Finite Group Databasebeta

← C7×#12795 #13190011

Subgroups — C7×#12795 #13190011

Order 1792.

600 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2321C211yesmaximal, characteristic
2C2×C7×AΣL(1,𝔽16)C212yesmaximal
3C7×#1800C211yesmaximal, characteristic, complemented
4C7×#1799C211yesmaximal, characteristic, complemented
5C7×#1390C212yesmaximal, complemented
6C2×C7×#120C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C22⋊C4C411yescharacteristic, complemented
9C7×#423—21no—
10C2×C7×#100C2212yes—
11C2×C7×#99C2212yes—
12C7×#390—21no—
13C7×#388—21no—
14C7×#388—21no—
15C7×#386—21no—
16C7×#386C2211yescharacteristic
17C7×#385C411yescharacteristic, complemented
18C7×#380—21no—
19C7×#379C411yescharacteristic, complemented
20C4×C7×C22⋊C4C411yescharacteristic, complemented
21C7×AΣL(1,𝔽16)—24no—
22C7×#343—21no—
23C7×#342—21no—
24#12795C711yesmaximal, characteristic, direct product factor, Sylow
25C25×C7C2×C411yescharacteristic
26C22×C7×D4—21no—
27C22×C7×D4C2311yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—22no—
32C7×#120—21no—
33C7×#120—21no—
34C7×#120D412yes—
35C7×#120Q812yes—
36C4×C7×D4—41no—
37C4×C7×D4—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—41no—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4C2×C411yescharacteristic
52C2×C7×C22⋊C4C2×C411yescharacteristic
53C2×C7×C22⋊C4D411yescharacteristic
54C2×C42×C7—21no—
55C2×C42×C7D411yescharacteristic
56C7×C4⋊C8—41no—
57C7×C4⋊C8—41no—
58C7×#100—22no—
59C7×#100—22no—
60C7×#99—22no—
61C7×#99—22no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7×#95—41no—
66#2321C2×C711yescharacteristic
67C2×AΣL(1,𝔽16)C2×C712yes—
68#1800C2×C711yescharacteristic, complemented
69#1799C2×C711yescharacteristic, complemented
70#1390C2×C712yescomplemented
71C24×C7—21no—
72C24×C7—21no—
73C24×C7—41no—
74C24×C7—41no—
75C24×C7—81no—
76C24×C7—81no—
77C24×C7C4⋊C412yescomplemented
78C24×C7C22×C411yescharacteristic
79C24×C7C2×D411yescharacteristic
80C2×C7×D4—22no—
81C2×C7×D4—22no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—81no—
85C2×C7×D4—81no—
86C2×C7×D4C2×D411yescharacteristic
87C2×C7×D4C2×Q811yescharacteristic
88C2×C7×D4#4111yescharacteristic
89C2×C7×D4#4111yescharacteristic
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—41no—
98C22×C4×C7—42no—
99C22×C4×C7—81no—
100C22×C4×C7—81no—
101C22×C4×C7—81no—
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22⋊C411yescharacteristic
104C22×C4×C7C22⋊C411yescharacteristic
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C7×M16—42no—
108C7×M16—42no—
109C2×C7×C8—42no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—81no—
112C7×C4⋊C4—82no—
113C7×C22⋊C4—21no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—22no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—42no—
123C7×C22⋊C4—42no—
124C7×C22⋊C4—81no—
125C7×C22⋊C4—81no—
126C7×C22⋊C4—81no—
127C7×C22⋊C4C4⋊C412yescomplemented
128C42×C7—21no—
129C42×C7—21no—
130C42×C7—81no—
131C2×#120C22×C711yescharacteristic
132C2×C4×D4C22×C711yescharacteristic
133C22×C22⋊C4C4×C711yescharacteristic, complemented
134#423—21no—
135C2×#100C22×C712yes—
136C2×#99C22×C712yes—
137#390—21no—
138#388—21no—
139#388—21no—
140#386—21no—
141#386C22×C711yescharacteristic
142#385C4×C711yescharacteristic, complemented
143#380—21no—
144#379C4×C711yescharacteristic, complemented
145C4×C22⋊C4C4×C711yescharacteristic, complemented
146AΣL(1,𝔽16)—24no—
147#343—21no—
148#342—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—22no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7—42no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—82no—
167C23×C7—82no—
168C23×C7—82no—
169C23×C7—82no—
170C23×C7—84no—
171C23×C7C2×C22⋊C411yescharacteristic
172C23×C7C2×C4⋊C411yescharacteristic
173C23×C7#11711yescharacteristic
174C23×C7#12111yescharacteristic
175C23×C7C22⋊Q811yescharacteristic
176C7×D4—41no—
177C7×D4—41no—
178C7×D4—41no—
179C7×D4—41no—
180C7×D4—82no—
181C7×D4—82no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—22no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—81no—
202C2×C4×C7—81no—
203C2×C4×C7—81no—
204C2×C4×C7—82no—
205C2×C4×C7—82no—
206C2×C4×C7—82no—
207C2×C4×C7—82no—
208C2×C4×C7#9911yescharacteristic
209C2×C4×C7#9911yescharacteristic
210C2×C4×C7#12111yescharacteristic
211C2×C4×C7C22⋊Q811yescharacteristic
212C7×C8—82no—
213C25C2×C4×C711yescharacteristic
214C22×D4—21no—
215C22×D4C23×C711yescharacteristic
216C23×C4C2×C4×C711yescharacteristic
217C23×C4C2×C4×C711yescharacteristic
218C23×C4C7×D411yescharacteristic
219C2×M16—22no—
220#120—21no—
221#120—21no—
222#120C7×D412yes—
223#120C7×Q812yes—
224C4×D4—41no—
225C4×D4—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C7×D411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4C2×C4×C711yescharacteristic
239C2×C22⋊C4C2×C4×C711yescharacteristic
240C2×C22⋊C4C2×C4×C711yescharacteristic
241C2×C22⋊C4C7×D411yescharacteristic
242C2×C42—21no—
243C2×C42C7×D411yescharacteristic
244C4⋊C8—41no—
245C4⋊C8—41no—
246#100—22no—
247#100—22no—
248#99—22no—
249#99—22no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—41no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—42no—
265C22×C7—42no—
266C22×C7—42no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—41no—
270C22×C7—41no—
271C22×C7—42no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—81no—
275C22×C7—84no—
276C22×C7—84no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7#38011yescharacteristic
280C22×C7C2×#9911yescharacteristic
281C22×C7#41011yescharacteristic
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—42no—
287C4×C7—41no—
288C4×C7—41no—
289C4×C7—41no—
290C4×C7—81no—
291C4×C7—81no—
292C4×C7—162no—
293C24—21no—
294C24—21no—
295C24—41no—
296C24—41no—
297C24—81no—
298C24—81no—
299C24C7×C4⋊C412yescomplemented
300C24C22×C4×C711yescharacteristic
301C24C2×C7×D411yescharacteristic
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4C2×C7×D411yescharacteristic
309C2×D4C2×C7×Q811yescharacteristic
310C2×D4C7×#4111yescharacteristic
311C2×D4C7×#4111yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4C7×C22⋊C411yescharacteristic
325C22×C4C7×C22⋊C411yescharacteristic
326C22×C4C7×C22⋊C411yescharacteristic
327C22×C4C7×C22⋊C411yescharacteristic
328C22×C4C2×C7×D411yescharacteristic
329M16—42no—
330M16—42no—
331C2×C8—42no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—82no—
335C22⋊C4—21no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4C7×C4⋊C412yescomplemented
350C42—21no—
351C42—21no—
352C42—81no—
353C2×C7—21no—
354C2×C7—21no—
355C2×C7—41no—
356C2×C7—41no—
357C2×C7—41no—
358C2×C7—41no—
359C2×C7—82no—
360C2×C7#179911yescharacteristic
361C2×C7#212111yescharacteristic
362C2×C7#212211yescharacteristic
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—84no—
385C23C2×C7×C22⋊C411yescharacteristic
386C23C2×C7×C4⋊C411yescharacteristic
387C23C7×#11711yescharacteristic
388C23C7×#12111yescharacteristic
389C23C7×C22⋊Q811yescharacteristic
390D4—41no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4C7×#9911yescharacteristic
423C2×C4C7×#9911yescharacteristic
424C2×C4C7×#12111yescharacteristic
425C2×C4C7×C22⋊Q811yescharacteristic
426C8—82no—
427C7#1279511yescharacteristic, direct product factor, Sylow
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—84no—
450C22—84no—
451C22—81no—
452C22—81no—
453C22C7×#38011yescharacteristic
454C22C2×C7×#9911yescharacteristic
455C22C7×#41011yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—81no—
465C4—81no—
466C4—162no—
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—82no—
474C2C7×#179911yescharacteristic
475C2C7×#212111yescharacteristic
476C2C7×#212211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC2×C7×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1800C2semidirect
4. C7×#1799C2semidirect
5. C7×#1390C2semidirect
8. C22×C7×C22⋊C4C4semidirect
17. C7×#385C4semidirect
19. C7×#379C4semidirect
20. C4×C7×C22⋊C4C4semidirect
24. #12795C7direct (N × H)
68. #1800C2×C7semidirect
69. #1799C2×C7semidirect
70. #1390C2×C7semidirect
77. C24×C7C4⋊C4semidirect
127. C7×C22⋊C4C4⋊C4semidirect
133. C22×C22⋊C4C4×C7semidirect
142. #385C4×C7semidirect
144. #379C4×C7semidirect
145. C4×C22⋊C4C4×C7semidirect
299. C24C7×C4⋊C4semidirect
349. C22⋊C4C7×C4⋊C4semidirect
427. C7#12795direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23212
C2×C7×AΣL(1,𝔽16)2
C7×#18002
C7×#17992
C7×#13902
#127957