Finite Group Databasebeta

← C7×#18319 #13195535

Subgroups — C7×#18319 #13195535

Order 1792.

1058 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
1C2×C7×#543C211yesmaximal, characteristic
2C7×#3195C212yesmaximal, complemented
3C7×#2561C211yesmaximal, characteristic
4C7×#1608C214yesmaximal, complemented
5C7×#1602C214yesmaximal, complemented
6C7×#1508C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic
8C7×#543—24no—
9C7×#543—22no—
10C7×#543C2214yes—
11C2×C7×C22⋊Q8—22no—
12C2×C7×C22⋊Q8—21no—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×C22⋊Q8C2211yescharacteristic
16C22×C7×C22⋊C4—21no—
17C22×C7×C22⋊C4C2211yescharacteristic
18C7×#483—22no—
19C7×#483C2214yescomplemented
20C7×#470—24no—
21C7×#451—22no—
22C7×#451C2214yescomplemented
23C7×#407C2212yes—
24C7×#397C2212yes—
25C7×#393C2212yes—
26C7×#388—22no—
27C7×#381—21no—
28C7×#379—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#328C2218yescomplemented
31C7×#323C2214yes—
32#18319C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7D411yescharacteristic
34C22×C7×Q8—21no—
35C22×C7×Q8D412yes—
36C22×C7×Q8C2311yescharacteristic
37C23×C4×C7—21no—
38C23×C4×C7—21no—
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C2×C7×Q16—24no—
42C2×C7×M16—22no—
43C7×C22⋊Q8—28no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8—28no—
47C7×C22⋊Q8—24no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8—28no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8D414yes—
54C7×C22⋊Q8C2314yes—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4D412yes—
72C2×C7×C22⋊C4C2312yes—
73C2×C42×C7—21no—
74C7×#106—24no—
75C7×#103—24no—
76C7×#103—24no—
77C7×C22⋊C8—22no—
78C7×C22⋊C8C2314yes—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95C2312yes—
82C2×#543C2×C711yescharacteristic
83#3195C2×C712yescomplemented
84#2561C2×C711yescharacteristic
85#1608C2×C714yescomplemented
86#1602C2×C714yescomplemented
87#1508C2×C712yescomplemented
88#1462C2×C711yescharacteristic
89C24×C7—21no—
90C24×C7—21no—
91C24×C7—24no—
92C24×C7—42no—
93C24×C7—81no—
94C24×C7C2×D411yescharacteristic
95C24×C7C2×D412yes—
96C2×C7×Q8—24no—
97C2×C7×Q8—22no—
98C2×C7×Q8—22no—
99C2×C7×Q8—81no—
100C2×C7×Q8—82no—
101C2×C7×Q8—82no—
102C2×C7×Q8C2×D414yes—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—24no—
107C22×C4×C7—24no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—42no—
113C22×C4×C7—81no—
114C22×C4×C7—82no—
115C22×C4×C7—82no—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D412yes—
120C22×C4×C7C2411yescharacteristic
121C7×Q16—84no—
122C7×M16—82no—
123C2×C7×C8—24no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4—28no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—42no—
131C7×C4⋊C4C2×D414yes—
132C7×C22⋊C4—28no—
133C7×C22⋊C4—24no—
134C7×C22⋊C4—24no—
135C7×C22⋊C4—24no—
136C7×C22⋊C4—24no—
137C7×C22⋊C4—42no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—44no—
141C7×C22⋊C4—82no—
142C42×C7—81no—
143#543—24no—
144#543—22no—
145#543C22×C714yes—
146C2×C22⋊Q8—22no—
147C2×C22⋊Q8—21no—
148C2×C22⋊Q8—22no—
149C2×C22⋊Q8C22×C712yes—
150C2×C22⋊Q8C22×C711yescharacteristic
151C22×C22⋊C4—21no—
152C22×C22⋊C4C22×C711yescharacteristic
153#483—22no—
154#483C22×C714yescomplemented
155#470—24no—
156#451—22no—
157#451C22×C714yescomplemented
158#407C22×C712yes—
159#397C22×C712yes—
160#393C22×C712yes—
161#388—22no—
162#381—21no—
163#379—21no—
164C4×C22⋊C4C22×C711yescharacteristic
165#328C22×C718yescomplemented
166#323C22×C714yes—
167C23×C7—24no—
168C23×C7—24no—
169C23×C7—22no—
170C23×C7—22no—
171C23×C7—22no—
172C23×C7—22no—
173C23×C7—41no—
174C23×C7—42no—
175C23×C7—42no—
176C23×C7—42no—
177C23×C7—42no—
178C23×C7—81no—
179C23×C7—81no—
180C23×C7—88no—
181C23×C7#12011yescharacteristic
182C23×C7#12011yescharacteristic
183C23×C7#13714yes—
184C23×C7C22×D411yescharacteristic
185C7×Q8—44no—
186C7×Q8—44no—
187C7×Q8—82no—
188C7×Q8—84no—
189C2×C4×C7—24no—
190C2×C4×C7—22no—
191C2×C4×C7—24no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—42no—
195C2×C4×C7—41no—
196C2×C4×C7—44no—
197C2×C4×C7—42no—
198C2×C4×C7—44no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—81no—
208C2×C4×C7—84no—
209C2×C4×C7—84no—
210C2×C4×C7#12012yes—
211C2×C4×C7C22×D412yes—
212C7×C8—44no—
213C25C7×D411yescharacteristic
214C22×Q8—21no—
215C22×Q8C7×D412yes—
216C22×Q8C23×C711yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C7×D411yescharacteristic
220C23×C4C23×C711yescharacteristic
221C2×Q16—24no—
222C2×M16—22no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—24no—
226C22⋊Q8—28no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—28no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8C7×D414yes—
234C22⋊Q8C23×C714yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C7×D412yes—
240C2×C4⋊C4C23×C711yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4C7×D412yes—
252C2×C22⋊C4C23×C712yes—
253C2×C42—21no—
254#106—24no—
255#103—24no—
256#103—24no—
257C22⋊C8—22no—
258C22⋊C8C23×C714yes—
259#95—22no—
260#95—22no—
261#95C23×C712yes—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—24no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—44no—
271C22×C7—44no—
272C22×C7—48no—
273C22×C7—42no—
274C22×C7—42no—
275C22×C7—82no—
276C22×C7—84no—
277C22×C7#45512yes—
278C22×C7C2×#12011yescharacteristic
279C22×C7C2×#13712yes—
280C4×C7—22no—
281C4×C7—22no—
282C4×C7—42no—
283C4×C7—44no—
284C4×C7—44no—
285C4×C7—82no—
286C4×C7—81no—
287C4×C7—81no—
288C24—21no—
289C24—21no—
290C24—24no—
291C24—42no—
292C24—81no—
293C24C2×C7×D411yescharacteristic
294C24C2×C7×D412yes—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—81no—
299C2×Q8—82no—
300C2×Q8—82no—
301C2×Q8C2×C7×D414yes—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4C2×C7×D412yes—
316C22×C4C2×C7×D411yescharacteristic
317C22×C4C2×C7×D412yes—
318C22×C4C2×C7×D412yes—
319C22×C4C24×C711yescharacteristic
320Q16—84no—
321M16—82no—
322C2×C8—24no—
323C4⋊C4—24no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4C2×C7×D414yes—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C42—81no—
342C2×C7—22no—
343C2×C7—22no—
344C2×C7—41no—
345C2×C7—44no—
346C2×C7#300411yescharacteristic
347C2×C7C2×#45511yescharacteristic
348C2×C7#302611yescharacteristic
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—88no—
363C23C7×#12011yescharacteristic
364C23C7×#12011yescharacteristic
365C23C7×#13714yes—
366C23C22×C7×D411yescharacteristic
367Q8—44no—
368Q8—44no—
369Q8—82no—
370Q8—84no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4C7×#12012yes—
393C2×C4C22×C7×D412yes—
394C8—44no—
395C7#1831911yescharacteristic, direct product factor, Sylow
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—24no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22C7×#45512yes—
412C22C2×C7×#12011yescharacteristic
413C22C2×C7×#13712yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—82no—
420C4—81no—
421C4—81no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2C7×#300411yescharacteristic
427C2C2×C7×#45511yescharacteristic
428C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC25×C7D4
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×#3195C2semidirect
4. C7×#1608C2semidirect
5. C7×#1602C2semidirect
6. C7×#1508C2semidirect
19. C7×#483C22semidirect
22. C7×#451C22semidirect
30. C7×#328C22semidirect
32. #18319C7direct (N × H)
83. #3195C2×C7semidirect
85. #1608C2×C7semidirect
86. #1602C2×C7semidirect
87. #1508C2×C7semidirect
154. #483C22×C7semidirect
157. #451C22×C7semidirect
165. #328C22×C7semidirect
395. C7#18319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5432
C7×#31952
C7×#25612
C7×#16082
C7×#16022
C7×#15082
C7×#14622
#183197