Finite Group Databasebeta

← C7×#18995 #13196211

Subgroups — C7×#18995 #13196211

Order 1792.

1024 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3529C211yesmaximal, characteristic, complemented
2C7×#3217C212yesmaximal, complemented
3C7×#3151C211yesmaximal, characteristic, complemented
4C7×#1657C212yesmaximal, complemented
5C7×#1656C214yesmaximal, complemented
6C7×#1655C212yesmaximal, complemented
7C7×#1536C211yesmaximal, characteristic, complemented
8C7×#1529C212yesmaximal, complemented
9C2×C7×AGL(1,ℤ/8)—21no—
10C2×C7×AGL(1,ℤ/8)—21no—
11C7×#563—21no—
12C7×#563C2211yescharacteristic, complemented
13C7×#562—22no—
14C7×#562C2212yescomplemented
15C7×H(3,𝔽4)C2211yescharacteristic, complemented
16C7×#560—21no—
17C7×#560—22no—
18C7×#532—21no—
19C7×#532C2211yescharacteristic
20C2×C7×#127C2211yescharacteristic, complemented
21C7×#529—22no—
22C7×#529C2212yes—
23C7×#525—21no—
24C7×#525—22no—
25C7×#496C2214yes—
26C7×#496C2212yes—
27C7×#496C2212yescomplemented
28C7×#493—21no—
29C7×#492—21no—
30C7×#490C2214yescomplemented
31C7×#460—22no—
32C7×#459—22no—
33C7×#450—22no—
34C7×#447—22no—
35C7×#432C2212yescomplemented
36C4×C7×M16C2211yescharacteristic, complemented
37C7×#330C2214yescomplemented
38C7×#329C2214yescomplemented
39C7×#329C2212yescomplemented
40C7×#329C2212yescomplemented
41#18995C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C2×C7×#41D411yescharacteristic
44C2×C7×#41D411yescharacteristic
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C23×C4×C7—21no—
48C7×AGL(1,ℤ/8)—44no—
49C7×AGL(1,ℤ/8)—44no—
50C2×C7×SD16—22no—
51C2×C7×SD16—22no—
52C2×C7×SD16—24no—
53C2×C7×D8—22no—
54C2×C7×D8—22no—
55C2×C7×D8—24no—
56C2×C7×M16—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D411yescharacteristic
59C7×#127—21no—
60C7×#127—21no—
61C7×#127—22no—
62C7×#127D411yescharacteristic
63C7×#127C2312yes—
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#125—22no—
68C7×#125C2312yes—
69C7×#124—21no—
70C7×#124—22no—
71C7×#124—21no—
72C7×#124C2312yes—
73C7×#124C2311yescharacteristic
74C7×#124C2311yescharacteristic
75C7×#123—22no—
76C7×#123—21no—
77C7×#123—22no—
78C7×#123—21no—
79C7×#123—22no—
80C7×#123—22no—
81C7×#123—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8D412yes—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—22no—
94C7×#121D412yes—
95C7×#120—21no—
96C7×#120—22no—
97C7×#120—21no—
98C7×#120—22no—
99C4×C7×Q8D412yes—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C4×C7×D4D412yes—
105C7×#117—21no—
106C7×#117—22no—
107C2×C42×C7—21no—
108C2×C42×C7C2311yescharacteristic
109C7×C4⋊C8—22no—
110C7×#102—24no—
111C7×#102—24no—
112C7×C22⋊C8—22no—
113C7×#97C2314yes—
114C7×#97C2312yes—
115C4×C7×C8—21no—
116#3529C2×C711yescharacteristic, complemented
117#3217C2×C712yescomplemented
118#3151C2×C711yescharacteristic, complemented
119#1657C2×C712yescomplemented
120#1656C2×C714yescomplemented
121#1655C2×C712yescomplemented
122#1536C2×C711yescharacteristic, complemented
123#1529C2×C712yescomplemented
124C24×C7—21no—
125C24×C7—41no—
126C7×#41—42no—
127C7×#41—42no—
128C7×#41—82no—
129C2×C7×Q8—21no—
130C2×C7×Q8C2×D412yes—
131C2×C7×Q8C2×D411yescharacteristic
132C2×C7×Q8C2×D411yescharacteristic
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—22no—
141C2×C7×D4—24no—
142C2×C7×D4—22no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4—42no—
146C2×C7×D4—41no—
147C2×C7×D4—84no—
148C2×C7×D4C2×D412yes—
149C2×C7×D4C2×D411yescharacteristic
150C2×C7×D4C2×D411yescharacteristic
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—82no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C7×SD16—44no—
161C7×SD16—44no—
162C7×SD16—84no—
163C7×D8—44no—
164C7×D8—44no—
165C7×D8—84no—
166C7×M16—24no—
167C2×C7×C8—22no—
168C2×C7×C8—24no—
169C7×C4⋊C4—22no—
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⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C22⋊C4—21no—
180C7×C22⋊C4—21no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—22no—
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D412yes—
197C42×C7C2411yescharacteristic
198C2×AGL(1,ℤ/8)—21no—
199C2×AGL(1,ℤ/8)—21no—
200#563—21no—
201#563C22×C711yescharacteristic, complemented
202#562—22no—
203#562C22×C712yescomplemented
204H(3,𝔽4)C22×C711yescharacteristic, complemented
205#560—21no—
206#560—22no—
207#532—21no—
208#532C22×C711yescharacteristic
209C2×#127C22×C711yescharacteristic, complemented
210#529—22no—
211#529C22×C712yes—
212#525—21no—
213#525—22no—
214#496C22×C712yes—
215#496C22×C714yes—
216#496C22×C712yescomplemented
217#493—21no—
218#492—21no—
219#490C22×C714yescomplemented
220#460—22no—
221#459—22no—
222#450—22no—
223#447—22no—
224#432C22×C712yescomplemented
225C4×M16C22×C711yescharacteristic, complemented
226#330C22×C714yescomplemented
227#329C22×C712yescomplemented
228#329C22×C712yescomplemented
229#329C22×C714yescomplemented
230C23×C7—21no—
231C23×C7—22no—
232C23×C7—21no—
233C23×C7—42no—
234C23×C7—82no—
235C23×C7—84no—
236C23×C7—81no—
237C23×C7—82no—
238C23×C7#12011yescharacteristic
239C7×Q8—41no—
240C7×Q8—41no—
241C7×Q8—42no—
242C7×Q8—81no—
243C7×D4—41no—
244C7×D4—41no—
245C7×D4—42no—
246C7×D4—42no—
247C7×D4—42no—
248C7×D4—44no—
249C7×D4—44no—
250C7×D4—82no—
251C7×D4—84no—
252C7×D4—82no—
253C7×D4—81no—
254C7×D4—81no—
255C7×D4—82no—
256C7×D4—82no—
257C7×D4—82no—
258C7×D4—81no—
259C7×D4—82no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—44no—
269C2×C4×C7—42no—
270C2×C4×C7—82no—
271C2×C4×C7—82no—
272C2×C4×C7—81no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12011yescharacteristic
275C2×C4×C7C22×D412yes—
276C2×C4×C7C22×D411yescharacteristic
277C7×C8—24no—
278C7×C8—44no—
279C2×#41—21no—
280C2×#41C7×D411yescharacteristic
281C2×#41C7×D411yescharacteristic
282C22×D4—21no—
283C22×D4—22no—
284C23×C4—21no—
285AGL(1,ℤ/8)—44no—
286AGL(1,ℤ/8)—44no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—24no—
290C2×D8—22no—
291C2×D8—22no—
292C2×D8—24no—
293C2×M16—21no—
294C4⋊Q8—21no—
295C4⋊Q8C7×D411yescharacteristic
296#127—21no—
297#127—22no—
298#127—21no—
299#127C7×D411yescharacteristic
300#127C23×C712yes—
301#126—22no—
302#126—22no—
303#126—22no—
304#125—22no—
305#125C23×C712yes—
306#124—21no—
307#124—22no—
308#124—21no—
309#124C23×C712yes—
310#124C23×C711yescharacteristic
311#124C23×C711yescharacteristic
312#123—22no—
313#123—21no—
314#123—22no—
315#123—21no—
316#123—22no—
317#123—22no—
318#123—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8C7×D412yes—
324#121—22no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121—22no—
330#121—22no—
331#121C7×D412yes—
332#120—21no—
333#120—22no—
334#120—21no—
335#120—22no—
336C4×Q8C7×D412yes—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4C7×D412yes—
342#117—21no—
343#117—22no—
344C2×C42—21no—
345C2×C42C23×C711yescharacteristic
346C4⋊C8—22no—
347#102—24no—
348#102—24no—
349C22⋊C8—22no—
350#97C23×C714yes—
351#97C23×C712yes—
352C4×C8—21no—
353C22×C7—21no—
354C22×C7—22no—
355C22×C7—41no—
356C22×C7—42no—
357C22×C7—41no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—84no—
363C22×C7—82no—
364C22×C7—84no—
365C22×C7—164no—
366C22×C7C2×#12011yescharacteristic
367C4×C7—21no—
368C4×C7—22no—
369C4×C7—22no—
370C4×C7—41no—
371C4×C7—42no—
372C4×C7—41no—
373C4×C7—42no—
374C4×C7—42no—
375C4×C7#45312yes—
376C24—21no—
377C24—41no—
378#41—42no—
379#41—42no—
380#41—82no—
381C2×Q8—21no—
382C2×Q8C2×C7×D412yes—
383C2×Q8C2×C7×D411yescharacteristic
384C2×Q8C2×C7×D411yescharacteristic
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—84no—
400C2×D4C2×C7×D412yes—
401C2×D4C2×C7×D411yescharacteristic
402C2×D4C2×C7×D411yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—82no—
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D412yes—
412SD16—44no—
413SD16—44no—
414SD16—84no—
415D8—44no—
416D8—44no—
417D8—84no—
418M16—24no—
419C2×C8—22no—
420C2×C8—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4C2×C7×D412yes—
430C4⋊C4C2×C7×D412yes—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42C2×C7×D411yescharacteristic
448C42C2×C7×D412yes—
449C42C24×C711yescharacteristic
450C2×C7—41no—
451C2×C7—81no—
452C2×C7—81no—
453C2×C7—82no—
454C2×C7—162no—
455C2×C7C2×#45311yescharacteristic
456C2×C7#302112yes—
457C23—21no—
458C23—22no—
459C23—21no—
460C23—42no—
461C23—82no—
462C23—84no—
463C23—81no—
464C23—82no—
465C23C7×#12011yescharacteristic
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—81no—
470D4—41no—
471D4—41no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—44no—
476D4—44no—
477D4—82no—
478D4—84no—
479D4—82no—
480D4—81no—
481D4—81no—
482D4—82no—
483D4—82no—
484D4—82no—
485D4—81no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4C7×#12012yes—
501C2×C4C7×#12011yescharacteristic
502C2×C4C22×C7×D412yes—
503C2×C4C22×C7×D411yescharacteristic
504C8—24no—
505C8—44no—
506C7#1899511yescharacteristic, direct product factor, Sylow
507C22—21no—
508C22—22no—
509C22—41no—
510C22—42no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—84no—
517C22—82no—
518C22—84no—
519C22—164no—
520C22C2×C7×#12011yescharacteristic
521C4—21no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4C7×#45312yes—
530C2—41no—
531C2—81no—
532C2—82no—
533C2—81no—
534C2—162no—
535C2C2×C7×#45311yescharacteristic
536C2C7×#302112yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
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×#3529C2semidirect
2. C7×#3217C2semidirect
3. C7×#3151C2semidirect
4. C7×#1657C2semidirect
5. C7×#1656C2semidirect
6. C7×#1655C2semidirect
7. C7×#1536C2semidirect
8. C7×#1529C2semidirect
12. C7×#563C22semidirect
14. C7×#562C22semidirect
15. C7×H(3,𝔽4)C22semidirect
20. C2×C7×#127C22semidirect
27. C7×#496C22semidirect
30. C7×#490C22semidirect
35. C7×#432C22semidirect
36. C4×C7×M16C22semidirect
37. C7×#330C22semidirect
38. C7×#329C22semidirect
39. C7×#329C22semidirect
40. C7×#329C22semidirect
41. #18995C7direct (N × H)
116. #3529C2×C7semidirect
117. #3217C2×C7semidirect
118. #3151C2×C7semidirect
119. #1657C2×C7semidirect
120. #1656C2×C7semidirect
121. #1655C2×C7semidirect
122. #1536C2×C7semidirect
123. #1529C2×C7semidirect
201. #563C22×C7semidirect
203. #562C22×C7semidirect
204. H(3,𝔽4)C22×C7semidirect
209. C2×#127C22×C7semidirect
216. #496C22×C7semidirect
219. #490C22×C7semidirect
224. #432C22×C7semidirect
225. C4×M16C22×C7semidirect
226. #330C22×C7semidirect
227. #329C22×C7semidirect
228. #329C22×C7semidirect
229. #329C22×C7semidirect
506. C7#18995direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35292
C7×#32172
C7×#31512
C7×#16572
C7×#16562
C7×#16552
C7×#15362
C7×#15292
#189957