Finite Group Databasebeta

← C7×#8382 #13185598

Subgroups — C7×#8382 #13185598

Order 1792.

780 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic
2C7×#1823C212yesmaximal, complemented
3C7×#1799C211yesmaximal, characteristic
4C7×#1520C212yesmaximal, complemented
5C7×#1459C211yesmaximal, characteristic
6C7×#552—24no—
7C7×#552—24no—
8C2×C7×#126—21no—
9C2×C7×#125—21no—
10C2×C7×#125C411yescharacteristic, complemented
11C2×C7×#123—21no—
12C2×C7×C22⋊Q8C411yescharacteristic, complemented
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C411yescharacteristic, complemented
15C2×C4×C7×D4C2211yescharacteristic
16C2×C7×#117—21no—
17C2×C7×#117C411yescharacteristic, complemented
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×C7×#99—21no—
20C2×C7×C22⋊C8C2212yes—
21C7×#380C2212yes—
22C7×#342—21no—
23C7×#340—24no—
24C7×#328—24no—
25C7×#324—21no—
26C7×#323—21no—
27#8382C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C22×C7×C8—22no—
34C7×#126—28no—
35C7×#125—28no—
36C7×#125—22no—
37C7×#125—22no—
38C7×#123—28no—
39C7×C22⋊Q8—22no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8—22no—
42C7×C22⋊Q8D414yes—
43C4×C7×Q8—22no—
44C4×C7×Q8—22no—
45C4×C7×D4—22no—
46C4×C7×D4—22no—
47C7×#117—28no—
48C7×#117—22no—
49C7×#117—22no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4C2×C411yescharacteristic
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C4⋊C4C2×C411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4D414yes—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4D411yescharacteristic
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C42×C7—21no—
70C2×C42×C7—21no—
71C2×C42×C7C2×C411yescharacteristic
72C7×#99—42no—
73C7×C22⋊C8—22no—
74C7×C22⋊C8—22no—
75C7×#95—22no—
76C2×#552C2×C711yescharacteristic
77#1823C2×C712yescomplemented
78#1799C2×C711yescharacteristic
79#1520C2×C712yescomplemented
80#1459C2×C711yescharacteristic
81C24×C7C22⋊C411yescharacteristic
82C24×C7C2×D411yescharacteristic
83C2×C7×Q8—21no—
84C2×C7×Q8—42no—
85C2×C7×Q8C22⋊C412yes—
86C2×C7×D4—22no—
87C2×C7×D4—22no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—24no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—42no—
98C22×C4×C7—42no—
99C22×C4×C7—42no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7—42no—
103C22×C4×C7C22⋊C411yescharacteristic
104C22×C4×C7C22⋊C411yescharacteristic
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C22×C411yescharacteristic
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2×D411yescharacteristic
110C2×C7×C8—44no—
111C2×C7×C8—42no—
112C7×C4⋊C4—24no—
113C7×C4⋊C4—24no—
114C7×C4⋊C4—24no—
115C7×C4⋊C4—24no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—44no—
123C7×C4⋊C4C22⋊C412yes—
124C7×C4⋊C4C22⋊C412yes—
125C7×C4⋊C4C22⋊C412yes—
126C7×C22⋊C4—24no—
127C7×C22⋊C4—24no—
128C7×C22⋊C4—24no—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C7×C22⋊C4—42no—
132C7×C22⋊C4—42no—
133C42×C7—24no—
134C42×C7—24no—
135C42×C7—21no—
136C42×C7—21no—
137#552—24no—
138#552—24no—
139C2×#126—21no—
140C2×#125—21no—
141C2×#125C4×C711yescharacteristic, complemented
142C2×#123—21no—
143C2×C22⋊Q8C4×C711yescharacteristic, complemented
144C2×C22⋊Q8C22×C711yescharacteristic
145C2×C4×Q8C4×C711yescharacteristic, complemented
146C2×C4×D4C22×C711yescharacteristic
147C2×#117—21no—
148C2×#117C4×C711yescharacteristic, complemented
149C22×C4⋊C4C22×C711yescharacteristic
150C2×#99—21no—
151C2×C22⋊C8C22×C712yes—
152#380C22×C712yes—
153#342—21no—
154#340—24no—
155#328—24no—
156#324—21no—
157#323—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7C2×C22⋊C411yescharacteristic
166C23×C7#12012yes—
167C23×C7#13512yes—
168C23×C7AGL(1,ℤ/8)11yescharacteristic
169C23×C7#13711yescharacteristic
170C7×Q8—42no—
171C7×Q8—42no—
172C7×D4—42no—
173C7×D4—42no—
174C2×C4×C7—24no—
175C2×C4×C7—22no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—21no—
182C2×C4×C7—22no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—44no—
195C2×C4×C7—44no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—82no—
200C2×C4×C7—82no—
201C2×C4×C7—82no—
202C2×C4×C7C2×C22⋊C411yescharacteristic
203C2×C4×C7C2×C22⋊C411yescharacteristic
204C2×C4×C7#12012yes—
205C7×C8—82no—
206C22×Q8C2×C4×C711yescharacteristic
207C22×D4C7×D411yescharacteristic
208C23×C4—21no—
209C23×C4C2×C4×C711yescharacteristic
210C23×C4C23×C711yescharacteristic
211C22×C8—22no—
212#126—28no—
213#125—28no—
214#125—22no—
215#125—22no—
216#123—28no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8C7×D414yes—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—28no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2×C4×C711yescharacteristic
237C2×C4⋊C4C2×C4×C711yescharacteristic
238C2×C4⋊C4C2×C4×C711yescharacteristic
239C2×C4⋊C4C7×D411yescharacteristic
240C2×C4⋊C4C7×D414yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C7×D411yescharacteristic
246C2×C22⋊C4C7×D411yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2×C4×C711yescharacteristic
250#99—42no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#95—22no—
254C22×C7—21no—
255C22×C7—22no—
256C22×C7—21no—
257C22×C7—22no—
258C22×C7—42no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7#37911yescharacteristic
267C22×C7#41611yescharacteristic
268C22×C7#41711yescharacteristic
269C22×C7#44912yes—
270C22×C7#45212yes—
271C4×C7—22no—
272C4×C7—21no—
273C4×C7—41no—
274C4×C7—42no—
275C4×C7—41no—
276C4×C7—42no—
277C4×C7—41no—
278C4×C7—42no—
279C4×C7—42no—
280C4×C7—162no—
281C4×C7#41012yes—
282C24C7×C22⋊C411yescharacteristic
283C24C2×C7×D411yescharacteristic
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8C7×C22⋊C412yes—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4C7×C22⋊C411yescharacteristic
305C22×C4C7×C22⋊C411yescharacteristic
306C22×C4C7×C22⋊C411yescharacteristic
307C22×C4C22×C4×C711yescharacteristic
308C22×C4C2×C7×D412yes—
309C22×C4C2×C7×D412yes—
310C22×C4C2×C7×D411yescharacteristic
311C2×C8—44no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—44no—
324C4⋊C4C7×C22⋊C412yes—
325C4⋊C4C7×C22⋊C412yes—
326C4⋊C4C7×C22⋊C412yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C42—24no—
335C42—24no—
336C42—21no—
337C42—21no—
338C2×C7—22no—
339C2×C7—21no—
340C2×C7—21no—
341C2×C7—22no—
342C2×C7—81no—
343C2×C7#178211yescharacteristic
344C2×C7#178811yescharacteristic
345C2×C7#179611yescharacteristic
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C2×C7×C22⋊C411yescharacteristic
354C23C7×#12012yes—
355C23C7×#13512yes—
356C23C7×AGL(1,ℤ/8)11yescharacteristic
357C23C7×#13711yescharacteristic
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—42no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4C2×C7×C22⋊C411yescharacteristic
391C2×C4C2×C7×C22⋊C411yescharacteristic
392C2×C4C7×#12012yes—
393C8—82no—
394C7#838211yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22C7×#37911yescharacteristic
408C22C7×#41611yescharacteristic
409C22C7×#41711yescharacteristic
410C22C7×#44912yes—
411C22C7×#45212yes—
412C4—22no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—162no—
422C4C7×#41012yes—
423C2—22no—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—81no—
428C2C7×#178211yescharacteristic
429C2C7×#178811yescharacteristic
430C2C7×#179611yescharacteristic
subgroupG/H
center, socleC22×C7#379
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×D4D4
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×#1823C2semidirect
4. C7×#1520C2semidirect
10. C2×C7×#125C4semidirect
12. C2×C7×C22⋊Q8C4semidirect
14. C2×C4×C7×Q8C4semidirect
17. C2×C7×#117C4semidirect
27. #8382C7direct (N × H)
77. #1823C2×C7semidirect
79. #1520C2×C7semidirect
141. C2×#125C4×C7semidirect
143. C2×C22⋊Q8C4×C7semidirect
145. C2×C4×Q8C4×C7semidirect
148. C2×#117C4×C7semidirect
394. C7#8382direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C7×#18232
C7×#17992
C7×#15202
C7×#14592
#83827