Finite Group Databasebeta

← C7×#9247 #13186463

Subgroups — C7×#9247 #13186463

Order 1792.

544 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#461C211yesmaximal, characteristic
2C7×D4×M16C211yesmaximal, characteristic
3C7×#1997C211yesmaximal, characteristic, complemented
4C7×#1548C211yesmaximal, characteristic, complemented
5C7×#1486C211yesmaximal, characteristic
6C7×#1292C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C22×C7×SD16C412yes—
9C22×C7×M16—21no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#461—21no—
13C7×#461—21no—
14C7×#461—21no—
15C7×#461—21no—
16C7×#436—21no—
17C7×#435—22no—
18C7×C8×D4—21no—
19C7×#431C2211yescharacteristic
20C7×#423C2211yescharacteristic
21C7×#423C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#103C412yes—
24C7×#407—21no—
25C4×C7×M16C2211yescharacteristic
26C7×#343—21no—
27C7×#343—21no—
28C7×#333—22no—
29C7×D4⋊C8—21no—
30C7×D4⋊C8—21no—
31#9247C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8C2×C412yes—
33C22×C7×D4C2×C411yescharacteristic
34C23×C4×C7—21no—
35C2×C7×SD16—22no—
36C2×C7×SD16—22no—
37C2×C7×SD16—44no—
38C2×C7×M16—22no—
39C2×C7×M16—21no—
40C2×C7×M16—21no—
41C2×C7×M16—42no—
42C2×C7×M16—42no—
43C2×C7×M16D411yescharacteristic
44C2×C7×M16D411yescharacteristic
45C22×C7×C8—22no—
46C22×C7×C8C2×C412yes—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4D411yescharacteristic
54C4×C7×D4D411yescharacteristic
55C4×C7×D4D411yescharacteristic
56C4×C7×D4D411yescharacteristic
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C42×C7C2311yescharacteristic
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×#103—22no—
70C7×#103—22no—
71C7×C22⋊C8—22no—
72C7×C22⋊C8—22no—
73C7×C22⋊C8—21no—
74C7×#97—21no—
75C4×C7×C8—21no—
76C4×C7×C8—21no—
77C2×#461C2×C711yescharacteristic
78D4×M16C2×C711yescharacteristic
79#1997C2×C711yescharacteristic, complemented
80#1548C2×C711yescharacteristic, complemented
81#1486C2×C711yescharacteristic
82#1292C2×C711yescharacteristic, complemented
83#1283C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C2×C7×Q8—22no—
86C2×C7×Q8—22no—
87C2×C7×Q8—42no—
88C2×C7×Q8—42no—
89C2×C7×D4—22no—
90C2×C7×D4—21no—
91C2×C7×D4—41no—
92C2×C7×D4C22⋊C411yescharacteristic
93C2×C7×D4C22⋊C411yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7C22×C411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7#4111yescharacteristic
107C7×SD16—44no—
108C7×SD16—44no—
109C7×M16—21no—
110C7×M16—21no—
111C7×M16—21no—
112C7×M16—44no—
113C7×M16—41no—
114C7×M16—41no—
115C7×M16—41no—
116C7×M16—41no—
117C7×M16SD1612yescomplemented
118C2×C7×C8—21no—
119C2×C7×C8—22no—
120C2×C7×C8—21no—
121C2×C7×C8—22no—
122C2×C7×C8—44no—
123C2×C7×C8—42no—
124C2×C7×C8—41no—
125C2×C7×C8—41no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4C22⋊C411yescharacteristic
132C7×C4⋊C4C22⋊C411yescharacteristic
133C7×C22⋊C4—21no—
134C7×C22⋊C4—21no—
135C7×C22⋊C4—21no—
136C7×C22⋊C4—21no—
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7#4111yescharacteristic
141C22×SD16C4×C712yes—
142C22×M16—21no—
143C2×C4⋊Q8C22×C711yescharacteristic
144C2×C4×D4C22×C711yescharacteristic
145#461—21no—
146#461—21no—
147#461—21no—
148#461—21no—
149#436—21no—
150#435—22no—
151C8×D4—21no—
152#431C22×C711yescharacteristic
153#423C22×C711yescharacteristic
154#423C22×C711yescharacteristic
155C2×C4⋊C8C22×C711yescharacteristic
156C2×#103C4×C712yes—
157#407—21no—
158C4×M16C22×C711yescharacteristic
159#343—21no—
160#343—21no—
161#333—22no—
162D4⋊C8—21no—
163D4⋊C8—21no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—42no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—81no—
170C23×C7C4×D411yescharacteristic
171C7×Q8—42no—
172C7×Q8—42no—
173C7×Q8—82no—
174C7×D4—21no—
175C7×D4—21no—
176C7×D4—21no—
177C7×D4—42no—
178C7×D4#10012yes—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—82no—
196C2×C4×C7C2×C22⋊C411yescharacteristic
197C2×C4×C7C4×D411yescharacteristic
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12311yescharacteristic
202C2×C4×C7C2×SD1611yescharacteristic
203C2×C4×C7#13511yescharacteristic
204C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C7#13711yescharacteristic
206C7×C8—22no—
207C7×C8—42no—
208C7×C8—42no—
209C7×C8—41no—
210C7×C8—81no—
211C7×C8—81no—
212C22×Q8C2×C4×C712yes—
213C22×D4C2×C4×C711yescharacteristic
214C23×C4—21no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16—44no—
218C2×M16—22no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—42no—
222C2×M16—42no—
223C2×M16C7×D411yescharacteristic
224C2×M16C7×D411yescharacteristic
225C22×C8—22no—
226C22×C8C2×C4×C712yes—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C7×D411yescharacteristic
234C4×D4C7×D411yescharacteristic
235C4×D4C7×D411yescharacteristic
236C4×D4C7×D411yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C2×C4×C711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C23×C711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8C7×D412yes—
249#103—22no—
250#103—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—21no—
254#97—21no—
255C4×C8—21no—
256C4×C8—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—22no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7—82no—
266C22×C7—82no—
267C22×C7#38611yescharacteristic
268C22×C7C4×SD1611yescharacteristic
269C22×C7#44011yescharacteristic
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—82no—
276C4×C7C2×#10011yescharacteristic
277C4×C7#41311yescharacteristic
278C4×C7#45011yescharacteristic
279C4×C7#45211yescharacteristic
280C4×C7#46111yescharacteristic
281C4×C7#46311yescharacteristic
282C24—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4C7×C22⋊C411yescharacteristic
291C2×D4C7×C22⋊C411yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C22×C4×C711yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304C22×C4C7×#4111yescharacteristic
305SD16—44no—
306SD16—44no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—41no—
312M16—41no—
313M16—41no—
314M16—41no—
315M16C7×SD1612yescomplemented
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4C7×C22⋊C411yescharacteristic
330C4⋊C4C7×C22⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C42C2×C7×D411yescharacteristic
336C42C2×C7×D411yescharacteristic
337C42C2×C7×D411yescharacteristic
338C42C7×#4111yescharacteristic
339C2×C7—21no—
340C2×C7—21no—
341C2×C7—42no—
342C2×C7—81no—
343C2×C7#187711yescharacteristic
344C2×C7#190011yescharacteristic
345C2×C7#190211yescharacteristic
346C23—21no—
347C23—21no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—81no—
352C23C4×C7×D411yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—82no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4C7×#10012yes—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4C2×C7×C22⋊C411yescharacteristic
379C2×C4C4×C7×D411yescharacteristic
380C2×C4C7×#12011yescharacteristic
381C2×C4C7×#12111yescharacteristic
382C2×C4C7×#12111yescharacteristic
383C2×C4C7×#12311yescharacteristic
384C2×C4C2×C7×SD1611yescharacteristic
385C2×C4C7×#13511yescharacteristic
386C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C7×#13711yescharacteristic
388C8—22no—
389C8—42no—
390C8—42no—
391C8—41no—
392C8—81no—
393C8—81no—
394C7#924711yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—82no—
404C22—82no—
405C22C7×#38611yescharacteristic
406C22C4×C7×SD1611yescharacteristic
407C22C7×#44011yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4C2×C7×#10011yescharacteristic
415C4C7×#41311yescharacteristic
416C4C7×#45011yescharacteristic
417C4C7×#45211yescharacteristic
418C4C7×#46111yescharacteristic
419C4C7×#46311yescharacteristic
420C2—21no—
421C2—21no—
422C2—42no—
423C2—81no—
424C2C7×#187711yescharacteristic
425C2C7×#190011yescharacteristic
426C2C7×#190211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1997C2semidirect
4. C7×#1548C2semidirect
6. C7×#1292C2semidirect
7. C7×#1283C2semidirect
31. #9247C7direct (N × H)
79. #1997C2×C7semidirect
80. #1548C2×C7semidirect
82. #1292C2×C7semidirect
83. #1283C2×C7semidirect
117. C7×M16SD16semidirect
315. M16C7×SD16semidirect
394. C7#9247direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4612
C7×D4×M162
C7×#19972
C7×#15482
C7×#14862
C7×#12922
C7×#12832
#92477