Finite Group Databasebeta

← C7×#18324 #13195540

Subgroups — C7×#18324 #13195540

Order 1792.

1256 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic, complemented
2C7×#3192C212yesmaximal, complemented
3C7×#2591C211yesmaximal, characteristic, complemented
4C7×#1616C214yesmaximal, complemented
5C7×#1598C214yesmaximal, 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×#121—22no—
14C2×C7×#121—21no—
15C2×C7×#121C2212yes—
16C2×C7×#121C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C7×#481—22no—
19C7×#481C2214yescomplemented
20C7×#469—24no—
21C7×#447—22no—
22C7×#447C2214yescomplemented
23C7×#407C2212yes—
24C7×#400C2212yescomplemented
25C7×#394C2212yescomplemented
26C7×#388—22no—
27C7×#386—21no—
28C7×#386—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#327C2218yescomplemented
31C7×#324C2214yescomplemented
32#18324C711yesmaximal, characteristic, direct product factor, Sylow
33C2×C7×#41—24no—
34C2×C7×#41—24no—
35C2×C7×#41—48no—
36C22×C7×Q8—21no—
37C22×C7×D4—21no—
38C22×C7×D4—22no—
39C22×C7×D4—21no—
40C22×C7×D4—21no—
41C22×C7×D4D412yes—
42C22×C7×D4C2311yescharacteristic
43C23×C4×C7—22no—
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C2×C7×D8—24no—
48C2×C7×M16—22no—
49C7×C22⋊Q8—28no—
50C7×C22⋊Q8—28no—
51C7×#121—24no—
52C7×#121—28no—
53C7×#121—28no—
54C7×#121—24no—
55C7×#121—22no—
56C7×#121—22no—
57C7×#121—22no—
58C7×#121—22no—
59C7×#121D414yes—
60C7×#121C2314yes—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4D412yes—
65C2×C7×C4⋊C4D412yes—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2312yes—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C42×C7—21no—
75C7×#106—24no—
76C7×#102—24no—
77C7×#102—24no—
78C7×C22⋊C8—22no—
79C7×C22⋊C8C2314yes—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95C2312yes—
83C2×#537C2×C711yescharacteristic, complemented
84#3192C2×C712yescomplemented
85#2591C2×C711yescharacteristic, complemented
86#1616C2×C714yescomplemented
87#1598C2×C714yescomplemented
88#1507C2×C712yescomplemented
89#1463C2×C711yescharacteristic, complemented
90C24×C7—22no—
91C24×C7—21no—
92C24×C7C2×D411yescharacteristic
93C7×#41—416no—
94C7×#41—48no—
95C7×#41—48no—
96C2×C7×Q8—24no—
97C2×C7×Q8—44no—
98C2×C7×D4—24no—
99C2×C7×D4—24no—
100C2×C7×D4—24no—
101C2×C7×D4—24no—
102C2×C7×D4—22no—
103C2×C7×D4—22no—
104C2×C7×D4—44no—
105C2×C7×D4—48no—
106C2×C7×D4—44no—
107C2×C7×D4—41no—
108C2×C7×D4—41no—
109C2×C7×D4—81no—
110C2×C7×D4—82no—
111C2×C7×D4—82no—
112C2×C7×D4C2×D414yes—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—24no—
116C22×C4×C7—28no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—48no—
123C22×C4×C7—81no—
124C22×C4×C7—81no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2411yescharacteristic
131C7×D8—84no—
132C7×M16—82no—
133C2×C7×C8—24no—
134C7×C4⋊C4—24no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4C2×D414yes—
140C7×C22⋊C4—28no—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—42no—
144C7×C22⋊C4—42no—
145C7×C22⋊C4—44no—
146C7×C22⋊C4—82no—
147C42×C7—81no—
148C22×#41—21no—
149#537—24no—
150#537—22no—
151#537C22×C714yes—
152C2×C22⋊Q8—22no—
153C2×#121—22no—
154C2×#121—21no—
155C2×#121C22×C712yes—
156C2×#121C22×C711yescharacteristic
157C22×C4⋊C4C22×C711yescharacteristic
158#481—22no—
159#481C22×C714yescomplemented
160#469—24no—
161#447—22no—
162#447C22×C714yescomplemented
163#407C22×C712yes—
164#400C22×C712yescomplemented
165#394C22×C712yescomplemented
166#388—22no—
167#386—21no—
168#386—21no—
169C4×C22⋊C4C22×C711yescharacteristic
170#327C22×C718yescomplemented
171#324C22×C714yescomplemented
172C23×C7—22no—
173C23×C7—22no—
174C23×C7—24no—
175C23×C7—41no—
176C23×C7—44no—
177C23×C7—44no—
178C23×C7—42no—
179C23×C7—42no—
180C23×C7—44no—
181C23×C7—41no—
182C23×C7—42no—
183C23×C7#12011yescharacteristic
184C23×C7#12011yescharacteristic
185C23×C7C22×D411yescharacteristic
186C7×Q8—48no—
187C7×D4—44no—
188C7×D4—48no—
189C7×D4—44no—
190C7×D4—48no—
191C7×D4—48no—
192C7×D4—82no—
193C7×D4—84no—
194C7×D4—82no—
195C7×D4—82no—
196C2×C4×C7—24no—
197C2×C4×C7—24no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—22no—
201C2×C4×C7—24no—
202C2×C4×C7—41no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—41no—
206C2×C4×C7—416no—
207C2×C4×C7—44no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7—81no—
216C2×C4×C7#12012yes—
217C2×C4×C7AGL(1,ℤ/8)14yes—
218C2×C4×C7C22×D412yes—
219C7×C8—44no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—48no—
223C22×Q8—21no—
224C22×D4—21no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C7×D412yes—
229C22×D4C23×C711yescharacteristic
230C23×C4—22no—
231C23×C4C7×D411yescharacteristic
232C23×C4C7×D411yescharacteristic
233C23×C4C23×C711yescharacteristic
234C2×D8—24no—
235C2×M16—22no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238#121—24no—
239#121—28no—
240#121—28no—
241#121—24no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121—22no—
246#121C7×D414yes—
247#121C23×C714yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4C7×D412yes—
252C2×C4⋊C4C7×D412yes—
253C2×C4⋊C4C23×C711yescharacteristic
254C2×C4⋊C4C23×C712yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C42—21no—
262#106—24no—
263#102—24no—
264#102—24no—
265C22⋊C8—22no—
266C22⋊C8C23×C714yes—
267#95—22no—
268#95—22no—
269#95C23×C712yes—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—44no—
278C22×C7—44no—
279C22×C7—44no—
280C22×C7—44no—
281C22×C7—88no—
282C22×C7—84no—
283C22×C7—81no—
284C22×C7—81no—
285C22×C7—81no—
286C22×C7#45412yes—
287C22×C7C2×#12011yescharacteristic
288C22×C7C2×AGL(1,ℤ/8)12yes—
289C4×C7—24no—
290C4×C7—22no—
291C4×C7—22no—
292C4×C7—44no—
293C4×C7—42no—
294C4×C7—82no—
295C4×C7—81no—
296C24—22no—
297C24—21no—
298C24C2×C7×D411yescharacteristic
299#41—416no—
300#41—48no—
301#41—48no—
302C2×Q8—24no—
303C2×Q8—44no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4C2×C7×D414yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—48no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C2×C7×D412yes—
332C22×C4C2×C7×D412yes—
333C22×C4C2×C7×D411yescharacteristic
334C22×C4C2×C7×D412yes—
335C22×C4C2×C7×D412yes—
336C22×C4C24×C711yescharacteristic
337D8—84no—
338M16—82no—
339C2×C8—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4C2×C7×D414yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—82no—
353C42—81no—
354C2×C7—22no—
355C2×C7—22no—
356C2×C7—41no—
357C2×C7—84no—
358C2×C7—161no—
359C2×C7#300311yescharacteristic
360C2×C7C2×#45411yescharacteristic
361C2×C7#302611yescharacteristic
362C23—22no—
363C23—22no—
364C23—24no—
365C23—41no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—41no—
372C23—42no—
373C23C7×#12011yescharacteristic
374C23C7×#12011yescharacteristic
375C23C22×C7×D411yescharacteristic
376Q8—48no—
377D4—44no—
378D4—48no—
379D4—44no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—82no—
385D4—82no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—416no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—81no—
406C2×C4C7×#12012yes—
407C2×C4C7×AGL(1,ℤ/8)14yes—
408C2×C4C22×C7×D412yes—
409C8—44no—
410C7#1832411yescharacteristic, direct product factor, Sylow
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—84no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22C7×#45412yes—
428C22C2×C7×#12011yescharacteristic
429C22C2×C7×AGL(1,ℤ/8)12yes—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—82no—
436C4—81no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—161no—
442C2C7×#300311yescharacteristic
443C2C2×C7×#45411yescharacteristic
444C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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
1. C2×C7×#537C2semidirect
2. C7×#3192C2semidirect
3. C7×#2591C2semidirect
4. C7×#1616C2semidirect
5. C7×#1598C2semidirect
6. C7×#1507C2semidirect
7. C7×#1463C2semidirect
19. C7×#481C22semidirect
22. C7×#447C22semidirect
24. C7×#400C22semidirect
25. C7×#394C22semidirect
30. C7×#327C22semidirect
31. C7×#324C22semidirect
32. #18324C7direct (N × H)
83. C2×#537C2×C7semidirect
84. #3192C2×C7semidirect
85. #2591C2×C7semidirect
86. #1616C2×C7semidirect
87. #1598C2×C7semidirect
88. #1507C2×C7semidirect
89. #1463C2×C7semidirect
159. #481C22×C7semidirect
162. #447C22×C7semidirect
164. #400C22×C7semidirect
165. #394C22×C7semidirect
170. #327C22×C7semidirect
171. #324C22×C7semidirect
410. C7#18324direct (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×#31922
C7×#25912
C7×#16162
C7×#15982
C7×#15072
C7×#14632
#183247