Finite Group Databasebeta

← C7×#61892 #13239108

Subgroups — C7×#61892 #13239108

Order 1792.

1770 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3502C211yesmaximal, characteristic, complemented
2C2×C7×#559C211yesmaximal, characteristic
3C7×#3400C218yesmaximal, complemented
4C7×#3375C218yesmaximal, complemented
5C7×#3215C214yesmaximal, complemented
6C7×#3208C212yesmaximal
7C7×#3110C214yesmaximal, complemented
8C7×#3104C212yesmaximal, complemented
9C7×#2983C211yesmaximal, characteristic, complemented
10C7×#559—24no—
11C7×#559C2218yes—
12C7×Q82C2214yescomplemented
13C7×#557—24no—
14C7×#557C2214yescomplemented
15C7×#544—21no—
16C7×#544—22no—
17C7×#533C2212yes—
18C7×#533C2214yes—
19C2×C7×#127C2211yescharacteristic
20C2×C7×#124—21no—
21C2×C7×#124C2212yes—
22C2×C7×#123—22no—
23C7×#519—21no—
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×Q8C2211yescharacteristic
26C2×C4×C7×D4—21no—
27C2×C4×C7×D4C2212yes—
28C7×#497C2218yescomplemented
29C7×#492C2214yes—
30C7×#489C2218yescomplemented
31C7×#486C2214yes—
32C7×#484C2214yes—
33C7×#477C2214yescomplemented
34C7×#474C2218yescomplemented
35C7×#464C22116yescomplemented
36C7×#462C2214yescomplemented
37C7×#461C2218yescomplemented
38C7×#460C2218yes—
39C7×#451C2214yes—
40C7×#450C2218yes—
41C7×#446C2218yescomplemented
42C7×#440C22116yescomplemented
43C4×C7×Q16C2218yescomplemented
44C7×#432C2214yes—
45C7×#431C2212yes—
46#61892C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8D411yescharacteristic
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4C2312yes—
52C23×C4×C7—22no—
53C2×C7×Q16C2318yes—
54C2×C7×SD16C23116yes—
55C7×C4⋊Q8—24no—
56C7×C4⋊Q8—28no—
57C7×C4⋊Q8C2312yes—
58C7×C4⋊Q8C2314yes—
59C7×C4⋊Q8C2314yes—
60C7×C4⋊Q8C2312yes—
61C7×#127—22no—
62C7×#127C2314yes—
63C7×#125—24no—
64C7×#125—28no—
65C7×#125—24no—
66C7×#125—22no—
67C7×#124—28no—
68C7×#124—24no—
69C7×#124C2318yes—
70C7×#123—216no—
71C7×C22⋊Q8—24no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8C2314yes—
74C4×C7×Q8—24no—
75C4×C7×Q8—24no—
76C4×C7×Q8—22no—
77C4×C7×Q8—24no—
78C4×C7×Q8—22no—
79C4×C7×Q8D414yes—
80C4×C7×Q8C2314yes—
81C4×C7×Q8C2314yes—
82C4×C7×Q8C2318yes—
83C4×C7×Q8C2314yes—
84C4×C7×D4—28no—
85C4×C7×D4—24no—
86C4×C7×D4C2318yes—
87C7×#117—24no—
88C7×#117—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—22no—
95C2×C7×C22⋊C4—22no—
96C2×C42×C7—21no—
97C2×C42×C7C2312yes—
98C2×C42×C7C2311yescharacteristic
99C7×#107C2318yes—
100C7×C4⋊C8C2318yes—
101C7×C4⋊C8C2314yes—
102C7×#103C2318yes—
103C7×#103C2318yes—
104C7×#102C23116yes—
105C7×C22⋊C8C2314yes—
106C7×#97C2318yes—
107C4×C7×C8C2314yes—
108#3502C2×C711yescharacteristic, complemented
109C2×#559C2×C711yescharacteristic
110#3400C2×C718yescomplemented
111#3375C2×C718yescomplemented
112#3215C2×C714yescomplemented
113#3208C2×C712yes—
114#3110C2×C714yescomplemented
115#3104C2×C712yescomplemented
116#2983C2×C711yescharacteristic, complemented
117C24×C7—22no—
118C2×C7×Q8—24no—
119C2×C7×Q8—21no—
120C2×C7×Q8—24no—
121C2×C7×Q8—22no—
122C2×C7×Q8—42no—
123C2×C7×Q8C2×D414yes—
124C2×C7×Q8C2412yes—
125C2×C7×Q8C2414yes—
126C2×C7×D4—22no—
127C2×C7×D4—24no—
128C2×C7×D4—24no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—82no—
132C2×C7×D4C2414yes—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—28no—
136C22×C4×C7—84no—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C7×Q16—48no—
143C7×SD16—416no—
144C2×C7×C8C2418yes—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—24no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4C2×D414yes—
157C7×C4⋊C4C2×D418yes—
158C7×C4⋊C4C2414yes—
159C7×C4⋊C4C2412yes—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—28no—
162C7×C22⋊C4—28no—
163C7×C22⋊C4—28no—
164C42×C7—24no—
165C42×C7—28no—
166C42×C7—24no—
167C42×C7C2×D414yes—
168C42×C7C2×D412yes—
169C42×C7C2412yes—
170C42×C7C2414yes—
171#559—24no—
172#559C22×C718yes—
173Q82C22×C714yescomplemented
174#557—24no—
175#557C22×C714yescomplemented
176#544—21no—
177#544—22no—
178#533C22×C712yes—
179#533C22×C714yes—
180C2×#127C22×C711yescharacteristic
181C2×#124—21no—
182C2×#124C22×C712yes—
183C2×#123—22no—
184#519—21no—
185C2×C4×Q8C22×C711yescharacteristic
186C2×C4×Q8C22×C711yescharacteristic
187C2×C4×D4—21no—
188C2×C4×D4C22×C712yes—
189#497C22×C718yescomplemented
190#492C22×C714yes—
191#489C22×C718yescomplemented
192#486C22×C714yes—
193#484C22×C714yes—
194#477C22×C714yescomplemented
195#474C22×C718yescomplemented
196#464C22×C7116yescomplemented
197#462C22×C714yescomplemented
198#461C22×C718yescomplemented
199#460C22×C718yes—
200#451C22×C714yes—
201#450C22×C718yes—
202#446C22×C718yescomplemented
203#440C22×C7116yescomplemented
204C4×Q16C22×C718yescomplemented
205#432C22×C714yes—
206#431C22×C712yes—
207C23×C7—24no—
208C23×C7—44no—
209C23×C7—44no—
210C23×C7—44no—
211C23×C7C22×D411yescharacteristic
212C7×Q8—28no—
213C7×Q8—28no—
214C7×Q8—24no—
215C7×Q8—42no—
216C7×Q8—84no—
217C7×Q8—82no—
218C7×D4—28no—
219C7×D4—44no—
220C7×D4—48no—
221C7×D4—84no—
222C2×C4×C7—24no—
223C2×C4×C7—22no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—42no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—88no—
233C2×C4×C7#13714yes—
234C2×C4×C7C22×D414yes—
235C2×C4×C7C22×D411yescharacteristic
236C2×C4×C7C22×D414yes—
237C2×C4×C7C22×D412yes—
238C2×C4×C7C22×D412yes—
239C2×C4×C7C2511yescharacteristic
240C7×C8—28no—
241C22×Q8C7×D411yescharacteristic
242C22×Q8C23×C711yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C23×C712yes—
246C23×C4—22no—
247C2×Q16C23×C718yes—
248C2×SD16C23×C7116yes—
249C4⋊Q8—24no—
250C4⋊Q8—28no—
251C4⋊Q8C23×C714yes—
252C4⋊Q8C23×C712yes—
253C4⋊Q8C23×C714yes—
254C4⋊Q8C23×C712yes—
255#127—22no—
256#127C23×C714yes—
257#125—24no—
258#125—28no—
259#125—24no—
260#125—22no—
261#124—28no—
262#124—24no—
263#124C23×C718yes—
264#123—216no—
265C22⋊Q8—24no—
266C22⋊Q8—22no—
267C22⋊Q8C23×C714yes—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8C7×D414yes—
274C4×Q8C23×C714yes—
275C4×Q8C23×C714yes—
276C4×Q8C23×C718yes—
277C4×Q8C23×C714yes—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4C23×C718yes—
281#117—24no—
282#117—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C7×D411yescharacteristic
285C2×C4⋊C4C7×D412yes—
286C2×C4⋊C4C23×C711yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—21no—
291C2×C42C23×C712yes—
292C2×C42C23×C711yescharacteristic
293#107C23×C718yes—
294C4⋊C8C23×C714yes—
295C4⋊C8C23×C718yes—
296#103C23×C718yes—
297#103C23×C718yes—
298#102C23×C7116yes—
299C22⋊C8C23×C714yes—
300#97C23×C718yes—
301C4×C8C23×C714yes—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—44no—
305C22×C7—44no—
306C22×C7—44no—
307C22×C7—88no—
308C22×C7C2×#13712yes—
309C22×C7C23×D411yescharacteristic
310C4×C7—24no—
311C4×C7—22no—
312C4×C7—21no—
313C4×C7—24no—
314C4×C7—42no—
315C4×C7—44no—
316C4×C7—42no—
317C4×C7—44no—
318C4×C7C2×#13714yes—
319C4×C7#58512yes—
320C24—22no—
321C2×Q8—24no—
322C2×Q8—21no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—42no—
326C2×Q8C2×C7×D414yes—
327C2×Q8C24×C712yes—
328C2×Q8C24×C714yes—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—82no—
335C2×D4C24×C714yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—28no—
339C22×C4—84no—
340C22×C4C2×C7×D412yes—
341C22×C4C2×C7×D412yes—
342C22×C4C2×C7×D411yescharacteristic
343C22×C4C2×C7×D411yescharacteristic
344C22×C4C24×C711yescharacteristic
345Q16—48no—
346SD16—416no—
347C2×C8C24×C718yes—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4C2×C7×D414yes—
360C4⋊C4C2×C7×D418yes—
361C4⋊C4C24×C714yes—
362C4⋊C4C24×C712yes—
363C22⋊C4—24no—
364C22⋊C4—28no—
365C22⋊C4—28no—
366C22⋊C4—28no—
367C42—24no—
368C42—28no—
369C42—24no—
370C42C2×C7×D414yes—
371C42C2×C7×D412yes—
372C42C24×C712yes—
373C42C24×C714yes—
374C2×C7—22no—
375C2×C7—84no—
376C2×C7#348711yescharacteristic
377C2×C7C22×#13711yescharacteristic
378C2×C7#358511yescharacteristic
379C23—24no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23C22×C7×D411yescharacteristic
384Q8—28no—
385Q8—28no—
386Q8—24no—
387Q8—42no—
388Q8—84no—
389Q8—82no—
390D4—28no—
391D4—44no—
392D4—48no—
393D4—84no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4C7×#13714yes—
406C2×C4C22×C7×D414yes—
407C2×C4C22×C7×D411yescharacteristic
408C2×C4C22×C7×D414yes—
409C2×C4C22×C7×D412yes—
410C2×C4C22×C7×D412yes—
411C2×C4C25×C711yescharacteristic
412C8—28no—
413C7#6189211yescharacteristic, direct product factor, Sylow
414C22—21no—
415C22—21no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C2×C7×#13712yes—
421C22C23×C7×D411yescharacteristic
422C4—24no—
423C4—24no—
424C4—22no—
425C4—21no—
426C4—42no—
427C4—44no—
428C4—42no—
429C4—44no—
430C4C2×C7×#13714yes—
431C4C7×#58512yes—
432C2—22no—
433C2—84no—
434C2C7×#348711yescharacteristic
435C2C22×C7×#13711yescharacteristic
436C2C7×#358511yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#127C22
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×#3400C2semidirect
4. C7×#3375C2semidirect
5. C7×#3215C2semidirect
7. C7×#3110C2semidirect
8. C7×#3104C2semidirect
9. C7×#2983C2semidirect
12. C7×Q82C22semidirect
14. C7×#557C22semidirect
28. C7×#497C22semidirect
30. C7×#489C22semidirect
33. C7×#477C22semidirect
34. C7×#474C22semidirect
35. C7×#464C22semidirect
36. C7×#462C22semidirect
37. C7×#461C22semidirect
41. C7×#446C22semidirect
42. C7×#440C22semidirect
43. C4×C7×Q16C22semidirect
46. #61892C7direct (N × H)
108. #3502C2×C7semidirect
110. #3400C2×C7semidirect
111. #3375C2×C7semidirect
112. #3215C2×C7semidirect
114. #3110C2×C7semidirect
115. #3104C2×C7semidirect
116. #2983C2×C7semidirect
173. Q82C22×C7semidirect
175. #557C22×C7semidirect
189. #497C22×C7semidirect
191. #489C22×C7semidirect
194. #477C22×C7semidirect
195. #474C22×C7semidirect
196. #464C22×C7semidirect
197. #462C22×C7semidirect
198. #461C22×C7semidirect
202. #446C22×C7semidirect
203. #440C22×C7semidirect
204. C4×Q16C22×C7semidirect
413. C7#61892direct (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×#5592
C7×#34002
C7×#33752
C7×#32152
C7×#32082
C7×#31102
C7×#31042
C7×#29832
#618927