Finite Group Databasebeta

← C2×C7×#3381 #13238570

Subgroups — C2×C7×#3381 #13238570

Order 1792.

1842 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×C7×#554C211yesmaximal, characteristic
3C7×#3381C2116yesmaximal, direct product factor
4C2×C7×#499C212yesmaximal, complemented
5C2×C7×#498C211yesmaximal, characteristic, complemented
6C2×C7×#479C212yesmaximal, complemented
7C2×C7×#477C211yesmaximal, characteristic, complemented
8C2×C7×#478C212yesmaximal
9C2×C7×#474C211yesmaximal, characteristic
10C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
11C2×C4×C7×SD16C211yesmaximal, characteristic
12C2×C4×C7×SD16C212yesmaximal
13C22×C7×SD16C2211yescharacteristic
14C7×#557—24no—
15C7×#557C2218yescomplemented
16C7×#554—24no—
17C7×#554C2218yes—
18C2×C7×C4⋊Q8—21no—
19C2×C7×C4⋊Q8C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C2×C7×#125—22no—
22C2×C7×#125C2212yes—
23C2×C7×#125C2212yes—
24C2×C7×C22⋊Q8—21no—
25C2×C7×C22⋊Q8—22no—
26C2×C4×C7×Q8—21no—
27C2×C4×C7×Q8C2211yescharacteristic
28C2×C4×C7×Q8C2212yes—
29C2×C4×C7×Q8C2211yescharacteristic
30C2×C4×C7×D4C2211yescharacteristic
31C2×C4×C7×D4C2212yes—
32C22×C7×C4⋊C4—21no—
33C7×#499C22116yescomplemented
34C7×#498C2218yescomplemented
35C7×#479C22116yescomplemented
36C7×#478C22116yes—
37C7×#477C2218yescomplemented
38C7×#474C2218yes—
39C7×C8×Q8C2218yescomplemented
40C4×C7×SD16C2218yes—
41C4×C7×SD16C22116yes—
42C2×C7×#107C2212yes—
43C2×C7×#107C2212yes—
44C2×C7×#106C2212yes—
45C2×C7×#106C2212yes—
46C2×C7×#106C2211yescharacteristic
47C2×C7×C4⋊C8C2211yescharacteristic
48C2×C7×C4⋊C8C2212yes—
49C2×C7×#103C2212yes—
50C2×C7×#103C2211yescharacteristic
51C2×C7×#102C2212yes—
52C2×C7×#102C2211yescharacteristic
53C2×C4×C7×C8C2211yescharacteristic
54C2×C4×C7×C8C2212yes—
55C2×#3381C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7—22no—
63C2×C7×SD16Q818yescomplemented
64C2×C7×SD16C2314yes—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2312yes—
67C22×C7×C8C2311yescharacteristic
68C7×C4⋊Q8—28no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8C2314yes—
72C7×C4⋊Q8C2314yes—
73C7×#125—216no—
74C7×#125—24no—
75C7×#125—24no—
76C7×#125C2318yes—
77C7×#125C2318yes—
78C7×C22⋊Q8—28no—
79C7×C22⋊Q8—216no—
80C4×C7×Q8—28no—
81C4×C7×Q8—24no—
82C4×C7×Q8—22no—
83C4×C7×Q8D414yes—
84C4×C7×Q8C2314yes—
85C4×C7×Q8C2318yes—
86C4×C7×Q8C2314yes—
87C4×C7×D4—24no—
88C4×C7×D4—22no—
89C4×C7×D4C2314yes—
90C4×C7×D4C2318yes—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4—22no—
97C2×C7×C4⋊C4—21no—
98C2×C7×C4⋊C4—22no—
99C2×C7×C4⋊C4—28no—
100C2×C7×C4⋊C4D411yescharacteristic
101C2×C7×C4⋊C4D412yes—
102C2×C7×C4⋊C4C2311yescharacteristic
103C2×C7×C4⋊C4C2312yes—
104C2×C7×C4⋊C4C2311yescharacteristic
105C2×C7×C4⋊C4C2312yes—
106C2×C7×C22⋊C4—21no—
107C2×C7×C22⋊C4—22no—
108C2×C42×C7—22no—
109C2×C42×C7—21no—
110C2×C42×C7C2312yes—
111C2×C42×C7C2311yescharacteristic
112C7×#107C2318yes—
113C7×#107C2318yes—
114C7×#106C2318yes—
115C7×#106C2318yes—
116C7×#106C2314yes—
117C7×C4⋊C8C2314yes—
118C7×C4⋊C8C2318yes—
119C7×#103C2318yes—
120C7×#103C2314yes—
121C7×#102C2318yes—
122C7×#102C2314yes—
123C4×C7×C8C2314yes—
124C4×C7×C8C2318yes—
125C2×#557C2×C711yescharacteristic, complemented
126C2×#554C2×C711yescharacteristic
127#3381C2×C7116yesdirect product factor
128C2×#499C2×C712yescomplemented
129C2×#498C2×C711yescharacteristic, complemented
130C2×#479C2×C712yescomplemented
131C2×#477C2×C711yescharacteristic, complemented
132C2×#478C2×C712yes—
133C2×#474C2×C711yescharacteristic
134C2×C8×Q8C2×C711yescharacteristic, complemented
135C2×C4×SD16C2×C712yes—
136C2×C4×SD16C2×C711yescharacteristic
137C24×C7—21no—
138C2×C7×Q8—24no—
139C2×C7×Q8—21no—
140C2×C7×Q8—24no—
141C2×C7×Q8—24no—
142C2×C7×Q8—22no—
143C2×C7×Q8—44no—
144C2×C7×Q8—82no—
145C2×C7×Q8C2×D414yes—
146C2×C7×Q8C2×Q814yes—
147C2×C7×Q8C2412yes—
148C2×C7×D4—21no—
149C2×C7×D4—22no—
150C2×C7×D4C2×Q814yes—
151C2×C7×D4C2412yes—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—24no—
158C22×C4×C7—28no—
159C22×C4×C7—44no—
160C22×C4×C7—84no—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2411yescharacteristic
166C7×SD16C2×Q8116yescomplemented
167C2×C7×C8—22no—
168C2×C7×C8—24no—
169C2×C7×C8C2×Q814yes—
170C2×C7×C8C2412yes—
171C2×C7×C8C2414yes—
172C2×C7×C8C2412yes—
173C7×C4⋊C4—28no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—28no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—28no—
178C7×C4⋊C4—28no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—28no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4C2×D414yes—
186C7×C4⋊C4C2×D418yes—
187C7×C4⋊C4C2414yes—
188C7×C4⋊C4C2412yes—
189C7×C4⋊C4C2414yes—
190C7×C4⋊C4C2412yes—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—28no—
193C42×C7—28no—
194C42×C7—24no—
195C42×C7C2×D414yes—
196C42×C7C2×D412yes—
197C42×C7C2414yes—
198C42×C7C2412yes—
199C22×SD16C22×C711yescharacteristic
200#557—24no—
201#557C22×C718yescomplemented
202#554—24no—
203#554C22×C718yes—
204C2×C4⋊Q8—21no—
205C2×C4⋊Q8C22×C711yescharacteristic
206C2×C4⋊Q8C22×C711yescharacteristic
207C2×#125—22no—
208C2×#125C22×C712yes—
209C2×#125C22×C712yes—
210C2×C22⋊Q8—21no—
211C2×C22⋊Q8—22no—
212C2×C4×Q8—21no—
213C2×C4×Q8C22×C711yescharacteristic
214C2×C4×Q8C22×C712yes—
215C2×C4×Q8C22×C711yescharacteristic
216C2×C4×D4C22×C712yes—
217C2×C4×D4C22×C711yescharacteristic
218C22×C4⋊C4—21no—
219#499C22×C7116yescomplemented
220#498C22×C718yescomplemented
221#479C22×C7116yescomplemented
222#478C22×C7116yes—
223#477C22×C718yescomplemented
224#474C22×C718yes—
225C8×Q8C22×C718yescomplemented
226C4×SD16C22×C718yes—
227C4×SD16C22×C7116yes—
228C2×#107C22×C712yes—
229C2×#107C22×C712yes—
230C2×#106C22×C712yes—
231C2×#106C22×C712yes—
232C2×#106C22×C711yescharacteristic
233C2×C4⋊C8C22×C712yes—
234C2×C4⋊C8C22×C711yescharacteristic
235C2×#103C22×C712yes—
236C2×#103C22×C711yescharacteristic
237C2×#102C22×C711yescharacteristic
238C2×#102C22×C712yes—
239C2×C4×C8C22×C711yescharacteristic
240C2×C4×C8C22×C712yes—
241C23×C7—24no—
242C23×C7—22no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7C22×D411yescharacteristic
246C7×Q8—24no—
247C7×Q8—28no—
248C7×Q8—22no—
249C7×Q8—48no—
250C7×Q8—84no—
251C7×Q8C22×Q814yes—
252C7×D4—24no—
253C7×D4—22no—
254C7×D4C22×Q814yes—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—48no—
269C2×C4×C7—88no—
270C2×C4×C7#13514yes—
271C2×C4×C7C22×D414yes—
272C2×C4×C7C22×D414yes—
273C2×C4×C7C22×D412yes—
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×D412yes—
276C2×C4×C7C22×Q812yes—
277C2×C4×C72−1+412yes—
278C2×C4×C7C2511yescharacteristic
279C7×C8—22no—
280C7×C8—24no—
281C7×C8C22×Q814yes—
282C22×Q8—21no—
283C22×Q8—21no—
284C22×Q8C7×D411yescharacteristic
285C22×Q8C23×C711yescharacteristic
286C22×D4C23×C711yescharacteristic
287C23×C4—21no—
288C23×C4—22no—
289C2×SD16C7×Q818yescomplemented
290C2×SD16C23×C714yes—
291C22×C8C23×C711yescharacteristic
292C22×C8C23×C712yes—
293C22×C8C23×C711yescharacteristic
294C4⋊Q8—28no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8C23×C714yes—
298C4⋊Q8C23×C714yes—
299#125—216no—
300#125—24no—
301#125—24no—
302#125C23×C718yes—
303#125C23×C718yes—
304C22⋊Q8—28no—
305C22⋊Q8—216no—
306C4×Q8—28no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8C7×D414yes—
310C4×Q8C23×C714yes—
311C4×Q8C23×C718yes—
312C4×Q8C23×C714yes—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4C23×C714yes—
316C4×D4C23×C718yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4C7×D411yescharacteristic
327C2×C4⋊C4C7×D412yes—
328C2×C4⋊C4C23×C711yescharacteristic
329C2×C4⋊C4C23×C712yes—
330C2×C4⋊C4C23×C711yescharacteristic
331C2×C4⋊C4C23×C712yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42C23×C712yes—
337C2×C42C23×C711yescharacteristic
338#107C23×C718yes—
339#107C23×C718yes—
340#106C23×C718yes—
341#106C23×C714yes—
342#106C23×C718yes—
343C4⋊C8C23×C714yes—
344C4⋊C8C23×C718yes—
345#103C23×C718yes—
346#103C23×C714yes—
347#102C23×C718yes—
348#102C23×C714yes—
349C4×C8C23×C714yes—
350C4×C8C23×C718yes—
351C22×C7—24no—
352C22×C7—48no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7D4×Q812yes—
356C22×C7C2×#13512yes—
357C22×C7#57712yes—
358C22×C7C23×D411yescharacteristic
359C4×C7—24no—
360C4×C7—24no—
361C4×C7—24no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—44no—
365C4×C7—42no—
366C4×C7—44no—
367C4×C7D4×Q812yes—
368C4×C7C2×#13514yes—
369C4×C7C23×Q811yescharacteristic
370C4×C7C2×2−1+411yescharacteristic
371C24—21no—
372C2×Q8—24no—
373C2×Q8—21no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—44no—
378C2×Q8—82no—
379C2×Q8C2×C7×D414yes—
380C2×Q8C2×C7×Q814yes—
381C2×Q8C24×C712yes—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4C2×C7×Q814yes—
385C2×D4C24×C712yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—44no—
394C22×C4—84no—
395C22×C4C2×C7×D412yes—
396C22×C4C2×C7×D412yes—
397C22×C4C2×C7×D411yescharacteristic
398C22×C4C2×C7×D411yescharacteristic
399C22×C4C24×C711yescharacteristic
400SD16C2×C7×Q8116yescomplemented
401C2×C8—22no—
402C2×C8—24no—
403C2×C8C2×C7×Q814yes—
404C2×C8C24×C712yes—
405C2×C8C24×C714yes—
406C2×C8C24×C712yes—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—28no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4C2×C7×D414yes—
420C4⋊C4C2×C7×D418yes—
421C4⋊C4C24×C714yes—
422C4⋊C4C24×C712yes—
423C4⋊C4C24×C714yes—
424C4⋊C4C24×C712yes—
425C22⋊C4—24no—
426C22⋊C4—28no—
427C42—28no—
428C42—24no—
429C42C2×C7×D414yes—
430C42C2×C7×D412yes—
431C42C24×C714yes—
432C42C24×C712yes—
433C2×C7—44no—
434C2×C7#338114yesdirect product factor
435C2×C7C2×D4×Q811yescharacteristic
436C2×C7C22×#13511yescharacteristic
437C2×C7C2×#57711yescharacteristic
438C23—22no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23C22×C7×D411yescharacteristic
443Q8—22no—
444Q8—24no—
445Q8—28no—
446Q8—48no—
447Q8—84no—
448Q8C22×C7×Q814yes—
449D4—22no—
450D4—24no—
451D4C22×C7×Q814yes—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—88no—
467C2×C4C7×#13514yes—
468C2×C4C22×C7×D414yes—
469C2×C4C22×C7×D414yes—
470C2×C4C22×C7×D412yes—
471C2×C4C22×C7×D411yescharacteristic
472C2×C4C22×C7×D412yes—
473C2×C4C22×C7×Q812yes—
474C2×C4C7×2−1+412yes—
475C2×C4C25×C711yescharacteristic
476C8—22no—
477C8—24no—
478C8C22×C7×Q814yes—
479C7C2×#338111yescharacteristic, direct product factor, Sylow
480C22—24no—
481C22—42no—
482C22—42no—
483C22—48no—
484C22C7×D4×Q812yes—
485C22C2×C7×#13512yes—
486C22C7×#57712yes—
487C22C23×C7×D411yescharacteristic
488C4—24no—
489C4—24no—
490C4—22no—
491C4—24no—
492C4—42no—
493C4—44no—
494C4—44no—
495C4—42no—
496C4C7×D4×Q812yes—
497C4C2×C7×#13514yes—
498C4C23×C7×Q811yescharacteristic
499C4C2×C7×2−1+411yescharacteristic
500C2—44no—
501C2C7×#338114yesdirect product factor
502C2C2×C7×D4×Q811yescharacteristic
503C2C22×C7×#13511yescharacteristic
504C2C2×C7×#57711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
3. C7×#3381C2direct (N × H)
4. C2×C7×#499C2semidirect
5. C2×C7×#498C2semidirect
6. C2×C7×#479C2semidirect
7. C2×C7×#477C2semidirect
10. C2×C7×C8×Q8C2semidirect
15. C7×#557C22semidirect
33. C7×#499C22semidirect
34. C7×#498C22semidirect
35. C7×#479C22semidirect
37. C7×#477C22semidirect
39. C7×C8×Q8C22semidirect
55. C2×#3381C7direct (N × H)
63. C2×C7×SD16Q8semidirect
125. C2×#557C2×C7semidirect
127. #3381C2×C7direct (N × H)
128. C2×#499C2×C7semidirect
129. C2×#498C2×C7semidirect
130. C2×#479C2×C7semidirect
131. C2×#477C2×C7semidirect
134. C2×C8×Q8C2×C7semidirect
166. C7×SD16C2×Q8semidirect
201. #557C22×C7semidirect
219. #499C22×C7semidirect
220. #498C22×C7semidirect
221. #479C22×C7semidirect
223. #477C22×C7semidirect
225. C8×Q8C22×C7semidirect
289. C2×SD16C7×Q8semidirect
400. SD16C2×C7×Q8semidirect
434. C2×C7#3381direct (N × H)
479. C7C2×#3381direct (N × H)
501. C2C7×#3381direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×C7×#5542
C7×#33812
C2×C7×#4992
C2×C7×#4982
C2×C7×#4792
C2×C7×#4772
C2×C7×#4782
C2×C7×#4742
C2×C7×C8×Q82
C2×C4×C7×SD162
C2×#33817