharacteristic
241C7×Q8—44no—
242C7×Q8—42no—
243C7×Q8—42no—
244C7×Q8—81no—
245C7×D4—42no—
246C7×D4—42no—
247C7×D4—44no—
248C7×D4—44no—
249C7×D4—42no—
250C7×D4—42no—
251C7×D4—84no—
252C7×D4—82no—
253C7×D4—81no—
254C7×D4—82no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—22no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—41no—
269C2×C4×C7—44no—
270C2×C4×C7—44no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—82no—
277C2×C4×C7#12011yescharacteristic
278C2×C4×C7#12012yes—
279C2×C4×C7#12011yescharacteristic
280C2×C4×C7AGL(1,ℤ/8)12yes—
281C2×C4×C7C22×D412yes—
282C7×C8—44no—
283C2×#41—21no—
284C22×Q8C7×D411yescharacteristic
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C23×C712yes—
288C2×SD16—24no—
289C2×D8—24no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292#127—22no—
293#127—22no—
294#126—22no—
295#126—22no—
296#126—22no—
297#126—22no—
298#125—21no—
299#125—22no—
300#125—22no—
301#125—22no—
302#125C7×D412yes—
303#124—22no—
304#124—21no—
305#124—44no—
306#124C7×D412yes—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—21no—
316C22⋊Q8—22no—
317C22⋊Q8C23×C712yes—
318#121—22no—
319#121—21no—
320#121C23×C712yes—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8C7×D412yes—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4C7×D412yes—
333#117—22no—
334#117—22no—
335#117—21no—
336#117—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C7×D411yescharacteristic
339C2×C4⋊C4C23×C711yescharacteristic
340C2×C4⋊C4C23×C711yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42C7×D411yescharacteristic
348C2×C42C7×D411yescharacteristic
349C2×C42C23×C711yescharacteristic
350C4⋊C8—22no—
351#102—24no—
352#102—24no—
353C22⋊C8C23×C714yes—
354#97—22no—
355#95C23×C712yes—
356C22×C7—21no—
357C22×C7—21no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—82no—
362C22×C7—82no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7C2×#12011yescharacteristic
369C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C7#57511yescharacteristic
371C4×C7—22no—
372C4×C7—22no—
373C4×C7—21no—
374C4×C7—22no—
375C4×C7—41no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—41no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—82no—
382C4×C7#45412yes—
383C24—22no—
384#41—42no—
385#41—42no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—41no—
391C2×Q8C2×C7×D412yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—24no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—82no—
404C2×D4C2×C7×D412yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D411yescharacteristic
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C2×C7×D411yescharacteristic
415C22×C4C24×C711yescharacteristic
416SD16—84no—
417D8—84no—
418C2×C8—24no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4C2×C7×D412yes—
436C4⋊C4C2×C7×D412yes—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C42—22no—
451C42—22no—
452C42—21no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—42no—
458C42C2×C7×D412yes—
459C42C2×C7×D412yes—
460C2×C7—21no—
461C2×C7—21no—
462C2×C7—82no—
463C2×C7—162no—
464C2×C7#300611yescharacteristic
465C2×C7C2×#45411yescharacteristic
466C2×C7#302711yescharacteristic
467C23—22no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23C22×C7×D411yescharacteristic
476Q8—44no—
477Q8—42no—
478Q8—42no—
479Q8—81no—
480D4—42no—
481D4—42no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—84no—
487D4—82no—
488D4—81no—
489D4—82no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4C7×#12011yescharacteristic
513C2×C4C7×#12012yes—
514C2×C4C7×#12011yescharacteristic
515C2×C4C7×AGL(1,ℤ/8)12yes—
516C2×C4C22×C7×D412yes—
517C8—44no—
518C7#1833711yescharacteristic, direct product factor, Sylow
519C22—21no—
520C22—21no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22C2×C7×#12011yescharacteristic
532C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
533C22C7×#57511yescharacteristic
534C4—22no—
535C4—22no—
536C4—21no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—42no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—82no—
545C4C7×#45412yes—
546C2—21no—
547C2—21no—
548C2—82no—
549C2—162no—
550C2C7×#300611yescharacteristic
551C2C2×C7×#45411yescharacteristic
552C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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
1. C7×#3461C2semidirect
2. C7×#3217C2semidirect
3. C7×#2548C2semidirect
4. C7×#1599C2semidirect
5. C7×#1598C2semidirect
6. C7×#1597C2semidirect
7. C7×#1509C2semidirect
8. C7×#1467C2semidirect
41. #18337C7direct (N × H)
115. #3461C2×C7semidirect
116. #3217C2×C7semidirect
117. #2548C2×C7semidirect
118. #1599C2×C7semidirect
119. #1598C2×C7semidirect
120. #1597C2×C7semidirect
121. #1509C2×C7semidirect
122. #1467C2×C7semidirect
518. C7#18337direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34612
C7×#32172
C7×#25482
C7×#15992
C7×#15982
C7×#15972
C7×#15092
C7×#14672
#183377