Finite Group Databasebeta

← C7×#18327 #13195543

Subgroups — C7×#18327 #13195543

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic
2C7×#3195C212yesmaximal
3C7×#2583C211yesmaximal, characteristic
4C7×#1611C214yesmaximal, complemented
5C7×#1605C214yesmaximal, complemented
6C7×#1508C212yesmaximal
7C7×#1463C211yesmaximal, characteristic
8C7×#544—24no—
9C7×#544—22no—
10C7×#544C2214yes—
11C2×C7×C4⋊Q8—22no—
12C2×C7×#125—22no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×C7×#117—21no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C7×#483—22no—
19C7×#483C2214yes—
20C7×#470—24no—
21C7×#451—22no—
22C7×#451C2214yes—
23C7×#407C2212yes—
24C7×#398C2212yes—
25C7×#395C2212yes—
26C7×#389—22no—
27C7×#385—21no—
28C7×#385—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#328C2218yes—
31C7×#324C2214yes—
32#18327C711yesmaximal, characteristic, direct product factor, Sylow
33C22×C7×Q8—21no—
34C22×C7×Q8D412yes—
35C22×C7×Q8C2311yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7C2311yescharacteristic
39C2×C7×Q16—24no—
40C2×C7×M16—22no—
41C7×C4⋊Q8—28no—
42C7×C4⋊Q8—28no—
43C7×#125—28no—
44C7×#125—28no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8—24no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8D414yes—
52C7×C22⋊Q8C2314yes—
53C7×#117—24no—
54C7×#117—24no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4D412yes—
66C2×C7×C4⋊C4D412yes—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7—22no—
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×#544C2×C711yescharacteristic
83#3195C2×C712yes—
84#2583C2×C711yescharacteristic
85#1611C2×C714yescomplemented
86#1605C2×C714yescomplemented
87#1508C2×C712yes—
88#1463C2×C711yescharacteristic
89C24×C7C2×D411yescharacteristic
90C2×C7×Q8—24no—
91C2×C7×Q8—22no—
92C2×C7×Q8—22no—
93C2×C7×Q8—81no—
94C2×C7×Q8—82no—
95C2×C7×Q8—82no—
96C2×C7×Q8C2×D414yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—24no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—81no—
108C22×C4×C7—81no—
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2×D412yes—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D412yes—
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2411yescharacteristic
115C7×Q16—84no—
116C7×M16—82no—
117C2×C7×C8—24no—
118C7×C4⋊C4—24no—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—28no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—24no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—42no—
129C7×C4⋊C4—44no—
130C7×C4⋊C4—82no—
131C7×C4⋊C4C2×D414yes—
132C7×C22⋊C4—24no—
133C7×C22⋊C4—24no—
134C7×C22⋊C4—24no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C42×C7—28no—
138C42×C7—81no—
139#544—24no—
140#544—22no—
141#544C22×C714yes—
142C2×C4⋊Q8—22no—
143C2×#125—22no—
144C2×C22⋊Q8—21no—
145C2×C22⋊Q8C22×C712yes—
146C2×C22⋊Q8C22×C711yescharacteristic
147C2×#117—21no—
148C22×C4⋊C4C22×C711yescharacteristic
149#483—22no—
150#483C22×C714yes—
151#470—24no—
152#451—22no—
153#451C22×C714yes—
154#407C22×C712yes—
155#398C22×C712yes—
156#395C22×C712yes—
157#389—22no—
158#385—21no—
159#385—21no—
160C4×C22⋊C4C22×C711yescharacteristic
161#328C22×C718yes—
162#324C22×C714yes—
163C23×C7—22no—
164C23×C7—22no—
165C23×C7—41no—
166C23×C7#12011yescharacteristic
167C23×C7#12011yescharacteristic
168C23×C7C22×D411yescharacteristic
169C7×Q8—44no—
170C7×Q8—44no—
171C7×Q8—82no—
172C7×Q8—84no—
173C2×C4×C7—24no—
174C2×C4×C7—24no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—24no—
178C2×C4×C7—24no—
179C2×C4×C7—24no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—44no—
188C2×C4×C7—42no—
189C2×C4×C7—44no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—81no—
196C2×C4×C7#12012yes—
197C2×C4×C7#13714yes—
198C2×C4×C7C22×D412yes—
199C7×C8—44no—
200C22×Q8—21no—
201C22×Q8C7×D412yes—
202C22×Q8C23×C711yescharacteristic
203C23×C4C7×D411yescharacteristic
204C23×C4C7×D411yescharacteristic
205C23×C4C23×C711yescharacteristic
206C2×Q16—24no—
207C2×M16—22no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210#125—28no—
211#125—28no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8C7×D414yes—
219C22⋊Q8C23×C714yes—
220#117—24no—
221#117—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C7×D412yes—
233C2×C4⋊C4C7×D412yes—
234C2×C4⋊C4C23×C711yescharacteristic
235C2×C4⋊C4C23×C712yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42—22no—
240C2×C42—21no—
241#106—24no—
242#103—24no—
243#103—24no—
244C22⋊C8—22no—
245C22⋊C8C23×C714yes—
246#95—22no—
247#95—22no—
248#95C23×C712yes—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—44no—
257C22×C7#45612yes—
258C22×C7C2×#12011yescharacteristic
259C22×C7C2×#13712yes—
260C4×C7—24no—
261C4×C7—22no—
262C4×C7—22no—
263C4×C7—44no—
264C4×C7—44no—
265C4×C7—42no—
266C4×C7—44no—
267C4×C7—81no—
268C4×C7—81no—
269C24C2×C7×D411yescharacteristic
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—81no—
274C2×Q8—82no—
275C2×Q8—82no—
276C2×Q8C2×C7×D414yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C2×C7×D412yes—
290C22×C4C2×C7×D412yes—
291C22×C4C2×C7×D411yescharacteristic
292C22×C4C2×C7×D412yes—
293C22×C4C2×C7×D412yes—
294C22×C4C24×C711yescharacteristic
295Q16—84no—
296M16—82no—
297C2×C8—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C4⋊C4C2×C7×D414yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—28no—
318C42—81no—
319C2×C7—22no—
320C2×C7—22no—
321C2×C7—41no—
322C2×C7#300411yescharacteristic
323C2×C7C2×#45611yescharacteristic
324C2×C7#302611yescharacteristic
325C23—22no—
326C23—22no—
327C23—41no—
328C23C7×#12011yescharacteristic
329C23C7×#12011yescharacteristic
330C23C22×C7×D411yescharacteristic
331Q8—44no—
332Q8—44no—
333Q8—82no—
334Q8—84no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4C7×#12012yes—
359C2×C4C7×#13714yes—
360C2×C4C22×C7×D412yes—
361C8—44no—
362C7#1832711yescharacteristic, direct product factor, Sylow
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22C7×#45612yes—
372C22C2×C7×#12011yescharacteristic
373C22C2×C7×#13712yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—44no—
379C4—42no—
380C4—44no—
381C4—81no—
382C4—81no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2C7×#300411yescharacteristic
387C2C2×C7×#45611yescharacteristic
388C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC24×C7C2×D4
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
4. C7×#1611C2semidirect
5. C7×#1605C2semidirect
32. #18327C7direct (N × H)
85. #1611C2×C7semidirect
86. #1605C2×C7semidirect
362. C7#18327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5442
C7×#31952
C7×#25832
C7×#16112
C7×#16052
C7×#15082
C7×#14632
#183277