Finite Group Databasebeta

← C7×#36692 #13213908

Subgroups — C7×#36692 #13213908

Order 1792.

1716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2837C214yesmaximal, complemented
2C7×#2836C214yesmaximal, complemented
3C7×#2829C214yesmaximal, complemented
4C7×#2828C214yesmaximal, complemented
5C7×#2402C211yesmaximal, characteristic, complemented
6C7×#2397C211yesmaximal, characteristic, complemented
7C7×#2395C214yesmaximal, complemented
8C7×#2392C214yesmaximal, complemented
9C7×#2392C212yesmaximal, complemented
10C7×#2298C211yesmaximal, characteristic, complemented
11C7×#2293C212yesmaximal, complemented
12C2×C4×C7×D4C2212yes—
13C2×C4×C7×D4C2214yes—
14C2×C4×C7×D4C2214yes—
15C2×C7×#117C2211yescharacteristic
16C2×C7×#117C2214yes—
17C7×#400C2214yes—
18C7×#400C2214yes—
19C7×#399C2214yes—
20C7×#399C2214yes—
21C7×#398C2214yes—
22C7×#398C2212yes—
23C7×#398C2214yes—
24C7×#398C2214yes—
25C7×#397C2214yes—
26C7×#397C2212yes—
27C7×#397C2214yes—
28C7×#397C2214yes—
29C7×#397C2214yes—
30C7×#396C2214yes—
31C7×#396C2214yes—
32C7×#396C2214yes—
33C7×#394C2214yes—
34C7×#394C2214yes—
35C7×#394C2214yes—
36C7×#394C2214yes—
37C7×#394C2212yes—
38C7×#394C2214yes—
39C7×#394C2214yes—
40C7×#392C2212yes—
41C7×#392C2214yes—
42C7×#392C2214yes—
43C7×#390C2214yes—
44C7×#389C2214yes—
45C7×#388C2214yes—
46C7×#388C2214yes—
47C7×#388C2214yes—
48C7×#387C2214yes—
49C4×C7×C4⋊C4C2214yes—
50C4×C7×C4⋊C4C2212yes—
51C4×C7×C4⋊C4C2212yes—
52C4×C7×C22⋊C4C2212yes—
53C4×C7×C22⋊C4C2214yes—
54C4×C7×C22⋊C4C2214yes—
55C7×#376C2211yescharacteristic
56C43×C7C2211yescharacteristic
57#36692C711yesmaximal, characteristic, direct product factor, Sylow
58C22×C7×D4C2312yes—
59C22×C7×D4C2314yes—
60C22×C7×D4C2314yes—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2314yes—
63C4×C7×D4—28no—
64C4×C7×D4—48no—
65C4×C7×D4—48no—
66C7×#117—24no—
67C7×#117—48no—
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C4⋊C4C2314yes—
71C2×C7×C4⋊C4C2314yes—
72C2×C7×C4⋊C4C2314yes—
73C2×C7×C4⋊C4C2314yes—
74C2×C7×C4⋊C4C2314yes—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C4⋊C4C2314yes—
77C2×C7×C4⋊C4C2314yes—
78C2×C7×C22⋊C4C2312yes—
79C2×C7×C22⋊C4C2314yes—
80C2×C7×C22⋊C4C2314yes—
81C2×C7×C22⋊C4C2314yes—
82C2×C7×C22⋊C4C2314yes—
83C2×C7×C22⋊C4C2314yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C7×C22⋊C4C2314yes—
86C2×C7×C22⋊C4C2314yes—
87C2×C7×C22⋊C4C2314yes—
88C2×C7×C22⋊C4C2314yes—
89C2×C7×C22⋊C4C2314yes—
90C2×C7×C22⋊C4C2314yes—
91C2×C42×C7C2312yes—
92C2×C42×C7C2312yes—
93C2×C42×C7C2311yescharacteristic
94C2×C42×C7C2314yes—
95C2×C42×C7C2314yes—
96C2×C42×C7C2311yescharacteristic
97C7×#95C2314yes—
98C7×#95C2314yes—
99C7×#95C2314yes—
100C7×#95C2314yes—
101C7×#95C2314yes—
102C7×#95C2314yes—
103C7×#95C2314yes—
104C7×#95C2312yes—
105C7×#95C2312yes—
106C7×#95C2314yes—
107C7×#95C2314yes—
108#2837C2×C714yescomplemented
109#2836C2×C714yescomplemented
110#2829C2×C714yescomplemented
111#2828C2×C714yescomplemented
112#2402C2×C711yescharacteristic, complemented
113#2397C2×C711yescharacteristic, complemented
114#2395C2×C714yescomplemented
115#2392C2×C712yescomplemented
116#2392C2×C714yescomplemented
117#2298C2×C711yescharacteristic, complemented
118#2293C2×C712yescomplemented
119C24×C7C2411yescharacteristic
120C24×C7C2414yes—
121C2×C7×D4—24no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C22×C4×C7—22no—
130C22×C4×C7—28no—
131C22×C4×C7—28no—
132C22×C4×C7—24no—
133C22×C4×C7—28no—
134C22×C4×C7C2412yes—
135C22×C4×C7C2414yes—
136C22×C4×C7C2414yes—
137C22×C4×C7C2414yes—
138C22×C4×C7C2414yes—
139C22×C4×C7C2414yes—
140C22×C4×C7C2411yescharacteristic
141C22×C4×C7C2412yes—
142C22×C4×C7C2411yescharacteristic
143C7×C4⋊C4—24no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—44no—
146C7×C4⋊C4—44no—
147C7×C4⋊C4—44no—
148C7×C4⋊C4—44no—
149C7×C4⋊C4—44no—
150C7×C4⋊C4—44no—
151C7×C4⋊C4—44no—
152C7×C4⋊C4—44no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—28no—
155C7×C22⋊C4—44no—
156C7×C22⋊C4—44no—
157C7×C22⋊C4—44no—
158C7×C22⋊C4—44no—
159C7×C22⋊C4—44no—
160C7×C22⋊C4—44no—
161C7×C22⋊C4—44no—
162C7×C22⋊C4—44no—
163C7×C22⋊C4—44no—
164C7×C22⋊C4—44no—
165C7×C22⋊C4—44no—
166C42×C7—22no—
167C42×C7—24no—
168C42×C7—28no—
169C42×C7—24no—
170C42×C7—44no—
171C42×C7#4114yes—
172C2×C4×D4C22×C712yes—
173C2×C4×D4C22×C714yes—
174C2×C4×D4C22×C714yes—
175C2×#117C22×C711yescharacteristic
176C2×#117C22×C714yes—
177#400C22×C714yes—
178#400C22×C714yes—
179#399C22×C714yes—
180#399C22×C714yes—
181#398C22×C714yes—
182#398C22×C714yes—
183#398C22×C712yes—
184#398C22×C714yes—
185#397C22×C714yes—
186#397C22×C714yes—
187#397C22×C714yes—
188#397C22×C714yes—
189#397C22×C712yes—
190#396C22×C714yes—
191#396C22×C714yes—
192#396C22×C714yes—
193#394C22×C714yes—
194#394C22×C714yes—
195#394C22×C714yes—
196#394C22×C714yes—
197#394C22×C714yes—
198#394C22×C714yes—
199#394C22×C712yes—
200#392C22×C712yes—
201#392C22×C714yes—
202#392C22×C714yes—
203#390C22×C714yes—
204#389C22×C714yes—
205#388C22×C714yes—
206#388C22×C714yes—
207#388C22×C714yes—
208#387C22×C714yes—
209C4×C4⋊C4C22×C714yes—
210C4×C4⋊C4C22×C712yes—
211C4×C4⋊C4C22×C712yes—
212C4×C22⋊C4C22×C712yes—
213C4×C22⋊C4C22×C714yes—
214C4×C22⋊C4C22×C714yes—
215#376C22×C711yescharacteristic
216C43C22×C711yescharacteristic
217C23×C7—21no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—22no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7C2511yescharacteristic
228C7×D4—48no—
229C7×D4—416no—
230C7×D4—416no—
231C2×C4×C7—24no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—21no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—44no—
252C2×C4×C7—416no—
253C2×C4×C7C2×#4114yes—
254C2×C4×C7C2×#4112yes—
255C2×C4×C7H(5,𝔽2)14yes—
256C22×D4C23×C712yes—
257C22×D4C23×C714yes—
258C22×D4C23×C714yes—
259C23×C4C23×C711yescharacteristic
260C23×C4C23×C714yes—
261C4×D4—28no—
262C4×D4—48no—
263C4×D4—48no—
264#117—24no—
265#117—48no—
266C2×C4⋊C4C23×C712yes—
267C2×C4⋊C4C23×C712yes—
268C2×C4⋊C4C23×C714yes—
269C2×C4⋊C4C23×C714yes—
270C2×C4⋊C4C23×C714yes—
271C2×C4⋊C4C23×C714yes—
272C2×C4⋊C4C23×C714yes—
273C2×C4⋊C4C23×C714yes—
274C2×C4⋊C4C23×C714yes—
275C2×C4⋊C4C23×C714yes—
276C2×C22⋊C4C23×C712yes—
277C2×C22⋊C4C23×C714yes—
278C2×C22⋊C4C23×C714yes—
279C2×C22⋊C4C23×C714yes—
280C2×C22⋊C4C23×C714yes—
281C2×C22⋊C4C23×C714yes—
282C2×C22⋊C4C23×C714yes—
283C2×C22⋊C4C23×C714yes—
284C2×C22⋊C4C23×C714yes—
285C2×C22⋊C4C23×C714yes—
286C2×C22⋊C4C23×C714yes—
287C2×C22⋊C4C23×C714yes—
288C2×C22⋊C4C23×C714yes—
289C2×C42C23×C712yes—
290C2×C42C23×C714yes—
291C2×C42C23×C711yescharacteristic
292C2×C42C23×C712yes—
293C2×C42C23×C714yes—
294C2×C42C23×C711yescharacteristic
295#95C23×C714yes—
296#95C23×C714yes—
297#95C23×C714yes—
298#95C23×C714yes—
299#95C23×C714yes—
300#95C23×C714yes—
301#95C23×C714yes—
302#95C23×C712yes—
303#95C23×C712yes—
304#95C23×C714yes—
305#95C23×C714yes—
306C22×C7—41no—
307C22×C7—44no—
308C22×C7—44no—
309C22×C7—42no—
310C22×C7—44no—
311C22×C7—44no—
312C22×C7—44no—
313C22×C7—44no—
314C22×C7—44no—
315C22×C7—44no—
316C22×C7C22×#4111yescharacteristic
317C22×C7C2×H(5,𝔽2)12yes—
318C22×C7#58514yes—
319C4×C7—24no—
320C4×C7—44no—
321C4×C7—41no—
322C4×C7—42no—
323C4×C7—44no—
324C4×C7—44no—
325C4×C7—44no—
326C4×C7—44no—
327C4×C7#54014yes—
328C24C24×C711yescharacteristic
329C24C24×C714yes—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4C24×C712yes—
344C22×C4C24×C714yes—
345C22×C4C24×C714yes—
346C22×C4C24×C714yes—
347C22×C4C24×C714yes—
348C22×C4C24×C714yes—
349C22×C4C24×C711yescharacteristic
350C22×C4C24×C712yes—
351C22×C4C24×C711yescharacteristic
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C22⋊C4—24no—
363C22⋊C4—28no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C42—22no—
376C42—24no—
377C42—28no—
378C42—24no—
379C42—44no—
380C42C7×#4114yes—
381C2×C7—84no—
382C2×C7—81no—
383C2×C7C2×#54011yescharacteristic
384C2×C7#345512yes—
385C2×C7#352914yes—
386C23—21no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23C25×C711yescharacteristic
397D4—48no—
398D4—416no—
399D4—416no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—416no—
422C2×C4C2×C7×#4114yes—
423C2×C4C2×C7×#4112yes—
424C2×C4C7×H(5,𝔽2)14yes—
425C7#3669211yescharacteristic, direct product factor, Sylow
426C22—41no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22C22×C7×#4111yescharacteristic
437C22C2×C7×H(5,𝔽2)12yes—
438C22C7×#58514yes—
439C4—24no—
440C4—44no—
441C4—41no—
442C4—42no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4C7×#54014yes—
448C2—84no—
449C2—81no—
450C2C2×C7×#54011yescharacteristic
451C2C7×#345512yes—
452C2C7×#352914yes—
subgroupG/H
centerC22×C4×C7C24
commutator, Frattini, cosocleC23C25×C7
socleC23×C7C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2837C2semidirect
2. C7×#2836C2semidirect
3. C7×#2829C2semidirect
4. C7×#2828C2semidirect
5. C7×#2402C2semidirect
6. C7×#2397C2semidirect
7. C7×#2395C2semidirect
8. C7×#2392C2semidirect
9. C7×#2392C2semidirect
10. C7×#2298C2semidirect
11. C7×#2293C2semidirect
57. #36692C7direct (N × H)
108. #2837C2×C7semidirect
109. #2836C2×C7semidirect
110. #2829C2×C7semidirect
111. #2828C2×C7semidirect
112. #2402C2×C7semidirect
113. #2397C2×C7semidirect
114. #2395C2×C7semidirect
115. #2392C2×C7semidirect
116. #2392C2×C7semidirect
117. #2298C2×C7semidirect
118. #2293C2×C7semidirect
425. C7#36692direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28372
C7×#28362
C7×#28292
C7×#28282
C7×#24022
C7×#23972
C7×#23952
C7×#23922
C7×#22982
C7×#22932
#366927