Finite Group Databasebeta

← C7×#18326 #13195542

Subgroups — C7×#18326 #13195542

Order 1792.

976 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic, complemented
2C7×#3193C212yesmaximal
3C7×#2591C211yesmaximal, characteristic
4C7×#1617C214yesmaximal, complemented
5C7×#1599C214yesmaximal, complemented
6C7×#1508C212yesmaximal, complemented
7C7×#1463C211yesmaximal, characteristic, complemented
8C7×#544—24no—
9C7×#544—22no—
10C7×#544C2214yes—
11C2×C7×C4⋊Q8—22no—
12C2×C7×#125—22no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×#121C2211yescharacteristic
16C2×C7×#117—21no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C7×#484C2214yes—
19C7×#480—22no—
20C7×#468—24no—
21C7×#450C2214yes—
22C7×#448—22no—
23C7×#407C2212yes—
24C7×#400C2212yes—
25C7×#394C2212yes—
26C7×#388—22no—
27C7×#386—21no—
28C7×#386—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#328C2218yes—
31C7×#324C2214yes—
32#18326C711yesmaximal, characteristic, direct product factor, Sylow
33C22×C7×Q8—21no—
34C22×C7×Q8D412yes—
35C22×C7×D4—21no—
36C22×C7×D4C2311yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C2×C7×SD16—24no—
41C2×C7×M16—22no—
42C7×C4⋊Q8—28no—
43C7×C4⋊Q8—28no—
44C7×#125—28no—
45C7×#125—28no—
46C7×C22⋊Q8—24no—
47C7×C22⋊Q8—24no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8D414yes—
51C7×C22⋊Q8C2314yes—
52C7×#121—22no—
53C7×#121—22no—
54C7×#117—24no—
55C7×#117—24no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4D412yes—
64C2×C7×C4⋊C4D412yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2312yes—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C42×C7—22no—
73C2×C42×C7—21no—
74C7×#107—24no—
75C7×#103—24no—
76C7×#102—24no—
77C7×C22⋊C8—22no—
78C7×C22⋊C8C2314yes—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95C2312yes—
82C2×#544C2×C711yescharacteristic, complemented
83#3193C2×C712yes—
84#2591C2×C711yescharacteristic
85#1617C2×C714yescomplemented
86#1599C2×C714yescomplemented
87#1508C2×C712yescomplemented
88#1463C2×C711yescharacteristic, complemented
89C24×C7—21no—
90C24×C7C2×D411yescharacteristic
91C2×C7×Q8—24no—
92C2×C7×Q8—22no—
93C2×C7×Q8—82no—
94C2×C7×Q8—82no—
95C2×C7×Q8C2×D414yes—
96C2×C7×D4—22no—
97C2×C7×D4—44no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—81no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—24no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—81no—
111C22×C4×C7—81no—
112C22×C4×C7C2×D412yes—
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D412yes—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2411yescharacteristic
118C7×SD16—84no—
119C7×M16—82no—
120C2×C7×C8—24no—
121C7×C4⋊C4—24no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—28no—
125C7×C4⋊C4—24no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4—24no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4C2×D414yes—
132C7×C22⋊C4—24no—
133C7×C22⋊C4—24no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—42no—
136C7×C22⋊C4—44no—
137C7×C22⋊C4—82no—
138C42×C7—28no—
139C42×C7—81no—
140#544—24no—
141#544—22no—
142#544C22×C714yes—
143C2×C4⋊Q8—22no—
144C2×#125—22no—
145C2×C22⋊Q8—21no—
146C2×C22⋊Q8C22×C712yes—
147C2×#121C22×C711yescharacteristic
148C2×#117—21no—
149C22×C4⋊C4C22×C711yescharacteristic
150#484C22×C714yes—
151#480—22no—
152#468—24no—
153#450C22×C714yes—
154#448—22no—
155#407C22×C712yes—
156#400C22×C712yes—
157#394C22×C712yes—
158#388—22no—
159#386—21no—
160#386—21no—
161C4×C22⋊C4C22×C711yescharacteristic
162#328C22×C718yes—
163#324C22×C714yes—
164C23×C7—22no—
165C23×C7—22no—
166C23×C7—41no—
167C23×C7—42no—
168C23×C7—42no—
169C23×C7—41no—
170C23×C7—42no—
171C23×C7#12011yescharacteristic
172C23×C7#12011yescharacteristic
173C23×C7C22×D411yescharacteristic
174C7×Q8—44no—
175C7×Q8—82no—
176C7×Q8—84no—
177C7×D4—44no—
178C7×D4—82no—
179C7×D4—82no—
180C2×C4×C7—24no—
181C2×C4×C7—24no—
182C2×C4×C7—22no—
183C2×C4×C7—22no—
184C2×C4×C7—24no—
185C2×C4×C7—24no—
186C2×C4×C7—22no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—44no—
193C2×C4×C7—44no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—81no—
201C2×C4×C7#12012yes—
202C2×C4×C7#13714yes—
203C2×C4×C7C22×D412yes—
204C7×C8—44no—
205C22×Q8—21no—
206C22×Q8C7×D412yes—
207C22×D4—21no—
208C22×D4C23×C711yescharacteristic
209C23×C4C7×D411yescharacteristic
210C23×C4C7×D411yescharacteristic
211C23×C4C23×C711yescharacteristic
212C2×SD16—24no—
213C2×M16—22no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216#125—28no—
217#125—28no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8C7×D414yes—
223C22⋊Q8C23×C714yes—
224#121—22no—
225#121—22no—
226#117—24no—
227#117—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C7×D412yes—
236C2×C4⋊C4C7×D412yes—
237C2×C4⋊C4C23×C711yescharacteristic
238C2×C4⋊C4C23×C712yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42—21no—
246#107—24no—
247#103—24no—
248#102—24no—
249C22⋊C8—22no—
250C22⋊C8C23×C714yes—
251#95—22no—
252#95—22no—
253#95C23×C712yes—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—44no—
262C22×C7—84no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7#45412yes—
267C22×C7C2×#12011yescharacteristic
268C22×C7C2×#13712yes—
269C4×C7—24no—
270C4×C7—22no—
271C4×C7—22no—
272C4×C7—44no—
273C4×C7—42no—
274C4×C7—44no—
275C4×C7—82no—
276C4×C7—81no—
277C24—21no—
278C24C2×C7×D411yescharacteristic
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8C2×C7×D414yes—
284C2×D4—22no—
285C2×D4—44no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C2×C7×D412yes—
301C22×C4C2×C7×D412yes—
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D412yes—
304C22×C4C2×C7×D412yes—
305C22×C4C24×C711yescharacteristic
306SD16—84no—
307M16—82no—
308C2×C8—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4C2×C7×D414yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C42—28no—
327C42—81no—
328C2×C7—22no—
329C2×C7—22no—
330C2×C7—41no—
331C2×C7—161no—
332C2×C7#300411yescharacteristic
333C2×C7C2×#45411yescharacteristic
334C2×C7#302611yescharacteristic
335C23—22no—
336C23—22no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23C7×#12011yescharacteristic
343C23C7×#12011yescharacteristic
344C23C22×C7×D411yescharacteristic
345Q8—44no—
346Q8—82no—
347Q8—84no—
348D4—44no—
349D4—82no—
350D4—82no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4C7×#12012yes—
373C2×C4C7×#13714yes—
374C2×C4C22×C7×D412yes—
375C8—44no—
376C7#1832611yescharacteristic, direct product factor, Sylow
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—84no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C7×#45412yes—
390C22C2×C7×#12011yescharacteristic
391C22C2×C7×#13712yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—82no—
399C4—81no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—161no—
404C2C7×#300411yescharacteristic
405C2C2×C7×#45411yescharacteristic
406C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
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
1. C2×C7×#544C2semidirect
4. C7×#1617C2semidirect
5. C7×#1599C2semidirect
6. C7×#1508C2semidirect
7. C7×#1463C2semidirect
32. #18326C7direct (N × H)
82. C2×#544C2×C7semidirect
85. #1617C2×C7semidirect
86. #1599C2×C7semidirect
87. #1508C2×C7semidirect
88. #1463C2×C7semidirect
376. C7#18326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5442
C7×#31932
C7×#25912
C7×#16172
C7×#15992
C7×#15082
C7×#14632
#183267