Finite Group Databasebeta

← C7×#9035 #13186251

Subgroups — C7×#9035 #13186251

Order 1792.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2905C211yesmaximal, characteristic, complemented
2C7×#2392C211yesmaximal, characteristic
3C7×#1486C212yesmaximal, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5C7×#1303C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#117C2211yescharacteristic
10C7×#431C2212yes—
11C7×#424—24no—
12C7×#423—21no—
13C7×#423C2212yes—
14C7×#398C411yescharacteristic
15C7×#397C411yescharacteristic
16C7×#396—21no—
17C7×#394—21no—
18C7×#394C411yescharacteristic
19C7×#392C411yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C4×C7×C22⋊C4—21no—
22C7×D4⋊C8—24no—
23C7×#324—21no—
24C7×#323—21no—
25#9035C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C2×C7×M16—22no—
31C2×C7×M16—22no—
32C2×C7×M16—42no—
33C2×C7×M16—42no—
34C22×C7×C8—22no—
35C4×C7×D4—22no—
36C4×C7×D4—22no—
37C4×C7×D4—42no—
38C4×C7×D4—42no—
39C7×#117—22no—
40C7×#117—22no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4C2×C411yescharacteristic
44C2×C7×C4⋊C4C2×C411yescharacteristic
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4C2×C411yescharacteristic
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C42×C7—21no—
54C2×C42×C7D411yescharacteristic
55C2×C42×C7C2311yescharacteristic
56C7×C4⋊C8—24no—
57C7×C4⋊C8—22no—
58C7×C4⋊C8—22no—
59C7×C4⋊C8D414yescomplemented
60C7×C22⋊C8—22no—
61C4×C7×C8—22no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95C2×C411yescharacteristic
65C7×#95C2×C411yescharacteristic
66#2905C2×C711yescharacteristic, complemented
67#2392C2×C711yescharacteristic
68#1486C2×C712yescomplemented
69#1458C2×C711yescharacteristic, complemented
70#1303C2×C712yescomplemented
71C24×C7—21no—
72C24×C7C22⋊C411yescharacteristic
73C2×C7×D4—22no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—42no—
77C2×C7×D4—42no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7D812yes—
95C22×C4×C7SD1612yes—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×M16—44no—
100C7×M16—42no—
101C7×M16—42no—
102C2×C7×C8—24no—
103C2×C7×C8—44no—
104C2×C7×C8—42no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—42no—
109C7×C4⋊C4—42no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C42×C7—21no—
118C42×C7—21no—
119C42×C7—41no—
120C42×C7—41no—
121C42×C7C2×D412yes—
122C42×C7#4112yes—
123C22×M16—21no—
124C2×C4×D4—21no—
125C2×C4×D4C22×C711yescharacteristic
126C2×#117C22×C711yescharacteristic
127#431C22×C712yes—
128#424—24no—
129#423—21no—
130#423C22×C712yes—
131#398C4×C711yescharacteristic
132#397C4×C711yescharacteristic
133#396—21no—
134#394—21no—
135#394C4×C711yescharacteristic
136#392C4×C711yescharacteristic
137C4×C4⋊C4C22×C711yescharacteristic
138C4×C22⋊C4—21no—
139D4⋊C8—24no—
140#324—21no—
141#323—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—22no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7#10212yes—
154C23×C7C2×C22⋊C411yescharacteristic
155C7×D4—24no—
156C7×D4—42no—
157C7×D4—48no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7#10212yes—
184C2×C4×C7C4×D412yes—
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12112yes—
187C2×C4×C7#12311yescharacteristic
188C2×C4×C7C2×D811yescharacteristic
189C2×C4×C7C2×SD1611yescharacteristic
190C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C7#13711yescharacteristic
192C7×C8—44no—
193C7×C8—82no—
194C22×D4—21no—
195C22×D4C2×C4×C711yescharacteristic
196C23×C4—21no—
197C23×C4C7×D411yescharacteristic
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—42no—
201C2×M16—42no—
202C22×C8—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207#117—22no—
208#117—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×C4×C711yescharacteristic
212C2×C4⋊C4C2×C4×C711yescharacteristic
213C2×C4⋊C4C7×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C711yescharacteristic
220C2×C22⋊C4C7×D411yescharacteristic
221C2×C42—21no—
222C2×C42C7×D411yescharacteristic
223C2×C42C23×C711yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8C7×D414yescomplemented
228C22⋊C8—22no—
229C4×C8—22no—
230#95—21no—
231#95—21no—
232#95C2×C4×C711yescharacteristic
233#95C2×C4×C711yescharacteristic
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—22no—
237C22×C7—42no—
238C22×C7—42no—
239C22×C7—42no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—42no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7#38611yescharacteristic
249C22×C7C2×#10211yescharacteristic
250C22×C7#41711yescharacteristic
251C4×C7—22no—
252C4×C7—22no—
253C4×C7—22no—
254C4×C7—42no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7#44711yescharacteristic
260C4×C7#44811yescharacteristic
261C4×C7#48011yescharacteristic
262C4×C7#48111yescharacteristic
263C24—21no—
264C24C7×C22⋊C411yescharacteristic
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C7×C22⋊C411yescharacteristic
284C22×C4C7×C22⋊C411yescharacteristic
285C22×C4C7×C22⋊C411yescharacteristic
286C22×C4C7×D812yes—
287C22×C4C7×SD1612yes—
288C22×C4C22×C4×C711yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291M16—44no—
292M16—42no—
293M16—42no—
294C2×C8—24no—
295C2×C8—44no—
296C2×C8—42no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—41no—
313C42C2×C7×D412yes—
314C42C7×#4112yes—
315C2×C7—21no—
316C2×C7—21no—
317C2×C7—42no—
318C2×C7—82no—
319C2×C7#187411yescharacteristic
320C2×C7#188411yescharacteristic
321C2×C7#189111yescharacteristic
322C23—21no—
323C23—21no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23C7×#10212yes—
334C23C2×C7×C22⋊C411yescharacteristic
335D4—24no—
336D4—42no—
337D4—48no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C7×#10212yes—
364C2×C4C4×C7×D412yes—
365C2×C4C7×#12011yescharacteristic
366C2×C4C7×#12112yes—
367C2×C4C7×#12311yescharacteristic
368C2×C4C2×C7×D811yescharacteristic
369C2×C4C2×C7×SD1611yescharacteristic
370C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C7×#13711yescharacteristic
372C8—44no—
373C8—82no—
374C7#903511yescharacteristic, direct product factor, Sylow
375C22—21no—
376C22—21no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C7×#38611yescharacteristic
390C22C2×C7×#10211yescharacteristic
391C22C7×#41711yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4C7×#44711yescharacteristic
401C4C7×#44811yescharacteristic
402C4C7×#48011yescharacteristic
403C4C7×#48111yescharacteristic
404C2—21no—
405C2—21no—
406C2—42no—
407C2—82no—
408C2C7×#187411yescharacteristic
409C2C7×#188411yescharacteristic
410C2C7×#189111yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC7×#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2905C2semidirect
3. C7×#1486C2semidirect
4. C7×#1458C2semidirect
5. C7×#1303C2semidirect
25. #9035C7direct (N × H)
59. C7×C4⋊C8D4semidirect
66. #2905C2×C7semidirect
68. #1486C2×C7semidirect
69. #1458C2×C7semidirect
70. #1303C2×C7semidirect
227. C4⋊C8C7×D4semidirect
374. C7#9035direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29052
C7×#23922
C7×#14862
C7×#14582
C7×#13032
#90357