Finite Group Databasebeta

← C2×C7×#3231 #13238060

Subgroups — C2×C7×#3231 #13238060

Order 1792.

1778 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic
2C2×C7×#533C211yesmaximal, characteristic
3C7×#3231C2116yesmaximal, direct product factor
4C2×C7×#500C211yesmaximal, characteristic, complemented
5C2×C7×#498C211yesmaximal, characteristic, complemented
6C2×C7×#487C211yesmaximal, characteristic, complemented
7C2×C7×#487C211yesmaximal, characteristic, complemented
8C2×C7×#485C214yesmaximal
9C2×C7×#479C214yesmaximal, complemented
10C2×C7×#431C211yesmaximal, characteristic
11C7×#533—24no—
12C7×#533—24no—
13C7×#533C2218yes—
14C7×#533C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C7×#125—21no—
20C2×C7×#125C2212yes—
21C2×C7×C22⋊Q8—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8—21no—
24C2×C7×C22⋊Q8C2212yes—
25C2×C4×C7×Q8—21no—
26C2×C4×C7×Q8—21no—
27C2×C4×C7×Q8—21no—
28C2×C4×C7×Q8C2212yes—
29C2×C7×#117—21no—
30C2×C7×#117C2212yes—
31C22×C42×C7C2211yescharacteristic
32C7×#500C2218yescomplemented
33C7×#498C2218yescomplemented
34C7×#487C2218yescomplemented
35C7×#487C2218yescomplemented
36C7×#485C22132yes—
37C7×#479C22132yescomplemented
38C7×#431C2218yes—
39C2×C7×#107C2214yes—
40C2×C7×#106C2214yes—
41C2×C7×C4⋊C8C2212yes—
42C2×C7×#103C2214yes—
43C2×C7×#103C2214yes—
44C2×C7×C22⋊C8C2212yes—
45C2×C4×C7×C8C2211yescharacteristic
46C2×C4×C7×C8C2211yescharacteristic
47C2×#3231C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8—21no—
51C22×C7×Q8C2312yes—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C22×C7×C8C2312yes—
56C22×C7×C8C2312yes—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8C2314yes—
62C7×C4⋊Q8C2314yes—
63C7×C4⋊Q8C2314yes—
64C7×C4⋊Q8C2314yes—
65C7×#125—28no—
66C7×#125—24no—
67C7×#125C2318yes—
68C7×C22⋊Q8—28no—
69C7×C22⋊Q8—28no—
70C7×C22⋊Q8—28no—
71C7×C22⋊Q8—24no—
72C7×C22⋊Q8C2318yes—
73C4×C7×Q8—28no—
74C4×C7×Q8—28no—
75C4×C7×Q8—28no—
76C4×C7×Q8—24no—
77C4×C7×Q8C2318yes—
78C7×#117—28no—
79C7×#117—24no—
80C7×#117C2318yes—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—22no—
95C2×C42×C7—22no—
96C2×C42×C7—22no—
97C2×C42×C7D411yescharacteristic
98C2×C42×C7D411yescharacteristic
99C2×C42×C7D414yes—
100C2×C42×C7C2311yescharacteristic
101C2×C42×C7C2311yescharacteristic
102C2×C42×C7C2314yes—
103C7×#107C23116yes—
104C7×#106C23116yes—
105C7×C4⋊C8C2318yes—
106C7×#103C23116yes—
107C7×#103C23116yes—
108C7×C22⋊C8C2318yes—
109C4×C7×C8C2314yes—
110C4×C7×C8C2314yes—
111C2×#533C2×C711yescharacteristic
112C2×#533C2×C711yescharacteristic
113#3231C2×C7116yesdirect product factor
114C2×#500C2×C711yescharacteristic, complemented
115C2×#498C2×C711yescharacteristic, complemented
116C2×#487C2×C711yescharacteristic, complemented
117C2×#487C2×C711yescharacteristic, complemented
118C2×#485C2×C714yes—
119C2×#479C2×C714yescomplemented
120C2×#431C2×C711yescharacteristic
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—22no—
123C2×C7×Q8—24no—
124C2×C7×Q8—24no—
125C2×C7×Q8—24no—
126C2×C7×Q8—28no—
127C2×C7×Q8—44no—
128C2×C7×Q8—44no—
129C2×C7×Q8—44no—
130C2×C7×Q8C2414yes—
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—22no—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D414yes—
140C22×C4×C7C2×D414yes—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2412yes—
147C22×C4×C7C2411yescharacteristic
148C2×C7×C8—24no—
149C2×C7×C8—24no—
150C2×C7×C8C2414yes—
151C2×C7×C8C2414yes—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—28no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4C2414yes—
165C7×C4⋊C4C2414yes—
166C7×C4⋊C4C2414yes—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—28no—
169C42×C7—28no—
170C42×C7—28no—
171C42×C7C2×D414yes—
172C42×C7C2×D414yes—
173C42×C7C2×D412yes—
174C42×C7C2×D412yes—
175C42×C7C2412yes—
176C42×C7C2412yes—
177#533—24no—
178#533—24no—
179#533C22×C718yes—
180#533C22×C718yes—
181C2×C4⋊Q8C22×C711yescharacteristic
182C2×C4⋊Q8C22×C711yescharacteristic
183C2×C4⋊Q8C22×C711yescharacteristic
184C2×C4⋊Q8C22×C711yescharacteristic
185C2×#125—21no—
186C2×#125C22×C712yes—
187C2×C22⋊Q8—21no—
188C2×C22⋊Q8—21no—
189C2×C22⋊Q8—21no—
190C2×C22⋊Q8C22×C712yes—
191C2×C4×Q8—21no—
192C2×C4×Q8—21no—
193C2×C4×Q8—21no—
194C2×C4×Q8C22×C712yes—
195C2×#117—21no—
196C2×#117C22×C712yes—
197C22×C42C22×C711yescharacteristic
198#500C22×C718yescomplemented
199#498C22×C718yescomplemented
200#487C22×C718yescomplemented
201#487C22×C718yescomplemented
202#485C22×C7132yes—
203#479C22×C7132yescomplemented
204#431C22×C718yes—
205C2×#107C22×C714yes—
206C2×#106C22×C714yes—
207C2×C4⋊C8C22×C712yes—
208C2×#103C22×C714yes—
209C2×#103C22×C714yes—
210C2×C22⋊C8C22×C712yes—
211C2×C4×C8C22×C711yescharacteristic
212C2×C4×C8C22×C711yescharacteristic
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7C22×D412yes—
217C23×C7C22×D411yescharacteristic
218C7×Q8—28no—
219C7×Q8—44no—
220C7×Q8—48no—
221C7×Q8—48no—
222C7×Q8—48no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—24no—
227C2×C4×C7—28no—
228C2×C4×C7—28no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—44no—
235C2×C4×C7—44no—
236C2×C4×C7—44no—
237C2×C4×C7—44no—
238C2×C4×C7#13514yes—
239C2×C4×C7#13714yes—
240C2×C4×C7C22×D412yes—
241C2×C4×C7C22×D412yes—
242C2×C4×C7C22×D412yes—
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×D412yes—
245C2×C4×C7C22×D412yes—
246C2×C4×C72−1+414yes—
247C2×C4×C7C2511yescharacteristic
248C7×C8—24no—
249C7×C8—24no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8C23×C712yes—
254C23×C4C7×D411yescharacteristic
255C23×C4C7×D411yescharacteristic
256C23×C4C23×C711yescharacteristic
257C22×C8C23×C712yes—
258C22×C8C23×C712yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8C23×C714yes—
264C4⋊Q8C23×C714yes—
265C4⋊Q8C23×C714yes—
266C4⋊Q8C23×C714yes—
267#125—28no—
268#125—24no—
269#125C23×C718yes—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8C23×C718yes—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—24no—
279C4×Q8C23×C718yes—
280#117—28no—
281#117—24no—
282#117C23×C718yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C23×C712yes—
293C2×C4⋊C4C23×C712yes—
294C2×C4⋊C4C23×C712yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C7×D411yescharacteristic
300C2×C42C7×D411yescharacteristic
301C2×C42C7×D414yes—
302C2×C42C23×C711yescharacteristic
303C2×C42C23×C711yescharacteristic
304C2×C42C23×C714yes—
305#107C23×C7116yes—
306#106C23×C7116yes—
307C4⋊C8C23×C718yes—
308#103C23×C7116yes—
309#103C23×C7116yes—
310C22⋊C8C23×C718yes—
311C4×C8C23×C714yes—
312C4×C8C23×C714yes—
313C22×C7—22no—
314C22×C7—22no—
315C22×C7—22no—
316C22×C7—44no—
317C22×C7#53612yes—
318C22×C7C2×#13512yes—
319C22×C7C2×#13712yes—
320C22×C7C23×D411yescharacteristic
321C4×C7—22no—
322C4×C7—22no—
323C4×C7—22no—
324C4×C7—44no—
325C4×C7—44no—
326C4×C7—44no—
327C4×C7—44no—
328C4×C7#53612yes—
329C4×C7C2×#13514yes—
330C4×C7C2×#13714yes—
331C4×C7C2×2−1+412yes—
332C24C2×C7×D411yescharacteristic
333C2×Q8—22no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—28no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8C24×C714yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×C7×D411yescharacteristic
350C22×C4C2×C7×D411yescharacteristic
351C22×C4C2×C7×D412yes—
352C22×C4C2×C7×D411yescharacteristic
353C22×C4C2×C7×D414yes—
354C22×C4C2×C7×D414yes—
355C22×C4C2×C7×D411yescharacteristic
356C22×C4C2×C7×D411yescharacteristic
357C22×C4C24×C712yes—
358C22×C4C24×C711yescharacteristic
359C2×C8—24no—
360C2×C8—24no—
361C2×C8C24×C714yes—
362C2×C8C24×C714yes—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4C24×C714yes—
376C4⋊C4C24×C714yes—
377C4⋊C4C24×C714yes—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C42—28no—
381C42—28no—
382C42C2×C7×D414yes—
383C42C2×C7×D414yes—
384C42C2×C7×D412yes—
385C42C2×C7×D412yes—
386C42C24×C712yes—
387C42C24×C712yes—
388C2×C7—42no—
389C2×C7#323114yesdirect product factor
390C2×C7C2×#53611yescharacteristic
391C2×C7C22×#13511yescharacteristic
392C2×C7C22×#13711yescharacteristic
393C23—22no—
394C23—22no—
395C23—22no—
396C23C22×C7×D412yes—
397C23C22×C7×D411yescharacteristic
398Q8—28no—
399Q8—44no—
400Q8—48no—
401Q8—48no—
402Q8—48no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4C7×#13514yes—
419C2×C4C7×#13714yes—
420C2×C4C22×C7×D412yes—
421C2×C4C22×C7×D412yes—
422C2×C4C22×C7×D412yes—
423C2×C4C22×C7×D411yescharacteristic
424C2×C4C22×C7×D412yes—
425C2×C4C22×C7×D412yes—
426C2×C4C7×2−1+414yes—
427C2×C4C25×C711yescharacteristic
428C8—24no—
429C8—24no—
430C7C2×#323111yescharacteristic, direct product factor, Sylow
431C22—22no—
432C22—22no—
433C22—22no—
434C22—44no—
435C22C7×#53612yes—
436C22C2×C7×#13512yes—
437C22C2×C7×#13712yes—
438C22C23×C7×D411yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—44no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4C7×#53612yes—
447C4C2×C7×#13514yes—
448C4C2×C7×#13714yes—
449C4C2×C7×2−1+412yes—
450C2—42no—
451C2C7×#323114yesdirect product factor
452C2C2×C7×#53611yescharacteristic
453C2C22×C7×#13511yescharacteristic
454C2C22×C7×#13711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC24×C7C2×D4
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
3. C7×#3231C2direct (N × H)
4. C2×C7×#500C2semidirect
5. C2×C7×#498C2semidirect
6. C2×C7×#487C2semidirect
7. C2×C7×#487C2semidirect
9. C2×C7×#479C2semidirect
32. C7×#500C22semidirect
33. C7×#498C22semidirect
34. C7×#487C22semidirect
35. C7×#487C22semidirect
37. C7×#479C22semidirect
47. C2×#3231C7direct (N × H)
113. #3231C2×C7direct (N × H)
114. C2×#500C2×C7semidirect
115. C2×#498C2×C7semidirect
116. C2×#487C2×C7semidirect
117. C2×#487C2×C7semidirect
119. C2×#479C2×C7semidirect
198. #500C22×C7semidirect
199. #498C22×C7semidirect
200. #487C22×C7semidirect
201. #487C22×C7semidirect
203. #479C22×C7semidirect
389. C2×C7#3231direct (N × H)
430. C7C2×#3231direct (N × H)
451. C2C7×#3231direct (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
C7×#32312
C2×C7×#5002
C2×C7×#4982
C2×C7×#4872
C2×C7×#4852
C2×C7×#4792
C2×C7×#4312
C2×#32317