Finite Group Databasebeta

← C7×#36700 #13213916

Subgroups — C7×#36700 #13213916

Order 1792.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2839C214yesmaximal, complemented
2C7×#2836C214yesmaximal
3C7×#2833C214yesmaximal, complemented
4C7×#2831C214yesmaximal
5C7×#2402C211yesmaximal, characteristic, complemented
6C7×#2402C214yesmaximal, complemented
7C7×#2401C214yesmaximal
8C7×#2399C211yesmaximal, characteristic, complemented
9C7×#2396C212yesmaximal
10C7×#2298C211yesmaximal, characteristic
11C7×#2295C212yesmaximal, complemented
12C2×C4×C7×Q8C2212yes—
13C2×C7×#117C2211yescharacteristic
14C7×#401C2214yes—
15C7×#400C2212yes—
16C7×#400C2214yes—
17C7×#400C2214yes—
18C7×#400C2214yes—
19C7×#400C2214yes—
20C7×#400C2214yes—
21C7×#400C2214yes—
22C7×#400C2214yes—
23C7×#400C2214yes—
24C7×#399C2212yes—
25C7×#398C2214yes—
26C7×#398C2214yes—
27C7×#398C2214yes—
28C7×#398C2214yes—
29C7×#398C2214yes—
30C7×#398C2214yes—
31C7×#398C2214yes—
32C7×#397C2214yes—
33C7×#397C2214yes—
34C7×#397C2214yes—
35C7×#396C2214yes—
36C7×#396C2214yes—
37C7×#395C2212yes—
38C7×#395C2214yes—
39C7×#393C2212yes—
40C7×#391C2214yes—
41C7×#389C2214yes—
42C7×#388C2214yes—
43C7×#387C2214yes—
44C7×#387C2214yes—
45C7×#387C2214yes—
46C4×C7×C4⋊C4C2214yes—
47C4×C7×C4⋊C4C2212yes—
48C4×C7×C4⋊C4C2214yes—
49C4×C7×C4⋊C4C2214yes—
50C4×C7×C4⋊C4C2214yes—
51C4×C7×C4⋊C4C2212yes—
52C4×C7×C22⋊C4C2214yes—
53C4×C7×C22⋊C4C2212yes—
54C7×#376C2211yescharacteristic
55C7×#376C2214yes—
56C43×C7C2211yescharacteristic
57#36700C711yesmaximal, characteristic, direct product factor, Sylow
58C22×C7×Q8C2312yes—
59C23×C4×C7C2311yescharacteristic
60C4×C7×Q8—28no—
61C7×#117—24no—
62C2×C7×C4⋊C4C2312yes—
63C2×C7×C4⋊C4C2314yes—
64C2×C7×C4⋊C4C2312yes—
65C2×C7×C4⋊C4C2314yes—
66C2×C7×C4⋊C4C2314yes—
67C2×C7×C4⋊C4C2314yes—
68C2×C7×C4⋊C4C2314yes—
69C2×C7×C4⋊C4C2314yes—
70C2×C7×C4⋊C4C2314yes—
71C2×C7×C4⋊C4C2314yes—
72C2×C7×C4⋊C4C2314yes—
73C2×C7×C4⋊C4C2314yes—
74C2×C7×C4⋊C4C2314yes—
75C2×C7×C22⋊C4C2312yes—
76C2×C7×C22⋊C4C2314yes—
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2314yes—
79C2×C42×C7C2312yes—
80C2×C42×C7C2312yes—
81C2×C42×C7C2314yes—
82C2×C42×C7C2314yes—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2314yes—
85C2×C42×C7C2311yescharacteristic
86C7×#95C2314yes—
87C7×#95C2314yes—
88C7×#95C2314yes—
89C7×#95C2314yes—
90C7×#95C2314yes—
91C7×#95C2314yes—
92C7×#95C2314yes—
93C7×#95C2314yes—
94C7×#95C2312yes—
95C7×#95C2312yes—
96C7×#95C2314yes—
97C7×#95C2314yes—
98C7×#95C2314yes—
99C7×#95C2314yes—
100C7×#95C2314yes—
101C7×#95C2314yes—
102C7×#95C2314yes—
103C7×#95C2314yes—
104C7×#95C2314yes—
105#2839C2×C714yescomplemented
106#2836C2×C714yes—
107#2833C2×C714yescomplemented
108#2831C2×C714yes—
109#2402C2×C711yescharacteristic, complemented
110#2402C2×C714yescomplemented
111#2401C2×C714yes—
112#2399C2×C711yescharacteristic, complemented
113#2396C2×C712yes—
114#2298C2×C711yescharacteristic
115#2295C2×C712yescomplemented
116C24×C7C2411yescharacteristic
117C2×C7×Q8—24no—
118C2×C7×Q8—44no—
119C22×C4×C7—22no—
120C22×C4×C7—24no—
121C22×C4×C7C2411yescharacteristic
122C22×C4×C7C2414yes—
123C22×C4×C7C2414yes—
124C22×C4×C7C2414yes—
125C22×C4×C7C2414yes—
126C22×C4×C7C2414yes—
127C22×C4×C7C2412yes—
128C22×C4×C7C2414yes—
129C22×C4×C7C2412yes—
130C22×C4×C7C2411yescharacteristic
131C7×C4⋊C4—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—28no—
134C7×C4⋊C4—44no—
135C7×C4⋊C4—44no—
136C7×C4⋊C4—44no—
137C7×C4⋊C4—44no—
138C7×C4⋊C4—44no—
139C7×C4⋊C4—44no—
140C7×C4⋊C4—44no—
141C7×C4⋊C4—44no—
142C7×C4⋊C4—44no—
143C7×C4⋊C4—44no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—44no—
146C7×C22⋊C4—44no—
147C7×C22⋊C4—44no—
148C42×C7—22no—
149C42×C7—28no—
150C42×C7—24no—
151C42×C7—24no—
152C42×C7—28no—
153C42×C7—44no—
154C42×C7#4114yes—
155C2×C4×Q8C22×C712yes—
156C2×#117C22×C711yescharacteristic
157#401C22×C714yes—
158#400C22×C714yes—
159#400C22×C714yes—
160#400C22×C714yes—
161#400C22×C714yes—
162#400C22×C714yes—
163#400C22×C714yes—
164#400C22×C714yes—
165#400C22×C712yes—
166#400C22×C714yes—
167#399C22×C712yes—
168#398C22×C714yes—
169#398C22×C714yes—
170#398C22×C714yes—
171#398C22×C714yes—
172#398C22×C714yes—
173#398C22×C714yes—
174#398C22×C714yes—
175#397C22×C714yes—
176#397C22×C714yes—
177#397C22×C714yes—
178#396C22×C714yes—
179#396C22×C714yes—
180#395C22×C714yes—
181#395C22×C712yes—
182#393C22×C712yes—
183#391C22×C714yes—
184#389C22×C714yes—
185#388C22×C714yes—
186#387C22×C714yes—
187#387C22×C714yes—
188#387C22×C714yes—
189C4×C4⋊C4C22×C714yes—
190C4×C4⋊C4C22×C712yes—
191C4×C4⋊C4C22×C714yes—
192C4×C4⋊C4C22×C714yes—
193C4×C4⋊C4C22×C714yes—
194C4×C4⋊C4C22×C712yes—
195C4×C22⋊C4C22×C714yes—
196C4×C22⋊C4C22×C712yes—
197#376C22×C711yescharacteristic
198#376C22×C714yes—
199C43C22×C711yescharacteristic
200C23×C7—21no—
201C23×C7—22no—
202C23×C7—24no—
203C23×C7C2511yescharacteristic
204C7×Q8—48no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—24no—
208C2×C4×C7—24no—
209C2×C4×C7—21no—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—24no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—44no—
229C2×C4×C7C2×#4114yes—
230C2×C4×C7C2×#4112yes—
231C2×C4×C72−1+414yes—
232C22×Q8C23×C712yes—
233C23×C4C23×C711yescharacteristic
234C4×Q8—28no—
235#117—24no—
236C2×C4⋊C4C23×C712yes—
237C2×C4⋊C4C23×C714yes—
238C2×C4⋊C4C23×C712yes—
239C2×C4⋊C4C23×C714yes—
240C2×C4⋊C4C23×C714yes—
241C2×C4⋊C4C23×C714yes—
242C2×C4⋊C4C23×C714yes—
243C2×C4⋊C4C23×C714yes—
244C2×C4⋊C4C23×C714yes—
245C2×C4⋊C4C23×C714yes—
246C2×C4⋊C4C23×C714yes—
247C2×C4⋊C4C23×C714yes—
248C2×C4⋊C4C23×C714yes—
249C2×C22⋊C4C23×C712yes—
250C2×C22⋊C4C23×C714yes—
251C2×C22⋊C4C23×C714yes—
252C2×C22⋊C4C23×C714yes—
253C2×C42C23×C711yescharacteristic
254C2×C42C23×C714yes—
255C2×C42C23×C712yes—
256C2×C42C23×C712yes—
257C2×C42C23×C714yes—
258C2×C42C23×C714yes—
259C2×C42C23×C711yescharacteristic
260#95C23×C714yes—
261#95C23×C714yes—
262#95C23×C714yes—
263#95C23×C714yes—
264#95C23×C714yes—
265#95C23×C714yes—
266#95C23×C714yes—
267#95C23×C714yes—
268#95C23×C712yes—
269#95C23×C712yes—
270#95C23×C714yes—
271#95C23×C714yes—
272#95C23×C714yes—
273#95C23×C714yes—
274#95C23×C714yes—
275#95C23×C714yes—
276#95C23×C714yes—
277#95C23×C714yes—
278#95C23×C714yes—
279C22×C7—41no—
280C22×C7—42no—
281C22×C7—44no—
282C22×C7C22×#4111yescharacteristic
283C22×C7C2×2−1+412yes—
284C22×C7#58514yes—
285C4×C7—24no—
286C4×C7—41no—
287C4×C7—42no—
288C4×C7—44no—
289C4×C7—44no—
290C4×C7—44no—
291C4×C7—44no—
292C4×C7—44no—
293C4×C7—44no—
294C4×C7#54114yes—
295C24C24×C711yescharacteristic
296C2×Q8—24no—
297C2×Q8—44no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4C24×C711yescharacteristic
301C22×C4C24×C714yes—
302C22×C4C24×C714yes—
303C22×C4C24×C714yes—
304C22×C4C24×C714yes—
305C22×C4C24×C714yes—
306C22×C4C24×C712yes—
307C22×C4C24×C714yes—
308C22×C4C24×C712yes—
309C22×C4C24×C711yescharacteristic
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—24no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C42—22no—
328C42—28no—
329C42—24no—
330C42—24no—
331C42—28no—
332C42—44no—
333C42C7×#4114yes—
334C2×C7—81no—
335C2×C7C2×#54111yescharacteristic
336C2×C7#345512yes—
337C2×C7#353114yes—
338C23—21no—
339C23—22no—
340C23—24no—
341C23C25×C711yescharacteristic
342Q8—48no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—44no—
367C2×C4C2×C7×#4114yes—
368C2×C4C2×C7×#4112yes—
369C2×C4C7×2−1+414yes—
370C7#3670011yescharacteristic, direct product factor, Sylow
371C22—41no—
372C22—42no—
373C22—44no—
374C22C22×C7×#4111yescharacteristic
375C22C2×C7×2−1+412yes—
376C22C7×#58514yes—
377C4—24no—
378C4—41no—
379C4—44no—
380C4—42no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4C7×#54114yes—
387C2—81no—
388C2C2×C7×#54111yescharacteristic
389C2C7×#345512yes—
390C2C7×#353114yes—
subgroupG/H
centerC22×C4×C7C24
commutator, Frattini, cosocleC23C25×C7
socleC23×C7C25
prime radicalC24×C7C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2839C2semidirect
3. C7×#2833C2semidirect
5. C7×#2402C2semidirect
6. C7×#2402C2semidirect
8. C7×#2399C2semidirect
11. C7×#2295C2semidirect
57. #36700C7direct (N × H)
105. #2839C2×C7semidirect
107. #2833C2×C7semidirect
109. #2402C2×C7semidirect
110. #2402C2×C7semidirect
112. #2399C2×C7semidirect
115. #2295C2×C7semidirect
370. C7#36700direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28392
C7×#28362
C7×#28332
C7×#28312
C7×#24022
C7×#24012
C7×#23992
C7×#23962
C7×#22982
C7×#22952
#367007