Finite Group Databasebeta

← C7×#60621 #13237837

Subgroups — C7×#60621 #13237837

Order 1792.

2078 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic, complemented
2C7×#3449C211yesmaximal, characteristic
3C7×#3142C214yesmaximal, complemented
4C7×#3141C214yesmaximal, complemented
5C7×#3137C218yesmaximal, complemented
6C7×#3136C218yesmaximal, complemented
7C7×#2896C212yesmaximal, complemented
8C2×C7×#417C212yesmaximal, complemented
9C7×#2873C211yesmaximal, characteristic, complemented
10C22×C7×#41C2212yes—
11C22×C7×M16C2212yes—
12C7×#544—24no—
13C7×#544C2218yescomplemented
14C7×#537—21no—
15C7×#537—21no—
16C7×#536C2214yes—
17C7×#535C2214yes—
18C7×#534—22no—
19C7×#532C2214yes—
20C2×C7×C4⋊Q8—21no—
21C2×C7×C4⋊Q8C2212yescomplemented
22C2×C7×#125—21no—
23C2×C7×#125C2212yescomplemented
24C7×#525—24no—
25C2×C7×C22⋊Q8—22no—
26C2×C7×#117C2211yescharacteristic
27C2×C7×#117C2211yescharacteristic
28C22×C7×C4⋊C4C2211yescharacteristic
29C7×#491C2218yescomplemented
30C7×#490C2218yescomplemented
31C7×#489C22116yescomplemented
32C7×#488C22116yescomplemented
33C7×#487C2218yescomplemented
34C7×#486C2218yescomplemented
35C7×#419C2214yescomplemented
36C7×#418C2214yescomplemented
37C7×#417C2218yes—
38C7×#417C2214yescomplemented
39C7×#417C2218yescomplemented
40C7×#417C2214yescomplemented
41C7×#416C2218yes—
42C2×C7×#103C2214yescomplemented
43C2×C7×#102C2214yescomplemented
44C7×#405C2218yescomplemented
45C4×C7×M16C2214yescomplemented
46#60621C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—28no—
50C2×C7×#41—28no—
51C2×C7×#41C2314yes—
52C2×C7×#41C2314yes—
53C22×C7×Q8—22no—
54C22×C7×Q8C2312yes—
55C22×C7×D4—22no—
56C22×C7×D4C2312yes—
57C23×C4×C7—21no—
58C23×C4×C7—22no—
59C23×C4×C7C2311yescharacteristic
60C2×C7×M16—28no—
61C2×C7×M16C2314yes—
62C2×C7×M16C2314yes—
63C22×C7×C8C2314yes—
64C7×C4⋊Q8—28no—
65C7×C4⋊Q8—24no—
66C7×C4⋊Q8C2314yes—
67C7×C4⋊Q8C2318yes—
68C7×#127C2314yes—
69C7×#125—28no—
70C7×#125—24no—
71C7×#125C2318yes—
72C7×#124C2318yes—
73C7×#123—28no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—24no—
76C7×C22⋊Q8—216no—
77C7×#121—24no—
78C7×#121—24no—
79C7×#121—24no—
80C7×#121—24no—
81C7×#120—28no—
82C4×C7×D4—24no—
83C4×C7×D4—24no—
84C7×#117—22no—
85C7×#117D414yes—
86C7×#117C2314yes—
87C7×#117C2314yes—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4D411yescharacteristic
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2314yes—
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4D412yes—
101C2×C42×C7—21no—
102C2×C42×C7C2312yes—
103C7×#103C2318yes—
104C7×#103C23116yes—
105C7×#103C2318yes—
106C7×#102C2318yes—
107C7×#102C23116yes—
108C7×#102C2318yes—
109C7×#97C2318yes—
110C4×C7×C8C2318yes—
111C2×#544C2×C711yescharacteristic, complemented
112#3449C2×C711yescharacteristic
113#3142C2×C714yescomplemented
114#3141C2×C714yescomplemented
115#3137C2×C718yescomplemented
116#3136C2×C718yescomplemented
117#2896C2×C712yescomplemented
118C2×#417C2×C712yescomplemented
119#2873C2×C711yescharacteristic, complemented
120C24×C7—22no—
121C24×C7C2×D411yescharacteristic
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—28no—
125C7×#41—28no—
126C7×#41—48no—
127C7×#41—48no—
128C2×C7×Q8—22no—
129C2×C7×Q8—28no—
130C2×C7×Q8—24no—
131C2×C7×Q8—24no—
132C2×C7×Q8—84no—
133C2×C7×Q8C2414yes—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—22no—
137C2×C7×D4—24no—
138C2×C7×D4—24no—
139C2×C7×D4—24no—
140C2×C7×D4—24no—
141C2×C7×D4—48no—
142C2×C7×D4C2414yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7—48no—
151C22×C4×C7—82no—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7#4118yes—
157C22×C4×C7C2411yescharacteristic
158C22×C4×C7C2412yes—
159C7×M16—28no—
160C7×M16—28no—
161C2×C7×C8—28no—
162C2×C7×C8C2418yes—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4C2×D414yes—
173C7×C4⋊C4C2×D414yes—
174C7×C4⋊C4C2414yes—
175C7×C4⋊C4C2412yes—
176C7×C4⋊C4C2412yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4C2×D418yes—
181C42×C7—24no—
182C42×C7C2×D414yes—
183C42×C7C2414yes—
184C22×#41C22×C712yes—
185C22×M16C22×C712yes—
186#544—24no—
187#544C22×C718yescomplemented
188#537—21no—
189#537—21no—
190#536C22×C714yes—
191#535C22×C714yes—
192#534—22no—
193#532C22×C714yes—
194C2×C4⋊Q8—21no—
195C2×C4⋊Q8C22×C712yescomplemented
196C2×#125—21no—
197C2×#125C22×C712yescomplemented
198#525—24no—
199C2×C22⋊Q8—22no—
200C2×#117C22×C711yescharacteristic
201C2×#117C22×C711yescharacteristic
202C22×C4⋊C4C22×C711yescharacteristic
203#491C22×C718yescomplemented
204#490C22×C718yescomplemented
205#489C22×C7116yescomplemented
206#488C22×C7116yescomplemented
207#487C22×C718yescomplemented
208#486C22×C718yescomplemented
209#419C22×C714yescomplemented
210#418C22×C714yescomplemented
211#417C22×C718yes—
212#417C22×C714yescomplemented
213#417C22×C718yescomplemented
214#417C22×C714yescomplemented
215#416C22×C718yes—
216C2×#103C22×C714yescomplemented
217C2×#102C22×C714yescomplemented
218#405C22×C718yescomplemented
219C4×M16C22×C714yescomplemented
220C23×C7—24no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—44no—
224C23×C7—82no—
225C23×C7—82no—
226C23×C7#12414yes—
227C23×C7C22×D412yes—
228C23×C7C22×D411yescharacteristic
229C7×Q8—24no—
230C7×Q8—24no—
231C7×Q8—44no—
232C7×Q8—88no—
233C7×D4—24no—
234C7×D4—24no—
235C7×D4—48no—
236C7×D4—48no—
237C7×D4—44no—
238C7×D4—84no—
239C7×D4—84no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—44no—
250C2×C4×C7—48no—
251C2×C4×C7—48no—
252C2×C4×C7—84no—
253C2×C4×C7#12418yes—
254C2×C4×C7#12414yes—
255C2×C4×C7C22×D414yes—
256C2×C4×C7C22×D414yes—
257C2×C4×C7C22×D412yes—
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C2×#4114yes—
260C2×C4×C7C2×#4114yes—
261C2×C4×C7C2×#4114yes—
262C2×C4×C7C2511yescharacteristic
263C7×C8—28no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—28no—
267C2×#41—28no—
268C2×#41C23×C714yes—
269C2×#41C23×C714yes—
270C22×Q8—22no—
271C22×Q8C23×C712yes—
272C22×D4—22no—
273C22×D4C23×C712yes—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4C23×C711yescharacteristic
277C2×M16—28no—
278C2×M16C23×C714yes—
279C2×M16C23×C714yes—
280C22×C8C23×C714yes—
281C4⋊Q8—28no—
282C4⋊Q8—24no—
283C4⋊Q8C23×C714yes—
284C4⋊Q8C23×C718yes—
285#127C23×C714yes—
286#125—28no—
287#125—24no—
288#125C23×C718yes—
289#124C23×C718yes—
290#123—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—216no—
294#121—24no—
295#121—24no—
296#121—24no—
297#121—24no—
298#120—28no—
299C4×D4—24no—
300C4×D4—24no—
301#117—22no—
302#117C7×D414yes—
303#117C23×C714yes—
304#117C23×C714yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C7×D411yescharacteristic
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C23×C712yes—
313C2×C4⋊C4C23×C714yes—
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C4⋊C4C23×C711yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4C7×D412yes—
318C2×C42—21no—
319C2×C42C23×C712yes—
320#103C23×C7116yes—
321#103C23×C718yes—
322#103C23×C718yes—
323#102C23×C7116yes—
324#102C23×C718yes—
325#102C23×C718yes—
326#97C23×C718yes—
327C4×C8C23×C718yes—
328C22×C7—21no—
329C22×C7—22no—
330C22×C7—21no—
331C22×C7—22no—
332C22×C7—44no—
333C22×C7—44no—
334C22×C7—44no—
335C22×C7—44no—
336C22×C7—88no—
337C22×C7C2×#12414yes—
338C22×C7C2×#12412yes—
339C22×C7C23×D411yescharacteristic
340C4×C7—24no—
341C4×C7—24no—
342C4×C7—44no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7—44no—
346C4×C7C2×#12412yes—
347C4×C7C2×#12414yes—
348C4×C7C22×#4112yes—
349C24—22no—
350C24C2×C7×D411yescharacteristic
351#41—28no—
352#41—28no—
353#41—28no—
354#41—28no—
355#41—48no—
356#41—48no—
357C2×Q8—22no—
358C2×Q8—28no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—84no—
362C2×Q8C24×C714yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—48no—
371C2×D4C24×C714yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—82no—
381C22×C4C2×C7×D412yes—
382C22×C4C2×C7×D412yes—
383C22×C4C2×C7×D411yescharacteristic
384C22×C4C2×C7×D412yes—
385C22×C4C7×#4118yes—
386C22×C4C24×C711yescharacteristic
387C22×C4C24×C712yes—
388M16—28no—
389M16—28no—
390C2×C8—28no—
391C2×C8C24×C718yes—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4C2×C7×D414yes—
402C4⋊C4C2×C7×D414yes—
403C4⋊C4C24×C714yes—
404C4⋊C4C24×C712yes—
405C4⋊C4C24×C712yes—
406C22⋊C4—24no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4C2×C7×D418yes—
410C42—24no—
411C42C2×C7×D414yes—
412C42C24×C714yes—
413C2×C7—24no—
414C2×C7—22no—
415C2×C7—84no—
416C2×C7C22×#12411yescharacteristic
417C2×C7#358511yescharacteristic
418C2×C7#358611yescharacteristic
419C23—24no—
420C23—41no—
421C23—41no—
422C23—44no—
423C23—82no—
424C23—82no—
425C23C7×#12414yes—
426C23C22×C7×D412yes—
427C23C22×C7×D411yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—44no—
431Q8—88no—
432D4—24no—
433D4—24no—
434D4—48no—
435D4—48no—
436D4—44no—
437D4—84no—
438D4—84no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—84no—
452C2×C4C7×#12418yes—
453C2×C4C7×#12414yes—
454C2×C4C22×C7×D414yes—
455C2×C4C22×C7×D414yes—
456C2×C4C22×C7×D412yes—
457C2×C4C22×C7×D411yescharacteristic
458C2×C4C2×C7×#4114yes—
459C2×C4C2×C7×#4114yes—
460C2×C4C2×C7×#4114yes—
461C2×C4C25×C711yescharacteristic
462C8—28no—
463C7#6062111yescharacteristic, direct product factor, Sylow
464C22—21no—
465C22—22no—
466C22—21no—
467C22—22no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—88no—
473C22C2×C7×#12414yes—
474C22C2×C7×#12412yes—
475C22C23×C7×D411yescharacteristic
476C4—24no—
477C4—24no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4C2×C7×#12412yes—
483C4C2×C7×#12414yes—
484C4C22×C7×#4112yes—
485C2—24no—
486C2—22no—
487C2—84no—
488C2C22×C7×#12411yescharacteristic
489C2C7×#358511yescharacteristic
490C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC7×#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#544C2semidirect
3. C7×#3142C2semidirect
4. C7×#3141C2semidirect
5. C7×#3137C2semidirect
6. C7×#3136C2semidirect
7. C7×#2896C2semidirect
8. C2×C7×#417C2semidirect
9. C7×#2873C2semidirect
13. C7×#544C22semidirect
21. C2×C7×C4⋊Q8C22semidirect
23. C2×C7×#125C22semidirect
29. C7×#491C22semidirect
30. C7×#490C22semidirect
31. C7×#489C22semidirect
32. C7×#488C22semidirect
33. C7×#487C22semidirect
34. C7×#486C22semidirect
35. C7×#419C22semidirect
36. C7×#418C22semidirect
38. C7×#417C22semidirect
39. C7×#417C22semidirect
40. C7×#417C22semidirect
42. C2×C7×#103C22semidirect
43. C2×C7×#102C22semidirect
44. C7×#405C22semidirect
45. C4×C7×M16C22semidirect
46. #60621C7direct (N × H)
111. C2×#544C2×C7semidirect
113. #3142C2×C7semidirect
114. #3141C2×C7semidirect
115. #3137C2×C7semidirect
116. #3136C2×C7semidirect
117. #2896C2×C7semidirect
118. C2×#417C2×C7semidirect
119. #2873C2×C7semidirect
187. #544C22×C7semidirect
195. C2×C4⋊Q8C22×C7semidirect
197. C2×#125C22×C7semidirect
203. #491C22×C7semidirect
204. #490C22×C7semidirect
205. #489C22×C7semidirect
206. #488C22×C7semidirect
207. #487C22×C7semidirect
208. #486C22×C7semidirect
209. #419C22×C7semidirect
210. #418C22×C7semidirect
212. #417C22×C7semidirect
213. #417C22×C7semidirect
214. #417C22×C7semidirect
216. C2×#103C22×C7semidirect
217. C2×#102C22×C7semidirect
218. #405C22×C7semidirect
219. C4×M16C22×C7semidirect
463. C7#60621direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5442
C7×#34492
C7×#31422
C7×#31412
C7×#31372
C7×#31362
C7×#28962
C2×C7×#4172
C7×#28732
#606217