Finite Group Databasebeta

← C7×#9203 #13186419

Subgroups — C7×#9203 #13186419

Order 1792.

762 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2617C211yesmaximal, characteristic
2C7×#2315C211yesmaximal, characteristic, complemented
3C7×#1519C212yesmaximal, complemented
4C2×C7×#323C211yesmaximal, characteristic, complemented
5C7×#1289C212yesmaximal, complemented
6C2×C7×#121C2211yescharacteristic
7C2×C7×#120—21no—
8C22×C7×C22⋊C4—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C2×C7×C22⋊C8C2212yes—
11C7×#397C412yescomplemented
12C7×#394—22no—
13C7×#392C412yescomplemented
14C7×#386—21no—
15C7×#385—24no—
16C7×#380—21no—
17C7×#380C2211yescharacteristic
18C7×#380C2212yes—
19C7×#327—24no—
20C7×#323—22no—
21#9203C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C22×C7×D4—21no—
24C22×C7×D4—22no—
25C22×C7×D4C2×C411yescharacteristic, complemented
26C23×C4×C7—22no—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C22×C7×C8—22no—
30C7×#121—22no—
31C7×#121—22no—
32C7×#120—42no—
33C7×#120—42no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—42no—
47C2×C7×C22⋊C4—42no—
48C2×C7×C22⋊C4C2×C412yes—
49C2×C7×C22⋊C4D412yes—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#95—22no—
53C7×#95—22no—
54C7×#95C2×C412yescomplemented
55C7×#95D414yescomplemented
56#2617C2×C711yescharacteristic
57#2315C2×C711yescharacteristic, complemented
58#1519C2×C712yescomplemented
59C2×#323C2×C711yescharacteristic, complemented
60#1289C2×C712yescomplemented
61C24×C7—21no—
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—81no—
67C24×C7—81no—
68C24×C7C22⋊C412yes—
69C24×C7D812yes—
70C24×C7SD1612yes—
71C24×C7C2×D411yescharacteristic
72C2×C7×D4—22no—
73C2×C7×D4—44no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—82no—
81C2×C7×D4—82no—
82C2×C7×D4—81no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—22no—
87C22×C4×C7—24no—
88C22×C4×C7—44no—
89C22×C4×C7—81no—
90C22×C4×C7—81no—
91C22×C4×C7—82no—
92C22×C4×C7—82no—
93C22×C4×C7C22⋊C412yes—
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7#4112yes—
98C2×C7×C8—42no—
99C2×C7×C8—82no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—82no—
102C7×C4⋊C4—81no—
103C7×C4⋊C4—82no—
104C7×C22⋊C4—24no—
105C7×C22⋊C4—24no—
106C7×C22⋊C4—44no—
107C7×C22⋊C4—44no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—82no—
114C2×#121C22×C711yescharacteristic
115C2×#120—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4C22×C711yescharacteristic
118C2×C22⋊C8C22×C712yes—
119#397C4×C712yescomplemented
120#394—22no—
121#392C4×C712yescomplemented
122#386—21no—
123#385—24no—
124#380—21no—
125#380C22×C711yescharacteristic
126#380C22×C712yes—
127#327—24no—
128#323—22no—
129C23×C7—22no—
130C23×C7—22no—
131C23×C7—22no—
132C23×C7—22no—
133C23×C7—22no—
134C23×C7—22no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—42no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—81no—
147C23×C7—81no—
148C23×C7—82no—
149C23×C7—82no—
150C23×C7—82no—
151C23×C7—82no—
152C23×C7#10214yes—
153C23×C7C2×C22⋊C411yescharacteristic
154C23×C7#12011yescharacteristic
155C23×C7#12311yescharacteristic
156C23×C7C2×D811yescharacteristic
157C23×C7C2×SD1611yescharacteristic
158C23×C7AGL(1,ℤ/8)11yescharacteristic
159C23×C7#13711yescharacteristic
160C7×D4—82no—
161C7×D4—88no—
162C7×D4—81no—
163C7×D4—81no—
164C7×D4—81no—
165C7×D4—81no—
166C2×C4×C7—22no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—44no—
175C2×C4×C7—44no—
176C2×C4×C7—42no—
177C2×C4×C7—41no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—44no—
182C2×C4×C7—81no—
183C2×C4×C7—88no—
184C2×C4×C7C4×D412yes—
185C2×C4×C7#12112yes—
186C7×C8—82no—
187C25C7×D411yescharacteristic
188C22×D4—21no—
189C22×D4—22no—
190C22×D4C2×C4×C711yescharacteristic, complemented
191C23×C4—22no—
192C23×C4C7×D411yescharacteristic
193C23×C4C23×C711yescharacteristic
194C22×C8—22no—
195#121—22no—
196#121—22no—
197#120—42no—
198#120—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4C2×C4×C712yes—
214C2×C22⋊C4C7×D412yes—
215C22⋊C8—22no—
216C22⋊C8—22no—
217#95—22no—
218#95—22no—
219#95C2×C4×C712yescomplemented
220#95C7×D414yescomplemented
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—24no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—44no—
233C22×C7—44no—
234C22×C7—44no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—82no—
238C22×C7—82no—
239C22×C7—82no—
240C22×C7—82no—
241C22×C7—82no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7#38611yescharacteristic
246C22×C7C2×#10211yescharacteristic
247C22×C7#41711yescharacteristic
248C22×C7#44711yescharacteristic
249C22×C7#44811yescharacteristic
250C22×C7#48011yescharacteristic
251C22×C7#48111yescharacteristic
252C4×C7—22no—
253C4×C7—22no—
254C4×C7—42no—
255C4×C7—84no—
256C4×C7—82no—
257C4×C7—81no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—81no—
264C24—81no—
265C24C7×C22⋊C412yes—
266C24C7×D812yes—
267C24C7×SD1612yes—
268C24C2×C7×D411yescharacteristic
269C2×D4—22no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—44no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4C7×C22⋊C412yes—
291C22×C4C22×C4×C711yescharacteristic
292C22×C4C2×C7×D412yes—
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C7×#4112yes—
295C2×C8—42no—
296C2×C8—82no—
297C4⋊C4—22no—
298C4⋊C4—82no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—82no—
311C2×C7—21no—
312C2×C7—21no—
313C2×C7—21no—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—21no—
317C2×C7—44no—
318C2×C7—161no—
319C2×C7#187411yescharacteristic
320C2×C7#191111yescharacteristic
321C2×C7#191311yescharacteristic
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23C7×#10214yes—
346C23C2×C7×C22⋊C411yescharacteristic
347C23C7×#12011yescharacteristic
348C23C7×#12311yescharacteristic
349C23C2×C7×D811yescharacteristic
350C23C2×C7×SD1611yescharacteristic
351C23C7×AGL(1,ℤ/8)11yescharacteristic
352C23C7×#13711yescharacteristic
353D4—82no—
354D4—88no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—81no—
376C2×C4—88no—
377C2×C4C4×C7×D412yes—
378C2×C4C7×#12112yes—
379C8—82no—
380C7#920311yescharacteristic, direct product factor, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C7×#38611yescharacteristic
406C22C2×C7×#10211yescharacteristic
407C22C7×#41711yescharacteristic
408C22C7×#44711yescharacteristic
409C22C7×#44811yescharacteristic
410C22C7×#48011yescharacteristic
411C22C7×#48111yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—84no—
416C4—82no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2C7×#187411yescharacteristic
427C2C7×#191111yescharacteristic
428C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#2617C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2315C2semidirect
3. C7×#1519C2semidirect
4. C2×C7×#323C2semidirect
5. C7×#1289C2semidirect
11. C7×#397C4semidirect
13. C7×#392C4semidirect
21. #9203C7direct (N × H)
25. C22×C7×D4C2×C4semidirect
37. C2×C7×C4⋊C4C2×C4semidirect
54. C7×#95C2×C4semidirect
55. C7×#95D4semidirect
57. #2315C2×C7semidirect
58. #1519C2×C7semidirect
59. C2×#323C2×C7semidirect
60. #1289C2×C7semidirect
119. #397C4×C7semidirect
121. #392C4×C7semidirect
190. C22×D4C2×C4×C7semidirect
202. C2×C4⋊C4C2×C4×C7semidirect
219. #95C2×C4×C7semidirect
220. #95C7×D4semidirect
380. C7#9203direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26172
C7×#23152
C7×#15192
C2×C7×#3232
C7×#12892
#92037