Finite Group Databasebeta

← C2×C7×#2981 #13237445

Subgroups — C2×C7×#2981 #13237445

Order 1792.

2230 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic
2C7×#2981C2116yesmaximal, direct product factor
3C2×C7×#436C212yesmaximal, complemented
4C2×C7×#435C214yesmaximal, complemented
5C2×C7×C8×D4C212yesmaximal, complemented
6C2×C7×#432C211yesmaximal, characteristic, complemented
7C2×C7×#431C211yesmaximal, characteristic, complemented
8C2×C7×#408C214yesmaximal, complemented
9C22×C7×#41C2212yes—
10C22×C7×M16C2214yes—
11C23×C7×C8C2214yes—
12C7×#532C418yes—
13C7×#532C2218yes—
14C2×C7×C4⋊Q8C411yescharacteristic
15C2×C7×#127C411yescharacteristic
16C2×C7×#124C412yes—
17C2×C7×#121C412yes—
18C2×C7×#121C412yes—
19C2×C4×C7×D4C2212yes—
20C2×C4×C7×D4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C7×#436C22116yescomplemented
23C7×#435C22132yescomplemented
24C7×C8×D4C22116yescomplemented
25C7×#432C2218yescomplemented
26C7×#431C2218yescomplemented
27C2×C7×C4⋊C8C2212yes—
28C2×C7×C4⋊C8C2212yes—
29C7×#408C22132yescomplemented
30C2×C7×C22⋊C8C2212yes—
31C2×C7×C22⋊C8C2212yes—
32C2×C7×C22⋊C8C2214yes—
33C2×C7×C22⋊C8C2214yes—
34C2×C7×#97C2212yes—
35C2×C4×C7×C8C2211yescharacteristic
36C2×C4×C7×C8C2211yescharacteristic
37C2×#2981C711yesmaximal, characteristic, direct product factor, Sylow
38C2×C7×#41—48no—
39C2×C7×#41C2×C418yes—
40C2×C7×#41C2318yes—
41C22×C7×Q8C2×C412yes—
42C22×C7×D4C2×C412yes—
43C22×C7×D4C2×C414yes—
44C22×C7×D4C2×C412yes—
45C22×C7×D4C2×C412yes—
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C411yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2314yes—
50C2×C7×M16—48no—
51C2×C7×M16C23116yes—
52C22×C7×C8—48no—
53C22×C7×C8C2312yes—
54C22×C7×C8C2312yes—
55C22×C7×C8C2314yes—
56C22×C7×C8C23116yes—
57C7×C4⋊Q8C2×C418yes—
58C7×#127C2×C418yes—
59C7×#124C2×C4116yes—
60C7×#121C2×C4116yes—
61C7×#121C2×C4116yes—
62C4×C7×D4C2×C418yes—
63C4×C7×D4C2×C418yes—
64C4×C7×D4C2318yes—
65C4×C7×D4C2318yes—
66C2×C7×C4⋊C4C2×C412yes—
67C2×C7×C4⋊C4C2×C412yes—
68C2×C7×C22⋊C4C2×C414yes—
69C2×C7×C22⋊C4C2×C414yes—
70C2×C42×C7C2×C411yescharacteristic
71C2×C42×C7C2×C411yescharacteristic
72C2×C42×C7C2×C414yes—
73C2×C42×C7C2314yes—
74C2×C42×C7C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C7×C4⋊C8C2318yes—
77C7×C4⋊C8C2318yes—
78C7×C22⋊C8C2318yes—
79C7×C22⋊C8C2318yes—
80C7×C22⋊C8C23116yes—
81C7×C22⋊C8C23116yes—
82C7×#97C2318yes—
83C4×C7×C8C2314yes—
84C4×C7×C8C2314yes—
85C2×#532C2×C711yescharacteristic
86#2981C2×C7116yesdirect product factor
87C2×#436C2×C712yescomplemented
88C2×#435C2×C714yescomplemented
89C2×C8×D4C2×C712yescomplemented
90C2×#432C2×C711yescharacteristic, complemented
91C2×#431C2×C711yescharacteristic, complemented
92C2×#408C2×C714yescomplemented
93C24×C7C22×C414yes—
94C24×C7C22×C411yescharacteristic
95C7×#41—416no—
96C7×#41—416no—
97C2×C7×Q8—44no—
98C2×C7×Q8C22×C418yes—
99C2×C7×D4—28no—
100C2×C7×D4—44no—
101C2×C7×D4—44no—
102C2×C7×D4—48no—
103C2×C7×D4C22×C418yes—
104C2×C7×D4C22×C418yes—
105C2×C7×D4C22×C4116yes—
106C2×C7×D4C22×C418yes—
107C22×C4×C7—24no—
108C22×C4×C7—28no—
109C22×C4×C7—28no—
110C22×C4×C7—24no—
111C22×C4×C7—42no—
112C22×C4×C7C22×C411yescharacteristic
113C22×C4×C7C22×C412yes—
114C22×C4×C7C22×C411yescharacteristic
115C22×C4×C7C22×C411yescharacteristic
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C22×C414yes—
118C22×C4×C7C22×C414yes—
119C22×C4×C7C22×C418yes—
120C22×C4×C7C22×C414yes—
121C22×C4×C7C22×C411yescharacteristic
122C22×C4×C7C2412yes—
123C22×C4×C7C2418yes—
124C22×C4×C7C2411yescharacteristic
125C7×M16—416no—
126C2×C7×C8—24no—
127C2×C7×C8—28no—
128C2×C7×C8—24no—
129C2×C7×C8—416no—
130C2×C7×C8C2414yes—
131C2×C7×C8C2414yes—
132C2×C7×C8C2418yes—
133C7×C4⋊C4C22×C418yes—
134C7×C4⋊C4C22×C418yes—
135C7×C22⋊C4C22×C4116yes—
136C7×C22⋊C4C22×C4116yes—
137C42×C7C22×C412yes—
138C42×C7C22×C412yes—
139C42×C7C22×C414yes—
140C42×C7C22×C414yes—
141C42×C7C2412yes—
142C42×C7C2412yes—
143C22×#41C22×C712yes—
144C22×M16C22×C714yes—
145C23×C8C22×C714yes—
146#532C4×C718yes—
147#532C22×C718yes—
148C2×C4⋊Q8C4×C711yescharacteristic
149C2×#127C4×C711yescharacteristic
150C2×#124C4×C712yes—
151C2×#121C4×C712yes—
152C2×#121C4×C712yes—
153C2×C4×D4C22×C712yes—
154C2×C4×D4C22×C712yes—
155C22×C42C22×C711yescharacteristic
156#436C22×C7116yescomplemented
157#435C22×C7132yescomplemented
158C8×D4C22×C7116yescomplemented
159#432C22×C718yescomplemented
160#431C22×C718yescomplemented
161C2×C4⋊C8C22×C712yes—
162C2×C4⋊C8C22×C712yes—
163#408C22×C7132yescomplemented
164C2×C22⋊C8C22×C712yes—
165C2×C22⋊C8C22×C712yes—
166C2×C22⋊C8C22×C714yes—
167C2×C22⋊C8C22×C714yes—
168C2×#97C22×C712yes—
169C2×C4×C8C22×C711yescharacteristic
170C2×C4×C8C22×C711yescharacteristic
171C23×C7—22no—
172C23×C7—28no—
173C23×C7—22no—
174C23×C7—28no—
175C23×C7—22no—
176C23×C7—28no—
177C23×C7C23×C418yes—
178C23×C7C23×C412yes—
179C23×C7C23×C411yescharacteristic
180C7×Q8—48no—
181C7×D4—216no—
182C7×D4—48no—
183C7×D4—48no—
184C7×D4—416no—
185C2×C4×C7—22no—
186C2×C4×C7—28no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—216no—
191C2×C4×C7—24no—
192C2×C4×C7—28no—
193C2×C4×C7—28no—
194C2×C4×C7—28no—
195C2×C4×C7—24no—
196C2×C4×C7—44no—
197C2×C4×C7#13114yes—
198C2×C4×C7C23×C412yes—
199C2×C4×C7C23×C412yes—
200C2×C4×C7C23×C412yes—
201C2×C4×C7C23×C412yes—
202C2×C4×C7C23×C418yes—
203C2×C4×C7C23×C411yescharacteristic
204C2×C4×C7C23×C412yes—
205C2×C4×C7H(5,𝔽2)14yes—
206C2×C4×C7C2511yescharacteristic
207C7×C8—24no—
208C7×C8—28no—
209C7×C8—24no—
210C2×#41—48no—
211C2×#41C2×C4×C718yes—
212C2×#41C23×C718yes—
213C22×Q8C2×C4×C712yes—
214C22×D4C2×C4×C712yes—
215C22×D4C2×C4×C712yes—
216C22×D4C2×C4×C714yes—
217C22×D4C2×C4×C712yes—
218C23×C4C2×C4×C711yescharacteristic
219C23×C4C2×C4×C711yescharacteristic
220C23×C4C23×C714yes—
221C23×C4C23×C711yescharacteristic
222C2×M16—48no—
223C2×M16C23×C7116yes—
224C22×C8—48no—
225C22×C8C23×C712yes—
226C22×C8C23×C714yes—
227C22×C8C23×C712yes—
228C22×C8C23×C7116yes—
229C4⋊Q8C2×C4×C718yes—
230#127C2×C4×C718yes—
231#124C2×C4×C7116yes—
232#121C2×C4×C7116yes—
233#121C2×C4×C7116yes—
234C4×D4C2×C4×C718yes—
235C4×D4C2×C4×C718yes—
236C4×D4C23×C718yes—
237C4×D4C23×C718yes—
238C2×C4⋊C4C2×C4×C712yes—
239C2×C4⋊C4C2×C4×C712yes—
240C2×C22⋊C4C2×C4×C714yes—
241C2×C22⋊C4C2×C4×C714yes—
242C2×C42C2×C4×C711yescharacteristic
243C2×C42C2×C4×C711yescharacteristic
244C2×C42C2×C4×C714yes—
245C2×C42C23×C711yescharacteristic
246C2×C42C23×C711yescharacteristic
247C2×C42C23×C714yes—
248C4⋊C8C23×C718yes—
249C4⋊C8C23×C718yes—
250C22⋊C8C23×C718yes—
251C22⋊C8C23×C718yes—
252C22⋊C8C23×C7116yes—
253C22⋊C8C23×C7116yes—
254#97C23×C718yes—
255C4×C8C23×C714yes—
256C4×C8C23×C714yes—
257C22×C7—22no—
258C22×C7—28no—
259C22×C7—22no—
260C22×C7—28no—
261C22×C7—22no—
262C22×C7—28no—
263C22×C7—44no—
264C22×C7—416no—
265C22×C7#51812yes—
266C22×C7C2×#13112yes—
267C22×C7#56812yes—
268C22×C7C24×C411yescharacteristic
269C4×C7—22no—
270C4×C7—28no—
271C4×C7—22no—
272C4×C7—22no—
273C4×C7—22no—
274C4×C7#51812yes—
275C4×C7C2×#13114yes—
276C4×C7C2×H(5,𝔽2)12yes—
277C24C22×C4×C714yes—
278C24C22×C4×C711yescharacteristic
279#41—416no—
280#41—416no—
281C2×Q8—44no—
282C2×Q8C22×C4×C718yes—
283C2×D4—28no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—44no—
287C2×D4C22×C4×C718yes—
288C2×D4C22×C4×C718yes—
289C2×D4C22×C4×C7116yes—
290C2×D4C22×C4×C718yes—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—42no—
296C22×C4C22×C4×C711yescharacteristic
297C22×C4C22×C4×C712yes—
298C22×C4C22×C4×C711yescharacteristic
299C22×C4C22×C4×C711yescharacteristic
300C22×C4C22×C4×C711yescharacteristic
301C22×C4C22×C4×C714yes—
302C22×C4C22×C4×C714yes—
303C22×C4C22×C4×C718yes—
304C22×C4C22×C4×C714yes—
305C22×C4C22×C4×C711yescharacteristic
306C22×C4C24×C712yes—
307C22×C4C24×C718yes—
308C22×C4C24×C711yescharacteristic
309M16—416no—
310C2×C8—24no—
311C2×C8—28no—
312C2×C8—24no—
313C2×C8—416no—
314C2×C8C24×C714yes—
315C2×C8C24×C714yes—
316C2×C8C24×C718yes—
317C4⋊C4C22×C4×C718yes—
318C4⋊C4C22×C4×C718yes—
319C22⋊C4C22×C4×C7116yes—
320C22⋊C4C22×C4×C7116yes—
321C42C22×C4×C712yes—
322C42C22×C4×C712yes—
323C42C22×C4×C714yes—
324C42C22×C4×C714yes—
325C42C24×C712yes—
326C42C24×C712yes—
327C2×C7—42no—
328C2×C7—48no—
329C2×C7#298114yesdirect product factor
330C2×C7C2×#51811yescharacteristic
331C2×C7C22×#13111yescharacteristic
332C2×C7C2×#56811yescharacteristic
333C23—22no—
334C23—28no—
335C23—22no—
336C23—28no—
337C23—22no—
338C23—28no—
339C23C23×C4×C718yes—
340C23C23×C4×C712yes—
341C23C23×C4×C711yescharacteristic
342Q8—48no—
343D4—216no—
344D4—48no—
345D4—48no—
346D4—416no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—44no—
359C2×C4C7×#13114yes—
360C2×C4C23×C4×C712yes—
361C2×C4C23×C4×C712yes—
362C2×C4C23×C4×C712yes—
363C2×C4C23×C4×C712yes—
364C2×C4C23×C4×C718yes—
365C2×C4C23×C4×C711yescharacteristic
366C2×C4C23×C4×C712yes—
367C2×C4C7×H(5,𝔽2)14yes—
368C2×C4C25×C711yescharacteristic
369C8—24no—
370C8—28no—
371C8—24no—
372C7C2×#298111yescharacteristic, direct product factor, Sylow
373C22—22no—
374C22—28no—
375C22—22no—
376C22—28no—
377C22—22no—
378C22—28no—
379C22—44no—
380C22—416no—
381C22C7×#51812yes—
382C22C2×C7×#13112yes—
383C22C7×#56812yes—
384C22C24×C4×C711yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—28no—
390C4C7×#51812yes—
391C4C2×C7×#13114yes—
392C4C2×C7×H(5,𝔽2)12yes—
393C2—42no—
394C2—48no—
395C2C7×#298114yesdirect product factor
396C2C2×C7×#51811yescharacteristic
397C2C22×C7×#13111yescharacteristic
398C2C2×C7×#56811yescharacteristic
subgroupG/H
centerC22×C4×C7C24
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC23×C7C23×C4
prime radicalC2×C7×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2981C2direct (N × H)
3. C2×C7×#436C2semidirect
4. C2×C7×#435C2semidirect
5. C2×C7×C8×D4C2semidirect
6. C2×C7×#432C2semidirect
7. C2×C7×#431C2semidirect
8. C2×C7×#408C2semidirect
22. C7×#436C22semidirect
23. C7×#435C22semidirect
24. C7×C8×D4C22semidirect
25. C7×#432C22semidirect
26. C7×#431C22semidirect
29. C7×#408C22semidirect
37. C2×#2981C7direct (N × H)
86. #2981C2×C7direct (N × H)
87. C2×#436C2×C7semidirect
88. C2×#435C2×C7semidirect
89. C2×C8×D4C2×C7semidirect
90. C2×#432C2×C7semidirect
91. C2×#431C2×C7semidirect
92. C2×#408C2×C7semidirect
156. #436C22×C7semidirect
157. #435C22×C7semidirect
158. C8×D4C22×C7semidirect
159. #432C22×C7semidirect
160. #431C22×C7semidirect
163. #408C22×C7semidirect
329. C2×C7#2981direct (N × H)
372. C7C2×#2981direct (N × H)
395. C2C7×#2981direct (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
C7×#29812
C2×C7×#4362
C2×C7×#4352
C2×C7×C8×D42
C2×C7×#4322
C2×C7×#4312
C2×C7×#4082
C2×#29817