Finite Group Databasebeta

← C7×#36607 #13213823

Subgroups — C7×#36607 #13213823

Order 1792.

2524 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×#2847C213yesmaximal, complemented
2C7×#2569C213yesmaximal, complemented
3C7×#2543C213yesmaximal, complemented
4C7×#2510C216yesmaximal, complemented
5C7×#2444C216yesmaximal, complemented
6C7×#2364C213yesmaximal, complemented
7C7×#2353C213yesmaximal, complemented
8C4×C7×#120C213yesmaximal, complemented
9C22×C7×#95C211yesmaximal, characteristic, complemented
10C24×C4×C7C2213yes—
11C2×C7×#121C2213yes—
12C2×C7×#121C2216yes—
13C2×C7×#120C2213yes—
14C2×C4×C7×D4C2213yes—
15C2×C4×C7×D4C2216yes—
16C7×#401C2212yescomplemented
17C7×#400C2216yescomplemented
18C7×#398C2213yescomplemented
19C7×#397C2216yescomplemented
20C7×#397C2213yescomplemented
21C7×#394C2213yescomplemented
22C7×#392C2211yescharacteristic, complemented
23C7×#388C2216yescomplemented
24C7×#388C2216yescomplemented
25C7×#387C2216yescomplemented
26C7×#387C2216yescomplemented
27C7×#386C2213yes—
28C7×#386C2216yes—
29C7×#386C2216yes—
30C7×#386C2216yes—
31C7×#386C2216yes—
32C7×#386C2216yescomplemented
33C7×#385C2216yes—
34C7×#385C2213yes—
35C7×#383C2213yescomplemented
36C7×#381C2213yescomplemented
37C7×#380C2216yes—
38C7×#380C2213yes—
39C7×#379C2211yescharacteristic
40C4×C7×C22⋊C4C2216yes—
41C4×C7×C22⋊C4C2213yes—
42C4×C7×C22⋊C4C2213yescomplemented
43C7×#376C2211yescharacteristic, complemented
44C2×C7×#95C2216yes—
45C2×C7×#95C2213yes—
46C2×C7×#95C2213yes—
47#36607C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7C2311yescharacteristic
49C22×C7×D4C2313yes—
50C22×C7×D4C2316yes—
51C22×C7×D4C2313yes—
52C23×C4×C7—212no—
53C23×C4×C7—412no—
54C23×C4×C7C2316yes—
55C23×C4×C7C2316yes—
56C23×C4×C7C2313yes—
57C23×C4×C7C2313yes—
58C23×C4×C7C2313yes—
59C7×#121—212no—
60C7×#121—412no—
61C7×#120—212no—
62C4×C7×D4—212no—
63C4×C7×D4—412no—
64C2×C7×C4⋊C4C2313yes—
65C2×C7×C4⋊C4C2313yes—
66C2×C7×C4⋊C4C2316yes—
67C2×C7×C4⋊C4C2316yes—
68C2×C7×C22⋊C4C2313yes—
69C2×C7×C22⋊C4C2313yes—
70C2×C7×C22⋊C4C2316yes—
71C2×C7×C22⋊C4C2313yes—
72C2×C7×C22⋊C4C2316yes—
73C2×C7×C22⋊C4C2313yes—
74C2×C7×C22⋊C4C2316yes—
75C2×C7×C22⋊C4C2316yes—
76C2×C7×C22⋊C4C2316yes—
77C2×C42×C7C2313yes—
78C2×C42×C7C2316yes—
79C7×#95C2311yescharacteristic
80C7×#95C2313yes—
81C7×#95C2316yes—
82C7×#95C2311yescharacteristic
83C7×#95C2313yes—
84C7×#95C2312yes—
85C7×#95C2313yes—
86C7×#95C2316yes—
87C7×#95C2316yes—
88C7×#95C2316yes—
89C7×#95C2313yes—
90C7×#95C2316yes—
91C7×#95C2316yes—
92#2847C2×C713yescomplemented
93#2569C2×C713yescomplemented
94#2543C2×C713yescomplemented
95#2510C2×C716yescomplemented
96#2444C2×C716yescomplemented
97#2364C2×C713yescomplemented
98#2353C2×C713yescomplemented
99C4×#120C2×C713yescomplemented
100C22×#95C2×C711yescharacteristic, complemented
101C24×C7—26no—
102C24×C7—26no—
103C24×C7#4114yes—
104C24×C7C2413yes—
105C24×C7C2413yes—
106C2×C7×D4—26no—
107C2×C7×D4—26no—
108C2×C7×D4—212no—
109C2×C7×D4—212no—
110C2×C7×D4—212no—
111C2×C7×D4—26no—
112C2×C7×D4—26no—
113C2×C7×D4—46no—
114C22×C4×C7—26no—
115C22×C4×C7—212no—
116C22×C4×C7—212no—
117C22×C4×C7—212no—
118C22×C4×C7—212no—
119C22×C4×C7—26no—
120C22×C4×C7—46no—
121C22×C4×C7—46no—
122C22×C4×C7—46no—
123C22×C4×C7—46no—
124C22×C4×C7—46no—
125C22×C4×C7—448no—
126C22×C4×C7#41112yes—
127C22×C4×C7C2416yes—
128C22×C4×C7C2411yescharacteristic
129C22×C4×C7C2413yes—
130C22×C4×C7C2416yes—
131C22×C4×C7C2413yes—
132C22×C4×C7C2413yes—
133C22×C4×C7C2413yes—
134C7×C4⋊C4—26no—
135C7×C4⋊C4—26no—
136C7×C4⋊C4—46no—
137C7×C4⋊C4—46no—
138C7×C22⋊C4—26no—
139C7×C22⋊C4—212no—
140C7×C22⋊C4—26no—
141C7×C22⋊C4—26no—
142C7×C22⋊C4—26no—
143C7×C22⋊C4—212no—
144C7×C22⋊C4—46no—
145C7×C22⋊C4—46no—
146C7×C22⋊C4—46no—
147C42×C7—26no—
148C42×C7—46no—
149C24×C4C22×C713yes—
150C2×#121C22×C713yes—
151C2×#121C22×C716yes—
152C2×#120C22×C713yes—
153C2×C4×D4C22×C713yes—
154C2×C4×D4C22×C716yes—
155#401C22×C712yescomplemented
156#400C22×C716yescomplemented
157#398C22×C713yescomplemented
158#397C22×C716yescomplemented
159#397C22×C713yescomplemented
160#394C22×C713yescomplemented
161#392C22×C711yescharacteristic, complemented
162#388C22×C716yescomplemented
163#388C22×C716yescomplemented
164#387C22×C716yescomplemented
165#387C22×C716yescomplemented
166#386C22×C716yes—
167#386C22×C716yes—
168#386C22×C716yes—
169#386C22×C713yes—
170#386C22×C716yes—
171#386C22×C716yescomplemented
172#385C22×C716yes—
173#385C22×C713yes—
174#383C22×C713yescomplemented
175#381C22×C713yescomplemented
176#380C22×C713yes—
177#380C22×C716yes—
178#379C22×C711yescharacteristic
179C4×C22⋊C4C22×C713yes—
180C4×C22⋊C4C22×C716yes—
181C4×C22⋊C4C22×C713yescomplemented
182#376C22×C711yescharacteristic, complemented
183C2×#95C22×C713yes—
184C2×#95C22×C713yes—
185C2×#95C22×C716yes—
186C23×C7—224no—
187C23×C7—28no—
188C23×C7—224no—
189C23×C7—26no—
190C23×C7—26no—
191C23×C7—23no—
192C23×C7—26no—
193C23×C7—23no—
194C23×C7—26no—
195C23×C7—23no—
196C23×C7C2×#4116yes—
197C23×C7C2×#4116yes—
198C23×C7H(5,𝔽2)16yes—
199C23×C7C2511yescharacteristic
200C7×D4—412no—
201C7×D4—424no—
202C7×D4—86no—
203C2×C4×C7—26no—
204C2×C4×C7—26no—
205C2×C4×C7—23no—
206C2×C4×C7—23no—
207C2×C4×C7—23no—
208C2×C4×C7—26no—
209C2×C4×C7—23no—
210C2×C4×C7—26no—
211C2×C4×C7—26no—
212C2×C4×C7—26no—
213C2×C4×C7—26no—
214C2×C4×C7—23no—
215C2×C4×C7—26no—
216C2×C4×C7—26no—
217C2×C4×C7—412no—
218C2×C4×C7—424no—
219C2×C4×C7—424no—
220C2×C4×C7—412no—
221C2×C4×C7—86no—
222C2×C4×C7C2×#4116yes—
223C2×C4×C7C2×#4116yes—
224C25C23×C711yescharacteristic
225C22×D4C23×C713yes—
226C22×D4C23×C716yes—
227C22×D4C23×C713yes—
228C23×C4—212no—
229C23×C4—412no—
230C23×C4C23×C716yes—
231C23×C4C23×C716yes—
232C23×C4C23×C713yes—
233C23×C4C23×C713yes—
234C23×C4C23×C713yes—
235#121—212no—
236#121—412no—
237#120—212no—
238C4×D4—212no—
239C4×D4—412no—
240C2×C4⋊C4C23×C713yes—
241C2×C4⋊C4C23×C713yes—
242C2×C4⋊C4C23×C716yes—
243C2×C4⋊C4C23×C716yes—
244C2×C22⋊C4C23×C713yes—
245C2×C22⋊C4C23×C713yes—
246C2×C22⋊C4C23×C713yes—
247C2×C22⋊C4C23×C713yes—
248C2×C22⋊C4C23×C716yes—
249C2×C22⋊C4C23×C716yes—
250C2×C22⋊C4C23×C716yes—
251C2×C22⋊C4C23×C716yes—
252C2×C22⋊C4C23×C716yes—
253C2×C42C23×C713yes—
254C2×C42C23×C716yes—
255#95C23×C711yescharacteristic
256#95C23×C713yes—
257#95C23×C716yes—
258#95C23×C711yescharacteristic
259#95C23×C713yes—
260#95C23×C712yes—
261#95C23×C713yes—
262#95C23×C716yes—
263#95C23×C716yes—
264#95C23×C716yes—
265#95C23×C713yes—
266#95C23×C716yes—
267#95C23×C716yes—
268C22×C7—232no—
269C22×C7—26no—
270C22×C7—212no—
271C22×C7—26no—
272C22×C7—212no—
273C22×C7—43no—
274C22×C7—43no—
275C22×C7—46no—
276C22×C7—43no—
277C22×C7—46no—
278C22×C7#553112yes—
279C22×C7C22×#4113yes—
280C22×C7C22×#4111yescharacteristic
281C22×C7C2×H(5,𝔽2)13yes—
282C4×C7—26no—
283C4×C7—26no—
284C4×C7—41no—
285C4×C7—43no—
286C4×C7—46no—
287C4×C7—46no—
288C4×C7—43no—
289C24—26no—
290C24—26no—
291C24C7×#4114yes—
292C24C24×C713yes—
293C24C24×C713yes—
294C2×D4—26no—
295C2×D4—26no—
296C2×D4—212no—
297C2×D4—212no—
298C2×D4—212no—
299C2×D4—26no—
300C2×D4—26no—
301C2×D4—46no—
302C22×C4—26no—
303C22×C4—212no—
304C22×C4—212no—
305C22×C4—212no—
306C22×C4—212no—
307C22×C4—26no—
308C22×C4—46no—
309C22×C4—46no—
310C22×C4—46no—
311C22×C4—46no—
312C22×C4—46no—
313C22×C4—448no—
314C22×C4C7×#41112yes—
315C22×C4C24×C716yes—
316C22×C4C24×C711yescharacteristic
317C22×C4C24×C713yes—
318C22×C4C24×C716yes—
319C22×C4C24×C713yes—
320C22×C4C24×C713yes—
321C22×C4C24×C713yes—
322C4⋊C4—26no—
323C4⋊C4—26no—
324C4⋊C4—46no—
325C4⋊C4—46no—
326C22⋊C4—26no—
327C22⋊C4—26no—
328C22⋊C4—212no—
329C22⋊C4—26no—
330C22⋊C4—26no—
331C22⋊C4—212no—
332C22⋊C4—46no—
333C22⋊C4—46no—
334C22⋊C4—46no—
335C42—26no—
336C42—46no—
337C2×C7—212no—
338C2×C7—83no—
339C2×C7#344411yescharacteristic
340C2×C7C2×#55313yes—
341C2×C7#349013yes—
342C23—224no—
343C23—28no—
344C23—26no—
345C23—26no—
346C23—224no—
347C23—23no—
348C23—26no—
349C23—23no—
350C23—26no—
351C23—23no—
352C23C2×C7×#4116yes—
353C23C2×C7×#4116yes—
354C23C7×H(5,𝔽2)16yes—
355C23C25×C711yescharacteristic
356D4—412no—
357D4—424no—
358D4—86no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—23no—
362C2×C4—23no—
363C2×C4—23no—
364C2×C4—26no—
365C2×C4—23no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—23no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—412no—
374C2×C4—424no—
375C2×C4—424no—
376C2×C4—412no—
377C2×C4—86no—
378C2×C4C2×C7×#4116yes—
379C2×C4C2×C7×#4116yes—
380C7#3660711yescharacteristic, direct product factor, Sylow
381C22—232no—
382C22—26no—
383C22—212no—
384C22—26no—
385C22—212no—
386C22—43no—
387C22—43no—
388C22—46no—
389C22—43no—
390C22—46no—
391C22C7×#553112yes—
392C22C22×C7×#4113yes—
393C22C22×C7×#4111yescharacteristic
394C22C2×C7×H(5,𝔽2)13yes—
395C4—26no—
396C4—26no—
397C4—41no—
398C4—43no—
399C4—46no—
400C4—46no—
401C4—43no—
402C2—212no—
403C2—83no—
404C2C7×#344411yescharacteristic
405C2C2×C7×#55313yes—
406C2C7×#349013yes—
subgroupG/H
center, socleC23×C7C25
commutator, Frattini, cosocleC23C25×C7
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2847C2semidirect
2. C7×#2569C2semidirect
3. C7×#2543C2semidirect
4. C7×#2510C2semidirect
5. C7×#2444C2semidirect
6. C7×#2364C2semidirect
7. C7×#2353C2semidirect
8. C4×C7×#120C2semidirect
9. C22×C7×#95C2semidirect
16. C7×#401C22semidirect
17. C7×#400C22semidirect
18. C7×#398C22semidirect
19. C7×#397C22semidirect
20. C7×#397C22semidirect
21. C7×#394C22semidirect
22. C7×#392C22semidirect
23. C7×#388C22semidirect
24. C7×#388C22semidirect
25. C7×#387C22semidirect
26. C7×#387C22semidirect
32. C7×#386C22semidirect
35. C7×#383C22semidirect
36. C7×#381C22semidirect
42. C4×C7×C22⋊C4C22semidirect
43. C7×#376C22semidirect
47. #36607C7direct (N × H)
92. #2847C2×C7semidirect
93. #2569C2×C7semidirect
94. #2543C2×C7semidirect
95. #2510C2×C7semidirect
96. #2444C2×C7semidirect
97. #2364C2×C7semidirect
98. #2353C2×C7semidirect
99. C4×#120C2×C7semidirect
100. C22×#95C2×C7semidirect
155. #401C22×C7semidirect
156. #400C22×C7semidirect
157. #398C22×C7semidirect
158. #397C22×C7semidirect
159. #397C22×C7semidirect
160. #394C22×C7semidirect
161. #392C22×C7semidirect
162. #388C22×C7semidirect
163. #388C22×C7semidirect
164. #387C22×C7semidirect
165. #387C22×C7semidirect
171. #386C22×C7semidirect
174. #383C22×C7semidirect
175. #381C22×C7semidirect
181. C4×C22⋊C4C22×C7semidirect
182. #376C22×C7semidirect
380. C7#36607direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28472
C7×#25692
C7×#25432
C7×#25102
C7×#24442
C7×#23642
C7×#23532
C4×C7×#1202
C22×C7×#952
#366077