Finite Group Databasebeta

← C2×C7×#3210 #13237996

Subgroups — C2×C7×#3210 #13237996

Order 1792.

2186 conjugacy classes of subgroups.

The subgroup lattice has 502 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×#3210C2116yesmaximal, direct product factor
4C2×C7×#494C211yesmaximal, characteristic, complemented
5C2×C7×#492C211yesmaximal, characteristic, complemented
6C2×C7×#487C211yesmaximal, characteristic, complemented
7C2×C7×#486C211yesmaximal, characteristic, complemented
8C2×C7×#464C214yesmaximal, complemented
9C2×C7×#452C214yesmaximal, complemented
10C2×C7×#431C211yesmaximal, characteristic, complemented
11C22×C7×#41C2212yes—
12C22×C7×Q16C2214yes—
13C22×C7×SD16C2214yes—
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×#124C2212yes—
24C2×C7×C22⋊Q8—21no—
25C2×C7×C22⋊Q8C2212yes—
26C2×C7×#121—21no—
27C2×C7×#121—21no—
28C2×C4×C7×Q8—21no—
29C2×C4×C7×Q8C2212yes—
30C2×C4×C7×D4—21no—
31C2×C4×C7×D4—21no—
32C2×C7×#117—21no—
33C22×C42×C7C2211yescharacteristic
34C7×#494C2218yescomplemented
35C7×#492C2218yescomplemented
36C7×#487C2218yescomplemented
37C7×#486C2218yescomplemented
38C7×#464C22132yescomplemented
39C7×#452C22132yescomplemented
40C7×#431C2218yescomplemented
41C2×C7×C4⋊C8C2212yes—
42C2×C7×#103C2214yes—
43C2×C7×#102C2214yes—
44C2×C7×C22⋊C8C2212yes—
45C2×C4×C7×C8C2211yescharacteristic
46C2×C4×C7×C8C2211yescharacteristic
47C2×#3210C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—24no—
49C2×C7×#41—48no—
50C2×C7×#41C2318yes—
51C22×C7×Q8—21no—
52C22×C7×Q8C2312yes—
53C22×C7×Q8C2312yes—
54C22×C7×D4—22no—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4C2312yes—
58C23×C4×C7—22no—
59C23×C4×C7D411yescharacteristic
60C23×C4×C7D411yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C2×C7×Q16—48no—
63C2×C7×Q16C23116yes—
64C2×C7×SD16—48no—
65C2×C7×SD16C23116yes—
66C22×C7×C8C2312yes—
67C22×C7×C8C2312yes—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8C2314yes—
72C7×C4⋊Q8C2314yes—
73C7×C4⋊Q8C2314yes—
74C7×#127—22no—
75C7×#127C2314yes—
76C7×#125—28no—
77C7×#124—24no—
78C7×#124C2318yes—
79C7×C22⋊Q8—28no—
80C7×C22⋊Q8—24no—
81C7×C22⋊Q8C2318yes—
82C7×#121—28no—
83C7×#121—28no—
84C4×C7×Q8—28no—
85C4×C7×Q8—24no—
86C4×C7×Q8C2318yes—
87C4×C7×D4—28no—
88C4×C7×D4—28no—
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—21no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4—22no—
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×C7D411yescharacteristic
104C2×C42×C7D414yes—
105C2×C42×C7C2311yescharacteristic
106C2×C42×C7C2311yescharacteristic
107C2×C42×C7C2314yes—
108C7×C4⋊C8C2318yes—
109C7×#103C23116yes—
110C7×#102C23116yes—
111C7×C22⋊C8C2318yes—
112C4×C7×C8C2314yes—
113C4×C7×C8C2314yes—
114C2×#533C2×C711yescharacteristic, complemented
115C2×#532C2×C711yescharacteristic
116#3210C2×C7116yesdirect product factor
117C2×#494C2×C711yescharacteristic, complemented
118C2×#492C2×C711yescharacteristic, complemented
119C2×#487C2×C711yescharacteristic, complemented
120C2×#486C2×C711yescharacteristic, complemented
121C2×#464C2×C714yescomplemented
122C2×#452C2×C714yescomplemented
123C2×#431C2×C711yescharacteristic, complemented
124C24×C7—22no—
125C24×C7C2×D411yescharacteristic
126C7×#41—416no—
127C7×#41—416no—
128C2×C7×Q8—22no—
129C2×C7×Q8—24no—
130C2×C7×Q8—22no—
131C2×C7×Q8—28no—
132C2×C7×Q8—28no—
133C2×C7×Q8—44no—
134C2×C7×Q8C2414yes—
135C2×C7×Q8C2414yes—
136C2×C7×D4—28no—
137C2×C7×D4—24no—
138C2×C7×D4—24no—
139C2×C7×D4—22no—
140C2×C7×D4—28no—
141C2×C7×D4—48no—
142C2×C7×D4—44no—
143C2×C7×D4—44no—
144C2×C7×D4C2414yes—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—28no—
152C22×C4×C7—48no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2412yes—
162C22×C4×C7C2411yescharacteristic
163C7×Q16—416no—
164C7×SD16—416no—
165C2×C7×C8—24no—
166C2×C7×C8—24no—
167C2×C7×C8C2414yes—
168C2×C7×C8C2414yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—28no—
175C7×C4⋊C4—28no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4C2414yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—28no—
181C42×C7—28no—
182C42×C7C2×D414yes—
183C42×C7C2×D414yes—
184C42×C7C2×D412yes—
185C42×C7C2×D412yes—
186C42×C7C2412yes—
187C42×C7C2412yes—
188C22×#41C22×C712yes—
189C22×Q16C22×C714yes—
190C22×SD16C22×C714yes—
191#533—24no—
192#533C22×C718yescomplemented
193#532—24no—
194#532C22×C718yes—
195C2×C4⋊Q8C22×C711yescharacteristic
196C2×C4⋊Q8C22×C711yescharacteristic
197C2×C4⋊Q8C22×C711yescharacteristic
198C2×#127C22×C711yescharacteristic
199C2×#125—21no—
200C2×#124C22×C712yes—
201C2×C22⋊Q8—21no—
202C2×C22⋊Q8C22×C712yes—
203C2×#121—21no—
204C2×#121—21no—
205C2×C4×Q8—21no—
206C2×C4×Q8C22×C712yes—
207C2×C4×D4—21no—
208C2×C4×D4—21no—
209C2×#117—21no—
210C22×C42C22×C711yescharacteristic
211#494C22×C718yescomplemented
212#492C22×C718yescomplemented
213#487C22×C718yescomplemented
214#486C22×C718yescomplemented
215#464C22×C7132yescomplemented
216#452C22×C7132yescomplemented
217#431C22×C718yescomplemented
218C2×C4⋊C8C22×C712yes—
219C2×#103C22×C714yes—
220C2×#102C22×C714yes—
221C2×C22⋊C8C22×C712yes—
222C2×C4×C8C22×C711yescharacteristic
223C2×C4×C8C22×C711yescharacteristic
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—24no—
228C23×C7—44no—
229C23×C7—44no—
230C23×C7—44no—
231C23×C7C22×D412yes—
232C23×C7C22×D411yescharacteristic
233C7×Q8—28no—
234C7×Q8—28no—
235C7×Q8—44no—
236C7×Q8—44no—
237C7×Q8—48no—
238C7×D4—28no—
239C7×D4—416no—
240C7×D4—44no—
241C7×D4—48no—
242C7×D4—48no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—28no—
248C2×C4×C7—28no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—416no—
254C2×C4×C7—44no—
255C2×C4×C7—44no—
256C2×C4×C7—44no—
257C2×C4×C7#13514yes—
258C2×C4×C7#13714yes—
259C2×C4×C7C22×D412yes—
260C2×C4×C7C22×D412yes—
261C2×C4×C7C22×D412yes—
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D412yes—
264C2×C4×C7C22×D412yes—
265C2×C4×C7H(5,𝔽2)14yes—
266C2×C4×C7C2511yescharacteristic
267C7×C8—24no—
268C7×C8—24no—
269C2×#41—24no—
270C2×#41—48no—
271C2×#41C23×C718yes—
272C22×Q8—21no—
273C22×Q8C23×C712yes—
274C22×Q8C23×C712yes—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C23×C712yes—
279C23×C4—22no—
280C23×C4C7×D411yescharacteristic
281C23×C4C7×D411yescharacteristic
282C23×C4C23×C711yescharacteristic
283C2×Q16—48no—
284C2×Q16C23×C7116yes—
285C2×SD16—48no—
286C2×SD16C23×C7116yes—
287C22×C8C23×C712yes—
288C22×C8C23×C712yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8C23×C714yes—
293C4⋊Q8C23×C714yes—
294C4⋊Q8C23×C714yes—
295#127—22no—
296#127C23×C714yes—
297#125—28no—
298#124—24no—
299#124C23×C718yes—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8C23×C718yes—
303#121—28no—
304#121—28no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8C23×C718yes—
308C4×D4—28no—
309C4×D4—28no—
310#117—28no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C23×C712yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—22no—
323C2×C42C7×D411yescharacteristic
324C2×C42C7×D411yescharacteristic
325C2×C42C7×D414yes—
326C2×C42C23×C711yescharacteristic
327C2×C42C23×C711yescharacteristic
328C2×C42C23×C714yes—
329C4⋊C8C23×C718yes—
330#103C23×C7116yes—
331#102C23×C7116yes—
332C22⋊C8C23×C718yes—
333C4×C8C23×C714yes—
334C4×C8C23×C714yes—
335C22×C7—22no—
336C22×C7—22no—
337C22×C7—22no—
338C22×C7—44no—
339C22×C7—44no—
340C22×C7—44no—
341C22×C7—44no—
342C22×C7—88no—
343C22×C7#53512yes—
344C22×C7C2×#13512yes—
345C22×C7C2×#13712yes—
346C22×C7C23×D411yescharacteristic
347C4×C7—22no—
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—44no—
351C4×C7—44no—
352C4×C7—44no—
353C4×C7#53512yes—
354C4×C7C2×#13514yes—
355C4×C7C2×#13714yes—
356C4×C7C2×H(5,𝔽2)12yes—
357C24—22no—
358C24C2×C7×D411yescharacteristic
359#41—416no—
360#41—416no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—44no—
367C2×Q8C24×C714yes—
368C2×Q8C24×C714yes—
369C2×D4—28no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4C24×C714yes—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—48no—
386C22×C4C2×C7×D411yescharacteristic
387C22×C4C2×C7×D411yescharacteristic
388C22×C4C2×C7×D412yes—
389C22×C4C2×C7×D411yescharacteristic
390C22×C4C2×C7×D414yes—
391C22×C4C2×C7×D414yes—
392C22×C4C2×C7×D411yescharacteristic
393C22×C4C2×C7×D411yescharacteristic
394C22×C4C24×C712yes—
395C22×C4C24×C711yescharacteristic
396Q16—416no—
397SD16—416no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8C24×C714yes—
401C2×C8C24×C714yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4C24×C714yes—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C42—28no—
415C42C2×C7×D414yes—
416C42C2×C7×D414yes—
417C42C2×C7×D412yes—
418C42C2×C7×D412yes—
419C42C24×C712yes—
420C42C24×C712yes—
421C2×C7—42no—
422C2×C7—84no—
423C2×C7#321014yesdirect product factor
424C2×C7C2×#53511yescharacteristic
425C2×C7C22×#13511yescharacteristic
426C2×C7C22×#13711yescharacteristic
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C22×C7×D412yes—
435C23C22×C7×D411yescharacteristic
436Q8—28no—
437Q8—28no—
438Q8—44no—
439Q8—44no—
440Q8—48no—
441D4—28no—
442D4—416no—
443D4—44no—
444D4—48no—
445D4—48no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—416no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4C7×#13514yes—
461C2×C4C7×#13714yes—
462C2×C4C22×C7×D412yes—
463C2×C4C22×C7×D412yes—
464C2×C4C22×C7×D412yes—
465C2×C4C22×C7×D411yescharacteristic
466C2×C4C22×C7×D412yes—
467C2×C4C22×C7×D412yes—
468C2×C4C7×H(5,𝔽2)14yes—
469C2×C4C25×C711yescharacteristic
470C8—24no—
471C8—24no—
472C7C2×#321011yescharacteristic, direct product factor, Sylow
473C22—22no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—88no—
481C22C7×#53512yes—
482C22C2×C7×#13512yes—
483C22C2×C7×#13712yes—
484C22C23×C7×D411yescharacteristic
485C4—22no—
486C4—22no—
487C4—22no—
488C4—44no—
489C4—44no—
490C4—44no—
491C4C7×#53512yes—
492C4C2×C7×#13514yes—
493C4C2×C7×#13714yes—
494C4C2×C7×H(5,𝔽2)12yes—
495C2—42no—
496C2—84no—
497C2C7×#321014yesdirect product factor
498C2C2×C7×#53511yescharacteristic
499C2C22×C7×#13511yescharacteristic
500C2C22×C7×#13711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#532C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×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×#3210C2direct (N × H)
4. C2×C7×#494C2semidirect
5. C2×C7×#492C2semidirect
6. C2×C7×#487C2semidirect
7. C2×C7×#486C2semidirect
8. C2×C7×#464C2semidirect
9. C2×C7×#452C2semidirect
10. C2×C7×#431C2semidirect
15. C7×#533C22semidirect
34. C7×#494C22semidirect
35. C7×#492C22semidirect
36. C7×#487C22semidirect
37. C7×#486C22semidirect
38. C7×#464C22semidirect
39. C7×#452C22semidirect
40. C7×#431C22semidirect
47. C2×#3210C7direct (N × H)
114. C2×#533C2×C7semidirect
116. #3210C2×C7direct (N × H)
117. C2×#494C2×C7semidirect
118. C2×#492C2×C7semidirect
119. C2×#487C2×C7semidirect
120. C2×#486C2×C7semidirect
121. C2×#464C2×C7semidirect
122. C2×#452C2×C7semidirect
123. C2×#431C2×C7semidirect
192. #533C22×C7semidirect
211. #494C22×C7semidirect
212. #492C22×C7semidirect
213. #487C22×C7semidirect
214. #486C22×C7semidirect
215. #464C22×C7semidirect
216. #452C22×C7semidirect
217. #431C22×C7semidirect
423. C2×C7#3210direct (N × H)
472. C7C2×#3210direct (N × H)
497. C2C7×#3210direct (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×#32102
C2×C7×#4942
C2×C7×#4922
C2×C7×#4872
C2×C7×#4862
C2×C7×#4642
C2×C7×#4522
C2×C7×#4312
C2×#32107