Finite Group Databasebeta

← C2×C7×#3487 #13240005

Subgroups — C2×C7×#3487 #13240005

Order 1792.

11510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×2−1+4C211yesmaximal, characteristic, complemented
2C23×C7×#41C211yesmaximal, characteristic, complemented
3C7×#3487C2132yesmaximal, direct product factor
4C2×C7×#550C218yesmaximal, complemented
5C2×C7×#548C218yesmaximal, complemented
6C2×C7×#535C216yesmaximal, complemented
7C2×C7×#525C216yesmaximal, complemented
8C2×C7×#519C211yesmaximal, characteristic, complemented
9C2×C7×2−1+4—216no—
10C2×C7×2−1+4C22116yescomplemented
11C22×C7×#41—216no—
12C22×C7×#41C2218yes—
13C22×C7×#41C2216yes—
14C22×C7×#41C22116yescomplemented
15C22×C7×#41C2214yescomplemented
16C23×C7×Q8C2211yescharacteristic
17C23×C7×Q8C2214yescomplemented
18C23×C7×D4C2213yes—
19C24×C4×C7C2213yes—
20C7×#550C221128yescomplemented
21C7×#548C221128yescomplemented
22C7×#535C22196yescomplemented
23C2×C7×#127C22112yescomplemented
24C2×C7×#124C22112yescomplemented
25C7×#525C22196yescomplemented
26C2×C7×#123C2214yes—
27C2×C7×C22⋊Q8C22112yescomplemented
28C2×C7×#121C22112yescomplemented
29C2×C7×#121C22124yescomplemented
30C2×C7×#120C22112yes—
31C7×#519C22116yescomplemented
32C2×C4×C7×Q8C2218yescomplemented
33C2×C4×C7×D4C22124yescomplemented
34C2×C7×#117C2216yes—
35C2×#3487C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7C2313yes—
37C7×2−1+4—2128no—
38C2×C7×#41—264no—
39C2×C7×#41—232no—
40C2×C7×#41—2128no—
41C2×C7×#41—248no—
42C2×C7×#41C23164yes—
43C2×C7×#41C23148yes—
44C2×C7×#41C23132yescomplemented
45C22×C7×Q8—232no—
46C22×C7×Q8D4116yescomplemented
47C22×C7×Q8C2316yes—
48C22×C7×Q8C2314yes—
49C22×C7×Q8C2318yes—
50C22×C7×Q8C23132yescomplemented
51C22×C7×D4—224no—
52C22×C7×D4C23112yes—
53C22×C7×D4C23124yes—
54C22×C7×D4C2314yes—
55C22×C7×D4C2316yes—
56C22×C7×D4C23124yes—
57C23×C4×C7—224no—
58C23×C4×C7C23112yes—
59C23×C4×C7C23112yes—
60C23×C4×C7C23124yes—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2313yes—
63C7×#127C23196yescomplemented
64C7×#124C23196yescomplemented
65C7×#123C23132yes—
66C7×C22⋊Q8C23196yescomplemented
67C7×#121C23196yescomplemented
68C7×#121C231192yescomplemented
69C7×#120C23196yes—
70C4×C7×Q8C23164yescomplemented
71C4×C7×D4C231192yescomplemented
72C7×#117C23148yes—
73C2×C7×C4⋊C4C23112yes—
74C2×C7×C4⋊C4C2318yes—
75C2×C7×C22⋊C4C2314yes—
76C2×C7×C22⋊C4C2318yes—
77C2×C7×C22⋊C4C23124yes—
78C2×C42×C7C23112yes—
79C22×2−1+4C2×C711yescharacteristic, complemented
80C23×#41C2×C711yescharacteristic, complemented
81#3487C2×C7132yesdirect product factor
82C2×#550C2×C718yescomplemented
83C2×#548C2×C718yescomplemented
84C2×#535C2×C716yescomplemented
85C2×#525C2×C716yescomplemented
86C2×#519C2×C711yescharacteristic, complemented
87C24×C7—212no—
88C24×C7—412no—
89C24×C7C2416yes—
90C24×C7C2414yes—
91C24×C7C2411yescharacteristic
92C24×C7C24112yes—
93C7×#41—2256no—
94C7×#41—2128no—
95C7×#41—2192no—
96C2×C7×Q8—2128no—
97C2×C7×Q8—224no—
98C2×C7×Q8C2×D4132yes—
99C2×C7×Q8C2×D4164yescomplemented
100C2×C7×Q8C24124yes—
101C2×C7×Q8C24116yes—
102C2×C7×D4—216no—
103C2×C7×D4—248no—
104C2×C7×D4—296no—
105C2×C7×D4—224no—
106C2×C7×D4—448no—
107C2×C7×D4C24116yes—
108C2×C7×D4C24148yes—
109C2×C7×D4C24124yes—
110C2×C7×D4C24196yes—
111C22×C4×C7—24no—
112C22×C4×C7—248no—
113C22×C4×C7—296no—
114C22×C4×C7—248no—
115C22×C4×C7C2×D4124yes—
116C22×C4×C7C24148yes—
117C22×C4×C7C24148yes—
118C22×C4×C7C2414yes—
119C22×C4×C7C2418yes—
120C22×C4×C7C2414yes—
121C22×C4×C7C2412yes—
122C22×C4×C7C24112yes—
123C22×C4×C7C2416yes—
124C7×C4⋊C4C24148yes—
125C7×C4⋊C4C24132yes—
126C7×C22⋊C4C24116yes—
127C7×C22⋊C4C24132yes—
128C7×C22⋊C4C24196yes—
129C42×C7C24148yes—
130C2×2−1+4—216no—
131C2×2−1+4C22×C7116yescomplemented
132C22×#41—216no—
133C22×#41C22×C718yes—
134C22×#41C22×C716yes—
135C22×#41C22×C7116yescomplemented
136C22×#41C22×C714yescomplemented
137C23×Q8C22×C711yescharacteristic
138C23×Q8C22×C714yescomplemented
139C23×D4C22×C713yes—
140C24×C4C22×C713yes—
141#550C22×C71128yescomplemented
142#548C22×C71128yescomplemented
143#535C22×C7196yescomplemented
144C2×#127C22×C7112yescomplemented
145C2×#124C22×C7112yescomplemented
146#525C22×C7196yescomplemented
147C2×#123C22×C714yes—
148C2×C22⋊Q8C22×C7112yescomplemented
149C2×#121C22×C7112yescomplemented
150C2×#121C22×C7124yescomplemented
151C2×#120C22×C7112yes—
152#519C22×C7116yescomplemented
153C2×C4×Q8C22×C718yescomplemented
154C2×C4×D4C22×C7124yescomplemented
155C2×#117C22×C716yes—
156C23×C7—24no—
157C23×C7—28no—
158C23×C7—224no—
159C23×C7—212no—
160C23×C7—216no—
161C23×C7—224no—
162C23×C7—424no—
163C23×C7—448no—
164C23×C7C22×D414yes—
165C23×C7C25112yes—
166C23×C7C2518yes—
167C23×C7C2512yes—
168C23×C7C2511yescharacteristic
169C7×Q8—248no—
170C7×Q8C22×D4164yes—
171C7×D4—232no—
172C7×D4—296no—
173C7×D4—248no—
174C7×D4—496no—
175C2×C4×C7—28no—
176C2×C4×C7—24no—
177C2×C4×C7—28no—
178C2×C4×C7—216no—
179C2×C4×C7—296no—
180C2×C4×C7—296no—
181C2×C4×C7C22×D4124yes—
182C2×C4×C7C22×D4148yes—
183C2×C4×C7C25116yes—
184C2×C4×C7C2518yes—
185C2×C4×C7C2514yes—
186C2×C4×C7C25112yes—
187C25C23×C713yes—
1882−1+4—2128no—
189C2×#41—264no—
190C2×#41—232no—
191C2×#41—2128no—
192C2×#41—248no—
193C2×#41C23×C7148yes—
194C2×#41C23×C7164yes—
195C2×#41C23×C7132yescomplemented
196C22×Q8—232no—
197C22×Q8C7×D4116yescomplemented
198C22×Q8C23×C716yes—
199C22×Q8C23×C714yes—
200C22×Q8C23×C718yes—
201C22×Q8C23×C7132yescomplemented
202C22×D4—224no—
203C22×D4C23×C714yes—
204C22×D4C23×C7112yes—
205C22×D4C23×C716yes—
206C22×D4C23×C7124yes—
207C22×D4C23×C7124yes—
208C23×C4—224no—
209C23×C4C23×C7112yes—
210C23×C4C23×C7112yes—
211C23×C4C23×C711yescharacteristic
212C23×C4C23×C7124yes—
213C23×C4C23×C713yes—
214#127C23×C7196yescomplemented
215#124C23×C7196yescomplemented
216#123C23×C7132yes—
217C22⋊Q8C23×C7196yescomplemented
218#121C23×C7196yescomplemented
219#121C23×C71192yescomplemented
220#120C23×C7196yes—
221C4×Q8C23×C7164yescomplemented
222C4×D4C23×C71192yescomplemented
223#117C23×C7148yes—
224C2×C4⋊C4C23×C7112yes—
225C2×C4⋊C4C23×C718yes—
226C2×C22⋊C4C23×C714yes—
227C2×C22⋊C4C23×C718yes—
228C2×C22⋊C4C23×C7124yes—
229C2×C42C23×C7112yes—
230C22×C7—24no—
231C22×C7—28no—
232C22×C7—28no—
233C22×C7—212no—
234C22×C7—216no—
235C22×C7—224no—
236C22×C7—448no—
237C22×C7—416no—
238C22×C7—424no—
239C22×C7C23×D414yes—
240C22×C7C23×D412yes—
241C22×C7#58514yes—
242C22×C7C2611yescharacteristic
243C4×C7—24no—
244C4×C7—28no—
245C4×C7—216no—
246C4×C7C23×D4124yes—
247C24—212no—
248C24—412no—
249C24C24×C716yes—
250C24C24×C714yes—
251C24C24×C711yescharacteristic
252C24C24×C7112yes—
253#41—2256no—
254#41—2128no—
255#41—2192no—
256C2×Q8—2128no—
257C2×Q8—224no—
258C2×Q8C2×C7×D4132yes—
259C2×Q8C2×C7×D4164yescomplemented
260C2×Q8C24×C7124yes—
261C2×Q8C24×C7116yes—
262C2×D4—216no—
263C2×D4—248no—
264C2×D4—296no—
265C2×D4—224no—
266C2×D4—448no—
267C2×D4C24×C7116yes—
268C2×D4C24×C7148yes—
269C2×D4C24×C7124yes—
270C2×D4C24×C7196yes—
271C22×C4—24no—
272C22×C4—248no—
273C22×C4—296no—
274C22×C4—248no—
275C22×C4C2×C7×D4124yes—
276C22×C4C24×C718yes—
277C22×C4C24×C714yes—
278C22×C4C24×C712yes—
279C22×C4C24×C7148yes—
280C22×C4C24×C7148yes—
281C22×C4C24×C714yes—
282C22×C4C24×C716yes—
283C22×C4C24×C7112yes—
284C4⋊C4C24×C7148yes—
285C4⋊C4C24×C7132yes—
286C22⋊C4C24×C7116yes—
287C22⋊C4C24×C7132yes—
288C22⋊C4C24×C7196yes—
289C42C24×C7148yes—
290C2×C7—24no—
291C2×C7—48no—
292C2×C7—412no—
293C2×C7#348714yesdirect product factor
294C2×C7C24×D411yescharacteristic
295C2×C7C2×#58512yes—
296C23—24no—
297C23—28no—
298C23—224no—
299C23—212no—
300C23—216no—
301C23—224no—
302C23—424no—
303C23—448no—
304C23C22×C7×D414yes—
305C23C25×C7112yes—
306C23C25×C718yes—
307C23C25×C712yes—
308C23C25×C711yescharacteristic
309Q8—248no—
310Q8C22×C7×D4164yes—
311D4—232no—
312D4—296no—
313D4—248no—
314D4—496no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—216no—
319C2×C4—296no—
320C2×C4—296no—
321C2×C4C22×C7×D4124yes—
322C2×C4C22×C7×D4148yes—
323C2×C4C25×C7116yes—
324C2×C4C25×C718yes—
325C2×C4C25×C714yes—
326C2×C4C25×C7112yes—
327C7C2×#348711yescharacteristic, direct product factor, Sylow
328C22—24no—
329C22—28no—
330C22—28no—
331C22—212no—
332C22—216no—
333C22—224no—
334C22—448no—
335C22—424no—
336C22—416no—
337C22C23×C7×D414yes—
338C22C23×C7×D412yes—
339C22C7×#58514yes—
340C22C26×C711yescharacteristic
341C4—24no—
342C4—28no—
343C4—216no—
344C4C23×C7×D4124yes—
345C2—24no—
346C2—412no—
347C2—48no—
348C2C7×#348714yesdirect product factor
349C2C24×C7×D411yescharacteristic
350C2C2×C7×#58512yes—
subgroupG/H
center, socleC23×C7C25
commutator, Frattini, cosocleC22C26×C7
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C7×2−1+4C2semidirect
2. C23×C7×#41C2semidirect
3. C7×#3487C2direct (N × H)
4. C2×C7×#550C2semidirect
5. C2×C7×#548C2semidirect
6. C2×C7×#535C2semidirect
7. C2×C7×#525C2semidirect
8. C2×C7×#519C2semidirect
10. C2×C7×2−1+4C22semidirect
14. C22×C7×#41C22semidirect
15. C22×C7×#41C22semidirect
17. C23×C7×Q8C22semidirect
20. C7×#550C22semidirect
21. C7×#548C22semidirect
22. C7×#535C22semidirect
23. C2×C7×#127C22semidirect
24. C2×C7×#124C22semidirect
25. C7×#525C22semidirect
27. C2×C7×C22⋊Q8C22semidirect
28. C2×C7×#121C22semidirect
29. C2×C7×#121C22semidirect
31. C7×#519C22semidirect
32. C2×C4×C7×Q8C22semidirect
33. C2×C4×C7×D4C22semidirect
35. C2×#3487C7direct (N × H)
44. C2×C7×#41C23semidirect
46. C22×C7×Q8D4semidirect
50. C22×C7×Q8C23semidirect
63. C7×#127C23semidirect
64. C7×#124C23semidirect
66. C7×C22⋊Q8C23semidirect
67. C7×#121C23semidirect
68. C7×#121C23semidirect
70. C4×C7×Q8C23semidirect
71. C4×C7×D4C23semidirect
79. C22×2−1+4C2×C7semidirect
80. C23×#41C2×C7semidirect
81. #3487C2×C7direct (N × H)
82. C2×#550C2×C7semidirect
83. C2×#548C2×C7semidirect
84. C2×#535C2×C7semidirect
85. C2×#525C2×C7semidirect
86. C2×#519C2×C7semidirect
99. C2×C7×Q8C2×D4semidirect
131. C2×2−1+4C22×C7semidirect
135. C22×#41C22×C7semidirect
136. C22×#41C22×C7semidirect
138. C23×Q8C22×C7semidirect
141. #550C22×C7semidirect
142. #548C22×C7semidirect
143. #535C22×C7semidirect
144. C2×#127C22×C7semidirect
145. C2×#124C22×C7semidirect
146. #525C22×C7semidirect
148. C2×C22⋊Q8C22×C7semidirect
149. C2×#121C22×C7semidirect
150. C2×#121C22×C7semidirect
152. #519C22×C7semidirect
153. C2×C4×Q8C22×C7semidirect
154. C2×C4×D4C22×C7semidirect
195. C2×#41C23×C7semidirect
197. C22×Q8C7×D4semidirect
201. C22×Q8C23×C7semidirect
214. #127C23×C7semidirect
215. #124C23×C7semidirect
217. C22⋊Q8C23×C7semidirect
218. #121C23×C7semidirect
219. #121C23×C7semidirect
221. C4×Q8C23×C7semidirect
222. C4×D4C23×C7semidirect
259. C2×Q8C2×C7×D4semidirect
293. C2×C7#3487direct (N × H)
327. C7C2×#3487direct (N × H)
348. C2C7×#3487direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×2−1+42
C23×C7×#412
C7×#34872
C2×C7×#5502
C2×C7×#5482
C2×C7×#5352
C2×C7×#5252
C2×C7×#5192
C2×#34877