Finite Group Databasebeta

← C7×#9320 #13186536

Subgroups — C7×#9320 #13186536

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3039C211yesmaximal, characteristic
2C7×#2989C211yesmaximal, characteristic
3C7×#1988C211yesmaximal, characteristic, complemented
4C7×#1806C211yesmaximal, characteristic
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1303C211yesmaximal, characteristic, complemented
7C7×#1292C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C4×C7×#41C2211yescharacteristic
11C7×#462—21no—
12C7×#462C412yescomplemented
13C7×#461—21no—
14C7×#461C412yescomplemented
15C7×#446—21no—
16C7×#436—21no—
17C7×#435—22no—
18C7×#432—21no—
19C7×#431C2211yescharacteristic
20C7×#423—21no—
21C7×#423C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×#104—21no—
25C7×#419—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#340—22no—
28C7×#339—21no—
29C7×#332—21no—
30C7×#332—21no—
31#9320C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41D411yescharacteristic
33C22×C7×Q8—21no—
34C7×#137—42no—
35C7×#137—42no—
36C2×C7×Q16—22no—
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C2×C7×M16—22no—
40C2×C7×M16—21no—
41C22×C7×C8D412yes—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8C2×C412yes—
46C4×C7×Q8C2×C411yescharacteristic
47C4×C7×Q8D411yescharacteristic
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C4×C7×D4C2×C411yescharacteristic
51C4×C7×D4D411yescharacteristic
52C7×#117—21no—
53C7×#117D411yescharacteristic
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C42×C7—21no—
58C2×C42×C7—21no—
59C2×C42×C7C2311yescharacteristic
60C7×C4⋊C8—22no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8C2×C412yes—
67C7×C4⋊C8D411yescharacteristic
68C7×C4⋊C8D411yescharacteristic
69C7×#104—42no—
70C7×#103—22no—
71C7×#103—22no—
72C7×C22⋊C8—22no—
73C7×C22⋊C8—21no—
74C7×#97—22no—
75C4×C7×C8—21no—
76C7×#95—21no—
77#3039C2×C711yescharacteristic
78#2989C2×C711yescharacteristic
79#1988C2×C711yescharacteristic, complemented
80#1806C2×C711yescharacteristic
81#1546C2×C711yescharacteristic, complemented
82#1303C2×C711yescharacteristic, complemented
83#1292C2×C711yescharacteristic, complemented
84C7×#41—21no—
85C7×#41—21no—
86C7×#41—21no—
87C7×#41—21no—
88C2×C7×Q8—22no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×Q8C22⋊C411yescharacteristic
95C2×C7×D4—21no—
96C2×C7×D4C22⋊C411yescharacteristic
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4111yescharacteristic
105C7×Q16—44no—
106C7×SD16—44no—
107C7×M16—41no—
108C7×M16—41no—
109C7×M16—41no—
110C7×M16—41no—
111C7×M16—82no—
112C2×C7×C8—22no—
113C2×C7×C8—21no—
114C2×C7×C8—21no—
115C2×C7×C8—22no—
116C2×C7×C8—42no—
117C2×C7×C8—41no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4C22⋊C411yescharacteristic
127C7×C4⋊C4C22⋊C411yescharacteristic
128C7×C22⋊C4—21no—
129C7×C22⋊C4—21no—
130C7×C22⋊C4—21no—
131C42×C7—21no—
132C42×C7—21no—
133C42×C7—21no—
134C42×C7—41no—
135C42×C7—41no—
136C42×C7C22×C411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7#4111yescharacteristic
140C2×#137—21no—
141C2×C4⋊Q8C22×C711yescharacteristic
142C4×#41C22×C711yescharacteristic
143#462—21no—
144#462C4×C712yescomplemented
145#461—21no—
146#461C4×C712yescomplemented
147#446—21no—
148#436—21no—
149#435—22no—
150#432—21no—
151#431C22×C711yescharacteristic
152#423—21no—
153#423C22×C711yescharacteristic
154#423C22×C711yescharacteristic
155C2×C4⋊C8C22×C711yescharacteristic
156C2×#104—21no—
157#419—21no—
158C4×C4⋊C4C22×C711yescharacteristic
159#340—22no—
160#339—21no—
161#332—21no—
162#332—21no—
163C23×C7—21no—
164C23×C7#12111yescharacteristic
165C7×Q8—21no—
166C7×Q8—21no—
167C7×Q8—42no—
168C7×Q8—42no—
169C7×Q8—81no—
170C7×Q8—81no—
171C7×D4—21no—
172C7×D4—21no—
173C7×D4—41no—
174C7×D4—41no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7C2×C22⋊C411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7C4×D411yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12311yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C7#13711yescharacteristic
202C7×C8—42no—
203C7×C8—42no—
204C7×C8—41no—
205C7×C8—41no—
206C7×C8—81no—
207C2×#41C7×D411yescharacteristic
208C22×Q8—21no—
209#137—42no—
210#137—42no—
211C2×Q16—22no—
212C2×SD16—22no—
213C2×M16—21no—
214C2×M16—22no—
215C2×M16—21no—
216C22×C8C7×D412yes—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8C2×C4×C712yes—
221C4×Q8C2×C4×C711yescharacteristic
222C4×Q8C7×D411yescharacteristic
223C4×D4—21no—
224C4×D4—21no—
225C4×D4C2×C4×C711yescharacteristic
226C4×D4C7×D411yescharacteristic
227#117—21no—
228#117C7×D411yescharacteristic
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C23×C711yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C2×C4×C712yes—
242C4⋊C8C7×D411yescharacteristic
243C4⋊C8C7×D411yescharacteristic
244#104—42no—
245#103—22no—
246#103—22no—
247C22⋊C8—22no—
248C22⋊C8—21no—
249#97—22no—
250C4×C8—21no—
251#95—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7#38611yescharacteristic
258C22×C7#46811yescharacteristic
259C22×C7#46911yescharacteristic
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—41no—
266C4×C7—82no—
267C4×C7—82no—
268C4×C7#41711yescharacteristic
269C4×C7#41811yescharacteristic
270C4×C7#45511yescharacteristic
271C4×C7#45611yescharacteristic
272C4×C7#46811yescharacteristic
273C4×C7#46911yescharacteristic
274#41—21no—
275#41—21no—
276#41—21no—
277#41—21no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8C7×C22⋊C411yescharacteristic
285C2×D4—21no—
286C2×D4C7×C22⋊C411yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C7×D411yescharacteristic
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C7×#4111yescharacteristic
295Q16—44no—
296SD16—44no—
297M16—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—82no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—42no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4C7×C22⋊C411yescharacteristic
317C4⋊C4C7×C22⋊C411yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—41no—
325C42—41no—
326C42C22×C4×C711yescharacteristic
327C42C2×C7×D411yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C7×#4111yescharacteristic
330C2×C7—21no—
331C2×C7—21no—
332C2×C7—81no—
333C2×C7#188211yescharacteristic
334C2×C7#189511yescharacteristic
335C2×C7#190211yescharacteristic
336C23—21no—
337C23C7×#12111yescharacteristic
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344D4—21no—
345D4—21no—
346D4—41no—
347D4—41no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4C2×C7×C22⋊C411yescharacteristic
366C2×C4C4×C7×D411yescharacteristic
367C2×C4C4×C7×D411yescharacteristic
368C2×C4C7×#12011yescharacteristic
369C2×C4C7×#12111yescharacteristic
370C2×C4C7×#12311yescharacteristic
371C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
373C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C7×#13711yescharacteristic
375C8—42no—
376C8—42no—
377C8—41no—
378C8—41no—
379C8—81no—
380C7#932011yescharacteristic, direct product factor, Sylow
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22C7×#38611yescharacteristic
387C22C7×#46811yescharacteristic
388C22C7×#46911yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4C7×#41711yescharacteristic
398C4C7×#41811yescharacteristic
399C4C7×#45511yescharacteristic
400C4C7×#45611yescharacteristic
401C4C7×#46811yescharacteristic
402C4C7×#46911yescharacteristic
403C2—21no—
404C2—21no—
405C2—81no—
406C2C7×#188211yescharacteristic
407C2C7×#189511yescharacteristic
408C2C7×#190211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#41D4
LCSG ▷ C42 ▷ 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×#1988C2semidirect
5. C7×#1546C2semidirect
6. C7×#1303C2semidirect
7. C7×#1292C2semidirect
12. C7×#462C4semidirect
14. C7×#461C4semidirect
31. #9320C7direct (N × H)
79. #1988C2×C7semidirect
81. #1546C2×C7semidirect
82. #1303C2×C7semidirect
83. #1292C2×C7semidirect
144. #462C4×C7semidirect
146. #461C4×C7semidirect
380. C7#9320direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30392
C7×#29892
C7×#19882
C7×#18062
C7×#15462
C7×#13032
C7×#12922
#93207