Finite Group Databasebeta

← C7×#61289 #13238505

Subgroups — C7×#61289 #13238505

Order 1792.

1774 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3512C212yesmaximal, complemented
2C7×#3351C218yesmaximal, complemented
3C7×#3339C214yesmaximal, complemented
4C7×#3336C214yesmaximal, complemented
5C2×C7×#499C211yesmaximal, characteristic, complemented
6C7×#3099C214yesmaximal, complemented
7C7×#3080C212yesmaximal, complemented
8C7×#3061C212yesmaximal, complemented
9C4×C7×#135C211yesmaximal, characteristic, complemented
10C7×#2931C211yesmaximal, characteristic, complemented
11C7×#2913C212yesmaximal, complemented
12C2×C7×#135C2211yescharacteristic
13C2×C7×#131C2212yes—
14C7×#553—28no—
15C7×#553C2218yes—
16C7×#552C2218yes—
17C7×#541—22no—
18C7×#540—22no—
19C7×#537C2214yes—
20C7×#532C2211yescharacteristic
21C7×#529—24no—
22C2×C7×#125C2212yes—
23C7×#520C2214yes—
24C4×C7×#41C2212yes—
25C7×#499C2214yes—
26C7×#499C2214yes—
27C7×#485C2218yes—
28C7×#483C2218yes—
29C7×#482C2218yes—
30C7×#481C2218yes—
31C7×#479C2218yes—
32C7×#478C2218yes—
33C7×#468C2218yes—
34C7×#467C2214yes—
35C7×#466C2214yes—
36C4×C7×Q16C2212yes—
37C4×C7×SD16C2214yes—
38C4×C7×D8C2212yes—
39C7×#436C2214yes—
40C7×C8×D4C2214yes—
41C7×#428C2214yes—
42C7×#428C2218yes—
43C7×#427C2211yescharacteristic
44C7×#427C2214yes—
45C2×C7×#107C2212yes—
46C2×C7×#107C2214yes—
47C2×C7×#106C2212yes—
48C7×#423C2211yescharacteristic
49C7×#417C2214yes—
50C7×#416C2212yes—
51C7×#408C2212yes—
52C2×C4×C7×C8C2211yescharacteristic
53#61289C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2312yes—
55C2×C7×#41C2312yes—
56C7×#135—24no—
57C2×C7×Q16C2311yescharacteristic
58C2×C7×SD16C2312yes—
59C2×C7×D8C2311yescharacteristic
60C7×#131—28no—
61C2×C7×M16C2314yes—
62C2×C7×M16C2312yes—
63C22×C7×C8C2314yes—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×C4⋊Q8D411yescharacteristic
67C7×#127D411yescharacteristic
68C7×#126—28no—
69C7×#126—28no—
70C7×#125—28no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125C2318yes—
74C7×#124D412yes—
75C7×#123—28no—
76C7×#123—28no—
77C7×C22⋊Q8—24no—
78C7×C22⋊Q8C2318yes—
79C7×#121—24no—
80C7×#121—28no—
81C7×#121—28no—
82C7×#121D414yes—
83C7×#121C2318yes—
84C4×C7×Q8—24no—
85C4×C7×Q8C2314yes—
86C4×C7×D4—24no—
87C4×C7×D4—28no—
88C4×C7×D4—24no—
89C4×C7×D4—28no—
90C4×C7×D4C2314yes—
91C4×C7×D4C2314yes—
92C7×#117—28no—
93C7×#117—22no—
94C7×#117C2318yes—
95C7×#117C2312yes—
96C2×C7×C4⋊C4—24no—
97C2×C7×C4⋊C4C2318yes—
98C2×C7×C4⋊C4C2314yes—
99C2×C42×C7—22no—
100C2×C42×C7C2311yescharacteristic
101C7×#107C2314yes—
102C7×#107C2314yes—
103C7×#107C2318yes—
104C7×#107C2312yes—
105C7×#106C2318yes—
106C7×#106C2318yes—
107C7×#106C2312yes—
108C7×C4⋊C8C2314yes—
109C7×#103C2318yes—
110C7×#103C2314yes—
111C7×#102C2318yes—
112C7×#102C2314yes—
113C7×C22⋊C8C2314yes—
114C7×C22⋊C8C2314yes—
115C4×C7×C8C2312yes—
116C4×C7×C8C2312yes—
117#3512C2×C712yescomplemented
118#3351C2×C718yescomplemented
119#3339C2×C714yescomplemented
120#3336C2×C714yescomplemented
121C2×#499C2×C711yescharacteristic, complemented
122#3099C2×C714yescomplemented
123#3080C2×C712yescomplemented
124#3061C2×C712yescomplemented
125C4×#135C2×C711yescharacteristic, complemented
126#2931C2×C711yescharacteristic, complemented
127#2913C2×C712yescomplemented
128C7×#41—24no—
129C7×#41—24no—
130C7×#41—24no—
131C7×#41—24no—
132C2×C7×Q8C2×D412yes—
133C2×C7×Q8C2412yes—
134C2×C7×D4—22no—
135C2×C7×D4—28no—
136C2×C7×D4C2×D414yes—
137C2×C7×D4C2×D412yes—
138C2×C7×D4C2×D414yes—
139C2×C7×D4C2412yes—
140C22×C4×C7—24no—
141C22×C4×C7—28no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2414yes—
146C22×C4×C7C2411yescharacteristic
147C7×Q16—22no—
148C7×SD16—24no—
149C7×D8—22no—
150C7×M16—24no—
151C7×M16—28no—
152C2×C7×C8—22no—
153C2×C7×C8—28no—
154C2×C7×C8#4114yes—
155C2×C7×C8C2412yes—
156C2×C7×C8C2414yes—
157C2×C7×C8C2411yescharacteristic
158C2×C7×C8C2411yescharacteristic
159C7×C4⋊C4—28no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—28no—
165C7×C4⋊C4C2×D414yes—
166C7×C4⋊C4C2414yes—
167C7×C4⋊C4C2418yes—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—22no—
170C7×C22⋊C4—28no—
171C7×C22⋊C4—24no—
172C7×C22⋊C4—28no—
173C7×C22⋊C4C2×D414yes—
174C7×C22⋊C4C2×D414yes—
175C42×C7—24no—
176C42×C7—22no—
177C42×C7—28no—
178C42×C7C2×D411yescharacteristic
179C42×C7C2×D411yescharacteristic
180C42×C7C2412yes—
181C2×#135C22×C711yescharacteristic
182C2×#131C22×C712yes—
183#553—28no—
184#553C22×C718yes—
185#552C22×C718yes—
186#541—22no—
187#540—22no—
188#537C22×C714yes—
189#532C22×C711yescharacteristic
190#529—24no—
191C2×#125C22×C712yes—
192#520C22×C714yes—
193C4×#41C22×C712yes—
194#499C22×C714yes—
195#499C22×C714yes—
196#485C22×C718yes—
197#483C22×C718yes—
198#482C22×C718yes—
199#481C22×C718yes—
200#479C22×C718yes—
201#478C22×C718yes—
202#468C22×C718yes—
203#467C22×C714yes—
204#466C22×C714yes—
205C4×Q16C22×C712yes—
206C4×SD16C22×C714yes—
207C4×D8C22×C712yes—
208#436C22×C714yes—
209C8×D4C22×C714yes—
210#428C22×C714yes—
211#428C22×C718yes—
212#427C22×C714yes—
213#427C22×C711yescharacteristic
214C2×#107C22×C714yes—
215C2×#107C22×C712yes—
216C2×#106C22×C712yes—
217#423C22×C711yescharacteristic
218#417C22×C714yes—
219#416C22×C712yes—
220#408C22×C712yes—
221C2×C4×C8C22×C711yescharacteristic
222C23×C7—22no—
223C23×C7C22×D414yes—
224C23×C7C22×D411yescharacteristic
225C7×Q8—24no—
226C7×Q8—22no—
227C7×Q8—22no—
228C7×D4—28no—
229C7×D4—24no—
230C7×D4—22no—
231C7×D4—22no—
232C7×D4—44no—
233C7×D4—44no—
234C7×D4—88no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—22no—
239C2×C4×C7—28no—
240C2×C4×C7—42no—
241C2×C4×C7—44no—
242C2×C4×C7—84no—
243C2×C4×C7—88no—
244C2×C4×C7—84no—
245C2×C4×C7C22×D412yes—
246C2×C4×C7C22×D414yes—
247C2×C4×C7C22×D412yes—
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C2×#4112yes—
250C2×C4×C72−1+412yes—
251C2×C4×C7C2511yescharacteristic
252C7×C8—24no—
253C7×C8—22no—
254C7×C8C2×#4112yes—
255C7×C8C2×#4112yes—
256C2×#41C23×C712yes—
257C2×#41C23×C712yes—
258#135—24no—
259C2×Q16C23×C711yescharacteristic
260C2×SD16C23×C712yes—
261C2×D8C23×C711yescharacteristic
262#131—28no—
263C2×M16C23×C712yes—
264C2×M16C23×C714yes—
265C22×C8C23×C714yes—
266C22×C8C23×C711yescharacteristic
267C22×C8C23×C711yescharacteristic
268C4⋊Q8C7×D411yescharacteristic
269#127C7×D411yescharacteristic
270#126—28no—
271#126—28no—
272#125—28no—
273#125—22no—
274#125—22no—
275#125C23×C718yes—
276#124C7×D412yes—
277#123—28no—
278#123—28no—
279C22⋊Q8—24no—
280C22⋊Q8C23×C718yes—
281#121—24no—
282#121—28no—
283#121—28no—
284#121C7×D414yes—
285#121C23×C718yes—
286C4×Q8—24no—
287C4×Q8C23×C714yes—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—28no—
292C4×D4C23×C714yes—
293C4×D4C23×C714yes—
294#117—28no—
295#117—22no—
296#117C23×C712yes—
297#117C23×C718yes—
298C2×C4⋊C4—24no—
299C2×C4⋊C4C23×C714yes—
300C2×C4⋊C4C23×C718yes—
301C2×C42—22no—
302C2×C42C23×C711yescharacteristic
303#107C23×C712yes—
304#107C23×C714yes—
305#107C23×C714yes—
306#107C23×C718yes—
307#106C23×C718yes—
308#106C23×C712yes—
309#106C23×C718yes—
310C4⋊C8C23×C714yes—
311#103C23×C714yes—
312#103C23×C718yes—
313#102C23×C714yes—
314#102C23×C718yes—
315C22⋊C8C23×C714yes—
316C22⋊C8C23×C714yes—
317C4×C8C23×C712yes—
318C4×C8C23×C712yes—
319C22×C7—22no—
320C22×C7—24no—
321C22×C7—24no—
322C22×C7—24no—
323C22×C7—44no—
324C22×C7—42no—
325C22×C7#54712yes—
326C22×C7C23×D411yescharacteristic
327C4×C7—22no—
328C4×C7—22no—
329C4×C7—24no—
330C4×C7—44no—
331C4×C7—48no—
332C4×C7—42no—
333C4×C7#54712yes—
334C4×C7C22×#4111yescharacteristic
335C4×C7C2×2−1+411yescharacteristic
336#41—24no—
337#41—24no—
338#41—24no—
339#41—24no—
340C2×Q8C2×C7×D412yes—
341C2×Q8C24×C712yes—
342C2×D4—22no—
343C2×D4—28no—
344C2×D4C2×C7×D414yes—
345C2×D4C2×C7×D412yes—
346C2×D4C2×C7×D414yes—
347C2×D4C24×C712yes—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4C2×C7×D412yes—
353C22×C4C24×C714yes—
354C22×C4C24×C711yescharacteristic
355Q16—22no—
356SD16—24no—
357D8—22no—
358M16—24no—
359M16—28no—
360C2×C8—22no—
361C2×C8—28no—
362C2×C8C7×#4114yes—
363C2×C8C24×C712yes—
364C2×C8C24×C714yes—
365C2×C8C24×C711yescharacteristic
366C2×C8C24×C711yescharacteristic
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4C2×C7×D414yes—
374C4⋊C4C24×C714yes—
375C4⋊C4C24×C718yes—
376C22⋊C4—28no—
377C22⋊C4—22no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—28no—
381C22⋊C4C2×C7×D414yes—
382C22⋊C4C2×C7×D414yes—
383C42—24no—
384C42—22no—
385C42—28no—
386C42C2×C7×D411yescharacteristic
387C42C2×C7×D411yescharacteristic
388C42C24×C712yes—
389C2×C7—22no—
390C2×C7—44no—
391C2×C7—82no—
392C2×C7C2×#54711yescharacteristic
393C2×C7#358412yes—
394C23—22no—
395C23C22×C7×D414yes—
396C23C22×C7×D411yescharacteristic
397Q8—24no—
398Q8—22no—
399Q8—22no—
400D4—28no—
401D4—22no—
402D4—22no—
403D4—24no—
404D4—44no—
405D4—44no—
406D4—88no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—84no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4C22×C7×D412yes—
418C2×C4C22×C7×D414yes—
419C2×C4C22×C7×D412yes—
420C2×C4C22×C7×D411yescharacteristic
421C2×C4C2×C7×#4112yes—
422C2×C4C7×2−1+412yes—
423C2×C4C25×C711yescharacteristic
424C8—24no—
425C8—22no—
426C8C2×C7×#4112yes—
427C8C2×C7×#4112yes—
428C7#6128911yescharacteristic, direct product factor, Sylow
429C22—22no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—44no—
434C22—42no—
435C22C7×#54712yes—
436C22C23×C7×D411yescharacteristic
437C4—22no—
438C4—22no—
439C4—24no—
440C4—44no—
441C4—48no—
442C4—42no—
443C4C7×#54712yes—
444C4C22×C7×#4111yescharacteristic
445C4C2×C7×2−1+411yescharacteristic
446C2—22no—
447C2—44no—
448C2—82no—
449C2C2×C7×#54711yescharacteristic
450C2C7×#358412yes—
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C7×#532 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3512C2semidirect
2. C7×#3351C2semidirect
3. C7×#3339C2semidirect
4. C7×#3336C2semidirect
5. C2×C7×#499C2semidirect
6. C7×#3099C2semidirect
7. C7×#3080C2semidirect
8. C7×#3061C2semidirect
9. C4×C7×#135C2semidirect
10. C7×#2931C2semidirect
11. C7×#2913C2semidirect
53. #61289C7direct (N × H)
117. #3512C2×C7semidirect
118. #3351C2×C7semidirect
119. #3339C2×C7semidirect
120. #3336C2×C7semidirect
121. C2×#499C2×C7semidirect
122. #3099C2×C7semidirect
123. #3080C2×C7semidirect
124. #3061C2×C7semidirect
125. C4×#135C2×C7semidirect
126. #2931C2×C7semidirect
127. #2913C2×C7semidirect
428. C7#61289direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35122
C7×#33512
C7×#33392
C7×#33362
C2×C7×#4992
C7×#30992
C7×#30802
C7×#30612
C4×C7×#1352
C7×#29312
C7×#29132
#612897