Finite Group Databasebeta

← C7×#24113 #13201329

Subgroups — C7×#24113 #13201329

Order 1792.

1122 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3491C211yesmaximal, characteristic
2C7×#2853C212yesmaximal, complemented
3C7×#2296C211yesmaximal, characteristic, complemented
4C7×#2062C212yesmaximal, complemented
5C7×#2044C212yesmaximal
6C7×#1899C212yesmaximal, complemented
7C7×#1867C214yesmaximal, complemented
8C7×#1820C211yesmaximal, characteristic, complemented
9C7×#552—24no—
10C7×#551—24no—
11C7×#543C2211yescharacteristic
12C7×#539—22no—
13C7×#534C2211yescharacteristic
14C7×#529—22no—
15C2×C7×#123C2212yes—
16C7×#518C2212yes—
17C2×C7×#117C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C2×C7×#99C2214yes—
21C7×#389—22no—
22C7×#388—24no—
23C7×#387—24no—
24C7×#386C2214yes—
25C7×#385C2212yes—
26C7×#385C2214yes—
27C7×#385C2212yes—
28C7×#381C2212yes—
29C4×C7×C4⋊C4—22no—
30C7×#342C2212yes—
31C7×#342C2214yes—
32C7×#342C2212yes—
33#24113C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×D4C2312yes—
35C23×C4×C7C2312yes—
36C23×C4×C7C2311yescharacteristic
37C7×#126—24no—
38C7×#126—24no—
39C7×#125—22no—
40C7×#125—24no—
41C7×#124—22no—
42C7×#123—24no—
43C7×#123—22no—
44C7×#123—22no—
45C7×#123—24no—
46C7×#123—22no—
47C7×C22⋊Q8—24no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—24no—
50C7×#121—22no—
51C7×#120—22no—
52C4×C7×Q8—22no—
53C4×C7×D4—22no—
54C4×C7×D4—24no—
55C4×C7×D4—24no—
56C7×#117—24no—
57C7×#117—24no—
58C7×#117—22no—
59C7×#117—24no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4D412yes—
65C2×C7×C4⋊C4D414yes—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4D414yes—
69C2×C7×C22⋊C4C2314yes—
70C2×C7×C22⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4C2312yes—
73C2×C7×C22⋊C4C2312yes—
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7—24no—
77C2×C42×C7—21no—
78C2×C42×C7D412yes—
79C7×#99—28no—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—24no—
85#3491C2×C711yescharacteristic
86#2853C2×C712yescomplemented
87#2296C2×C711yescharacteristic, complemented
88#2062C2×C712yescomplemented
89#2044C2×C712yes—
90#1899C2×C712yescomplemented
91#1867C2×C714yescomplemented
92#1820C2×C711yescharacteristic, complemented
93C24×C7C2×D412yes—
94C24×C7C2411yescharacteristic
95C2×C7×Q8—22no—
96C2×C7×D4—22no—
97C2×C7×D4—22no—
98C2×C7×D4—24no—
99C2×C7×D4—24no—
100C2×C7×D4—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—24no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—24no—
107C22×C4×C7—24no—
108C22×C4×C7—44no—
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2×D414yes—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2×D412yes—
115C22×C4×C7C2×D414yes—
116C7×C4⋊C4—24no—
117C7×C4⋊C4—24no—
118C7×C4⋊C4—24no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—24no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—48no—
125C7×C4⋊C4—44no—
126C7×C4⋊C4—44no—
127C7×C22⋊C4—24no—
128C7×C22⋊C4—24no—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C7×C22⋊C4—22no—
132C7×C22⋊C4—22no—
133C7×C22⋊C4—22no—
134C7×C22⋊C4—24no—
135C7×C22⋊C4—24no—
136C7×C22⋊C4—24no—
137C7×C22⋊C4—24no—
138C7×C22⋊C4—48no—
139C7×C22⋊C4#4114yes—
140C42×C7—24no—
141C42×C7—24no—
142C42×C7—22no—
143C42×C7—22no—
144C42×C7—42no—
145C42×C7—42no—
146#552—24no—
147#551—24no—
148#543C22×C711yescharacteristic
149#539—22no—
150#534C22×C711yescharacteristic
151#529—22no—
152C2×#123C22×C712yes—
153#518C22×C712yes—
154C2×#117C22×C711yescharacteristic
155C2×#117C22×C711yescharacteristic
156C22×C42C22×C711yescharacteristic
157C2×#99C22×C714yes—
158#389—22no—
159#388—24no—
160#387—24no—
161#386C22×C714yes—
162#385C22×C714yes—
163#385C22×C712yes—
164#385C22×C712yes—
165#381C22×C712yes—
166C4×C4⋊C4—22no—
167#342C22×C712yes—
168#342C22×C714yes—
169#342C22×C712yes—
170C23×C7—22no—
171C23×C7—24no—
172C23×C7—42no—
173C23×C7—44no—
174C23×C7C22×D411yescharacteristic
175C23×C7C22×D412yes—
176C23×C7C2×#4112yes—
177C23×C72−1+412yes—
178C7×Q8—82no—
179C7×D4—42no—
180C7×D4—42no—
181C7×D4—44no—
182C7×D4—82no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—28no—
188C2×C4×C7—28no—
189C2×C4×C7—24no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—24no—
194C2×C4×C7—24no—
195C2×C4×C7—24no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—44no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—44no—
203C2×C4×C7—42no—
204C2×C4×C7—44no—
205C2×C4×C7—44no—
206C2×C4×C7—44no—
207C2×C4×C7—84no—
208C2×C4×C7#12012yes—
209C2×C4×C7#12012yes—
210C22×D4C23×C712yes—
211C23×C4C23×C711yescharacteristic
212C23×C4C23×C712yes—
213#126—24no—
214#126—24no—
215#125—22no—
216#125—24no—
217#124—22no—
218#123—24no—
219#123—22no—
220#123—22no—
221#123—24no—
222#123—22no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8—24no—
226#121—22no—
227#120—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—24no—
231C4×D4—24no—
232#117—24no—
233#117—24no—
234#117—22no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4C7×D412yes—
241C2×C4⋊C4C7×D414yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4C7×D414yes—
245C2×C22⋊C4C23×C714yes—
246C2×C22⋊C4C23×C711yescharacteristic
247C2×C22⋊C4C23×C711yescharacteristic
248C2×C22⋊C4C23×C712yes—
249C2×C22⋊C4C23×C712yes—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42C7×D412yes—
255#99—28no—
256#95—22no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261C22×C7—22no—
262C22×C7—24no—
263C22×C7—42no—
264C22×C7—42no—
265C22×C7—42no—
266C22×C7—44no—
267C22×C7—84no—
268C22×C7C2×#12011yescharacteristic
269C22×C7#52511yescharacteristic
270C22×C7#53611yescharacteristic
271C22×C7#54714yes—
272C4×C7—22no—
273C4×C7—22no—
274C4×C7—44no—
275C4×C7—42no—
276C4×C7—44no—
277C4×C7—42no—
278C4×C7—44no—
279C4×C7—42no—
280C4×C7—84no—
281C24C2×C7×D412yes—
282C24C24×C711yescharacteristic
283C2×Q8—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4C2×C7×D412yes—
298C22×C4C2×C7×D414yes—
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D411yescharacteristic
301C22×C4C2×C7×D412yes—
302C22×C4C2×C7×D412yes—
303C22×C4C2×C7×D414yes—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—48no—
327C22⋊C4C7×#4114yes—
328C42—24no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C2×C7—21no—
335C2×C7—24no—
336C2×C7—21no—
337C2×C7—82no—
338C2×C7#240611yescharacteristic
339C2×C7#302512yes—
340C23—22no—
341C23—24no—
342C23—42no—
343C23—44no—
344C23C22×C7×D411yescharacteristic
345C23C22×C7×D412yes—
346C23C2×C7×#4112yes—
347C23C7×2−1+412yes—
348Q8—82no—
349D4—42no—
350D4—42no—
351D4—44no—
352D4—82no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4C7×#12012yes—
379C2×C4C7×#12012yes—
380C7#2411311yescharacteristic, direct product factor, Sylow
381C22—22no—
382C22—24no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—44no—
387C22—84no—
388C22C2×C7×#12011yescharacteristic
389C22C7×#52511yescharacteristic
390C22C7×#53611yescharacteristic
391C22C7×#54714yes—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—42no—
396C4—44no—
397C4—42no—
398C4—44no—
399C4—42no—
400C4—84no—
401C2—21no—
402C2—24no—
403C2—21no—
404C2—82no—
405C2C7×#240611yescharacteristic
406C2C7×#302512yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#534C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2853C2semidirect
3. C7×#2296C2semidirect
4. C7×#2062C2semidirect
6. C7×#1899C2semidirect
7. C7×#1867C2semidirect
8. C7×#1820C2semidirect
33. #24113C7direct (N × H)
86. #2853C2×C7semidirect
87. #2296C2×C7semidirect
88. #2062C2×C7semidirect
90. #1899C2×C7semidirect
91. #1867C2×C7semidirect
92. #1820C2×C7semidirect
380. C7#24113direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34912
C7×#28532
C7×#22962
C7×#20622
C7×#20442
C7×#18992
C7×#18672
C7×#18202
#241137