Finite Group Databasebeta

← C7×#60520 #13237736

Subgroups — C7×#60520 #13237736

Order 1792.

2170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C7×#3428C211yesmaximal, characteristic, complemented
3C7×#3099C218yesmaximal, complemented
4C7×#3098C218yesmaximal, complemented
5C7×#3095C214yesmaximal, complemented
6C7×#3094C214yesmaximal, complemented
7C2×C7×#428C212yesmaximal, complemented
8C7×#2896C212yesmaximal, complemented
9C7×#2879C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C22×C7×M16C2212yes—
13C7×#537—24no—
14C7×#537C2218yes—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8C2212yes—
17C2×C7×#121—21no—
18C2×C7×#121—22no—
19C2×C7×#121C2212yes—
20C7×#520C2214yescomplemented
21C7×#520C2214yescomplemented
22C7×#519—21no—
23C7×#518—21no—
24C7×#518—22no—
25C4×C7×#41—24no—
26C2×C7×#117C2212yes—
27C2×C7×#117C2214yes—
28C22×C7×C4⋊C4C2211yescharacteristic
29C7×#485C22116yescomplemented
30C7×#484C2218yescomplemented
31C7×#483C2218yescomplemented
32C7×#482C22116yescomplemented
33C7×#481C2218yescomplemented
34C7×#480C2218yescomplemented
35C7×#428C2218yes—
36C7×#428C2218yescomplemented
37C2×C7×#107C2214yescomplemented
38C2×C7×#106C2214yescomplemented
39C7×#419C2214yes—
40C7×#418C2214yes—
41C7×#417C2214yescomplemented
42C7×#417C2214yescomplemented
43C7×#416C2218yescomplemented
44C7×#408C2218yescomplemented
45C7×#407C2214yes—
46#60520C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—28no—
49C2×C7×#41—28no—
50C2×C7×#41D414yes—
51C2×C7×#41C2314yes—
52C2×C7×#41C2314yes—
53C22×C7×Q8D411yescharacteristic
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4D412yes—
58C22×C7×D4D411yescharacteristic
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7—21no—
61C23×C4×C7—21no—
62C23×C4×C7C2312yes—
63C23×C4×C7C2311yescharacteristic
64C2×C7×M16—28no—
65C2×C7×M16C2314yes—
66C2×C7×M16C2314yes—
67C22×C7×C8C2314yes—
68C7×C22⋊Q8—28no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8C2318yes—
71C7×#121—28no—
72C7×#121—216no—
73C7×#121—24no—
74C7×#121C2318yes—
75C4×C7×Q8—24no—
76C4×C7×Q8—24no—
77C4×C7×D4—28no—
78C4×C7×D4—24no—
79C4×C7×D4—28no—
80C4×C7×D4—24no—
81C7×#117—24no—
82C7×#117—22no—
83C7×#117—24no—
84C7×#117—22no—
85C7×#117C2318yes—
86C7×#117C2314yes—
87C7×#117C2314yes—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4C2314yes—
92C2×C7×C4⋊C4C2314yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2314yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C22⋊C4—24no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4—22no—
101C2×C42×C7—24no—
102C2×C42×C7—22no—
103C7×#107C23116yes—
104C7×#106C23116yes—
105C7×#103C2318yes—
106C7×#103C2318yes—
107C7×#102C2318yes—
108C7×#102C2318yes—
109C7×C22⋊C8C2318yes—
110C7×C22⋊C8C2318yes—
111C2×#537C2×C711yescharacteristic
112#3428C2×C711yescharacteristic, complemented
113#3099C2×C718yescomplemented
114#3098C2×C718yescomplemented
115#3095C2×C714yescomplemented
116#3094C2×C714yescomplemented
117C2×#428C2×C712yescomplemented
118#2896C2×C712yescomplemented
119#2879C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7C2×D412yes—
122C24×C7C2×D411yescharacteristic
123C7×#41—216no—
124C7×#41—28no—
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41—48no—
129C2×C7×Q8—24no—
130C2×C7×Q8—21no—
131C2×C7×Q8—24no—
132C2×C7×Q8C2×D414yes—
133C2×C7×Q8C2412yes—
134C2×C7×D4—28no—
135C2×C7×D4—24no—
136C2×C7×D4—28no—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—24no—
140C2×C7×D4—44no—
141C2×C7×D4—84no—
142C2×C7×D4C2×D418yes—
143C2×C7×D4C2×D414yes—
144C2×C7×D4C2412yes—
145C22×C4×C7—28no—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—24no—
154C22×C4×C7—44no—
155C22×C4×C7—82no—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7#4118yes—
161C22×C4×C7C2414yes—
162C22×C4×C7C2412yes—
163C22×C4×C7C2411yescharacteristic
164C7×M16—28no—
165C7×M16—28no—
166C2×C7×C8—28no—
167C2×C7×C8C2418yes—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4C2414yes—
176C7×C4⋊C4C2414yes—
177C7×C4⋊C4C2414yes—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—28no—
182C7×C22⋊C4—28no—
183C42×C7—24no—
184C42×C7—28no—
185C42×C7—24no—
186C22×#41C22×C711yescharacteristic
187C22×#41C22×C711yescharacteristic
188C22×M16C22×C712yes—
189#537—24no—
190#537C22×C718yes—
191C2×C22⋊Q8—21no—
192C2×C22⋊Q8C22×C712yes—
193C2×#121—21no—
194C2×#121—22no—
195C2×#121C22×C712yes—
196#520C22×C714yescomplemented
197#520C22×C714yescomplemented
198#519—21no—
199#518—21no—
200#518—22no—
201C4×#41—24no—
202C2×#117C22×C714yes—
203C2×#117C22×C712yes—
204C22×C4⋊C4C22×C711yescharacteristic
205#485C22×C7116yescomplemented
206#484C22×C718yescomplemented
207#483C22×C718yescomplemented
208#482C22×C7116yescomplemented
209#481C22×C718yescomplemented
210#480C22×C718yescomplemented
211#428C22×C718yes—
212#428C22×C718yescomplemented
213C2×#107C22×C714yescomplemented
214C2×#106C22×C714yescomplemented
215#419C22×C714yes—
216#418C22×C714yes—
217#417C22×C714yescomplemented
218#417C22×C714yescomplemented
219#416C22×C718yescomplemented
220#408C22×C718yescomplemented
221#407C22×C714yes—
222C23×C7—24no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7#12314yes—
232C23×C7C22×D414yes—
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D412yes—
235C7×Q8—28no—
236C7×Q8—24no—
237C7×Q8—42no—
238C7×D4—216no—
239C7×D4—28no—
240C7×D4—24no—
241C7×D4—48no—
242C7×D4—42no—
243C7×D4—88no—
244C2×C4×C7—24no—
245C2×C4×C7—28no—
246C2×C4×C7—28no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—44no—
252C2×C4×C7—42no—
253C2×C4×C7—48no—
254C2×C4×C7—82no—
255C2×C4×C7—84no—
256C2×C4×C7—82no—
257C2×C4×C7—84no—
258C2×C4×C7—84no—
259C2×C4×C7#12314yes—
260C2×C4×C7#12318yes—
261C2×C4×C7C22×D414yes—
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D412yes—
264C2×C4×C7C2×#4114yes—
265C2×C4×C7C2×#4114yes—
266C2×C4×C7C2×#4114yes—
267C2×C4×C7C2511yescharacteristic
268C7×C8—28no—
269C2×#41—22no—
270C2×#41—28no—
271C2×#41—28no—
272C2×#41C7×D414yes—
273C2×#41C23×C714yes—
274C2×#41C23×C714yes—
275C22×Q8C7×D411yescharacteristic
276C22×Q8C23×C711yescharacteristic
277C22×D4—21no—
278C22×D4—22no—
279C22×D4C7×D412yes—
280C22×D4C7×D411yescharacteristic
281C22×D4C23×C711yescharacteristic
282C23×C4—21no—
283C23×C4—21no—
284C23×C4C23×C712yes—
285C23×C4C23×C711yescharacteristic
286C2×M16—28no—
287C2×M16C23×C714yes—
288C2×M16C23×C714yes—
289C22×C8C23×C714yes—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8C23×C718yes—
293#121—28no—
294#121—216no—
295#121—24no—
296#121C23×C718yes—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×D4—28no—
300C4×D4—24no—
301C4×D4—28no—
302C4×D4—24no—
303#117—24no—
304#117—22no—
305#117—24no—
306#117—22no—
307#117C23×C714yes—
308#117C23×C714yes—
309#117C23×C718yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C23×C712yes—
314C2×C4⋊C4C23×C714yes—
315C2×C4⋊C4C23×C712yes—
316C2×C4⋊C4C23×C712yes—
317C2×C4⋊C4C23×C714yes—
318C2×C4⋊C4C23×C714yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—24no—
324C2×C42—22no—
325#107C23×C7116yes—
326#106C23×C7116yes—
327#103C23×C718yes—
328#103C23×C718yes—
329#102C23×C718yes—
330#102C23×C718yes—
331C22⋊C8C23×C718yes—
332C22⋊C8C23×C718yes—
333C22×C7—21no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7—21no—
337C22×C7—24no—
338C22×C7—24no—
339C22×C7—24no—
340C22×C7—44no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—84no—
345C22×C7—84no—
346C22×C7C2×#12312yes—
347C22×C7C2×#12314yes—
348C22×C7C23×D411yescharacteristic
349C4×C7—24no—
350C4×C7—44no—
351C4×C7—44no—
352C4×C7—44no—
353C4×C7—42no—
354C4×C7C2×#12312yes—
355C4×C7C2×#12314yes—
356C4×C7C22×#4112yes—
357C24—21no—
358C24C2×C7×D412yes—
359C24C2×C7×D411yescharacteristic
360#41—216no—
361#41—28no—
362#41—28no—
363#41—28no—
364#41—28no—
365#41—48no—
366C2×Q8—24no—
367C2×Q8—21no—
368C2×Q8—24no—
369C2×Q8C2×C7×D414yes—
370C2×Q8C24×C712yes—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—21no—
376C2×D4—24no—
377C2×D4—44no—
378C2×D4—84no—
379C2×D4C2×C7×D418yes—
380C2×D4C2×C7×D414yes—
381C2×D4C24×C712yes—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4—82no—
393C22×C4C2×C7×D414yes—
394C22×C4C2×C7×D412yes—
395C22×C4C2×C7×D412yes—
396C22×C4C2×C7×D411yescharacteristic
397C22×C4C7×#4118yes—
398C22×C4C24×C714yes—
399C22×C4C24×C712yes—
400C22×C4C24×C711yescharacteristic
401M16—28no—
402M16—28no—
403C2×C8—28no—
404C2×C8C24×C718yes—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4C24×C714yes—
413C4⋊C4C24×C714yes—
414C4⋊C4C24×C714yes—
415C22⋊C4—24no—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C42—24no—
421C42—28no—
422C42—24no—
423C2×C7—22no—
424C2×C7—24no—
425C2×C7—44no—
426C2×C7—82no—
427C2×C7C22×#12311yescharacteristic
428C2×C7#358511yescharacteristic
429C2×C7#358611yescharacteristic
430C23—24no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23C7×#12314yes—
440C23C22×C7×D414yes—
441C23C22×C7×D411yescharacteristic
442C23C22×C7×D412yes—
443Q8—28no—
444Q8—24no—
445Q8—42no—
446D4—216no—
447D4—24no—
448D4—28no—
449D4—48no—
450D4—42no—
451D4—88no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—48no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C7×#12314yes—
468C2×C4C7×#12318yes—
469C2×C4C22×C7×D414yes—
470C2×C4C22×C7×D411yescharacteristic
471C2×C4C22×C7×D412yes—
472C2×C4C2×C7×#4114yes—
473C2×C4C2×C7×#4114yes—
474C2×C4C2×C7×#4114yes—
475C2×C4C25×C711yescharacteristic
476C8—28no—
477C7#6052011yescharacteristic, direct product factor, Sylow
478C22—21no—
479C22—22no—
480C22—22no—
481C22—21no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—84no—
490C22—84no—
491C22C2×C7×#12312yes—
492C22C2×C7×#12314yes—
493C22C23×C7×D411yescharacteristic
494C4—24no—
495C4—44no—
496C4—44no—
497C4—44no—
498C4—42no—
499C4C2×C7×#12312yes—
500C4C2×C7×#12314yes—
501C4C22×C7×#4112yes—
502C2—22no—
503C2—24no—
504C2—44no—
505C2—82no—
506C2C22×C7×#12311yescharacteristic
507C2C7×#358511yescharacteristic
508C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C7×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#3428C2semidirect
3. C7×#3099C2semidirect
4. C7×#3098C2semidirect
5. C7×#3095C2semidirect
6. C7×#3094C2semidirect
7. C2×C7×#428C2semidirect
8. C7×#2896C2semidirect
9. C7×#2879C2semidirect
20. C7×#520C22semidirect
21. C7×#520C22semidirect
29. C7×#485C22semidirect
30. C7×#484C22semidirect
31. C7×#483C22semidirect
32. C7×#482C22semidirect
33. C7×#481C22semidirect
34. C7×#480C22semidirect
36. C7×#428C22semidirect
37. C2×C7×#107C22semidirect
38. C2×C7×#106C22semidirect
41. C7×#417C22semidirect
42. C7×#417C22semidirect
43. C7×#416C22semidirect
44. C7×#408C22semidirect
46. #60520C7direct (N × H)
112. #3428C2×C7semidirect
113. #3099C2×C7semidirect
114. #3098C2×C7semidirect
115. #3095C2×C7semidirect
116. #3094C2×C7semidirect
117. C2×#428C2×C7semidirect
118. #2896C2×C7semidirect
119. #2879C2×C7semidirect
196. #520C22×C7semidirect
197. #520C22×C7semidirect
205. #485C22×C7semidirect
206. #484C22×C7semidirect
207. #483C22×C7semidirect
208. #482C22×C7semidirect
209. #481C22×C7semidirect
210. #480C22×C7semidirect
212. #428C22×C7semidirect
213. C2×#107C22×C7semidirect
214. C2×#106C22×C7semidirect
217. #417C22×C7semidirect
218. #417C22×C7semidirect
219. #416C22×C7semidirect
220. #408C22×C7semidirect
477. C7#60520direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C7×#34282
C7×#30992
C7×#30982
C7×#30952
C7×#30942
C2×C7×#4282
C7×#28962
C7×#28792
#605207