Finite Group Databasebeta

← C7×#9360 #13186576

Subgroups — C7×#9360 #13186576

Order 1792.

538 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3038C211yesmaximal, characteristic
2C7×#2991C211yesmaximal, characteristic, complemented
3C7×#1965C211yesmaximal, characteristic, complemented
4C7×#1545C211yesmaximal, characteristic, complemented
5C7×#1489C211yesmaximal, characteristic, complemented
6C7×#1309C211yesmaximal, characteristic, complemented
7C7×#1299C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)C412yes—
9C2×C7×#127C2211yescharacteristic
10C4×C7×#41C2211yescharacteristic
11C7×#460—21no—
12C7×#460—21no—
13C7×#459—21no—
14C7×#459—21no—
15C7×C8×Q8—21no—
16C7×#436—22no—
17C7×#436—21no—
18C7×#431—21no—
19C7×#431C2211yescharacteristic
20C7×#423—21no—
21C7×#423C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C7×#418C412yes—
25C2×C7×#100—22no—
26C4×C7×M16C2211yescharacteristic
27C7×#331—21no—
28C7×#331—21no—
29C7×Q8⋊C8—21no—
30C7×D4⋊C8—21no—
31#9360C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41C2×C411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4C2×C412yes—
36C7×AGL(1,ℤ/8)—44no—
37C2×C7×SD16—22no—
38C2×C7×D8—22no—
39C2×C7×M16—22no—
40C2×C7×M16—21no—
41C2×C7×M16—21no—
42C2×C7×M16C2×C412yes—
43C2×C7×M16D412yes—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127—21no—
48C4×C7×Q8D411yescharacteristic
49C4×C7×Q8D411yescharacteristic
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4D411yescharacteristic
53C4×C7×D4D411yescharacteristic
54C7×#117—21no—
55C7×#117C2×C411yescharacteristic
56C2×C42×C7—21no—
57C2×C42×C7C2311yescharacteristic
58C7×C4⋊C8—22no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D411yescharacteristic
66C7×C4⋊C8D411yescharacteristic
67C7×#102—22no—
68C7×#102—22no—
69C7×#100—44no—
70C7×C22⋊C8—22no—
71C7×C22⋊C8—21no—
72C7×#97—21no—
73C4×C7×C8—22no—
74C4×C7×C8—21no—
75C4×C7×C8—21no—
76#3038C2×C711yescharacteristic
77#2991C2×C711yescharacteristic, complemented
78#1965C2×C711yescharacteristic, complemented
79#1545C2×C711yescharacteristic, complemented
80#1489C2×C711yescharacteristic, complemented
81#1309C2×C711yescharacteristic, complemented
82#1299C2×C711yescharacteristic, complemented
83C24×C7—22no—
84C7×#41—22no—
85C7×#41—41no—
86C2×C7×Q8C22⋊C411yescharacteristic
87C2×C7×D4—21no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—42no—
91C2×C7×D4—42no—
92C2×C7×D4—42no—
93C2×C7×D4—42no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—81no—
99C2×C7×D4—81no—
100C2×C7×D4C22⋊C411yescharacteristic
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7C22×C411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7#4111yescharacteristic
107C7×SD16—44no—
108C7×D8—44no—
109C7×M16—22no—
110C7×M16—41no—
111C7×M16—42no—
112C7×M16—42no—
113C7×M16—82no—
114C7×M16—81no—
115C2×C7×C8—22no—
116C2×C7×C8—21no—
117C2×C7×C8—22no—
118C2×C7×C8—21no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4C22⋊C411yescharacteristic
124C7×C4⋊C4C22⋊C411yescharacteristic
125C7×C22⋊C4—21no—
126C7×C22⋊C4—21no—
127C7×C22⋊C4—21no—
128C42×C7—21no—
129C42×C7—21no—
130C42×C7—21no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135C2×AGL(1,ℤ/8)C4×C712yes—
136C2×#127C22×C711yescharacteristic
137C4×#41C22×C711yescharacteristic
138#460—21no—
139#460—21no—
140#459—21no—
141#459—21no—
142C8×Q8—21no—
143#436—22no—
144#436—21no—
145#431—21no—
146#431C22×C711yescharacteristic
147#423—21no—
148#423C22×C711yescharacteristic
149#423C22×C711yescharacteristic
150#423C22×C711yescharacteristic
151#418C4×C712yes—
152C2×#100—22no—
153C4×M16C22×C711yescharacteristic
154#331—21no—
155#331—21no—
156Q8⋊C8—21no—
157D4⋊C8—21no—
158C23×C7—21no—
159C23×C7—44no—
160C23×C7—42no—
161C23×C7—82no—
162C23×C7—82no—
163C23×C7C4×D411yescharacteristic
164C7×Q8—21no—
165C7×Q8#10012yescomplemented
166C7×D4—21no—
167C7×D4—21no—
168C7×D4—42no—
169C7×D4—42no—
170C7×D4—42no—
171C7×D4—82no—
172C7×D4—82no—
173C7×D4—82no—
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—81no—
177C7×D4—81no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—42no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7C2×C22⋊C411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12311yescharacteristic
198C7×C8—42no—
199C7×C8—42no—
200C7×C8—42no—
201C7×C8—41no—
202C7×C8—41no—
203C7×C8—81no—
204C2×#41C2×C4×C711yescharacteristic
205C22×D4—21no—
206C22×D4—21no—
207C22×D4C2×C4×C712yes—
208AGL(1,ℤ/8)—44no—
209C2×SD16—22no—
210C2×D8—22no—
211C2×M16—22no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16C2×C4×C712yes—
215C2×M16C7×D412yes—
216#127—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220C4×Q8C7×D411yescharacteristic
221C4×Q8C7×D411yescharacteristic
222C4×D4—21no—
223C4×D4—21no—
224C4×D4C7×D411yescharacteristic
225C4×D4C7×D411yescharacteristic
226#117—21no—
227#117C2×C4×C711yescharacteristic
228C2×C42—21no—
229C2×C42C23×C711yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C7×D411yescharacteristic
238C4⋊C8C7×D411yescharacteristic
239#102—22no—
240#102—22no—
241#100—44no—
242C22⋊C8—22no—
243C22⋊C8—21no—
244#97—21no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—82no—
255C22×C7—82no—
256C22×C7—82no—
257C22×C7—164no—
258C22×C7#38611yescharacteristic
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—22no—
262C4×C7—21no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7C2×#10011yescharacteristic
266C4×C7#41311yescharacteristic
267C4×C7#45311yescharacteristic
268C4×C7#45411yescharacteristic
269C4×C7#47111yescharacteristic
270C4×C7#47211yescharacteristic
271C24—22no—
272#41—22no—
273#41—41no—
274C2×Q8C7×C22⋊C411yescharacteristic
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4C7×C22⋊C411yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4C22×C4×C711yescharacteristic
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C7×#4111yescharacteristic
295SD16—44no—
296D8—44no—
297M16—22no—
298M16—41no—
299M16—42no—
300M16—42no—
301M16—82no—
302M16—81no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4C7×C22⋊C411yescharacteristic
312C4⋊C4C7×C22⋊C411yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42C2×C7×D411yescharacteristic
320C42C2×C7×D411yescharacteristic
321C42C2×C7×D411yescharacteristic
322C42C7×#4111yescharacteristic
323C2×C7—41no—
324C2×C7—81no—
325C2×C7—162no—
326C2×C7#188311yescharacteristic
327C2×C7#190011yescharacteristic
328C2×C7#190711yescharacteristic
329C23—21no—
330C23—44no—
331C23—42no—
332C23—82no—
333C23—82no—
334C23C4×C7×D411yescharacteristic
335Q8—21no—
336Q8C7×#10012yescomplemented
337D4—21no—
338D4—21no—
339D4—42no—
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4C2×C7×C22⋊C411yescharacteristic
364C2×C4C4×C7×D411yescharacteristic
365C2×C4C7×#12011yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12111yescharacteristic
368C2×C4C7×#12311yescharacteristic
369C8—42no—
370C8—42no—
371C8—42no—
372C8—41no—
373C8—41no—
374C8—81no—
375C7#936011yescharacteristic, direct product factor, Sylow
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—164no—
386C22C7×#38611yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4C2×C7×#10011yescharacteristic
394C4C7×#41311yescharacteristic
395C4C7×#45311yescharacteristic
396C4C7×#45411yescharacteristic
397C4C7×#47111yescharacteristic
398C4C7×#47211yescharacteristic
399C2—41no—
400C2—81no—
401C2—162no—
402C2C7×#188311yescharacteristic
403C2C7×#190011yescharacteristic
404C2C7×#190711yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3038C2
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
2. C7×#2991C2semidirect
3. C7×#1965C2semidirect
4. C7×#1545C2semidirect
5. C7×#1489C2semidirect
6. C7×#1309C2semidirect
7. C7×#1299C2semidirect
31. #9360C7direct (N × H)
77. #2991C2×C7semidirect
78. #1965C2×C7semidirect
79. #1545C2×C7semidirect
80. #1489C2×C7semidirect
81. #1309C2×C7semidirect
82. #1299C2×C7semidirect
165. C7×Q8#100semidirect
336. Q8C7×#100semidirect
375. C7#9360direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30382
C7×#29912
C7×#19652
C7×#15452
C7×#14892
C7×#13092
C7×#12992
#93607