Finite Group Databasebeta

← C2×C7×#1615 #13195322

Subgroups — C2×C7×#1615 #13195322

Order 1792.

1016 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
1C2×C7×#539C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic, complemented
3C2×C7×#481C211yesmaximal, characteristic
4C2×C7×#400C211yesmaximal, characteristic, complemented
5C7×#1615C218yesmaximal, direct product factor
6C2×C7×#328C211yesmaximal, characteristic, complemented
7C2×C7×#327C211yesmaximal, characteristic
8C2×C7×#324C211yesmaximal, characteristic, complemented
9C7×#539—22no—
10C7×#539—22no—
11C7×#539—22no—
12C7×#539C2214yes—
13C2×C7×#126—21no—
14C2×C7×#125—21no—
15C2×C7×#123—21no—
16C2×C7×#123—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C22×C7×C4⋊C4C2211yescharacteristic
22C22×C7×C4⋊C4C2211yescharacteristic
23C7×#484—22no—
24C7×#484C2214yescomplemented
25C7×#481—22no—
26C7×#481C2214yes—
27C2×C7×#107—21no—
28C2×C7×#106—21no—
29C2×C7×#103—21no—
30C2×C7×#102—21no—
31C2×C7×C22⋊C8C2211yescharacteristic
32C2×C7×C22⋊C8C2211yescharacteristic
33C7×#400—22no—
34C7×#400C2214yescomplemented
35C2×C7×#95—21no—
36C2×C7×#95—21no—
37C2×C7×#95C2211yescharacteristic
38C7×#328C2214yescomplemented
39C7×#327C2214yes—
40C7×#324C2214yescomplemented
41C2×#1615C711yesmaximal, characteristic, direct product factor, Sylow
42C22×C7×Q8D411yescharacteristic
43C22×C7×D4—21no—
44C22×C7×D4D411yescharacteristic
45C23×C4×C7—21no—
46C23×C4×C7—21no—
47C23×C4×C7—21no—
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C22×C7×C8—21no—
52C22×C7×C8—21no—
53C7×#126—24no—
54C7×#126—24no—
55C7×#125—24no—
56C7×#125—24no—
57C7×#123—24no—
58C7×#123—24no—
59C7×#123—24no—
60C7×#123—24no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—21no—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8D412yes—
66C7×C22⋊Q8C2312yes—
67C7×#121—21no—
68C7×#121—21no—
69C7×#121D412yes—
70C7×#121C2312yes—
71C4×C7×D4—24no—
72C4×C7×D4—24no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4D412yes—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7—21no—
97C7×#107—24no—
98C7×#106—24no—
99C7×#103—24no—
100C7×#102—24no—
101C7×C22⋊C8—21no—
102C7×C22⋊C8—21no—
103C7×C22⋊C8C2312yes—
104C7×C22⋊C8C2312yes—
105C7×#95—24no—
106C7×#95—24no—
107C7×#95—21no—
108C7×#95C2312yes—
109C2×#539C2×C711yescharacteristic
110C2×#484C2×C711yescharacteristic, complemented
111C2×#481C2×C711yescharacteristic
112C2×#400C2×C711yescharacteristic, complemented
113#1615C2×C718yesdirect product factor
114C2×#328C2×C711yescharacteristic, complemented
115C2×#327C2×C711yescharacteristic
116C2×#324C2×C711yescharacteristic, complemented
117C24×C7—21no—
118C24×C7C2×D411yescharacteristic
119C2×C7×Q8—21no—
120C2×C7×Q8—42no—
121C2×C7×Q8C2×D412yes—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—21no—
125C2×C7×D4—42no—
126C2×C7×D4—42no—
127C2×C7×D4—42no—
128C2×C7×D4C2×D412yes—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—44no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7—42no—
149C22×C4×C7—82no—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—42no—
161C2×C7×C8—42no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—41no—
176C7×C4⋊C4—41no—
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C42×C7—24no—
187#539—22no—
188#539—22no—
189#539—22no—
190#539C22×C714yes—
191C2×#126—21no—
192C2×#125—21no—
193C2×#123—21no—
194C2×#123—21no—
195C2×C22⋊Q8—21no—
196C2×C22⋊Q8C22×C711yescharacteristic
197C2×#121C22×C711yescharacteristic
198C2×C4×D4—21no—
199C22×C4⋊C4C22×C711yescharacteristic
200C22×C4⋊C4C22×C711yescharacteristic
201#484—22no—
202#484C22×C714yescomplemented
203#481—22no—
204#481C22×C714yes—
205C2×#107—21no—
206C2×#106—21no—
207C2×#103—21no—
208C2×#102—21no—
209C2×C22⋊C8C22×C711yescharacteristic
210C2×C22⋊C8C22×C711yescharacteristic
211#400—22no—
212#400C22×C714yescomplemented
213C2×#95—21no—
214C2×#95—21no—
215C2×#95C22×C711yescharacteristic
216#328C22×C714yescomplemented
217#327C22×C714yes—
218#324C22×C714yescomplemented
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7#12011yescharacteristic
226C23×C7#12011yescharacteristic
227C23×C7#13512yes—
228C23×C7#13712yes—
229C23×C7C22×D411yescharacteristic
230C7×Q8—42no—
231C7×Q8—81no—
232C7×D4—42no—
233C7×D4—42no—
234C7×D4—42no—
235C7×D4—44no—
236C7×D4—81no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—44no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—81no—
263C2×C4×C7—84no—
264C2×C4×C7—84no—
265C2×C4×C7—82no—
266C2×C4×C7—82no—
267C2×C4×C7#12011yescharacteristic
268C2×C4×C7#12011yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C7×C8—42no—
272C7×C8—42no—
273C22×Q8C7×D411yescharacteristic
274C22×D4—21no—
275C22×D4C7×D411yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4C7×D411yescharacteristic
280C23×C4C7×D411yescharacteristic
281C23×C4C23×C711yescharacteristic
282C22×C8—21no—
283C22×C8—21no—
284#126—24no—
285#126—24no—
286#125—24no—
287#125—24no—
288#123—24no—
289#123—24no—
290#123—24no—
291#123—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8C7×D412yes—
297C22⋊Q8C23×C712yes—
298#121—21no—
299#121—21no—
300#121C7×D412yes—
301#121C23×C712yes—
302C4×D4—24no—
303C4×D4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C7×D412yes—
318C2×C4⋊C4C7×D411yescharacteristic
319C2×C4⋊C4C7×D411yescharacteristic
320C2×C4⋊C4C23×C712yes—
321C2×C4⋊C4C23×C712yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C42—21no—
328#107—24no—
329#106—24no—
330#103—24no—
331#102—24no—
332C22⋊C8—21no—
333C22⋊C8—21no—
334C22⋊C8C23×C712yes—
335C22⋊C8C23×C712yes—
336#95—24no—
337#95—24no—
338#95—21no—
339#95C23×C712yes—
340C22×C7—24no—
341C22×C7—24no—
342C22×C7—21no—
343C22×C7—21no—
344C22×C7—21no—
345C22×C7—21no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—84no—
350C22×C7#45212yes—
351C22×C7#45212yes—
352C22×C7#45412yes—
353C22×C7#45812yes—
354C22×C7C2×#12011yescharacteristic
355C22×C7C2×#13511yescharacteristic
356C22×C7C2×#13711yescharacteristic
357C4×C7—21no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—21no—
361C4×C7—42no—
362C4×C7—42no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—82no—
367C4×C7—82no—
368C24—21no—
369C24C2×C7×D411yescharacteristic
370C2×Q8—21no—
371C2×Q8—42no—
372C2×Q8C2×C7×D412yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4C2×C7×D412yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—82no—
401C22×C4C2×C7×D412yes—
402C22×C4C2×C7×D411yescharacteristic
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×D412yes—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C24×C711yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4C2×C7×D412yes—
429C4⋊C4C2×C7×D412yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C42—24no—
438C2×C7—21no—
439C2×C7—21no—
440C2×C7—22no—
441C2×C7—82no—
442C2×C7#161514yesdirect product factor
443C2×C7C2×#45211yescharacteristic
444C2×C7C2×#45411yescharacteristic
445C2×C7C2×#45811yescharacteristic
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C7×#12011yescharacteristic
453C23C7×#12011yescharacteristic
454C23C7×#13512yes—
455C23C7×#13712yes—
456C23C22×C7×D411yescharacteristic
457Q8—42no—
458Q8—81no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—44no—
463D4—81no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4C7×#12011yescharacteristic
495C2×C4C7×#12011yescharacteristic
496C2×C4C22×C7×D411yescharacteristic
497C2×C4C22×C7×D411yescharacteristic
498C8—42no—
499C8—42no—
500C7C2×#161511yescharacteristic, direct product factor, Sylow
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22C7×#45212yes—
512C22C7×#45212yes—
513C22C7×#45412yes—
514C22C7×#45812yes—
515C22C2×C7×#12011yescharacteristic
516C22C2×C7×#13511yescharacteristic
517C22C2×C7×#13711yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—82no—
529C2—21no—
530C2—21no—
531C2—22no—
532C2—82no—
533C2C7×#161514yesdirect product factor
534C2C2×C7×#45211yescharacteristic
535C2C2×C7×#45411yescharacteristic
536C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#484C2semidirect
4. C2×C7×#400C2semidirect
5. C7×#1615C2direct (N × H)
6. C2×C7×#328C2semidirect
8. C2×C7×#324C2semidirect
24. C7×#484C22semidirect
34. C7×#400C22semidirect
38. C7×#328C22semidirect
40. C7×#324C22semidirect
41. C2×#1615C7direct (N × H)
110. C2×#484C2×C7semidirect
112. C2×#400C2×C7semidirect
113. #1615C2×C7direct (N × H)
114. C2×#328C2×C7semidirect
116. C2×#324C2×C7semidirect
202. #484C22×C7semidirect
212. #400C22×C7semidirect
216. #328C22×C7semidirect
218. #324C22×C7semidirect
442. C2×C7#1615direct (N × H)
500. C7C2×#1615direct (N × H)
533. C2C7×#1615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5392
C2×C7×#4842
C2×C7×#4812
C2×C7×#4002
C7×#16152
C2×C7×#3282
C2×C7×#3272
C2×C7×#3242
C2×#16157