Finite Group Databasebeta

← C2×C7×#2970 #13237418

Subgroups — C2×C7×#2970 #13237418

Order 1792.

2254 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×#41C211yesmaximal, characteristic
2C7×#2970C2116yesmaximal, direct product factor
3C2×C7×#446C212yesmaximal, complemented
4C2×C7×#435C214yesmaximal, complemented
5C2×C7×C8×D4C212yesmaximal, complemented
6C2×C7×#432C212yesmaximal, complemented
7C2×C7×#432C211yesmaximal, characteristic, complemented
8C2×C4×C7×M16C211yesmaximal, characteristic, complemented
9C22×C7×#97C212yesmaximal, complemented
10C22×C7×#41C411yescharacteristic
11C22×C7×M16C2212yes—
12C23×C7×C8C2214yes—
13C4×C7×#41C418yes—
14C4×C7×#41C2218yes—
15C2×C4×C7×Q8C411yescharacteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×D4C412yes—
18C2×C4×C7×D4C411yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2212yes—
21C2×C7×#117C411yescharacteristic
22C2×C7×#117C412yes—
23C22×C42×C7C2212yes—
24C22×C42×C7C2211yescharacteristic
25C7×#446C22116yescomplemented
26C7×#435C22132yescomplemented
27C7×C8×D4C22116yescomplemented
28C7×#432C2218yescomplemented
29C7×#432C22116yescomplemented
30C2×C7×C4⋊C8C2212yes—
31C2×C7×C4⋊C8C2214yes—
32C2×C7×C22⋊C8C2214yes—
33C2×C7×C22⋊C8C2212yes—
34C4×C7×M16C2218yescomplemented
35C2×C7×#97C2212yes—
36C2×C7×#97C2211yescharacteristic
37C2×C7×#97C2214yes—
38C2×C7×#97C2211yescharacteristic
39C2×C7×#97C22116yescomplemented
40C2×C4×C7×C8C2212yes—
41C2×#2970C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—28no—
43C2×C7×#41C2×C418yes—
44C22×C7×Q8C2×C411yescharacteristic
45C22×C7×D4C2×C412yes—
46C22×C7×D4C2×C411yescharacteristic
47C23×C4×C7C2×C412yes—
48C23×C4×C7C2×C411yescharacteristic
49C23×C4×C7C2×C411yescharacteristic
50C23×C4×C7C2×C412yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C2×C7×M16—28no—
54C2×C7×M16C2318yes—
55C22×C7×C8—216no—
56C22×C7×C8C23116yes—
57C22×C7×C8C2312yes—
58C22×C7×C8C2314yes—
59C22×C7×C8C2312yes—
60C4×C7×Q8C2×C418yes—
61C4×C7×Q8C2×C414yes—
62C4×C7×Q8C2314yes—
63C4×C7×D4C2×C4116yes—
64C4×C7×D4C2×C418yes—
65C4×C7×D4C2×C418yes—
66C4×C7×D4C2×C414yes—
67C4×C7×D4C2314yes—
68C4×C7×D4C2318yes—
69C7×#117C2×C418yes—
70C7×#117C2×C4116yes—
71C2×C7×C4⋊C4C2×C412yes—
72C2×C7×C4⋊C4C2×C411yescharacteristic
73C2×C7×C4⋊C4C2×C412yes—
74C2×C7×C4⋊C4C2×C411yescharacteristic
75C2×C7×C22⋊C4C2×C412yes—
76C2×C7×C22⋊C4C2×C412yes—
77C2×C7×C22⋊C4C2×C412yes—
78C2×C42×C7C2×C412yes—
79C2×C42×C7C2×C411yescharacteristic
80C2×C42×C7C2×C412yes—
81C2×C42×C7C2×C411yescharacteristic
82C2×C42×C7C2×C418yes—
83C2×C42×C7C2×C414yes—
84C2×C42×C7C2318yes—
85C2×C42×C7C2314yes—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2312yes—
89C7×C4⋊C8C2318yes—
90C7×C4⋊C8C23116yes—
91C7×C22⋊C8C23116yes—
92C7×C22⋊C8C2318yes—
93C7×#97C2314yes—
94C7×#97C2318yes—
95C7×#97C23116yes—
96C7×#97C2314yes—
97C4×C7×C8C2318yes—
98C2×C4×#41C2×C711yescharacteristic
99#2970C2×C7116yesdirect product factor
100C2×#446C2×C712yescomplemented
101C2×#435C2×C714yescomplemented
102C2×C8×D4C2×C712yescomplemented
103C2×#432C2×C711yescharacteristic, complemented
104C2×#432C2×C712yescomplemented
105C2×C4×M16C2×C711yescharacteristic, complemented
106C22×#97C2×C712yescomplemented
107C24×C7C22×C411yescharacteristic
108C24×C7C22×C412yes—
109C7×#41—232no—
110C2×C7×Q8—24no—
111C2×C7×Q8C22×C414yes—
112C2×C7×D4—28no—
113C2×C7×D4M1618yescomplemented
114C2×C7×D4C22×C418yes—
115C2×C7×D4C22×C414yes—
116C22×C4×C7—28no—
117C22×C4×C7—24no—
118C22×C4×C7—28no—
119C22×C4×C7—24no—
120C22×C4×C7—44no—
121C22×C4×C7—42no—
122C22×C4×C7C22×C411yescharacteristic
123C22×C4×C7C22×C411yescharacteristic
124C22×C4×C7C22×C412yes—
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C412yes—
127C22×C4×C7C22×C414yes—
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C22×C412yes—
130C22×C4×C7C22×C412yes—
131C22×C4×C7C22×C418yes—
132C22×C4×C7C22×C414yes—
133C22×C4×C7C22×C414yes—
134C22×C4×C7C22×C418yes—
135C22×C4×C7C22×C411yescharacteristic
136C22×C4×C7C2414yes—
137C22×C4×C7C2412yes—
138C22×C4×C7C2411yescharacteristic
139C7×M16—216no—
140C2×C7×C8—28no—
141C2×C7×C8—24no—
142C2×C7×C8—232no—
143C2×C7×C8#4118yescomplemented
144C2×C7×C8C2414yes—
145C2×C7×C8C2418yes—
146C2×C7×C8C2414yes—
147C7×C4⋊C4C22×C418yes—
148C7×C4⋊C4C22×C414yes—
149C7×C4⋊C4C22×C418yes—
150C7×C4⋊C4C22×C414yes—
151C7×C22⋊C4C22×C418yes—
152C7×C22⋊C4C22×C418yes—
153C7×C22⋊C4C22×C418yes—
154C42×C7C22×C412yes—
155C42×C7C22×C418yes—
156C42×C7C22×C414yes—
157C42×C7C22×C418yes—
158C42×C7C22×C414yes—
159C42×C7C22×C414yes—
160C42×C7C22×C412yes—
161C42×C7C2412yes—
162C42×C7C2412yes—
163C42×C7C2414yes—
164C22×#41C4×C711yescharacteristic
165C22×M16C22×C712yes—
166C23×C8C22×C714yes—
167C4×#41C4×C718yes—
168C4×#41C22×C718yes—
169C2×C4×Q8C4×C711yescharacteristic
170C2×C4×Q8C22×C711yescharacteristic
171C2×C4×D4C4×C712yes—
172C2×C4×D4C4×C711yescharacteristic
173C2×C4×D4C22×C711yescharacteristic
174C2×C4×D4C22×C712yes—
175C2×#117C4×C711yescharacteristic
176C2×#117C4×C712yes—
177C22×C42C22×C712yes—
178C22×C42C22×C711yescharacteristic
179#446C22×C7116yescomplemented
180#435C22×C7132yescomplemented
181C8×D4C22×C7116yescomplemented
182#432C22×C718yescomplemented
183#432C22×C7116yescomplemented
184C2×C4⋊C8C22×C712yes—
185C2×C4⋊C8C22×C714yes—
186C2×C22⋊C8C22×C714yes—
187C2×C22⋊C8C22×C712yes—
188C4×M16C22×C718yescomplemented
189C2×#97C22×C711yescharacteristic
190C2×#97C22×C712yes—
191C2×#97C22×C714yes—
192C2×#97C22×C711yescharacteristic
193C2×#97C22×C7116yescomplemented
194C2×C4×C8C22×C712yes—
195C23×C7—24no—
196C23×C7—24no—
197C23×C7—22no—
198C23×C7—22no—
199C23×C7—22no—
200C23×C7C2×M1618yes—
201C23×C7C23×C412yes—
202C23×C7C23×C414yes—
203C23×C7C23×C411yescharacteristic
204C7×Q8—28no—
205C7×D4—216no—
206C7×D4C2×M16116yescomplemented
207C2×C4×C7—24no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—24no—
211C2×C4×C7—216no—
212C2×C4×C7—22no—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—28no—
216C2×C4×C7—24no—
217C2×C4×C7—28no—
218C2×C4×C7—28no—
219C2×C4×C7—24no—
220C2×C4×C7—48no—
221C2×C4×C7—44no—
222C2×C4×C7C2×M1614yes—
223C2×C4×C7C23×C412yes—
224C2×C4×C7C23×C412yes—
225C2×C4×C7C23×C414yes—
226C2×C4×C7C23×C412yes—
227C2×C4×C7C23×C412yes—
228C2×C4×C7C23×C414yes—
229C2×C4×C7C23×C414yes—
230C2×C4×C7C23×C411yescharacteristic
231C2×C4×C7C23×C412yes—
232C2×C4×C7C2×#4114yes—
233C2×C4×C7C2511yescharacteristic
234C7×C8—28no—
235C7×C8—24no—
236C7×C8C2×#4118yescomplemented
237C2×#41—28no—
238C2×#41C2×C4×C718yes—
239C22×Q8C2×C4×C711yescharacteristic
240C22×D4C2×C4×C712yes—
241C22×D4C2×C4×C711yescharacteristic
242C23×C4C2×C4×C712yes—
243C23×C4C2×C4×C711yescharacteristic
244C23×C4C2×C4×C711yescharacteristic
245C23×C4C2×C4×C712yes—
246C23×C4C23×C712yes—
247C23×C4C23×C711yescharacteristic
248C2×M16—28no—
249C2×M16C23×C718yes—
250C22×C8—216no—
251C22×C8C23×C714yes—
252C22×C8C23×C7116yes—
253C22×C8C23×C712yes—
254C22×C8C23×C712yes—
255C4×Q8C2×C4×C718yes—
256C4×Q8C2×C4×C714yes—
257C4×Q8C23×C714yes—
258C4×D4C2×C4×C7116yes—
259C4×D4C2×C4×C718yes—
260C4×D4C2×C4×C718yes—
261C4×D4C2×C4×C714yes—
262C4×D4C23×C714yes—
263C4×D4C23×C718yes—
264#117C2×C4×C718yes—
265#117C2×C4×C7116yes—
266C2×C4⋊C4C2×C4×C712yes—
267C2×C4⋊C4C2×C4×C711yescharacteristic
268C2×C4⋊C4C2×C4×C712yes—
269C2×C4⋊C4C2×C4×C711yescharacteristic
270C2×C22⋊C4C2×C4×C712yes—
271C2×C22⋊C4C2×C4×C712yes—
272C2×C22⋊C4C2×C4×C712yes—
273C2×C42C2×C4×C712yes—
274C2×C42C2×C4×C711yescharacteristic
275C2×C42C2×C4×C712yes—
276C2×C42C2×C4×C711yescharacteristic
277C2×C42C2×C4×C718yes—
278C2×C42C2×C4×C714yes—
279C2×C42C23×C711yescharacteristic
280C2×C42C23×C712yes—
281C2×C42C23×C718yes—
282C2×C42C23×C714yes—
283C2×C42C23×C711yescharacteristic
284C4⋊C8C23×C718yes—
285C4⋊C8C23×C7116yes—
286C22⋊C8C23×C7116yes—
287C22⋊C8C23×C718yes—
288#97C23×C714yes—
289#97C23×C718yes—
290#97C23×C7116yes—
291#97C23×C714yes—
292C4×C8C23×C718yes—
293C22×C7—24no—
294C22×C7—216no—
295C22×C7—24no—
296C22×C7—22no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—44no—
300C22×C7C4×#4112yes—
301C22×C7C22×M1618yes—
302C22×C7C22×M1612yes—
303C22×C7#56812yes—
304C22×C7C24×C411yescharacteristic
305C4×C7—24no—
306C4×C7—22no—
307C4×C7—22no—
308C4×C7—24no—
309C4×C7—22no—
310C4×C7—24no—
311C4×C7—22no—
312C4×C7C4×#4112yes—
313C4×C7C22×M1614yes—
314C4×C7C22×#4112yes—
315C24C22×C4×C711yescharacteristic
316C24C22×C4×C712yes—
317#41—232no—
318C2×Q8—24no—
319C2×Q8C22×C4×C714yes—
320C2×D4—28no—
321C2×D4C7×M1618yescomplemented
322C2×D4C22×C4×C718yes—
323C2×D4C22×C4×C714yes—
324C22×C4—28no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4C22×C4×C711yescharacteristic
331C22×C4C22×C4×C711yescharacteristic
332C22×C4C22×C4×C712yes—
333C22×C4C22×C4×C711yescharacteristic
334C22×C4C22×C4×C712yes—
335C22×C4C22×C4×C714yes—
336C22×C4C22×C4×C711yescharacteristic
337C22×C4C22×C4×C712yes—
338C22×C4C22×C4×C712yes—
339C22×C4C22×C4×C718yes—
340C22×C4C22×C4×C714yes—
341C22×C4C22×C4×C714yes—
342C22×C4C22×C4×C718yes—
343C22×C4C22×C4×C711yescharacteristic
344C22×C4C24×C714yes—
345C22×C4C24×C712yes—
346C22×C4C24×C711yescharacteristic
347M16—216no—
348C2×C8—28no—
349C2×C8—24no—
350C2×C8—232no—
351C2×C8C7×#4118yescomplemented
352C2×C8C24×C714yes—
353C2×C8C24×C718yes—
354C2×C8C24×C714yes—
355C4⋊C4C22×C4×C718yes—
356C4⋊C4C22×C4×C714yes—
357C4⋊C4C22×C4×C718yes—
358C4⋊C4C22×C4×C714yes—
359C22⋊C4C22×C4×C718yes—
360C22⋊C4C22×C4×C718yes—
361C22⋊C4C22×C4×C718yes—
362C42C22×C4×C712yes—
363C42C22×C4×C718yes—
364C42C22×C4×C714yes—
365C42C22×C4×C718yes—
366C42C22×C4×C714yes—
367C42C22×C4×C714yes—
368C42C22×C4×C712yes—
369C42C24×C712yes—
370C42C24×C712yes—
371C42C24×C714yes—
372C2×C7—28no—
373C2×C7—42no—
374C2×C7#297014yesdirect product factor
375C2×C7C2×C4×#4111yescharacteristic
376C2×C7C23×M1611yescharacteristic
377C2×C7C2×#56811yescharacteristic
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23C2×C7×M1618yes—
384C23C23×C4×C712yes—
385C23C23×C4×C714yes—
386C23C23×C4×C711yescharacteristic
387Q8—28no—
388D4—216no—
389D4C2×C7×M16116yescomplemented
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—216no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—28no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4C2×C7×M1614yes—
406C2×C4C23×C4×C712yes—
407C2×C4C23×C4×C712yes—
408C2×C4C23×C4×C714yes—
409C2×C4C23×C4×C712yes—
410C2×C4C23×C4×C712yes—
411C2×C4C23×C4×C714yes—
412C2×C4C23×C4×C714yes—
413C2×C4C23×C4×C711yescharacteristic
414C2×C4C23×C4×C712yes—
415C2×C4C2×C7×#4114yes—
416C2×C4C25×C711yescharacteristic
417C8—28no—
418C8—24no—
419C8C2×C7×#4118yescomplemented
420C7C2×#297011yescharacteristic, direct product factor, Sylow
421C22—24no—
422C22—216no—
423C22—24no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—44no—
428C22C4×C7×#4112yes—
429C22C22×C7×M1618yes—
430C22C22×C7×M1612yes—
431C22C7×#56812yes—
432C22C24×C4×C711yescharacteristic
433C4—22no—
434C4—24no—
435C4—22no—
436C4—24no—
437C4—22no—
438C4—24no—
439C4—22no—
440C4C4×C7×#4112yes—
441C4C22×C7×M1614yes—
442C4C22×C7×#4112yes—
443C2—28no—
444C2—42no—
445C2C7×#297014yesdirect product factor
446C2C2×C4×C7×#4111yescharacteristic
447C2C23×C7×M1611yescharacteristic
448C2C2×C7×#56811yescharacteristic
subgroupG/H
centerC22×C4×C7C24
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC23×C7C23×C4
prime radicalC22×C7×#41C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2970C2direct (N × H)
3. C2×C7×#446C2semidirect
4. C2×C7×#435C2semidirect
5. C2×C7×C8×D4C2semidirect
6. C2×C7×#432C2semidirect
7. C2×C7×#432C2semidirect
8. C2×C4×C7×M16C2semidirect
9. C22×C7×#97C2semidirect
25. C7×#446C22semidirect
26. C7×#435C22semidirect
27. C7×C8×D4C22semidirect
28. C7×#432C22semidirect
29. C7×#432C22semidirect
34. C4×C7×M16C22semidirect
39. C2×C7×#97C22semidirect
41. C2×#2970C7direct (N × H)
99. #2970C2×C7direct (N × H)
100. C2×#446C2×C7semidirect
101. C2×#435C2×C7semidirect
102. C2×C8×D4C2×C7semidirect
103. C2×#432C2×C7semidirect
104. C2×#432C2×C7semidirect
105. C2×C4×M16C2×C7semidirect
106. C22×#97C2×C7semidirect
113. C2×C7×D4M16semidirect
143. C2×C7×C8#41semidirect
179. #446C22×C7semidirect
180. #435C22×C7semidirect
181. C8×D4C22×C7semidirect
182. #432C22×C7semidirect
183. #432C22×C7semidirect
188. C4×M16C22×C7semidirect
193. C2×#97C22×C7semidirect
206. C7×D4C2×M16semidirect
236. C7×C8C2×#41semidirect
321. C2×D4C7×M16semidirect
351. C2×C8C7×#41semidirect
374. C2×C7#2970direct (N × H)
389. D4C2×C7×M16semidirect
419. C8C2×C7×#41semidirect
420. C7C2×#2970direct (N × H)
445. C2C7×#2970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×#412
C7×#29702
C2×C7×#4462
C2×C7×#4352
C2×C7×C8×D42
C2×C7×#4322
C2×C4×C7×M162
C22×C7×#972
C2×#29707