Finite Group Databasebeta

← C7×#18325 #13195541

Subgroups — C7×#18325 #13195541

Order 1792.

1230 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C7×#3193C212yesmaximal, complemented
3C7×#2583C211yesmaximal, characteristic, complemented
4C7×#1610C214yesmaximal, complemented
5C7×#1604C214yesmaximal, complemented
6C7×#1507C212yesmaximal, complemented
7C7×#1463C211yesmaximal, characteristic, complemented
8C22×C7×#41—21no—
9C7×#537—24no—
10C7×#537—22no—
11C7×#537C2214yes—
12C2×C7×C22⋊Q8—22no—
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C7×#121—22no—
15C2×C7×#121—21no—
16C2×C7×#121C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C7×#484—22no—
19C7×#480C2214yescomplemented
20C7×#468—24no—
21C7×#450—22no—
22C7×#448C2214yescomplemented
23C7×#407C2212yes—
24C7×#398C2212yescomplemented
25C7×#395C2212yescomplemented
26C7×#389—22no—
27C7×#385—21no—
28C7×#385—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#327C2218yescomplemented
31C7×#324C2214yescomplemented
32#18325C711yesmaximal, characteristic, direct product factor, Sylow
33C2×C7×#41—24no—
34C2×C7×#41—24no—
35C2×C7×#41—48no—
36C22×C7×Q8—21no—
37C22×C7×Q8C2311yescharacteristic
38C22×C7×D4—21no—
39C22×C7×D4—22no—
40C22×C7×D4—21no—
41C22×C7×D4D412yes—
42C23×C4×C7—22no—
43C23×C4×C7D411yescharacteristic
44C23×C4×C7D411yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C2×C7×SD16—24no—
47C2×C7×M16—22no—
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8—28no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×#121—24no—
53C7×#121—28no—
54C7×#121—28no—
55C7×#121—24no—
56C7×#121—22no—
57C7×#121—22no—
58C7×#121D414yes—
59C7×#121C2314yes—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4D412yes—
67C2×C7×C4⋊C4D412yes—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C7×#107—24no—
76C7×#103—24no—
77C7×#102—24no—
78C7×C22⋊C8—22no—
79C7×C22⋊C8C2314yes—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95C2312yes—
83C2×#537C2×C711yescharacteristic
84#3193C2×C712yescomplemented
85#2583C2×C711yescharacteristic, complemented
86#1610C2×C714yescomplemented
87#1604C2×C714yescomplemented
88#1507C2×C712yescomplemented
89#1463C2×C711yescharacteristic, complemented
90C24×C7—22no—
91C24×C7C2×D411yescharacteristic
92C7×#41—416no—
93C7×#41—48no—
94C7×#41—48no—
95C2×C7×Q8—24no—
96C2×C7×Q8—22no—
97C2×C7×Q8—44no—
98C2×C7×Q8—81no—
99C2×C7×D4—24no—
100C2×C7×D4—24no—
101C2×C7×D4—24no—
102C2×C7×D4—24no—
103C2×C7×D4—22no—
104C2×C7×D4—44no—
105C2×C7×D4—48no—
106C2×C7×D4—82no—
107C2×C7×D4—82no—
108C2×C7×D4C2×D414yes—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—24no—
112C22×C4×C7—28no—
113C22×C4×C7—21no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—48no—
120C22×C4×C7—81no—
121C22×C4×C7—81no—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2411yescharacteristic
128C7×SD16—84no—
129C7×M16—82no—
130C2×C7×C8—24no—
131C7×C4⋊C4—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—24no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—42no—
137C7×C4⋊C4—44no—
138C7×C4⋊C4—82no—
139C7×C4⋊C4C2×D414yes—
140C7×C22⋊C4—24no—
141C7×C22⋊C4—28no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—41no—
145C7×C22⋊C4—41no—
146C42×C7—81no—
147C22×#41—21no—
148#537—24no—
149#537—22no—
150#537C22×C714yes—
151C2×C22⋊Q8—22no—
152C2×C22⋊Q8C22×C711yescharacteristic
153C2×#121—22no—
154C2×#121—21no—
155C2×#121C22×C712yes—
156C22×C4⋊C4C22×C711yescharacteristic
157#484—22no—
158#480C22×C714yescomplemented
159#468—24no—
160#450—22no—
161#448C22×C714yescomplemented
162#407C22×C712yes—
163#398C22×C712yescomplemented
164#395C22×C712yescomplemented
165#389—22no—
166#385—21no—
167#385—21no—
168C4×C22⋊C4C22×C711yescharacteristic
169#327C22×C718yescomplemented
170#324C22×C714yescomplemented
171C23×C7—22no—
172C23×C7—22no—
173C23×C7—24no—
174C23×C7—41no—
175C23×C7—44no—
176C23×C7—44no—
177C23×C7—44no—
178C23×C7#12011yescharacteristic
179C23×C7#12011yescharacteristic
180C23×C7C22×D411yescharacteristic
181C7×Q8—48no—
182C7×Q8—44no—
183C7×D4—44no—
184C7×D4—48no—
185C7×D4—48no—
186C7×D4—48no—
187C7×D4—82no—
188C7×D4—84no—
189C2×C4×C7—24no—
190C2×C4×C7—24no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—24no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—41no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—41no—
200C2×C4×C7—416no—
201C2×C4×C7—42no—
202C2×C4×C7—44no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—42no—
210C2×C4×C7—81no—
211C2×C4×C7#12012yes—
212C2×C4×C7AGL(1,ℤ/8)14yes—
213C2×C4×C7C22×D412yes—
214C7×C8—44no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—48no—
218C22×Q8—21no—
219C22×Q8C23×C711yescharacteristic
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4C7×D412yes—
224C23×C4—22no—
225C23×C4C7×D411yescharacteristic
226C23×C4C7×D411yescharacteristic
227C23×C4C23×C711yescharacteristic
228C2×SD16—24no—
229C2×M16—22no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234#121—24no—
235#121—28no—
236#121—28no—
237#121—24no—
238#121—22no—
239#121—22no—
240#121C7×D414yes—
241#121C23×C714yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C7×D412yes—
249C2×C4⋊C4C7×D412yes—
250C2×C4⋊C4C23×C711yescharacteristic
251C2×C4⋊C4C23×C712yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257#107—24no—
258#103—24no—
259#102—24no—
260C22⋊C8—22no—
261C22⋊C8C23×C714yes—
262#95—22no—
263#95—22no—
264#95C23×C712yes—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—44no—
273C22×C7—44no—
274C22×C7—44no—
275C22×C7—44no—
276C22×C7—88no—
277C22×C7#45612yes—
278C22×C7C2×#12011yescharacteristic
279C22×C7C2×AGL(1,ℤ/8)12yes—
280C4×C7—24no—
281C4×C7—22no—
282C4×C7—22no—
283C4×C7—44no—
284C4×C7—44no—
285C4×C7—42no—
286C4×C7—81no—
287C4×C7—81no—
288C24—22no—
289C24C2×C7×D411yescharacteristic
290#41—416no—
291#41—48no—
292#41—48no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×Q8—81no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—48no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4C2×C7×D414yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—48no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4C2×C7×D412yes—
321C22×C4C2×C7×D412yes—
322C22×C4C2×C7×D411yescharacteristic
323C22×C4C2×C7×D412yes—
324C22×C4C2×C7×D412yes—
325C22×C4C24×C711yescharacteristic
326SD16—84no—
327M16—82no—
328C2×C8—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—82no—
337C4⋊C4C2×C7×D414yes—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—81no—
345C2×C7—22no—
346C2×C7—22no—
347C2×C7—41no—
348C2×C7—84no—
349C2×C7#300311yescharacteristic
350C2×C7C2×#45611yescharacteristic
351C2×C7#302611yescharacteristic
352C23—22no—
353C23—22no—
354C23—24no—
355C23—41no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23C7×#12011yescharacteristic
360C23C7×#12011yescharacteristic
361C23C22×C7×D411yescharacteristic
362Q8—48no—
363Q8—44no—
364D4—44no—
365D4—48no—
366D4—48no—
367D4—48no—
368D4—82no—
369D4—84no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4C7×#12012yes—
393C2×C4C7×AGL(1,ℤ/8)14yes—
394C2×C4C22×C7×D412yes—
395C8—44no—
396C7#1832511yescharacteristic, direct product factor, Sylow
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—88no—
409C22C7×#45612yes—
410C22C2×C7×#12011yescharacteristic
411C22C2×C7×AGL(1,ℤ/8)12yes—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—44no—
417C4—42no—
418C4—81no—
419C4—81no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—84no—
424C2C7×#300311yescharacteristic
425C2C2×C7×#45611yescharacteristic
426C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3193C2semidirect
3. C7×#2583C2semidirect
4. C7×#1610C2semidirect
5. C7×#1604C2semidirect
6. C7×#1507C2semidirect
7. C7×#1463C2semidirect
19. C7×#480C22semidirect
22. C7×#448C22semidirect
24. C7×#398C22semidirect
25. C7×#395C22semidirect
30. C7×#327C22semidirect
31. C7×#324C22semidirect
32. #18325C7direct (N × H)
84. #3193C2×C7semidirect
85. #2583C2×C7semidirect
86. #1610C2×C7semidirect
87. #1604C2×C7semidirect
88. #1507C2×C7semidirect
89. #1463C2×C7semidirect
158. #480C22×C7semidirect
161. #448C22×C7semidirect
163. #398C22×C7semidirect
164. #395C22×C7semidirect
169. #327C22×C7semidirect
170. #324C22×C7semidirect
396. C7#18325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C7×#31932
C7×#25832
C7×#16102
C7×#16042
C7×#15072
C7×#14632
#183257