Finite Group Databasebeta

← C7×#61889 #13239105

Subgroups — C7×#61889 #13239105

Order 1792.

1706 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3502C211yesmaximal, characteristic, complemented
2C2×C7×#556C211yesmaximal, characteristic
3C7×#3398C218yesmaximal, complemented
4C7×#3369C218yesmaximal, complemented
5C7×#3215C214yesmaximal
6C7×#3209C212yesmaximal, complemented
7C7×#3113C214yesmaximal, complemented
8C7×#3101C212yesmaximal
9C7×#2983C211yesmaximal, characteristic, complemented
10C7×Q82C2214yescomplemented
11C7×#557—24no—
12C7×#557C2214yescomplemented
13C7×#556—24no—
14C7×#556C2218yes—
15C7×#544—21no—
16C7×#544—22no—
17C7×#533C2212yes—
18C7×#533C2214yes—
19C2×C7×C4⋊Q8C2211yescharacteristic
20C2×C7×#126—22no—
21C2×C7×#124C2212yes—
22C2×C7×C22⋊Q8—21no—
23C7×#519—21no—
24C2×C4×C7×Q8C2212yes—
25C2×C4×C7×Q8C2211yescharacteristic
26C2×C4×C7×D4C2211yescharacteristic
27C2×C7×#117—21no—
28C7×#497C2218yes—
29C7×#494C2214yescomplemented
30C7×#489C2218yes—
31C7×#487C2214yescomplemented
32C7×#483C2214yes—
33C7×#476C2214yes—
34C7×#475C2218yescomplemented
35C7×#464C22116yes—
36C7×#462C2218yescomplemented
37C7×#462C2218yescomplemented
38C7×#461C2214yes—
39C7×#451C2218yes—
40C7×#450C2214yes—
41C7×#446C2218yescomplemented
42C7×#441C22116yescomplemented
43C4×C7×SD16C2218yes—
44C7×#432C2214yes—
45C7×#431C2212yes—
46#61889C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8D411yescharacteristic
48C22×C7×Q8C2312yes—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C2×C7×Q16C23116yes—
52C2×C7×SD16C2318yes—
53C7×C4⋊Q8—24no—
54C7×C4⋊Q8—28no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8C2314yes—
57C7×C4⋊Q8C2312yes—
58C7×C4⋊Q8C2314yes—
59C7×C4⋊Q8C2314yes—
60C7×C4⋊Q8C2312yes—
61C7×#126—216no—
62C7×#125—24no—
63C7×#125—28no—
64C7×#125—22no—
65C7×#125—24no—
66C7×#124—24no—
67C7×#124C2318yes—
68C7×C22⋊Q8—28no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8C2314yes—
72C4×C7×Q8—24no—
73C4×C7×Q8—24no—
74C4×C7×Q8—24no—
75C4×C7×Q8—22no—
76C4×C7×Q8—24no—
77C4×C7×Q8D414yes—
78C4×C7×Q8C2318yes—
79C4×C7×Q8C2314yes—
80C4×C7×Q8C2318yes—
81C4×C7×Q8C2314yes—
82C4×C7×D4—22no—
83C4×C7×D4C2314yes—
84C7×#117—24no—
85C7×#117—22no—
86C7×#117—28no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4D412yes—
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—22no—
96C2×C42×C7—22no—
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2312yes—
99C7×#106C2318yes—
100C7×C4⋊C8C2318yes—
101C7×C4⋊C8C2314yes—
102C7×#103C23116yes—
103C7×#103C2318yes—
104C7×#102C2318yes—
105C7×C22⋊C8C2314yes—
106C7×#97C2318yes—
107C4×C7×C8C2314yes—
108#3502C2×C711yescharacteristic, complemented
109C2×#556C2×C711yescharacteristic
110#3398C2×C718yescomplemented
111#3369C2×C718yescomplemented
112#3215C2×C714yes—
113#3209C2×C712yescomplemented
114#3113C2×C714yescomplemented
115#3101C2×C712yes—
116#2983C2×C711yescharacteristic, complemented
117C24×C7—21no—
118C2×C7×Q8—24no—
119C2×C7×Q8—22no—
120C2×C7×Q8—24no—
121C2×C7×Q8—22no—
122C2×C7×Q8—44no—
123C2×C7×Q8C2×D414yes—
124C2×C7×Q8C2414yes—
125C2×C7×Q8C2414yes—
126C2×C7×D4—21no—
127C2×C7×D4—42no—
128C2×C7×D4C2412yes—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—24no—
132C22×C4×C7—44no—
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C7×Q16—416no—
139C7×SD16—48no—
140C2×C7×C8C2418yes—
141C7×C4⋊C4—24no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—24no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—24no—
150C7×C4⋊C4—24no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4C2×D414yes—
155C7×C4⋊C4C2×D418yes—
156C7×C4⋊C4C2414yes—
157C7×C4⋊C4C2412yes—
158C7×C22⋊C4—24no—
159C7×C22⋊C4—28no—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—28no—
162C42×C7—28no—
163C42×C7—28no—
164C42×C7—24no—
165C42×C7C2×D412yes—
166C42×C7C2×D414yes—
167C42×C7C2412yes—
168C42×C7C2414yes—
169Q82C22×C714yescomplemented
170#557—24no—
171#557C22×C714yescomplemented
172#556—24no—
173#556C22×C718yes—
174#544—21no—
175#544—22no—
176#533C22×C712yes—
177#533C22×C714yes—
178C2×C4⋊Q8C22×C711yescharacteristic
179C2×#126—22no—
180C2×#124C22×C712yes—
181C2×C22⋊Q8—21no—
182#519—21no—
183C2×C4×Q8C22×C712yes—
184C2×C4×Q8C22×C711yescharacteristic
185C2×C4×D4C22×C711yescharacteristic
186C2×#117—21no—
187#497C22×C718yes—
188#494C22×C714yescomplemented
189#489C22×C718yes—
190#487C22×C714yescomplemented
191#483C22×C714yes—
192#476C22×C714yes—
193#475C22×C718yescomplemented
194#464C22×C7116yes—
195#462C22×C718yescomplemented
196#462C22×C718yescomplemented
197#461C22×C714yes—
198#451C22×C718yes—
199#450C22×C714yes—
200#446C22×C718yescomplemented
201#441C22×C7116yescomplemented
202C4×SD16C22×C718yes—
203#432C22×C714yes—
204#431C22×C712yes—
205C23×C7—22no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7C22×D411yescharacteristic
210C7×Q8—28no—
211C7×Q8—28no—
212C7×Q8—28no—
213C7×Q8—44no—
214C7×Q8—84no—
215C7×Q8—82no—
216C7×D4—24no—
217C7×D4—42no—
218C2×C4×C7—24no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—42no—
225C2×C4×C7—41no—
226C2×C4×C7—44no—
227C2×C4×C7—42no—
228C2×C4×C7—48no—
229C2×C4×C7#13714yes—
230C2×C4×C7C22×D414yes—
231C2×C4×C7C22×D414yes—
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C22×D412yes—
234C2×C4×C7C22×D412yes—
235C2×C4×C7C2511yescharacteristic
236C7×C8—28no—
237C22×Q8C7×D411yescharacteristic
238C22×Q8C23×C712yes—
239C22×D4C23×C711yescharacteristic
240C23×C4—21no—
241C2×Q16C23×C7116yes—
242C2×SD16C23×C718yes—
243C4⋊Q8—24no—
244C4⋊Q8—28no—
245C4⋊Q8—22no—
246C4⋊Q8C23×C714yes—
247C4⋊Q8C23×C712yes—
248C4⋊Q8C23×C714yes—
249C4⋊Q8C23×C714yes—
250C4⋊Q8C23×C712yes—
251#126—216no—
252#125—24no—
253#125—28no—
254#125—22no—
255#125—24no—
256#124—24no—
257#124C23×C718yes—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8C23×C714yes—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—24no—
267C4×Q8C7×D414yes—
268C4×Q8C23×C718yes—
269C4×Q8C23×C714yes—
270C4×Q8C23×C718yes—
271C4×Q8C23×C714yes—
272C4×D4—22no—
273C4×D4C23×C714yes—
274#117—24no—
275#117—22no—
276#117—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C7×D411yescharacteristic
281C2×C4⋊C4C7×D412yes—
282C2×C4⋊C4C23×C711yescharacteristic
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C42—22no—
287C2×C42C23×C711yescharacteristic
288C2×C42C23×C712yes—
289#106C23×C718yes—
290C4⋊C8C23×C714yes—
291C4⋊C8C23×C718yes—
292#103C23×C718yes—
293#103C23×C7116yes—
294#102C23×C718yes—
295C22⋊C8C23×C714yes—
296#97C23×C718yes—
297C4×C8C23×C714yes—
298C22×C7—21no—
299C22×C7—21no—
300C22×C7—42no—
301C22×C7—42no—
302C22×C7—42no—
303C22×C7—84no—
304C22×C7C2×#13712yes—
305C22×C7C23×D411yescharacteristic
306C4×C7—24no—
307C4×C7—22no—
308C4×C7—21no—
309C4×C7—24no—
310C4×C7—44no—
311C4×C7—44no—
312C4×C7—44no—
313C4×C7—42no—
314C4×C7C2×#13714yes—
315C4×C7#58512yes—
316C24—21no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—44no—
322C2×Q8C2×C7×D414yes—
323C2×Q8C24×C714yes—
324C2×Q8C24×C714yes—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4C24×C712yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—44no—
332C22×C4C2×C7×D412yes—
333C22×C4C2×C7×D412yes—
334C22×C4C2×C7×D411yescharacteristic
335C22×C4C2×C7×D411yescharacteristic
336C22×C4C24×C711yescharacteristic
337Q16—416no—
338SD16—48no—
339C2×C8C24×C718yes—
340C4⋊C4—24no—
341C4⋊C4—28no—
342C4⋊C4—24no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4C2×C7×D414yes—
354C4⋊C4C2×C7×D418yes—
355C4⋊C4C24×C714yes—
356C4⋊C4C24×C712yes—
357C22⋊C4—24no—
358C22⋊C4—28no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C42—28no—
362C42—28no—
363C42—24no—
364C42C2×C7×D412yes—
365C42C2×C7×D414yes—
366C42C24×C712yes—
367C42C24×C714yes—
368C2×C7—22no—
369C2×C7—82no—
370C2×C7#348711yescharacteristic
371C2×C7C22×#13711yescharacteristic
372C2×C7#358611yescharacteristic
373C23—22no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23C22×C7×D411yescharacteristic
378Q8—28no—
379Q8—28no—
380Q8—28no—
381Q8—44no—
382Q8—84no—
383Q8—82no—
384D4—24no—
385D4—42no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4C7×#13714yes—
398C2×C4C22×C7×D414yes—
399C2×C4C22×C7×D414yes—
400C2×C4C22×C7×D411yescharacteristic
401C2×C4C22×C7×D412yes—
402C2×C4C22×C7×D412yes—
403C2×C4C25×C711yescharacteristic
404C8—28no—
405C7#6188911yescharacteristic, direct product factor, Sylow
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—84no—
412C22C2×C7×#13712yes—
413C22C23×C7×D411yescharacteristic
414C4—24no—
415C4—24no—
416C4—22no—
417C4—21no—
418C4—42no—
419C4—44no—
420C4—44no—
421C4—44no—
422C4C2×C7×#13714yes—
423C4C7×#58512yes—
424C2—22no—
425C2—82no—
426C2C7×#348711yescharacteristic
427C2C22×C7×#13711yescharacteristic
428C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3502C2semidirect
3. C7×#3398C2semidirect
4. C7×#3369C2semidirect
6. C7×#3209C2semidirect
7. C7×#3113C2semidirect
9. C7×#2983C2semidirect
10. C7×Q82C22semidirect
12. C7×#557C22semidirect
29. C7×#494C22semidirect
31. C7×#487C22semidirect
34. C7×#475C22semidirect
36. C7×#462C22semidirect
37. C7×#462C22semidirect
41. C7×#446C22semidirect
42. C7×#441C22semidirect
46. #61889C7direct (N × H)
108. #3502C2×C7semidirect
110. #3398C2×C7semidirect
111. #3369C2×C7semidirect
113. #3209C2×C7semidirect
114. #3113C2×C7semidirect
116. #2983C2×C7semidirect
169. Q82C22×C7semidirect
171. #557C22×C7semidirect
188. #494C22×C7semidirect
190. #487C22×C7semidirect
193. #475C22×C7semidirect
195. #462C22×C7semidirect
196. #462C22×C7semidirect
200. #446C22×C7semidirect
201. #441C22×C7semidirect
405. C7#61889direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35022
C2×C7×#5562
C7×#33982
C7×#33692
C7×#32152
C7×#32092
C7×#31132
C7×#31012
C7×#29832
#618897