Finite Group Databasebeta

← C7×#20347 #13197563

Subgroups — C7×#20347 #13197563

Order 1792.

2270 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C7×M16C211yesmaximal, characteristic
2C22×C7×#117C212yesmaximal
3C7×#1736C218yesmaximal, complemented
4C7×#1735C214yesmaximal, complemented
5C24×C4×C7C2211yescharacteristic
6C22×C7×M16—28no—
7C22×C7×M16C418yescomplemented
8C22×C7×M16C2214yes—
9C23×C7×C8C2212yes—
10C2×C7×#117—28no—
11C2×C7×#117—22no—
12C2×C7×#117—22no—
13C2×C7×#117C4116yescomplemented
14C2×C7×#117C2214yes—
15C2×C7×#117C2214yes—
16C22×C7×C4⋊C4C2214yes—
17C22×C7×C22⋊C4—22no—
18C22×C42×C7—22no—
19C7×#340C22116yescomplemented
20#20347C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7—24no—
23C23×C4×C7C2×C414yes—
24C23×C4×C7C2×C418yes—
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D412yes—
27C23×C4×C7D418yes—
28C23×C4×C7Q814yes—
29C23×C4×C7C2312yes—
30C23×C4×C7C2311yescharacteristic
31C2×C7×M16—232no—
32C2×C7×M16—28no—
33C2×C7×M16—28no—
34C2×C7×M16C2×C4116yescomplemented
35C22×C7×C8—28no—
36C22×C7×C8C2×C418yescomplemented
37C22×C7×C8C2314yes—
38C7×#117—216no—
39C7×#117—216no—
40C7×#117—28no—
41C7×#117—28no—
42C7×#117C2×C4116yes—
43C7×#117C2×C4116yescomplemented
44C2×C7×C4⋊C4—24no—
45C2×C7×C4⋊C4—28no—
46C2×C7×C4⋊C4C2×C4116yescomplemented
47C2×C7×C4⋊C4C2318yes—
48C2×C7×C22⋊C4—216no—
49C2×C7×C22⋊C4—24no—
50C2×C7×C22⋊C4—24no—
51C2×C42×C7—216no—
52C2×C42×C7—24no—
53C2×C42×C7—24no—
54C23×M16C2×C711yescharacteristic
55C22×#117C2×C712yes—
56#1736C2×C718yescomplemented
57#1735C2×C714yescomplemented
58C24×C7—44no—
59C24×C7C22⋊C418yes—
60C24×C7C4⋊C414yes—
61C24×C7C2×D411yescharacteristic
62C24×C7C2×D412yes—
63C22×C4×C7—216no—
64C22×C4×C7—28no—
65C22×C4×C7—416no—
66C22×C4×C7C42116yes—
67C22×C4×C7C22⋊C418yes—
68C22×C4×C7C22⋊C418yes—
69C22×C4×C7C22⋊C418yes—
70C22×C4×C7C22⋊C4116yes—
71C22×C4×C7C4⋊C414yes—
72C22×C4×C7C4⋊C418yes—
73C22×C4×C7C4⋊C4132yes—
74C22×C4×C7C22×C412yes—
75C22×C4×C7C22×C414yes—
76C22×C4×C7C22×C414yes—
77C22×C4×C7C22×C414yes—
78C22×C4×C7C22×C414yes—
79C22×C4×C7C2×D412yes—
80C22×C4×C7C2×D414yes—
81C22×C4×C7C2×D414yes—
82C22×C4×C7C2×D414yes—
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×Q814yes—
85C22×C4×C7C2×Q812yes—
86C22×C4×C7C2411yescharacteristic
87C7×M16—232no—
88C2×C7×C8—216no—
89C2×C7×C8—28no—
90C2×C7×C8C22×C418yes—
91C7×C4⋊C4—216no—
92C7×C4⋊C4—28no—
93C7×C4⋊C4C22×C4116yes—
94C7×C22⋊C4—216no—
95C7×C22⋊C4—216no—
96C42×C7—216no—
97C42×C7—216no—
98C24×C4C22×C711yescharacteristic
99C22×M16—28no—
100C22×M16C4×C718yescomplemented
101C22×M16C22×C714yes—
102C23×C8C22×C712yes—
103C2×#117—28no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117C4×C7116yescomplemented
107C2×#117C22×C714yes—
108C2×#117C22×C714yes—
109C22×C4⋊C4C22×C714yes—
110C22×C22⋊C4—22no—
111C22×C42—22no—
112#340C22×C7116yescomplemented
113C23×C7—28no—
114C23×C7—24no—
115C23×C7—416no—
116C23×C7—48no—
117C23×C7#95116yes—
118C23×C7C2×C22⋊C414yes—
119C23×C7C2×C22⋊C414yes—
120C23×C7C2×C22⋊C414yes—
121C23×C7C2×C4⋊C412yes—
122C23×C7C2×C4⋊C414yes—
123C23×C7C22×D411yescharacteristic
124C2×C4×C7—216no—
125C2×C4×C7—432no—
126C2×C4×C7—416no—
127C2×C4×C7#95132yes—
128C2×C4×C7#95116yes—
129C2×C4×C7C2×C4214yes—
130C2×C4×C7C2×C4218yes—
131C2×C4×C7C2×C22⋊C414yes—
132C2×C4×C7C2×C22⋊C418yes—
133C2×C4×C7C2×C22⋊C414yes—
134C2×C4×C7C2×C22⋊C414yes—
135C2×C4×C7C2×C22⋊C414yes—
136C2×C4×C7C2×C4⋊C418yes—
137C2×C4×C7C2×C4⋊C418yes—
138C2×C4×C7C2×C4⋊C418yes—
139C2×C4×C7C2×C4⋊C414yes—
140C2×C4×C7C2×C4⋊C412yes—
141C2×C4×C7C23×C411yescharacteristic
142C2×C4×C7C23×C412yes—
143C2×C4×C7C22×D412yes—
144C2×C4×C7C22×Q811yescharacteristic
145C7×C8—28no—
146C25C7×D411yescharacteristic
147C23×C4—24no—
148C23×C4C2×C4×C714yes—
149C23×C4C2×C4×C718yes—
150C23×C4C7×D411yescharacteristic
151C23×C4C7×D412yes—
152C23×C4C7×D418yes—
153C23×C4C7×Q814yes—
154C23×C4C23×C712yes—
155C23×C4C23×C711yescharacteristic
156C2×M16—232no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16C2×C4×C7116yescomplemented
160C22×C8—28no—
161C22×C8C2×C4×C718yescomplemented
162C22×C8C23×C714yes—
163#117—216no—
164#117—216no—
165#117—28no—
166#117—28no—
167#117C2×C4×C7116yes—
168#117C2×C4×C7116yescomplemented
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4C2×C4×C7116yescomplemented
172C2×C4⋊C4C23×C718yes—
173C2×C22⋊C4—216no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C42—216no—
177C2×C42—24no—
178C2×C42—24no—
179C22×C7—28no—
180C22×C7—28no—
181C22×C7—28no—
182C22×C7—28no—
183C22×C7—24no—
184C22×C7—416no—
185C22×C7C2×#9514yes—
186C22×C7C2×#9518yes—
187C22×C7C22×C22⋊C412yes—
188C22×C7C22×C4⋊C411yescharacteristic
189C22×C7#57514yes—
190C4×C7—416no—
191C4×C7C2×#9514yes—
192C4×C7C2×#9518yes—
193C4×C7C22×C4211yescharacteristic
194C4×C7C22×C22⋊C411yescharacteristic
195C4×C7C22×C4⋊C412yes—
196C24—44no—
197C24C7×C22⋊C418yes—
198C24C7×C4⋊C414yes—
199C24C2×C7×D411yescharacteristic
200C24C2×C7×D412yes—
201C22×C4—216no—
202C22×C4—28no—
203C22×C4—416no—
204C22×C4C42×C7116yes—
205C22×C4C7×C22⋊C418yes—
206C22×C4C7×C22⋊C418yes—
207C22×C4C7×C22⋊C418yes—
208C22×C4C7×C22⋊C4116yes—
209C22×C4C7×C4⋊C414yes—
210C22×C4C7×C4⋊C418yes—
211C22×C4C7×C4⋊C4132yes—
212C22×C4C22×C4×C712yes—
213C22×C4C22×C4×C714yes—
214C22×C4C22×C4×C714yes—
215C22×C4C22×C4×C714yes—
216C22×C4C22×C4×C714yes—
217C22×C4C2×C7×D411yescharacteristic
218C22×C4C2×C7×D412yes—
219C22×C4C2×C7×D414yes—
220C22×C4C2×C7×D414yes—
221C22×C4C2×C7×D414yes—
222C22×C4C2×C7×Q814yes—
223C22×C4C2×C7×Q812yes—
224C22×C4C24×C711yescharacteristic
225M16—232no—
226C2×C8—216no—
227C2×C8—28no—
228C2×C8C22×C4×C718yes—
229C4⋊C4—216no—
230C4⋊C4—28no—
231C4⋊C4C22×C4×C7116yes—
232C22⋊C4—216no—
233C22⋊C4—216no—
234C42—216no—
235C42—216no—
236C2×C7—28no—
237C2×C7—24no—
238C2×C7C22×#9511yescharacteristic
239C2×C7#289614yes—
240C2×C7#291112yes—
241C23—28no—
242C23—24no—
243C23—416no—
244C23—48no—
245C23C7×#95116yes—
246C23C2×C7×C22⋊C414yes—
247C23C2×C7×C22⋊C414yes—
248C23C2×C7×C22⋊C414yes—
249C23C2×C7×C4⋊C412yes—
250C23C2×C7×C4⋊C414yes—
251C23C22×C7×D411yescharacteristic
252C2×C4—216no—
253C2×C4—432no—
254C2×C4—416no—
255C2×C4C7×#95132yes—
256C2×C4C7×#95116yes—
257C2×C4C2×C42×C714yes—
258C2×C4C2×C42×C718yes—
259C2×C4C2×C7×C22⋊C414yes—
260C2×C4C2×C7×C22⋊C414yes—
261C2×C4C2×C7×C22⋊C414yes—
262C2×C4C2×C7×C22⋊C414yes—
263C2×C4C2×C7×C22⋊C418yes—
264C2×C4C2×C7×C4⋊C414yes—
265C2×C4C2×C7×C4⋊C412yes—
266C2×C4C2×C7×C4⋊C418yes—
267C2×C4C2×C7×C4⋊C418yes—
268C2×C4C2×C7×C4⋊C418yes—
269C2×C4C23×C4×C711yescharacteristic
270C2×C4C23×C4×C712yes—
271C2×C4C22×C7×D412yes—
272C2×C4C22×C7×Q811yescharacteristic
273C8—28no—
274C7#2034711yescharacteristic, direct product factor, Sylow
275C22—28no—
276C22—24no—
277C22—28no—
278C22—28no—
279C22—28no—
280C22—416no—
281C22C2×C7×#9514yes—
282C22C2×C7×#9518yes—
283C22C22×C7×C22⋊C412yes—
284C22C22×C7×C4⋊C411yescharacteristic
285C22C7×#57514yes—
286C4—416no—
287C4C2×C7×#9514yes—
288C4C2×C7×#9518yes—
289C4C22×C42×C711yescharacteristic
290C4C22×C7×C22⋊C411yescharacteristic
291C4C22×C7×C4⋊C412yes—
292C2—28no—
293C2—24no—
294C2C22×C7×#9511yescharacteristic
295C2C7×#289614yes—
296C2C7×#291112yes—
subgroupG/H
centerC22×C4×C7C2×D4
commutatorC4C22×C42×C7
Frattini, cosocleC22×C4C24×C7
socleC23×C7C22×D4
prime radicalC25×C7D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1736C2semidirect
4. C7×#1735C2semidirect
7. C22×C7×M16C4semidirect
13. C2×C7×#117C4semidirect
19. C7×#340C22semidirect
20. #20347C7direct (N × H)
34. C2×C7×M16C2×C4semidirect
36. C22×C7×C8C2×C4semidirect
43. C7×#117C2×C4semidirect
46. C2×C7×C4⋊C4C2×C4semidirect
56. #1736C2×C7semidirect
57. #1735C2×C7semidirect
100. C22×M16C4×C7semidirect
106. C2×#117C4×C7semidirect
112. #340C22×C7semidirect
159. C2×M16C2×C4×C7semidirect
161. C22×C8C2×C4×C7semidirect
168. #117C2×C4×C7semidirect
171. C2×C4⋊C4C2×C4×C7semidirect
274. C7#20347direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×M162
C22×C7×#1172
C7×#17362
C7×#17352
#203477