Finite Group Databasebeta

← C7×#18293 #13195509

Subgroups — C7×#18293 #13195509

Order 1792.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C7×#3187C212yesmaximal, complemented
3C7×#2627C211yesmaximal, characteristic, complemented
4C7×#1616C214yesmaximal, complemented
5C7×#1604C214yesmaximal, complemented
6C7×#1519C212yesmaximal, complemented
7C2×C7×#324C211yesmaximal, 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×#483—22no—
19C7×#481C2214yescomplemented
20C7×#465—24no—
21C7×#450—22no—
22C7×#448C2214yescomplemented
23C2×C7×C22⋊C8C2212yes—
24C7×#400C2212yescomplemented
25C7×#398—22no—
26C7×#395C2212yescomplemented
27C7×#385—21no—
28C7×#381—21no—
29C7×#380C2211yescharacteristic
30C7×#327C2218yescomplemented
31C7×#324C2214yescomplemented
32#18293C711yesmaximal, characteristic, direct product factor, Sylow
33C2×C7×#41—24no—
34C2×C7×#41—24no—
35C2×C7×#41—44no—
36C2×C7×#41—44no—
37C22×C7×Q8—21no—
38C22×C7×Q8C2311yescharacteristic
39C22×C7×D4—21no—
40C22×C7×D4—22no—
41C22×C7×D4—21no—
42C22×C7×D4D412yes—
43C23×C4×C7—22no—
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C2×C7×SD16—24no—
48C22×C7×C8—22no—
49C7×C22⋊Q8—28no—
50C7×C22⋊Q8—28no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×#121—24no—
54C7×#121—28no—
55C7×#121—28no—
56C7×#121—24no—
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⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4D412yes—
68C2×C7×C4⋊C4D412yes—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C7×#106—24no—
76C7×#103—24no—
77C7×#102—24no—
78C7×C22⋊C8—22no—
79C7×C22⋊C8C2314yes—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—22no—
83C7×#95C2312yes—
84C2×#537C2×C711yescharacteristic
85#3187C2×C712yescomplemented
86#2627C2×C711yescharacteristic, complemented
87#1616C2×C714yescomplemented
88#1604C2×C714yescomplemented
89#1519C2×C712yescomplemented
90C2×#324C2×C711yescharacteristic, complemented
91C24×C7—22no—
92C24×C7C2×D411yescharacteristic
93C7×#41—48no—
94C7×#41—416no—
95C7×#41—48no—
96C2×C7×Q8—24no—
97C2×C7×Q8—22no—
98C2×C7×Q8—44no—
99C2×C7×Q8—81no—
100C2×C7×D4—24no—
101C2×C7×D4—24no—
102C2×C7×D4—24no—
103C2×C7×D4—24no—
104C2×C7×D4—22no—
105C2×C7×D4—44no—
106C2×C7×D4—44no—
107C2×C7×D4—44no—
108C2×C7×D4—82no—
109C2×C7×D4—82no—
110C2×C7×D4C2×D414yes—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—28no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—48no—
122C22×C4×C7—81no—
123C22×C4×C7—81no—
124C22×C4×C7SD1614yescomplemented
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2411yescharacteristic
131C7×SD16—84no—
132C2×C7×C8—24no—
133C2×C7×C8—82no—
134C7×C4⋊C4—24no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4—81no—
141C7×C4⋊C4—82no—
142C7×C4⋊C4C2×D414yes—
143C7×C22⋊C4—28no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—42no—
149C22×#41—21no—
150#537—24no—
151#537—22no—
152#537C22×C714yes—
153C2×C22⋊Q8—22no—
154C2×C22⋊Q8C22×C711yescharacteristic
155C2×#121—22no—
156C2×#121—21no—
157C2×#121C22×C712yes—
158C22×C4⋊C4C22×C711yescharacteristic
159#483—22no—
160#481C22×C714yescomplemented
161#465—24no—
162#450—22no—
163#448C22×C714yescomplemented
164C2×C22⋊C8C22×C712yes—
165#400C22×C712yescomplemented
166#398—22no—
167#395C22×C712yescomplemented
168#385—21no—
169#381—21no—
170#380C22×C711yescharacteristic
171#327C22×C718yescomplemented
172#324C22×C714yescomplemented
173C23×C7—24no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7—44no—
177C23×C7—44no—
178C23×C7—44no—
179C23×C7#12011yescharacteristic
180C23×C7#12011yescharacteristic
181C23×C7C2×SD1612yes—
182C23×C7AGL(1,ℤ/8)12yes—
183C23×C7C22×D411yescharacteristic
184C7×Q8—48no—
185C7×Q8—44no—
186C7×D4—44no—
187C7×D4—48no—
188C7×D4—44no—
189C7×D4—44no—
190C7×D4—48no—
191C7×D4—82no—
192C7×D4—84no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—24no—
197C2×C4×C7—22no—
198C2×C4×C7—24no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—41no—
203C2×C4×C7—416no—
204C2×C4×C7—42no—
205C2×C4×C7—44no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—81no—
215C2×C4×C7#12012yes—
216C2×C4×C7C2×SD1614yes—
217C2×C4×C7C22×D412yes—
218C7×C8—44no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8C23×C711yescharacteristic
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4C7×D412yes—
229C23×C4—22no—
230C23×C4C7×D411yescharacteristic
231C23×C4C7×D411yescharacteristic
232C23×C4C23×C711yescharacteristic
233C2×SD16—24no—
234C22×C8—22no—
235C22⋊Q8—28no—
236C22⋊Q8—28no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239#121—24no—
240#121—28no—
241#121—28no—
242#121—24no—
243#121—22no—
244#121—22no—
245#121C7×D414yes—
246#121C23×C714yes—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C7×D412yes—
254C2×C4⋊C4C7×D412yes—
255C2×C4⋊C4C23×C711yescharacteristic
256C2×C4⋊C4C23×C712yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261#106—24no—
262#103—24no—
263#102—24no—
264C22⋊C8—22no—
265C22⋊C8C23×C714yes—
266#95—22no—
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—42no—
277C22×C7—42no—
278C22×C7—44no—
279C22×C7—44no—
280C22×C7—44no—
281C22×C7—88no—
282C22×C7#45012yes—
283C22×C7#45012yes—
284C22×C7C2×#12011yescharacteristic
285C22×C7C22×SD1611yescharacteristic
286C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C7—24no—
288C4×C7—22no—
289C4×C7—22no—
290C4×C7—44no—
291C4×C7—42no—
292C4×C7—82no—
293C4×C7—81no—
294C4×C7—81no—
295C24—22no—
296C24C2×C7×D411yescharacteristic
297#41—48no—
298#41—416no—
299#41—48no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—81no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C7×D414yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—48no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4C7×SD1614yescomplemented
329C22×C4C2×C7×D412yes—
330C22×C4C2×C7×D412yes—
331C22×C4C2×C7×D411yescharacteristic
332C22×C4C2×C7×D412yes—
333C22×C4C2×C7×D412yes—
334C22×C4C24×C711yescharacteristic
335SD16—84no—
336C2×C8—24no—
337C2×C8—82no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—81no—
345C4⋊C4—82no—
346C4⋊C4C2×C7×D414yes—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C2×C7—21no—
354C2×C7—22no—
355C2×C7—21no—
356C2×C7—22no—
357C2×C7—84no—
358C2×C7C2×#45011yescharacteristic
359C2×C7#302011yescharacteristic
360C2×C7#302611yescharacteristic
361C23—24no—
362C23—41no—
363C23—41no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23C7×#12011yescharacteristic
368C23C7×#12011yescharacteristic
369C23C2×C7×SD1612yes—
370C23C7×AGL(1,ℤ/8)12yes—
371C23C22×C7×D411yescharacteristic
372Q8—48no—
373Q8—44no—
374D4—44no—
375D4—48no—
376D4—44no—
377D4—44no—
378D4—48no—
379D4—82no—
380D4—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—416no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4C7×#12012yes—
404C2×C4C2×C7×SD1614yes—
405C2×C4C22×C7×D412yes—
406C8—44no—
407C7#1829311yescharacteristic, direct product factor, Sylow
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C7×#45012yes—
421C22C7×#45012yes—
422C22C2×C7×#12011yescharacteristic
423C22C22×C7×SD1611yescharacteristic
424C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
425C4—24no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—42no—
430C4—82no—
431C4—81no—
432C4—81no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2—22no—
437C2—84no—
438C2C2×C7×#45011yescharacteristic
439C2C7×#302011yescharacteristic
440C2C7×#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×#3187C2semidirect
3. C7×#2627C2semidirect
4. C7×#1616C2semidirect
5. C7×#1604C2semidirect
6. C7×#1519C2semidirect
7. C2×C7×#324C2semidirect
19. C7×#481C22semidirect
22. C7×#448C22semidirect
24. C7×#400C22semidirect
26. C7×#395C22semidirect
30. C7×#327C22semidirect
31. C7×#324C22semidirect
32. #18293C7direct (N × H)
85. #3187C2×C7semidirect
86. #2627C2×C7semidirect
87. #1616C2×C7semidirect
88. #1604C2×C7semidirect
89. #1519C2×C7semidirect
90. C2×#324C2×C7semidirect
124. C22×C4×C7SD16semidirect
160. #481C22×C7semidirect
163. #448C22×C7semidirect
165. #400C22×C7semidirect
167. #395C22×C7semidirect
171. #327C22×C7semidirect
172. #324C22×C7semidirect
328. C22×C4C7×SD16semidirect
407. C7#18293direct (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×#31872
C7×#26272
C7×#16162
C7×#16042
C7×#15192
C2×C7×#3242
#182937