Finite Group Databasebeta

← C2×C7×#3234 #13238063

Subgroups — C2×C7×#3234 #13238063

Order 1792.

2050 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic
2C22×C7×#125C211yesmaximal, characteristic, complemented
3C7×#3234C2116yesmaximal, direct product factor
4C2×C7×#501C212yesmaximal, complemented
5C2×C7×#490C212yesmaximal, complemented
6C2×C7×#481C212yesmaximal, complemented
7C2×C7×#480C212yesmaximal, complemented
8C2×C7×#478C214yesmaximal, complemented
9C2×C7×#432C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#532—24no—
12C7×#532C2218yes—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×#127C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×#125—21no—
17C2×C7×#125—24no—
18C2×C7×#125C2212yes—
19C2×C7×#125C2212yes—
20C2×C7×#125C2218yescomplemented
21C2×C7×#124—21no—
22C2×C7×#121—21no—
23C2×C7×#121C2212yes—
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2212yes—
26C22×C7×C4⋊C4—21no—
27C22×C7×C4⋊C4—21no—
28C22×C7×C4⋊C4C2211yescharacteristic
29C22×C7×C4⋊C4C2211yescharacteristic
30C22×C42×C7C2211yescharacteristic
31C7×#501C22116yescomplemented
32C7×#490C22116yescomplemented
33C7×#481C22116yescomplemented
34C7×#480C22116yescomplemented
35C7×#478C22132yescomplemented
36C7×#432C2218yescomplemented
37C2×C7×#107C2214yes—
38C2×C7×#106C2214yes—
39C2×C7×C4⋊C8C2212yes—
40C2×C7×#102C2214yes—
41C2×C7×#102C2214yes—
42C2×C7×C22⋊C8C2212yes—
43C2×C7×#97C2212yes—
44C2×#3234C711yesmaximal, characteristic, direct product factor, Sylow
45C2×C7×#41—28no—
46C2×C7×#41—48no—
47C22×C7×Q8—21no—
48C22×C7×D4—22no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4C2312yes—
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C23×C4×C7—22no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C22×C7×C8C2314yes—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8C2314yes—
61C7×#127—22no—
62C7×#127C2314yes—
63C7×#125—24no—
64C7×#125—24no—
65C7×#125—28no—
66C7×#125—28no—
67C7×#125C2318yes—
68C7×#125C2318yes—
69C7×#124—28no—
70C7×#121—28no—
71C7×#121—24no—
72C7×#121C2318yes—
73C4×C7×D4—28no—
74C4×C7×D4—24no—
75C4×C7×D4C2318yes—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—28no—
81C2×C7×C4⋊C4—28no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4C2314yes—
88C2×C7×C4⋊C4C2314yes—
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C42×C7D411yescharacteristic
95C2×C42×C7D411yescharacteristic
96C2×C42×C7D414yes—
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2314yes—
100C7×#107C23116yes—
101C7×#106C23116yes—
102C7×C4⋊C8C2318yes—
103C7×#102C23116yes—
104C7×#102C23116yes—
105C7×C22⋊C8C2318yes—
106C7×#97C2318yes—
107C2×#532C2×C711yescharacteristic
108C22×#125C2×C711yescharacteristic, complemented
109#3234C2×C7116yesdirect product factor
110C2×#501C2×C712yescomplemented
111C2×#490C2×C712yescomplemented
112C2×#481C2×C712yescomplemented
113C2×#480C2×C712yescomplemented
114C2×#478C2×C714yescomplemented
115C2×#432C2×C711yescharacteristic, complemented
116C24×C7—22no—
117C24×C7C2×D411yescharacteristic
118C7×#41—432no—
119C2×C7×Q8—24no—
120C2×C7×Q8—44no—
121C2×C7×D4—28no—
122C2×C7×D4—22no—
123C2×C7×D4—24no—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—44no—
127C2×C7×D4—44no—
128C2×C7×D4—48no—
129C2×C7×D4C2414yes—
130C22×C4×C7—24no—
131C22×C4×C7—24no—
132C22×C4×C7—24no—
133C22×C4×C7—24no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—28no—
138C22×C4×C7—42no—
139C22×C4×C7—48no—
140C22×C4×C7—44no—
141C22×C4×C7—44no—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D414yes—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2412yes—
151C22×C4×C7C2411yescharacteristic
152C2×C7×C8—28no—
153C2×C7×C8C2418yes—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—28no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4C2414yes—
165C7×C4⋊C4C2414yes—
166C7×C4⋊C4C2414yes—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—28no—
169C42×C7C2×D414yes—
170C42×C7C2×D412yes—
171C42×C7C2×D414yes—
172C42×C7C2×D412yes—
173C42×C7C2412yes—
174C42×C7C2412yes—
175C22×#41—21no—
176#532—24no—
177#532C22×C718yes—
178C2×C4⋊Q8C22×C711yescharacteristic
179C2×#127C22×C711yescharacteristic
180C2×#125—21no—
181C2×#125—21no—
182C2×#125—24no—
183C2×#125C22×C712yes—
184C2×#125C22×C712yes—
185C2×#125C22×C718yescomplemented
186C2×#124—21no—
187C2×#121—21no—
188C2×#121C22×C712yes—
189C2×C4×D4—21no—
190C2×C4×D4C22×C712yes—
191C22×C4⋊C4—21no—
192C22×C4⋊C4—21no—
193C22×C4⋊C4C22×C711yescharacteristic
194C22×C4⋊C4C22×C711yescharacteristic
195C22×C42C22×C711yescharacteristic
196#501C22×C7116yescomplemented
197#490C22×C7116yescomplemented
198#481C22×C7116yescomplemented
199#480C22×C7116yescomplemented
200#478C22×C7132yescomplemented
201#432C22×C718yescomplemented
202C2×#107C22×C714yes—
203C2×#106C22×C714yes—
204C2×C4⋊C8C22×C712yes—
205C2×#102C22×C714yes—
206C2×#102C22×C714yes—
207C2×C22⋊C8C22×C712yes—
208C2×#97C22×C712yes—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—24no—
212C23×C7—44no—
213C23×C7—44no—
214C23×C7—44no—
215C23×C7AGL(1,ℤ/8)14yes—
216C23×C7C22×D412yes—
217C23×C7C22×D411yescharacteristic
218C7×Q8—48no—
219C7×D4—28no—
220C7×D4—48no—
221C7×D4—48no—
222C7×D4—416no—
223C7×D4—44no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—28no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—44no—
236C2×C4×C7—416no—
237C2×C4×C7—48no—
238C2×C4×C7—48no—
239C2×C4×C7—44no—
240C2×C4×C7—44no—
241C2×C4×C7—44no—
242C2×C4×C7C22×D412yes—
243C2×C4×C7C22×D412yes—
244C2×C4×C7C22×D412yes—
245C2×C4×C7C22×D412yes—
246C2×C4×C7C22×D412yes—
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C72−1+414yes—
249C2×C4×C7C2511yescharacteristic
250C7×C8—28no—
251C2×#41—28no—
252C2×#41—48no—
253C22×Q8—21no—
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C23×C712yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C7×D411yescharacteristic
262C23×C4C7×D411yescharacteristic
263C23×C4C23×C711yescharacteristic
264C22×C8C23×C714yes—
265C4⋊Q8—22no—
266C4⋊Q8C23×C714yes—
267#127—22no—
268#127C23×C714yes—
269#125—24no—
270#125—24no—
271#125—28no—
272#125—28no—
273#125C23×C718yes—
274#125C23×C718yes—
275#124—28no—
276#121—28no—
277#121—24no—
278#121C23×C718yes—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4C23×C718yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C23×C712yes—
294C2×C4⋊C4C23×C712yes—
295C2×C4⋊C4C23×C714yes—
296C2×C4⋊C4C23×C714yes—
297C2×C4⋊C4C23×C712yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C42C7×D411yescharacteristic
301C2×C42C7×D411yescharacteristic
302C2×C42C7×D414yes—
303C2×C42C23×C711yescharacteristic
304C2×C42C23×C711yescharacteristic
305C2×C42C23×C714yes—
306#107C23×C7116yes—
307#106C23×C7116yes—
308C4⋊C8C23×C718yes—
309#102C23×C7116yes—
310#102C23×C7116yes—
311C22⋊C8C23×C718yes—
312#97C23×C718yes—
313C22×C7—22no—
314C22×C7—28no—
315C22×C7—22no—
316C22×C7—44no—
317C22×C7—44no—
318C22×C7—44no—
319C22×C7—88no—
320C22×C7#53612yes—
321C22×C7C2×AGL(1,ℤ/8)14yes—
322C22×C7C2×AGL(1,ℤ/8)12yes—
323C22×C7#57512yes—
324C22×C7C23×D411yescharacteristic
325C4×C7—22no—
326C4×C7—22no—
327C4×C7—22no—
328C4×C7—22no—
329C4×C7—22no—
330C4×C7—44no—
331C4×C7—44no—
332C4×C7—44no—
333C4×C7#53612yes—
334C4×C7C2×2−1+412yes—
335C24—22no—
336C24C2×C7×D411yescharacteristic
337#41—432no—
338C2×Q8—24no—
339C2×Q8—44no—
340C2×D4—28no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4C24×C714yes—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4C2×C7×D411yescharacteristic
362C22×C4C2×C7×D412yes—
363C22×C4C2×C7×D411yescharacteristic
364C22×C4C2×C7×D411yescharacteristic
365C22×C4C2×C7×D411yescharacteristic
366C22×C4C2×C7×D411yescharacteristic
367C22×C4C2×C7×D414yes—
368C22×C4C2×C7×D414yes—
369C22×C4C24×C712yes—
370C22×C4C24×C711yescharacteristic
371C2×C8—28no—
372C2×C8C24×C718yes—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4C24×C714yes—
384C4⋊C4C24×C714yes—
385C4⋊C4C24×C714yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C42C2×C7×D414yes—
389C42C2×C7×D412yes—
390C42C2×C7×D414yes—
391C42C2×C7×D412yes—
392C42C24×C712yes—
393C42C24×C712yes—
394C2×C7—24no—
395C2×C7—84no—
396C2×C7#323414yesdirect product factor
397C2×C7C2×#53611yescharacteristic
398C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
399C2×C7C2×#57511yescharacteristic
400C23—22no—
401C23—22no—
402C23—24no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23C7×AGL(1,ℤ/8)14yes—
407C23C22×C7×D412yes—
408C23C22×C7×D411yescharacteristic
409Q8—48no—
410D4—28no—
411D4—48no—
412D4—48no—
413D4—416no—
414D4—44no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—44no—
427C2×C4—416no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4C22×C7×D412yes—
434C2×C4C22×C7×D412yes—
435C2×C4C22×C7×D412yes—
436C2×C4C22×C7×D412yes—
437C2×C4C22×C7×D412yes—
438C2×C4C22×C7×D411yescharacteristic
439C2×C4C7×2−1+414yes—
440C2×C4C25×C711yescharacteristic
441C8—28no—
442C7C2×#323411yescharacteristic, direct product factor, Sylow
443C22—22no—
444C22—28no—
445C22—22no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—88no—
450C22C7×#53612yes—
451C22C2×C7×AGL(1,ℤ/8)14yes—
452C22C2×C7×AGL(1,ℤ/8)12yes—
453C22C7×#57512yes—
454C22C23×C7×D411yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—44no—
461C4—44no—
462C4—44no—
463C4C7×#53612yes—
464C4C2×C7×2−1+412yes—
465C2—24no—
466C2—84no—
467C2C7×#323414yesdirect product factor
468C2C2×C7×#53611yescharacteristic
469C2C22×C7×AGL(1,ℤ/8)11yescharacteristic
470C2C2×C7×#57511yescharacteristic
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
2. C22×C7×#125C2semidirect
3. C7×#3234C2direct (N × H)
4. C2×C7×#501C2semidirect
5. C2×C7×#490C2semidirect
6. C2×C7×#481C2semidirect
7. C2×C7×#480C2semidirect
8. C2×C7×#478C2semidirect
9. C2×C7×#432C2semidirect
20. C2×C7×#125C22semidirect
31. C7×#501C22semidirect
32. C7×#490C22semidirect
33. C7×#481C22semidirect
34. C7×#480C22semidirect
35. C7×#478C22semidirect
36. C7×#432C22semidirect
44. C2×#3234C7direct (N × H)
108. C22×#125C2×C7semidirect
109. #3234C2×C7direct (N × H)
110. C2×#501C2×C7semidirect
111. C2×#490C2×C7semidirect
112. C2×#481C2×C7semidirect
113. C2×#480C2×C7semidirect
114. C2×#478C2×C7semidirect
115. C2×#432C2×C7semidirect
185. C2×#125C22×C7semidirect
196. #501C22×C7semidirect
197. #490C22×C7semidirect
198. #481C22×C7semidirect
199. #480C22×C7semidirect
200. #478C22×C7semidirect
201. #432C22×C7semidirect
396. C2×C7#3234direct (N × H)
442. C7C2×#3234direct (N × H)
467. C2C7×#3234direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5322
C22×C7×#1252
C7×#32342
C2×C7×#5012
C2×C7×#4902
C2×C7×#4812
C2×C7×#4802
C2×C7×#4782
C2×C7×#4322
C2×#32347