Finite Group Databasebeta

← C2×C7×#3167 #13237885

Subgroups — C2×C7×#3167 #13237885

Order 1792.

2286 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×C7×#532C211yesmaximal, characteristic
3C7×#3167C2116yesmaximal, direct product factor
4C2×C7×#498C211yesmaximal, characteristic, complemented
5C2×C7×#492C211yesmaximal, characteristic, complemented
6C2×C7×#487C211yesmaximal, characteristic, complemented
7C2×C7×#486C211yesmaximal, characteristic, complemented
8C2×C7×#465C214yesmaximal, complemented
9C2×C4×C7×SD16C214yesmaximal, complemented
10C22×C4×C7×C8C211yesmaximal, characteristic, complemented
11C22×C7×#41—21no—
12C22×C7×SD16C2214yes—
13C23×C7×C8C2212yes—
14C7×#533—24no—
15C7×#533C2218yescomplemented
16C7×#532—24no—
17C7×#532C2218yes—
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C7×C4⋊Q8C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C2×C7×#127C2211yescharacteristic
22C2×C7×#125—21no—
23C2×C7×#124—21no—
24C2×C7×C22⋊Q8—21no—
25C2×C7×C22⋊Q8C2212yes—
26C2×C7×#121—21no—
27C2×C7×#121C2212yes—
28C2×C4×C7×Q8—21no—
29C2×C4×C7×Q8C2212yes—
30C2×C4×C7×D4—21no—
31C2×C4×C7×D4C2212yes—
32C2×C7×#117—21no—
33C22×C42×C7C2211yescharacteristic
34C7×#498C2218yescomplemented
35C7×#492C2218yescomplemented
36C7×#487C2218yescomplemented
37C7×#486C2218yescomplemented
38C7×#465C22132yescomplemented
39C4×C7×SD16C22132yescomplemented
40C2×C7×#106C2214yes—
41C2×C7×#103C2214yes—
42C2×C7×#102C2214yes—
43C2×C4×C7×C8C2211yescharacteristic
44C2×C4×C7×C8C2212yes—
45C2×C4×C7×C8C2211yescharacteristic
46C2×C4×C7×C8C2218yescomplemented
47C2×#3167C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—28no—
49C2×C7×#41—48no—
50C22×C7×Q8—21no—
51C22×C7×Q8—21no—
52C22×C7×Q8C2312yes—
53C22×C7×D4—22no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4C2312yes—
57C23×C4×C7—22no—
58C23×C4×C7D411yescharacteristic
59C23×C4×C7D411yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C2×C7×SD16—216no—
62C2×C7×SD16C23116yes—
63C22×C7×C8—28no—
64C22×C7×C8C2312yes—
65C22×C7×C8C2318yes—
66C22×C7×C8C2312yes—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8C2314yes—
71C7×C4⋊Q8C2314yes—
72C7×C4⋊Q8C2314yes—
73C7×#127—22no—
74C7×#127C2314yes—
75C7×#125—28no—
76C7×#124—28no—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8—24no—
79C7×C22⋊Q8C2318yes—
80C7×#121—28no—
81C7×#121—24no—
82C7×#121C2318yes—
83C4×C7×Q8—28no—
84C4×C7×Q8—24no—
85C4×C7×Q8C2318yes—
86C4×C7×D4—28no—
87C4×C7×D4—24no—
88C4×C7×D4C2318yes—
89C7×#117—28no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4—22no—
101C2×C42×C7—22no—
102C2×C42×C7D411yescharacteristic
103C2×C42×C7D414yes—
104C2×C42×C7D411yescharacteristic
105C2×C42×C7C2311yescharacteristic
106C2×C42×C7C2314yes—
107C2×C42×C7C2311yescharacteristic
108C7×#106C23116yes—
109C7×#103C23116yes—
110C7×#102C23116yes—
111C4×C7×C8C2314yes—
112C4×C7×C8C2314yes—
113C4×C7×C8C2318yes—
114C2×#533C2×C711yescharacteristic, complemented
115C2×#532C2×C711yescharacteristic
116#3167C2×C7116yesdirect product factor
117C2×#498C2×C711yescharacteristic, complemented
118C2×#492C2×C711yescharacteristic, complemented
119C2×#487C2×C711yescharacteristic, complemented
120C2×#486C2×C711yescharacteristic, complemented
121C2×#465C2×C714yescomplemented
122C2×C4×SD16C2×C714yescomplemented
123C22×C4×C8C2×C711yescharacteristic, complemented
124C24×C7—22no—
125C24×C7C2×D411yescharacteristic
126C7×#41—432no—
127C2×C7×Q8—22no—
128C2×C7×Q8—24no—
129C2×C7×Q8—24no—
130C2×C7×Q8—28no—
131C2×C7×Q8—44no—
132C2×C7×Q8—44no—
133C2×C7×Q8C2414yes—
134C2×C7×D4—28no—
135C2×C7×D4—22no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—28no—
139C2×C7×D4—48no—
140C2×C7×D4—44no—
141C2×C7×D4—44no—
142C2×C7×D4C2414yes—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—28no—
149C22×C4×C7—48no—
150C22×C4×C7SD1618yescomplemented
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2412yes—
160C22×C4×C7C2411yescharacteristic
161C7×SD16—232no—
162C2×C7×C8—24no—
163C2×C7×C8—216no—
164C2×C7×C8#4118yescomplemented
165C2×C7×C8C2414yes—
166C2×C7×C8C2414yes—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—28no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4C2414yes—
176C7×C4⋊C4C2414yes—
177C7×C22⋊C4—28no—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—28no—
180C42×C7—28no—
181C42×C7C2×D414yes—
182C42×C7C2×D412yes—
183C42×C7C2×D414yes—
184C42×C7C2×D412yes—
185C42×C7C2412yes—
186C42×C7C2412yes—
187C22×#41—21no—
188C22×SD16C22×C714yes—
189C23×C8C22×C712yes—
190#533—24no—
191#533C22×C718yescomplemented
192#532—24no—
193#532C22×C718yes—
194C2×C4⋊Q8C22×C711yescharacteristic
195C2×C4⋊Q8C22×C711yescharacteristic
196C2×C4⋊Q8C22×C711yescharacteristic
197C2×#127C22×C711yescharacteristic
198C2×#125—21no—
199C2×#124—21no—
200C2×C22⋊Q8—21no—
201C2×C22⋊Q8C22×C712yes—
202C2×#121—21no—
203C2×#121C22×C712yes—
204C2×C4×Q8—21no—
205C2×C4×Q8C22×C712yes—
206C2×C4×D4—21no—
207C2×C4×D4C22×C712yes—
208C2×#117—21no—
209C22×C42C22×C711yescharacteristic
210#498C22×C718yescomplemented
211#492C22×C718yescomplemented
212#487C22×C718yescomplemented
213#486C22×C718yescomplemented
214#465C22×C7132yescomplemented
215C4×SD16C22×C7132yescomplemented
216C2×#106C22×C714yes—
217C2×#103C22×C714yes—
218C2×#102C22×C714yes—
219C2×C4×C8C22×C711yescharacteristic
220C2×C4×C8C22×C711yescharacteristic
221C2×C4×C8C22×C712yes—
222C2×C4×C8C22×C718yescomplemented
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—24no—
226C23×C7—44no—
227C23×C7—44no—
228C23×C7—44no—
229C23×C7C2×SD1614yes—
230C23×C7C22×D412yes—
231C23×C7C22×D411yescharacteristic
232C7×Q8—28no—
233C7×Q8—44no—
234C7×Q8—48no—
235C7×Q8—48no—
236C7×D4—28no—
237C7×D4—416no—
238C7×D4—44no—
239C7×D4—48no—
240C7×D4—48no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—28no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—44no—
250C2×C4×C7—44no—
251C2×C4×C7—44no—
252C2×C4×C7—416no—
253C2×C4×C7C2×SD1614yes—
254C2×C4×C7C2×SD1614yes—
255C2×C4×C7C2×SD16116yescomplemented
256C2×C4×C7#13514yes—
257C2×C4×C7C22×D412yes—
258C2×C4×C7C22×D412yes—
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C22×D412yes—
261C2×C4×C7C22×D412yes—
262C2×C4×C7C22×D412yes—
263C2×C4×C7C2×#4114yes—
264C2×C4×C7C2511yescharacteristic
265C7×C8—24no—
266C7×C8C2×#4118yescomplemented
267C2×#41—28no—
268C2×#41—48no—
269C22×Q8—21no—
270C22×Q8—21no—
271C22×Q8C23×C712yes—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C23×C712yes—
276C23×C4—22no—
277C23×C4C7×D411yescharacteristic
278C23×C4C7×D411yescharacteristic
279C23×C4C23×C711yescharacteristic
280C2×SD16—216no—
281C2×SD16C23×C7116yes—
282C22×C8—28no—
283C22×C8C23×C712yes—
284C22×C8C23×C718yes—
285C22×C8C23×C712yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8C23×C714yes—
290C4⋊Q8C23×C714yes—
291C4⋊Q8C23×C714yes—
292#127—22no—
293#127C23×C714yes—
294#125—28no—
295#124—28no—
296C22⋊Q8—28no—
297C22⋊Q8—24no—
298C22⋊Q8C23×C718yes—
299#121—28no—
300#121—24no—
301#121C23×C718yes—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8C23×C718yes—
305C4×D4—28no—
306C4×D4—24no—
307C4×D4C23×C718yes—
308#117—28no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C23×C712yes—
316C2×C4⋊C4C23×C712yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C42—22no—
321C2×C42C7×D411yescharacteristic
322C2×C42C7×D411yescharacteristic
323C2×C42C7×D414yes—
324C2×C42C23×C711yescharacteristic
325C2×C42C23×C711yescharacteristic
326C2×C42C23×C714yes—
327#106C23×C7116yes—
328#103C23×C7116yes—
329#102C23×C7116yes—
330C4×C8C23×C718yes—
331C4×C8C23×C714yes—
332C4×C8C23×C714yes—
333C22×C7—28no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7—44no—
337C22×C7—44no—
338C22×C7—44no—
339C22×C7—88no—
340C22×C7#53212yes—
341C22×C7C22×SD1614yes—
342C22×C7C22×SD1612yes—
343C22×C7C2×#13512yes—
344C22×C7C23×D411yescharacteristic
345C4×C7—22no—
346C4×C7—22no—
347C4×C7—44no—
348C4×C7—44no—
349C4×C7—44no—
350C4×C7#53212yes—
351C4×C7C22×SD1614yes—
352C4×C7C22×SD1614yes—
353C4×C7C2×#13514yes—
354C4×C7C22×#4112yes—
355C24—22no—
356C24C2×C7×D411yescharacteristic
357#41—432no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—28no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8C24×C714yes—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—28no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4C24×C714yes—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—48no—
381C22×C4C7×SD1618yescomplemented
382C22×C4C2×C7×D411yescharacteristic
383C22×C4C2×C7×D412yes—
384C22×C4C2×C7×D411yescharacteristic
385C22×C4C2×C7×D414yes—
386C22×C4C2×C7×D411yescharacteristic
387C22×C4C2×C7×D411yescharacteristic
388C22×C4C2×C7×D414yes—
389C22×C4C2×C7×D411yescharacteristic
390C22×C4C24×C712yes—
391C22×C4C24×C711yescharacteristic
392SD16—232no—
393C2×C8—24no—
394C2×C8—216no—
395C2×C8C7×#4118yescomplemented
396C2×C8C24×C714yes—
397C2×C8C24×C714yes—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4C24×C714yes—
407C4⋊C4C24×C714yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C42—28no—
412C42C2×C7×D414yes—
413C42C2×C7×D412yes—
414C42C2×C7×D414yes—
415C42C2×C7×D412yes—
416C42C24×C712yes—
417C42C24×C712yes—
418C2×C7—24no—
419C2×C7—84no—
420C2×C7#316714yesdirect product factor
421C2×C7C2×#53211yescharacteristic
422C2×C7C23×SD1611yescharacteristic
423C2×C7C22×#13511yescharacteristic
424C23—22no—
425C23—22no—
426C23—24no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23C2×C7×SD1614yes—
431C23C22×C7×D412yes—
432C23C22×C7×D411yescharacteristic
433Q8—28no—
434Q8—44no—
435Q8—48no—
436Q8—48no—
437D4—28no—
438D4—416no—
439D4—44no—
440D4—48no—
441D4—48no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—416no—
454C2×C4C2×C7×SD1614yes—
455C2×C4C2×C7×SD1614yes—
456C2×C4C2×C7×SD16116yescomplemented
457C2×C4C7×#13514yes—
458C2×C4C22×C7×D412yes—
459C2×C4C22×C7×D412yes—
460C2×C4C22×C7×D411yescharacteristic
461C2×C4C22×C7×D412yes—
462C2×C4C22×C7×D412yes—
463C2×C4C22×C7×D412yes—
464C2×C4C2×C7×#4114yes—
465C2×C4C25×C711yescharacteristic
466C8—24no—
467C8C2×C7×#4118yescomplemented
468C7C2×#316711yescharacteristic, direct product factor, Sylow
469C22—28no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—88no—
476C22C7×#53212yes—
477C22C22×C7×SD1614yes—
478C22C22×C7×SD1612yes—
479C22C2×C7×#13512yes—
480C22C23×C7×D411yescharacteristic
481C4—22no—
482C4—22no—
483C4—44no—
484C4—44no—
485C4—44no—
486C4C7×#53212yes—
487C4C22×C7×SD1614yes—
488C4C22×C7×SD1614yes—
489C4C2×C7×#13514yes—
490C4C22×C7×#4112yes—
491C2—24no—
492C2—84no—
493C2C7×#316714yesdirect product factor
494C2C2×C7×#53211yescharacteristic
495C2C23×C7×SD1611yescharacteristic
496C2C22×C7×#13511yescharacteristic
subgroupG/H
centerC22×C4×C7C2×D4
commutator, Frattini, cosocleC2×C4C25×C7
socleC23×C7C22×D4
prime radicalC2×C7×#532C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
3. C7×#3167C2direct (N × H)
4. C2×C7×#498C2semidirect
5. C2×C7×#492C2semidirect
6. C2×C7×#487C2semidirect
7. C2×C7×#486C2semidirect
8. C2×C7×#465C2semidirect
9. C2×C4×C7×SD16C2semidirect
10. C22×C4×C7×C8C2semidirect
15. C7×#533C22semidirect
34. C7×#498C22semidirect
35. C7×#492C22semidirect
36. C7×#487C22semidirect
37. C7×#486C22semidirect
38. C7×#465C22semidirect
39. C4×C7×SD16C22semidirect
46. C2×C4×C7×C8C22semidirect
47. C2×#3167C7direct (N × H)
114. C2×#533C2×C7semidirect
116. #3167C2×C7direct (N × H)
117. C2×#498C2×C7semidirect
118. C2×#492C2×C7semidirect
119. C2×#487C2×C7semidirect
120. C2×#486C2×C7semidirect
121. C2×#465C2×C7semidirect
122. C2×C4×SD16C2×C7semidirect
123. C22×C4×C8C2×C7semidirect
150. C22×C4×C7SD16semidirect
164. C2×C7×C8#41semidirect
191. #533C22×C7semidirect
210. #498C22×C7semidirect
211. #492C22×C7semidirect
212. #487C22×C7semidirect
213. #486C22×C7semidirect
214. #465C22×C7semidirect
215. C4×SD16C22×C7semidirect
222. C2×C4×C8C22×C7semidirect
255. C2×C4×C7C2×SD16semidirect
266. C7×C8C2×#41semidirect
381. C22×C4C7×SD16semidirect
395. C2×C8C7×#41semidirect
420. C2×C7#3167direct (N × H)
456. C2×C4C2×C7×SD16semidirect
467. C8C2×C7×#41semidirect
468. C7C2×#3167direct (N × H)
493. C2C7×#3167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×C7×#5322
C7×#31672
C2×C7×#4982
C2×C7×#4922
C2×C7×#4872
C2×C7×#4862
C2×C7×#4652
C2×C4×C7×SD162
C22×C4×C7×C82
C2×#31677