Finite Group Databasebeta

← C7×#61288 #13238504

Subgroups — C7×#61288 #13238504

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3511C212yesmaximal, complemented
2C7×#3356C214yesmaximal, complemented
3C7×#3354C214yesmaximal, complemented
4C7×#3340C214yesmaximal, complemented
5C7×#3335C214yesmaximal, complemented
6C2×C7×#498C211yesmaximal, characteristic, complemented
7C7×#3099C214yesmaximal, complemented
8C7×#3079C212yesmaximal, complemented
9C7×#3060C212yesmaximal, complemented
10C4×C7×#135C211yesmaximal, characteristic, complemented
11C7×#2931C211yesmaximal, characteristic, complemented
12C7×#2912C212yesmaximal, complemented
13C2×C7×#135C2211yescharacteristic
14C2×C7×#131C2212yes—
15C7×#556C2214yes—
16C7×#555C2214yes—
17C7×#554C2214yes—
18C7×#553—24no—
19C7×#548—24no—
20C7×#547C2214yes—
21C7×#542—24no—
22C7×#537C2214yes—
23C7×#533—22no—
24C7×#532—22no—
25C7×#532C2211yescharacteristic
26C2×C7×C4⋊Q8C2212yes—
27C7×#520C2214yes—
28C4×C7×#41C2212yes—
29C7×#498C2214yes—
30C7×#498C2214yes—
31C7×#485C2218yes—
32C7×#483C2218yes—
33C7×#482C2218yes—
34C7×#481C2218yes—
35C7×#477C2214yes—
36C7×#476C2214yes—
37C7×#475C2214yes—
38C7×#474C2214yes—
39C7×#470C2214yes—
40C7×#469C2214yes—
41C7×#465C2218yes—
42C4×C7×Q16C2212yes—
43C4×C7×SD16C2212yes—
44C4×C7×SD16C2212yes—
45C4×C7×D8C2212yes—
46C7×#436C2214yes—
47C7×C8×D4C2214yes—
48C7×#428C2214yes—
49C7×#428C2218yes—
50C7×#427C2211yescharacteristic
51C7×#427C2214yes—
52C2×C7×#106C2212yes—
53C2×C7×#106C2212yes—
54C2×C7×#106C2214yes—
55C7×#423C2211yescharacteristic
56C7×#417C2214yes—
57C7×#416C2212yes—
58C7×#408C2212yes—
59C2×C4×C7×C8C2211yescharacteristic
60#61288C711yesmaximal, characteristic, direct product factor, Sylow
61C2×C7×#41—24no—
62C2×C7×#41C2312yes—
63C2×C7×#41C2312yes—
64C22×C7×Q8—22no—
65C7×#135—24no—
66C2×C7×Q16C2311yescharacteristic
67C2×C7×SD16C2312yes—
68C2×C7×D8C2311yescharacteristic
69C7×#131—28no—
70C2×C7×M16C2314yes—
71C2×C7×M16C2312yes—
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2314yes—
74C22×C7×C8C2311yescharacteristic
75C7×C4⋊Q8—22no—
76C7×C4⋊Q8—22no—
77C7×C4⋊Q8D411yescharacteristic
78C7×C4⋊Q8C2314yes—
79C7×C4⋊Q8C2314yes—
80C7×#127D411yescharacteristic
81C7×#126—28no—
82C7×#125—24no—
83C7×#124—24no—
84C7×#124—28no—
85C7×#124D412yes—
86C7×#123—28no—
87C7×C22⋊Q8—24no—
88C7×C22⋊Q8—24no—
89C7×C22⋊Q8—28no—
90C7×C22⋊Q8C2318yes—
91C7×#121—24no—
92C7×#121—24no—
93C7×#121—28no—
94C7×#121D414yes—
95C7×#121C2318yes—
96C4×C7×Q8—24no—
97C4×C7×Q8—24no—
98C4×C7×Q8C2312yes—
99C4×C7×Q8C2312yes—
100C4×C7×D4—24no—
101C4×C7×D4—28no—
102C4×C7×D4—24no—
103C4×C7×D4—22no—
104C4×C7×D4—22no—
105C4×C7×D4C2312yes—
106C4×C7×D4C2312yes—
107C4×C7×D4C2314yes—
108C7×#117—24no—
109C7×#117—22no—
110C7×#117C2318yes—
111C7×#117C2312yes—
112C2×C7×C4⋊C4—22no—
113C2×C7×C4⋊C4C2318yes—
114C2×C7×C4⋊C4C2314yes—
115C2×C42×C7—22no—
116C2×C42×C7C2311yescharacteristic
117C7×#107C2318yes—
118C7×#107C2312yes—
119C7×#106C2314yes—
120C7×#106C2318yes—
121C7×#106C2318yes—
122C7×#106C2314yes—
123C7×#106C2311yescharacteristic
124C7×#106C2311yescharacteristic
125C7×C4⋊C8C2314yes—
126C7×#103C2318yes—
127C7×#103C2312yes—
128C7×#103C2312yes—
129C7×#102C2318yes—
130C7×#102C2312yes—
131C7×#102C2312yes—
132C7×C22⋊C8C2318yes—
133C4×C7×C8C2312yes—
134C4×C7×C8C2312yes—
135#3511C2×C712yescomplemented
136#3356C2×C714yescomplemented
137#3354C2×C714yescomplemented
138#3340C2×C714yescomplemented
139#3335C2×C714yescomplemented
140C2×#498C2×C711yescharacteristic, complemented
141#3099C2×C714yescomplemented
142#3079C2×C712yescomplemented
143#3060C2×C712yescomplemented
144C4×#135C2×C711yescharacteristic, complemented
145#2931C2×C711yescharacteristic, complemented
146#2912C2×C712yescomplemented
147C7×#41—24no—
148C7×#41—24no—
149C7×#41—24no—
150C7×#41—24no—
151C7×#41—88no—
152C2×C7×Q8—24no—
153C2×C7×Q8—24no—
154C2×C7×Q8—84no—
155C2×C7×Q8C2×D412yes—
156C2×C7×Q8C2412yes—
157C2×C7×D4—22no—
158C2×C7×D4—28no—
159C2×C7×D4C2×D414yes—
160C2×C7×D4C2×D412yes—
161C2×C7×D4C2×D414yes—
162C2×C7×D4C2412yes—
163C22×C4×C7—24no—
164C22×C4×C7—28no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2414yes—
170C22×C4×C7C2411yescharacteristic
171C7×Q16—22no—
172C7×SD16—24no—
173C7×D8—22no—
174C7×M16—24no—
175C7×M16—28no—
176C2×C7×C8—22no—
177C2×C7×C8—28no—
178C2×C7×C8#4114yescomplemented
179C2×C7×C8C2412yes—
180C2×C7×C8C2414yes—
181C2×C7×C8C2411yescharacteristic
182C2×C7×C8C2411yescharacteristic
183C7×C4⋊C4—24no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—28no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—28no—
190C7×C4⋊C4C2×D414yes—
191C7×C4⋊C4C2412yes—
192C7×C4⋊C4C2418yes—
193C7×C4⋊C4C2412yes—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—28no—
196C7×C22⋊C4—28no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—28no—
200C7×C22⋊C4C2×D418yes—
201C42×C7—22no—
202C42×C7—22no—
203C42×C7—22no—
204C42×C7—28no—
205C42×C7C2×D411yescharacteristic
206C42×C7C2×D411yescharacteristic
207C42×C7C2412yes—
208C2×#135C22×C711yescharacteristic
209C2×#131C22×C712yes—
210#556C22×C714yes—
211#555C22×C714yes—
212#554C22×C714yes—
213#553—24no—
214#548—24no—
215#547C22×C714yes—
216#542—24no—
217#537C22×C714yes—
218#533—22no—
219#532—22no—
220#532C22×C711yescharacteristic
221C2×C4⋊Q8C22×C712yes—
222#520C22×C714yes—
223C4×#41C22×C712yes—
224#498C22×C714yes—
225#498C22×C714yes—
226#485C22×C718yes—
227#483C22×C718yes—
228#482C22×C718yes—
229#481C22×C718yes—
230#477C22×C714yes—
231#476C22×C714yes—
232#475C22×C714yes—
233#474C22×C714yes—
234#470C22×C714yes—
235#469C22×C714yes—
236#465C22×C718yes—
237C4×Q16C22×C712yes—
238C4×SD16C22×C712yes—
239C4×SD16C22×C712yes—
240C4×D8C22×C712yes—
241#436C22×C714yes—
242C8×D4C22×C714yes—
243#428C22×C714yes—
244#428C22×C718yes—
245#427C22×C714yes—
246#427C22×C711yescharacteristic
247C2×#106C22×C714yes—
248C2×#106C22×C712yes—
249C2×#106C22×C712yes—
250#423C22×C711yescharacteristic
251#417C22×C714yes—
252#416C22×C712yes—
253#408C22×C712yes—
254C2×C4×C8C22×C711yescharacteristic
255C23×C7—22no—
256C23×C7C22×D414yes—
257C23×C7C22×D411yescharacteristic
258C7×Q8—24no—
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—48no—
262C7×Q8—84no—
263C7×D4—28no—
264C7×D4—24no—
265C7×D4—28no—
266C7×D4—22no—
267C7×D4—22no—
268C7×D4—44no—
269C7×D4—88no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—28no—
277C2×C4×C7—44no—
278C2×C4×C7—44no—
279C2×C4×C7—44no—
280C2×C4×C7—84no—
281C2×C4×C7—88no—
282C2×C4×C7C22×D414yes—
283C2×C4×C7C22×D412yes—
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D412yes—
286C2×C4×C7C2×#4112yes—
287C2×C4×C72−1+412yes—
288C2×C4×C7C2511yescharacteristic
289C7×C8—24no—
290C7×C8—22no—
291C7×C8C2×#4112yes—
292C7×C8C2×#4112yes—
293C2×#41—24no—
294C2×#41C23×C712yes—
295C2×#41C23×C712yes—
296C22×Q8—22no—
297#135—24no—
298C2×Q16C23×C711yescharacteristic
299C2×SD16C23×C712yes—
300C2×D8C23×C711yescharacteristic
301#131—28no—
302C2×M16C23×C712yes—
303C2×M16C23×C714yes—
304C22×C8C23×C714yes—
305C22×C8C23×C711yescharacteristic
306C22×C8C23×C711yescharacteristic
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8C7×D411yescharacteristic
310C4⋊Q8C23×C714yes—
311C4⋊Q8C23×C714yes—
312#127C7×D411yescharacteristic
313#126—28no—
314#125—24no—
315#124—24no—
316#124—28no—
317#124C7×D412yes—
318#123—28no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—28no—
322C22⋊Q8C23×C718yes—
323#121—24no—
324#121—24no—
325#121—28no—
326#121C7×D414yes—
327#121C23×C718yes—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×Q8C23×C712yes—
331C4×Q8C23×C712yes—
332C4×D4—24no—
333C4×D4—28no—
334C4×D4—24no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4C23×C712yes—
338C4×D4C23×C712yes—
339C4×D4C23×C714yes—
340#117—24no—
341#117—22no—
342#117C23×C712yes—
343#117C23×C718yes—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C23×C714yes—
346C2×C4⋊C4C23×C718yes—
347C2×C42—22no—
348C2×C42C23×C711yescharacteristic
349#107C23×C712yes—
350#107C23×C718yes—
351#106C23×C711yescharacteristic
352#106C23×C714yes—
353#106C23×C714yes—
354#106C23×C718yes—
355#106C23×C718yes—
356#106C23×C711yescharacteristic
357C4⋊C8C23×C714yes—
358#103C23×C712yes—
359#103C23×C712yes—
360#103C23×C718yes—
361#102C23×C712yes—
362#102C23×C712yes—
363#102C23×C718yes—
364C22⋊C8C23×C718yes—
365C4×C8C23×C712yes—
366C4×C8C23×C712yes—
367C22×C7—22no—
368C22×C7—24no—
369C22×C7—28no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7#54712yes—
374C22×C7C23×D411yescharacteristic
375C4×C7—24no—
376C4×C7—22no—
377C4×C7—42no—
378C4×C7—42no—
379C4×C7—42no—
380C4×C7—48no—
381C4×C7#54712yes—
382C4×C7#57714yes—
383C4×C7C22×#4111yescharacteristic
384C4×C7C2×2−1+411yescharacteristic
385#41—24no—
386#41—24no—
387#41—24no—
388#41—24no—
389#41—88no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—84no—
393C2×Q8C2×C7×D412yes—
394C2×Q8C24×C712yes—
395C2×D4—22no—
396C2×D4—28no—
397C2×D4C2×C7×D414yes—
398C2×D4C2×C7×D412yes—
399C2×D4C2×C7×D414yes—
400C2×D4C24×C712yes—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4C2×C7×D412yes—
407C22×C4C24×C714yes—
408C22×C4C24×C711yescharacteristic
409Q16—22no—
410SD16—24no—
411D8—22no—
412M16—24no—
413M16—28no—
414C2×C8—22no—
415C2×C8—28no—
416C2×C8C7×#4114yescomplemented
417C2×C8C24×C712yes—
418C2×C8C24×C714yes—
419C2×C8C24×C711yescharacteristic
420C2×C8C24×C711yescharacteristic
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—28no—
428C4⋊C4C2×C7×D414yes—
429C4⋊C4C24×C712yes—
430C4⋊C4C24×C718yes—
431C4⋊C4C24×C712yes—
432C22⋊C4—22no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—28no—
438C22⋊C4C2×C7×D418yes—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—28no—
443C42C2×C7×D411yescharacteristic
444C42C2×C7×D411yescharacteristic
445C42C24×C712yes—
446C2×C7—22no—
447C2×C7—44no—
448C2×C7—82no—
449C2×C7C2×#54711yescharacteristic
450C2×C7C2×#57712yes—
451C23—22no—
452C23C22×C7×D414yes—
453C23C22×C7×D411yescharacteristic
454Q8—24no—
455Q8—22no—
456Q8—22no—
457Q8—48no—
458Q8—84no—
459D4—28no—
460D4—22no—
461D4—22no—
462D4—24no—
463D4—28no—
464D4—44no—
465D4—88no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—28no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—84no—
477C2×C4—88no—
478C2×C4C22×C7×D414yes—
479C2×C4C22×C7×D412yes—
480C2×C4C22×C7×D411yescharacteristic
481C2×C4C22×C7×D412yes—
482C2×C4C2×C7×#4112yes—
483C2×C4C7×2−1+412yes—
484C2×C4C25×C711yescharacteristic
485C8—24no—
486C8—22no—
487C8C2×C7×#4112yes—
488C8C2×C7×#4112yes—
489C7#6128811yescharacteristic, direct product factor, Sylow
490C22—22no—
491C22—28no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C7×#54712yes—
497C22C23×C7×D411yescharacteristic
498C4—22no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—48no—
503C4—42no—
504C4C7×#54712yes—
505C4C7×#57714yes—
506C4C22×C7×#4111yescharacteristic
507C4C2×C7×2−1+411yescharacteristic
508C2—22no—
509C2—44no—
510C2—82no—
511C2C2×C7×#54711yescharacteristic
512C2C2×C7×#57712yes—
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×#3511C2semidirect
2. C7×#3356C2semidirect
3. C7×#3354C2semidirect
4. C7×#3340C2semidirect
5. C7×#3335C2semidirect
6. C2×C7×#498C2semidirect
7. C7×#3099C2semidirect
8. C7×#3079C2semidirect
9. C7×#3060C2semidirect
10. C4×C7×#135C2semidirect
11. C7×#2931C2semidirect
12. C7×#2912C2semidirect
60. #61288C7direct (N × H)
135. #3511C2×C7semidirect
136. #3356C2×C7semidirect
137. #3354C2×C7semidirect
138. #3340C2×C7semidirect
139. #3335C2×C7semidirect
140. C2×#498C2×C7semidirect
141. #3099C2×C7semidirect
142. #3079C2×C7semidirect
143. #3060C2×C7semidirect
144. C4×#135C2×C7semidirect
145. #2931C2×C7semidirect
146. #2912C2×C7semidirect
178. C2×C7×C8#41semidirect
416. C2×C8C7×#41semidirect
489. C7#61288direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35112
C7×#33562
C7×#33542
C7×#33402
C7×#33352
C2×C7×#4982
C7×#30992
C7×#30792
C7×#30602
C4×C7×#1352
C7×#29312
C7×#29122
#612887