Finite Group Databasebeta

← C7×#12725 #13189941

Subgroups — C7×#12725 #13189941

Order 1792.

846 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3490C211yesmaximal, characteristic
2C7×#2121C212yesmaximal, complemented
3C2×C7×AΣL(1,𝔽16)C212yesmaximal, complemented
4C7×#1829C211yesmaximal, characteristic, complemented
5C7×#1828C211yesmaximal, characteristic, complemented
6C7×#551—22no—
7C7×#551—22no—
8C7×#546—22no—
9C7×#546—22no—
10C7×#543C411yescharacteristic, complemented
11C7×#538—22no—
12C7×#534C2211yescharacteristic
13C7×#525—22no—
14C2×C7×#120C411yescharacteristic, complemented
15C2×C7×#120C2211yescharacteristic
16C7×#518C411yescharacteristic, complemented
17C7×#518C2211yescharacteristic
18C22×C7×C22⋊C4C411yescharacteristic, complemented
19C2×C7×#100C2212yes—
20C2×C7×#99C2212yes—
21C7×#352—22no—
22C7×AΣL(1,𝔽16)—22no—
23C7×AΣL(1,𝔽16)—22no—
24C7×AΣL(1,𝔽16)—22no—
25C7×#344—22no—
26C7×#342—22no—
27#12725C711yesmaximal, characteristic, direct product factor, Sylow
28C25×C7C2×C411yescharacteristic
29C2×C7×#41—22no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4C2×C411yescharacteristic
33C22×C7×D4C2311yescharacteristic
34C23×C4×C7—21no—
35C2×C7×M16—22no—
36C7×#126—22no—
37C7×#126—22no—
38C7×#124—22no—
39C7×#124—22no—
40C7×#123—22no—
41C7×#123—22no—
42C7×#123—22no—
43C7×#123—22no—
44C7×#123—21no—
45C7×#123D412yes—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—22no—
50C7×#121—22no—
51C7×#121—22no—
52C7×#121—22no—
53C7×#121—22no—
54C7×#120—21no—
55C7×#120—21no—
56C7×#120—21no—
57C7×#120—22no—
58C7×#120—22no—
59C7×#120D412yes—
60C7×#120D412yes—
61C7×#120D412yes—
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C7×#117—22no—
69C7×#117—21no—
70C7×#117D412yes—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4C2×C411yescharacteristic
79C2×C7×C22⋊C4C2×C411yescharacteristic
80C2×C7×C22⋊C4C2×C411yescharacteristic
81C2×C7×C22⋊C4C2×C411yescharacteristic
82C2×C7×C22⋊C4D412yes—
83C7×#100—22no—
84C7×#100—22no—
85C7×#99—22no—
86C7×#99—22no—
87#3490C2×C711yescharacteristic
88#2121C2×C712yescomplemented
89C2×AΣL(1,𝔽16)C2×C712yescomplemented
90#1829C2×C711yescharacteristic, complemented
91#1828C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C24×C7—21no—
94C24×C7—42no—
95C24×C7—82no—
96C24×C7C22⋊C411yescharacteristic
97C24×C7C22⋊C411yescharacteristic
98C24×C7C22⋊C411yescharacteristic
99C24×C7C22×C411yescharacteristic
100C24×C7C2×D411yescharacteristic
101C7×#41—84no—
102C2×C7×Q8—22no—
103C2×C7×D4—22no—
104C2×C7×D4—21no—
105C2×C7×D4—21no—
106C2×C7×D4—22no—
107C2×C7×D4—22no—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—42no—
111C2×C7×D4—42no—
112C2×C7×D4—82no—
113C2×C7×D4—82no—
114C2×C7×D4C2×D412yes—
115C2×C7×D4C2×D412yes—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—42no—
125C22×C4×C7—82no—
126C22×C4×C7C22⋊C411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C7×M16—42no—
129C7×M16—42no—
130C2×C7×C8—42no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C22⋊C4—22no—
138C7×C22⋊C4—22no—
139C7×C22⋊C4—22no—
140C7×C22⋊C4—22no—
141C7×C22⋊C4—22no—
142C7×C22⋊C4—21no—
143C7×C22⋊C4—21no—
144C7×C22⋊C4—21no—
145C7×C22⋊C4—21no—
146C7×C22⋊C4—22no—
147C7×C22⋊C4—22no—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—22no—
150C7×C22⋊C4—22no—
151C7×C22⋊C4—42no—
152C7×C22⋊C4—42no—
153C7×C22⋊C4C22⋊C412yes—
154C7×C22⋊C4C22⋊C412yes—
155C7×C22⋊C4C22⋊C412yes—
156C7×C22⋊C4C22⋊C412yes—
157C42×C7—22no—
158C42×C7—22no—
159#551—22no—
160#551—22no—
161#546—22no—
162#546—22no—
163#543C4×C711yescharacteristic, complemented
164#538—22no—
165#534C22×C711yescharacteristic
166#525—22no—
167C2×#120C4×C711yescharacteristic, complemented
168C2×#120C22×C711yescharacteristic
169#518C4×C711yescharacteristic, complemented
170#518C22×C711yescharacteristic
171C22×C22⋊C4C4×C711yescharacteristic, complemented
172C2×#100C22×C712yes—
173C2×#99C22×C712yes—
174#352—22no—
175AΣL(1,𝔽16)—22no—
176AΣL(1,𝔽16)—22no—
177AΣL(1,𝔽16)—22no—
178#344—22no—
179#342—22no—
180C23×C7—21no—
181C23×C7—22no—
182C23×C7—22no—
183C23×C7—21no—
184C23×C7—21no—
185C23×C7—42no—
186C23×C7—41no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—42no—
190C23×C7—41no—
191C23×C7—41no—
192C23×C7—42no—
193C23×C7—81no—
194C23×C7—81no—
195C23×C7—81no—
196C23×C7—81no—
197C23×C7—81no—
198C23×C7—81no—
199C23×C7—84no—
200C23×C7—84no—
201C23×C7C2×C22⋊C411yescharacteristic
202C23×C7C2×C22⋊C411yescharacteristic
203C23×C7C2×C22⋊C411yescharacteristic
204C23×C7#12012yes—
205C7×Q8—82no—
206C7×D4—42no—
207C7×D4—42no—
208C7×D4—42no—
209C7×D4—44no—
210C7×D4—44no—
211C7×D4—42no—
212C7×D4—82no—
213C7×D4—82no—
214C7×D4—82no—
215C2×C4×C7—22no—
216C2×C4×C7—21no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—44no—
222C2×C4×C7—41no—
223C2×C4×C7—81no—
224C2×C4×C7—82no—
225C2×C4×C7—81no—
226C2×C4×C7—81no—
227C2×C4×C7—81no—
228C2×C4×C7—82no—
229C2×C4×C7—82no—
230C2×C4×C7—82no—
231C2×C4×C7—162no—
232C2×C4×C7#12012yes—
233C7×C8—82no—
234C25C2×C4×C711yescharacteristic
235C2×#41—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×C4×C711yescharacteristic
239C22×D4C23×C711yescharacteristic
240C23×C4—21no—
241C2×M16—22no—
242#126—22no—
243#126—22no—
244#124—22no—
245#124—22no—
246#123—22no—
247#123—22no—
248#123—22no—
249#123—22no—
250#123—21no—
251#123C7×D412yes—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—22no—
260#120—21no—
261#120—21no—
262#120—21no—
263#120—22no—
264#120—22no—
265#120C7×D412yes—
266#120C7×D412yes—
267#120C7×D412yes—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274#117—22no—
275#117—21no—
276#117C7×D412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4C2×C4×C711yescharacteristic
285C2×C22⋊C4C2×C4×C711yescharacteristic
286C2×C22⋊C4C2×C4×C711yescharacteristic
287C2×C22⋊C4C2×C4×C711yescharacteristic
288C2×C22⋊C4C7×D412yes—
289#100—22no—
290#100—22no—
291#99—22no—
292#99—22no—
293C22×C7—22no—
294C22×C7—21no—
295C22×C7—22no—
296C22×C7—21no—
297C22×C7—41no—
298C22×C7—41no—
299C22×C7—42no—
300C22×C7—41no—
301C22×C7—41no—
302C22×C7—42no—
303C22×C7—41no—
304C22×C7—41no—
305C22×C7—42no—
306C22×C7—42no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—82no—
311C22×C7—82no—
312C22×C7—84no—
313C22×C7—84no—
314C22×C7—82no—
315C22×C7—161no—
316C22×C7#37911yescharacteristic
317C22×C7#41012yes—
318C4×C7—22no—
319C4×C7—42no—
320C4×C7—42no—
321C4×C7—42no—
322C4×C7—42no—
323C4×C7—41no—
324C4×C7—162no—
325C24—21no—
326C24—21no—
327C24—42no—
328C24—82no—
329C24C7×C22⋊C411yescharacteristic
330C24C7×C22⋊C411yescharacteristic
331C24C7×C22⋊C411yescharacteristic
332C24C22×C4×C711yescharacteristic
333C24C2×C7×D411yescharacteristic
334#41—84no—
335C2×Q8—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4C2×C7×D412yes—
348C2×D4C2×C7×D412yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—82no—
359C22×C4C7×C22⋊C411yescharacteristic
360C22×C4C2×C7×D411yescharacteristic
361M16—42no—
362M16—42no—
363C2×C8—42no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4C7×C22⋊C412yes—
387C22⋊C4C7×C22⋊C412yes—
388C22⋊C4C7×C22⋊C412yes—
389C22⋊C4C7×C22⋊C412yes—
390C42—22no—
391C42—22no—
392C2×C7—21no—
393C2×C7—22no—
394C2×C7—42no—
395C2×C7—42no—
396C2×C7—41no—
397C2×C7—41no—
398C2×C7—81no—
399C2×C7—81no—
400C2×C7#178211yescharacteristic
401C23—21no—
402C23—22no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—84no—
421C23—84no—
422C23C2×C7×C22⋊C411yescharacteristic
423C23C2×C7×C22⋊C411yescharacteristic
424C23C2×C7×C22⋊C411yescharacteristic
425C23C7×#12012yes—
426Q8—82no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—44no—
431D4—44no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—162no—
453C2×C4C7×#12012yes—
454C8—82no—
455C7#1272511yescharacteristic, direct product factor, Sylow
456C22—22no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—82no—
474C22—82no—
475C22—84no—
476C22—84no—
477C22—82no—
478C22—161no—
479C22C7×#37911yescharacteristic
480C22C7×#41012yes—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—41no—
487C4—162no—
488C2—21no—
489C2—22no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2—81no—
495C2—81no—
496C2C7×#178211yescharacteristic
subgroupG/H
center, socleC2×C7#1782
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC7×#3490C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C7×#518 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2121C2semidirect
3. C2×C7×AΣL(1,𝔽16)C2semidirect
4. C7×#1829C2semidirect
5. C7×#1828C2semidirect
10. C7×#543C4semidirect
14. C2×C7×#120C4semidirect
16. C7×#518C4semidirect
18. C22×C7×C22⋊C4C4semidirect
27. #12725C7direct (N × H)
88. #2121C2×C7semidirect
89. C2×AΣL(1,𝔽16)C2×C7semidirect
90. #1829C2×C7semidirect
91. #1828C2×C7semidirect
163. #543C4×C7semidirect
167. C2×#120C4×C7semidirect
169. #518C4×C7semidirect
171. C22×C22⋊C4C4×C7semidirect
455. C7#12725direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34902
C7×#21212
C2×C7×AΣL(1,𝔽16)2
C7×#18292
C7×#18282
#127257