Finite Group Databasebeta

← C7×#10250 #13187466

Subgroups — C7×#10250 #13187466

Order 1792.

604 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3208C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C7×#2548C211yesmaximal, characteristic
4C7×#1545C211yesmaximal, characteristic
5C7×#1503C211yesmaximal, characteristic, complemented
6C7×#1479C211yesmaximal, characteristic, complemented
7C7×#1303C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×#127C2211yescharacteristic
10C2×C7×#124—21no—
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#492—21no—
13C7×#486—21no—
14C7×#479—22no—
15C7×#477—21no—
16C7×#475—21no—
17C7×#460—22no—
18C7×#450—22no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428—21no—
22C7×#423C2211yescharacteristic
23C7×#419—21no—
24C7×#394—22no—
25C7×#390—22no—
26C7×#390—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C7×#331—22no—
29C7×#328—22no—
30C7×Q8⋊C8—22no—
31#10250C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4D412yes—
36C2×C7×SD16—24no—
37C2×C7×M16—21no—
38C7×C4⋊Q8D411yescharacteristic
39C7×C4⋊Q8D411yescharacteristic
40C7×#127—21no—
41C7×#127—21no—
42C7×#127—21no—
43C7×#127—21no—
44C7×#125—21no—
45C7×#124—42no—
46C7×#124—42no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C7×#117—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C42×C7—21no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×#107—22no—
65C7×#106—22no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8D412yes—
70C7×#103—22no—
71C7×#103—22no—
72C7×#102—24no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C4×C7×C8—21no—
76C4×C7×C8—21no—
77C7×#95D412yes—
78#3208C2×C711yescharacteristic
79#3078C2×C711yescharacteristic, complemented
80#2548C2×C711yescharacteristic
81#1545C2×C711yescharacteristic
82#1503C2×C711yescharacteristic, complemented
83#1479C2×C711yescharacteristic, complemented
84#1303C2×C711yescharacteristic, complemented
85C24×C7—22no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8SD1612yescomplemented
94C2×C7×D4—22no—
95C2×C7×D4—22no—
96C2×C7×D4—44no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—41no—
105C2×C7×D4—82no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—21no—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C7×SD16—84no—
113C7×M16—41no—
114C7×M16—41no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—41no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—42no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C22⋊C4—48no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—84no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7—41no—
136C42×C7—41no—
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7#4111yescharacteristic
141#533C22×C711yescharacteristic
142C2×#127C22×C711yescharacteristic
143C2×#124—21no—
144C2×C4×Q8C22×C711yescharacteristic
145#492—21no—
146#486—21no—
147#479—22no—
148#477—21no—
149#475—21no—
150#460—22no—
151#450—22no—
152#431C22×C711yescharacteristic
153#431C22×C711yescharacteristic
154#428—21no—
155#423C22×C711yescharacteristic
156#419—21no—
157#394—22no—
158#390—22no—
159#390—21no—
160C4×C4⋊C4C22×C711yescharacteristic
161#331—22no—
162#328—22no—
163Q8⋊C8—22no—
164C23×C7—44no—
165C23×C7—44no—
166C23×C7—42no—
167C23×C7—42no—
168C23×C7—42no—
169C23×C7#12011yescharacteristic
170C7×Q8—22no—
171C7×Q8—22no—
172C7×Q8—41no—
173C7×Q8—41no—
174C7×Q8—42no—
175C7×Q8—81no—
176C7×D4—44no—
177C7×D4—82no—
178C7×D4—82no—
179C7×D4—82no—
180C7×D4—81no—
181C7×D4—81no—
182C7×D4—81no—
183C7×D4—81no—
184C2×C4×C7—22no—
185C2×C4×C7—22no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7#12011yescharacteristic
197C2×C4×C7#12011yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12711yescharacteristic
202C2×C4×C7C2×SD1611yescharacteristic
203C2×C4×C7#13511yescharacteristic
204C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C7#13711yescharacteristic
206C7×C8—42no—
207C7×C8—42no—
208C7×C8—81no—
209C22×Q8C7×D411yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C7×D412yes—
213C2×SD16—24no—
214C2×M16—21no—
215C4⋊Q8C7×D411yescharacteristic
216C4⋊Q8C7×D411yescharacteristic
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221#125—21no—
222#124—42no—
223#124—42no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42C23×C711yescharacteristic
241#107—22no—
242#106—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8C7×D412yes—
246C4⋊C8C7×D412yes—
247#103—22no—
248#103—22no—
249#102—24no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95C7×D412yes—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—84no—
258C22×C7—84no—
259C22×C7—82no—
260C22×C7—82no—
261C22×C7—82no—
262C22×C7#39211yescharacteristic
263C22×C7#45011yescharacteristic
264C22×C7#45211yescharacteristic
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—42no—
269C4×C7—82no—
270C4×C7—81no—
271C4×C7—81no—
272C4×C7—81no—
273C4×C7#44811yescharacteristic
274C4×C7#44911yescharacteristic
275C4×C7#45512yes—
276C4×C7#46511yescharacteristic
277C4×C7#46811yescharacteristic
278C24—22no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8C7×SD1612yescomplemented
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—44no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—82no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304C22×C4C2×C7×D411yescharacteristic
305SD16—84no—
306M16—41no—
307M16—41no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C22⋊C4—48no—
323C22⋊C4—41no—
324C22⋊C4—84no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42C2×C7×D411yescharacteristic
331C42C2×C7×D411yescharacteristic
332C42C2×C7×D411yescharacteristic
333C42C7×#4111yescharacteristic
334C2×C7—21no—
335C2×C7—21no—
336C2×C7—162no—
337C2×C7#200011yescharacteristic
338C2×C7#200411yescharacteristic
339C2×C7#200811yescharacteristic
340C23—44no—
341C23—44no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23C7×#12011yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350Q8—42no—
351Q8—81no—
352D4—44no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12011yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12111yescharacteristic
376C2×C4C7×#12111yescharacteristic
377C2×C4C7×#12711yescharacteristic
378C2×C4C2×C7×SD1611yescharacteristic
379C2×C4C7×#13511yescharacteristic
380C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C7×#13711yescharacteristic
382C8—42no—
383C8—42no—
384C8—81no—
385C7#1025011yescharacteristic, direct product factor, Sylow
386C22—21no—
387C22—21no—
388C22—84no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22C7×#39211yescharacteristic
394C22C7×#45011yescharacteristic
395C22C7×#45211yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—82no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4C7×#44811yescharacteristic
405C4C7×#44911yescharacteristic
406C4C7×#45512yes—
407C4C7×#46511yescharacteristic
408C4C7×#46811yescharacteristic
409C2—21no—
410C2—21no—
411C2—162no—
412C2C7×#200011yescharacteristic
413C2C7×#200411yescharacteristic
414C2C7×#200811yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C2×C42 ▷ 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×#3078C2semidirect
5. C7×#1503C2semidirect
6. C7×#1479C2semidirect
7. C7×#1303C2semidirect
31. #10250C7direct (N × H)
79. #3078C2×C7semidirect
82. #1503C2×C7semidirect
83. #1479C2×C7semidirect
84. #1303C2×C7semidirect
93. C2×C7×Q8SD16semidirect
286. C2×Q8C7×SD16semidirect
385. C7#10250direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32082
C7×#30782
C7×#25482
C7×#15452
C7×#15032
C7×#14792
C7×#13032
#102507