Finite Group Databasebeta

← C2×C7×#2951 #13237381

Subgroups — C2×C7×#2951 #13237381

Order 1792.

2806 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#135C211yesmaximal, characteristic
2C2×C7×#520C212yesmaximal, complemented
3C7×#2951C2116yesmaximal, direct product factor
4C2×C7×#440C214yesmaximal, complemented
5C2×C4×C7×Q16C212yesmaximal, complemented
6C2×C4×C7×D8C212yesmaximal, complemented
7C22×C7×#107C211yesmaximal, characteristic, complemented
8C2×C7×#417C212yesmaximal, complemented
9C2×C7×#405C211yesmaximal, characteristic, complemented
10C22×C7×#41C2212yes—
11C2×C7×#135C4116yescomplemented
12C2×C7×#135C2218yes—
13C22×C7×Q16C2211yescharacteristic
14C22×C7×SD16C2212yes—
15C22×C7×D8C2211yescharacteristic
16C22×C7×M16C2211yescharacteristic
17C23×C7×C8C2211yescharacteristic
18C7×#520—28no—
19C7×#520C22116yescomplemented
20C2×C4×C7×Q8C2214yes—
21C2×C4×C7×D4—24no—
22C2×C4×C7×D4C2214yes—
23C2×C7×#117—22no—
24C2×C7×#117C2211yescharacteristic
25C22×C7×C4⋊C4—22no—
26C22×C7×C4⋊C4C2212yes—
27C7×#440C22132yescomplemented
28C4×C7×Q16C22116yescomplemented
29C4×C7×D8C22116yescomplemented
30C2×C7×#107C2212yes—
31C2×C7×#107C2212yes—
32C2×C7×#107C2218yescomplemented
33C7×#417C22116yescomplemented
34C2×C7×#103C2214yes—
35C2×C7×#102C2214yes—
36C7×#405C2218yescomplemented
37C2×C7×#97C2212yes—
38C2×C4×C7×C8C2212yes—
39C2×#2951C711yesmaximal, characteristic, direct product factor, Sylow
40C2×C7×#41—24no—
41C2×C7×#41—28no—
42C2×C7×#41C2×C4116yescomplemented
43C2×C7×#41C2318yes—
44C22×C7×Q8C2312yes—
45C22×C7×D4—22no—
46C22×C7×D4C2312yes—
47C23×C4×C7—22no—
48C23×C4×C7—22no—
49C23×C4×C7—24no—
50C23×C4×C7C2311yescharacteristic
51C7×#135C2×C4164yescomplemented
52C2×C7×Q16C2×C418yescomplemented
53C2×C7×Q16C2314yes—
54C2×C7×SD16C2×C4116yescomplemented
55C2×C7×SD16C2318yes—
56C2×C7×D8C2×C418yescomplemented
57C2×C7×D8C2314yes—
58C2×C7×M16—24no—
59C2×C7×M16C2314yes—
60C22×C7×C8C2×C418yescomplemented
61C22×C7×C8C2312yes—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2314yes—
65C4×C7×Q8—28no—
66C4×C7×Q8C23116yes—
67C4×C7×D4—232no—
68C4×C7×D4—28no—
69C4×C7×D4C23116yes—
70C7×#117—216no—
71C7×#117D414yes—
72C7×#117C2314yes—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—24no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4—216no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2314yes—
80C2×C7×C4⋊C4C2318yes—
81C2×C7×C22⋊C4—24no—
82C2×C7×C22⋊C4D412yes—
83C2×C42×C7—24no—
84C2×C42×C7C2312yes—
85C7×#107C2318yes—
86C7×#107C2318yes—
87C7×#103C23116yes—
88C7×#102C23116yes—
89C7×#97C2318yes—
90C4×C7×C8C2318yes—
91C22×#135C2×C711yescharacteristic
92C2×#520C2×C712yescomplemented
93#2951C2×C7116yesdirect product factor
94C2×#440C2×C714yescomplemented
95C2×C4×Q16C2×C712yescomplemented
96C2×C4×D8C2×C712yescomplemented
97C22×#107C2×C711yescharacteristic, complemented
98C2×#417C2×C712yescomplemented
99C2×#405C2×C711yescharacteristic, complemented
100C24×C7—22no—
101C24×C7C2×D411yescharacteristic
102C7×#41—232no—
103C7×#41—216no—
104C7×#41C22×C4132yescomplemented
105C2×C7×Q8—22no—
106C2×C7×Q8—24no—
107C2×C7×Q8C22×C418yes—
108C2×C7×Q8C2414yes—
109C2×C7×D4—216no—
110C2×C7×D4—28no—
111C2×C7×D4—22no—
112C2×C7×D4—24no—
113C2×C7×D4C22×C418yes—
114C2×C7×D4C2414yes—
115C22×C4×C7—22no—
116C22×C4×C7—216no—
117C22×C4×C7—28no—
118C22×C4×C7—24no—
119C22×C4×C7—28no—
120C22×C4×C7—216no—
121C22×C4×C7—88no—
122C22×C4×C7—84no—
123C22×C4×C7C22×C414yes—
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7#4114yes—
129C22×C4×C7C2412yes—
130C22×C4×C7C2411yescharacteristic
131C7×Q16C22×C4116yescomplemented
132C7×SD16C22×C4132yescomplemented
133C7×D8C22×C4116yescomplemented
134C7×M16—28no—
135C2×C7×C8—24no—
136C2×C7×C8C22×C414yes—
137C2×C7×C8C22×C414yes—
138C2×C7×C8C22×C4116yescomplemented
139C2×C7×C8C2414yes—
140C2×C7×C8C2412yes—
141C2×C7×C8C2412yes—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—216no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4C2×D414yes—
146C7×C4⋊C4C2×D414yes—
147C7×C4⋊C4C2418yes—
148C7×C22⋊C4—216no—
149C7×C22⋊C4C2×D418yes—
150C42×C7—216no—
151C42×C7C2×D414yes—
152C42×C7C2414yes—
153C22×#41C22×C712yes—
154C2×#135C4×C7116yescomplemented
155C2×#135C22×C718yes—
156C22×Q16C22×C711yescharacteristic
157C22×SD16C22×C712yes—
158C22×D8C22×C711yescharacteristic
159C22×M16C22×C711yescharacteristic
160C23×C8C22×C711yescharacteristic
161#520—28no—
162#520C22×C7116yescomplemented
163C2×C4×Q8C22×C714yes—
164C2×C4×D4—24no—
165C2×C4×D4C22×C714yes—
166C2×#117—22no—
167C2×#117C22×C711yescharacteristic
168C22×C4⋊C4—22no—
169C22×C4⋊C4C22×C712yes—
170#440C22×C7132yescomplemented
171C4×Q16C22×C7116yescomplemented
172C4×D8C22×C7116yescomplemented
173C2×#107C22×C712yes—
174C2×#107C22×C712yes—
175C2×#107C22×C718yescomplemented
176#417C22×C7116yescomplemented
177C2×#103C22×C714yes—
178C2×#102C22×C714yes—
179#405C22×C718yescomplemented
180C2×#97C22×C712yes—
181C2×C4×C8C22×C712yes—
182C23×C7—22no—
183C23×C7—22no—
184C23×C7—28no—
185C23×C7—24no—
186C23×C7—44no—
187C23×C7—44no—
188C23×C7C4×D414yes—
189C23×C7C22×D412yes—
190C23×C7C22×D411yescharacteristic
191C7×Q8—28no—
192C7×Q8—24no—
193C7×Q8C23×C418yes—
194C7×D4—232no—
195C7×D4—28no—
196C7×D4—24no—
197C7×D4C23×C418yes—
198C2×C4×C7—28no—
199C2×C4×C7—24no—
200C2×C4×C7—24no—
201C2×C4×C7—24no—
202C2×C4×C7—232no—
203C2×C4×C7—28no—
204C2×C4×C7—48no—
205C2×C4×C7—816no—
206C2×C4×C7—88no—
207C2×C4×C7C4×D414yes—
208C2×C4×C7C4×D418yes—
209C2×C4×C7C23×C412yes—
210C2×C4×C7C23×C414yes—
211C2×C4×C7C22×D414yes—
212C2×C4×C7C22×D414yes—
213C2×C4×C7C22×D411yescharacteristic
214C2×C4×C7C22×D412yes—
215C2×C4×C7C2×#4112yes—
216C2×C4×C7C2×#4114yes—
217C2×C4×C7C2511yescharacteristic
218C7×C8—24no—
219C7×C8C23×C414yes—
220C7×C8C23×C414yes—
221C2×#41—24no—
222C2×#41—28no—
223C2×#41C2×C4×C7116yescomplemented
224C2×#41C23×C718yes—
225C22×Q8C23×C712yes—
226C22×D4—22no—
227C22×D4C23×C712yes—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4C23×C711yescharacteristic
232#135C2×C4×C7164yescomplemented
233C2×Q16C2×C4×C718yescomplemented
234C2×Q16C23×C714yes—
235C2×SD16C2×C4×C7116yescomplemented
236C2×SD16C23×C718yes—
237C2×D8C2×C4×C718yescomplemented
238C2×D8C23×C714yes—
239C2×M16—24no—
240C2×M16C23×C714yes—
241C22×C8C2×C4×C718yescomplemented
242C22×C8C23×C712yes—
243C22×C8C23×C711yescharacteristic
244C22×C8C23×C711yescharacteristic
245C22×C8C23×C714yes—
246C4×Q8—28no—
247C4×Q8C23×C7116yes—
248C4×D4—232no—
249C4×D4—28no—
250C4×D4C23×C7116yes—
251#117—216no—
252#117C7×D414yes—
253#117C23×C714yes—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4C7×D411yescharacteristic
259C2×C4⋊C4C7×D411yescharacteristic
260C2×C4⋊C4C23×C714yes—
261C2×C4⋊C4C23×C718yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4C7×D412yes—
264C2×C42—24no—
265C2×C42C23×C712yes—
266#107C23×C718yes—
267#107C23×C718yes—
268#103C23×C7116yes—
269#102C23×C7116yes—
270#97C23×C718yes—
271C4×C8C23×C718yes—
272C22×C7—28no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7—28no—
276C22×C7—416no—
277C22×C7—44no—
278C22×C7—44no—
279C22×C7C2×C4×D414yes—
280C22×C7C2×C4×D412yes—
281C22×C7#57612yes—
282C22×C7#57812yes—
283C22×C7C23×D411yescharacteristic
284C4×C7—28no—
285C4×C7—24no—
286C4×C7—24no—
287C4×C7—48no—
288C4×C7C2×C4×D414yes—
289C4×C7C2×C4×D412yes—
290C4×C7C24×C411yescharacteristic
291C4×C7C22×#4111yescharacteristic
292C24—22no—
293C24C2×C7×D411yescharacteristic
294#41—232no—
295#41—216no—
296#41C22×C4×C7132yescomplemented
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8C22×C4×C718yes—
300C2×Q8C24×C714yes—
301C2×D4—216no—
302C2×D4—28no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4C22×C4×C718yes—
306C2×D4C24×C714yes—
307C22×C4—22no—
308C22×C4—216no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—216no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4C22×C4×C714yes—
316C22×C4C2×C7×D412yes—
317C22×C4C2×C7×D412yes—
318C22×C4C2×C7×D411yescharacteristic
319C22×C4C2×C7×D412yes—
320C22×C4C7×#4114yes—
321C22×C4C24×C711yescharacteristic
322C22×C4C24×C712yes—
323Q16C22×C4×C7116yescomplemented
324SD16C22×C4×C7132yescomplemented
325D8C22×C4×C7116yescomplemented
326M16—28no—
327C2×C8—24no—
328C2×C8C22×C4×C714yes—
329C2×C8C22×C4×C714yes—
330C2×C8C22×C4×C7116yescomplemented
331C2×C8C24×C714yes—
332C2×C8C24×C712yes—
333C2×C8C24×C712yes—
334C4⋊C4—28no—
335C4⋊C4—216no—
336C4⋊C4—24no—
337C4⋊C4C2×C7×D414yes—
338C4⋊C4C2×C7×D414yes—
339C4⋊C4C24×C718yes—
340C22⋊C4—216no—
341C22⋊C4C2×C7×D418yes—
342C42—216no—
343C42C2×C7×D414yes—
344C42C24×C714yes—
345C2×C7—24no—
346C2×C7—48no—
347C2×C7#295114yesdirect product factor
348C2×C7C22×C4×D411yescharacteristic
349C2×C7C2×#57611yescharacteristic
350C2×C7C2×#57811yescharacteristic
351C23—22no—
352C23—22no—
353C23—24no—
354C23—28no—
355C23—44no—
356C23—44no—
357C23C4×C7×D414yes—
358C23C22×C7×D412yes—
359C23C22×C7×D411yescharacteristic
360Q8—24no—
361Q8—28no—
362Q8C23×C4×C718yes—
363D4—232no—
364D4—24no—
365D4—28no—
366D4C23×C4×C718yes—
367C2×C4—24no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—232no—
372C2×C4—28no—
373C2×C4—48no—
374C2×C4—816no—
375C2×C4—88no—
376C2×C4C4×C7×D414yes—
377C2×C4C4×C7×D418yes—
378C2×C4C23×C4×C712yes—
379C2×C4C23×C4×C714yes—
380C2×C4C22×C7×D414yes—
381C2×C4C22×C7×D414yes—
382C2×C4C22×C7×D411yescharacteristic
383C2×C4C22×C7×D412yes—
384C2×C4C2×C7×#4112yes—
385C2×C4C2×C7×#4114yes—
386C2×C4C25×C711yescharacteristic
387C8—24no—
388C8C23×C4×C714yes—
389C8C23×C4×C714yes—
390C7C2×#295111yescharacteristic, direct product factor, Sylow
391C22—22no—
392C22—22no—
393C22—28no—
394C22—28no—
395C22—44no—
396C22—44no—
397C22—416no—
398C22C2×C4×C7×D414yes—
399C22C2×C4×C7×D412yes—
400C22C7×#57612yes—
401C22C7×#57812yes—
402C22C23×C7×D411yescharacteristic
403C4—24no—
404C4—24no—
405C4—28no—
406C4—48no—
407C4C2×C4×C7×D414yes—
408C4C2×C4×C7×D412yes—
409C4C24×C4×C711yescharacteristic
410C4C22×C7×#4111yescharacteristic
411C2—24no—
412C2—48no—
413C2C7×#295114yesdirect product factor
414C2C22×C4×C7×D411yescharacteristic
415C2C2×C7×#57611yescharacteristic
416C2C2×C7×#57811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC4C24×C4×C7
Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×#135C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#520C2semidirect
3. C7×#2951C2direct (N × H)
4. C2×C7×#440C2semidirect
5. C2×C4×C7×Q16C2semidirect
6. C2×C4×C7×D8C2semidirect
7. C22×C7×#107C2semidirect
8. C2×C7×#417C2semidirect
9. C2×C7×#405C2semidirect
11. C2×C7×#135C4semidirect
19. C7×#520C22semidirect
27. C7×#440C22semidirect
28. C4×C7×Q16C22semidirect
29. C4×C7×D8C22semidirect
32. C2×C7×#107C22semidirect
33. C7×#417C22semidirect
36. C7×#405C22semidirect
39. C2×#2951C7direct (N × H)
42. C2×C7×#41C2×C4semidirect
51. C7×#135C2×C4semidirect
52. C2×C7×Q16C2×C4semidirect
54. C2×C7×SD16C2×C4semidirect
56. C2×C7×D8C2×C4semidirect
60. C22×C7×C8C2×C4semidirect
92. C2×#520C2×C7semidirect
93. #2951C2×C7direct (N × H)
94. C2×#440C2×C7semidirect
95. C2×C4×Q16C2×C7semidirect
96. C2×C4×D8C2×C7semidirect
97. C22×#107C2×C7semidirect
98. C2×#417C2×C7semidirect
99. C2×#405C2×C7semidirect
104. C7×#41C22×C4semidirect
131. C7×Q16C22×C4semidirect
132. C7×SD16C22×C4semidirect
133. C7×D8C22×C4semidirect
138. C2×C7×C8C22×C4semidirect
154. C2×#135C4×C7semidirect
162. #520C22×C7semidirect
170. #440C22×C7semidirect
171. C4×Q16C22×C7semidirect
172. C4×D8C22×C7semidirect
175. C2×#107C22×C7semidirect
176. #417C22×C7semidirect
179. #405C22×C7semidirect
223. C2×#41C2×C4×C7semidirect
232. #135C2×C4×C7semidirect
233. C2×Q16C2×C4×C7semidirect
235. C2×SD16C2×C4×C7semidirect
237. C2×D8C2×C4×C7semidirect
241. C22×C8C2×C4×C7semidirect
296. #41C22×C4×C7semidirect
323. Q16C22×C4×C7semidirect
324. SD16C22×C4×C7semidirect
325. D8C22×C4×C7semidirect
330. C2×C8C22×C4×C7semidirect
347. C2×C7#2951direct (N × H)
390. C7C2×#2951direct (N × H)
413. C2C7×#2951direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1352
C2×C7×#5202
C7×#29512
C2×C7×#4402
C2×C4×C7×Q162
C2×C4×C7×D82
C22×C7×#1072
C2×C7×#4172
C2×C7×#4052
C2×#29517