Finite Group Databasebeta

← C7×#20335 #13197551

Subgroups — C7×#20335 #13197551

Order 1792.

2206 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C7×M16C212yesmaximal
2C22×C7×#117C211yesmaximal, characteristic
3C7×#1729C218yesmaximal, complemented
4C7×#1728C214yesmaximal, complemented
5C24×C4×C7C2211yescharacteristic
6C22×C7×M16—28no—
7C22×C7×M16—28no—
8C22×C7×M16C4116yescomplemented
9C22×C7×M16C2214yes—
10C22×C7×M16C2214yes—
11C23×C7×C8C2214yes—
12C2×C7×#117—28no—
13C2×C7×#117C414yes—
14C2×C7×#117C2214yes—
15C22×C7×C4⋊C4C412yes—
16C22×C7×C22⋊C4C412yes—
17C22×C42×C7C2212yes—
18C7×#336C22116yescomplemented
19#20335C711yesmaximal, characteristic, direct product factor, Sylow
20C25×C7C2×C411yescharacteristic
21C23×C4×C7C2×C414yes—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7C2×C412yes—
24C23×C4×C7C2×C418yes—
25C23×C4×C7D414yes—
26C23×C4×C7D418yes—
27C23×C4×C7Q814yes—
28C23×C4×C7C2311yescharacteristic
29C23×C4×C7C2312yes—
30C2×C7×M16—264no—
31C2×C7×M16—28no—
32C2×C7×M16—28no—
33C2×C7×M16—28no—
34C2×C7×M16—28no—
35C2×C7×M16C2×C4116yes—
36C2×C7×M16C2×C4116yescomplemented
37C22×C7×C8—28no—
38C22×C7×C8—28no—
39C22×C7×C8C2×C4116yescomplemented
40C22×C7×C8C2318yes—
41C7×#117—216no—
42C7×#117—28no—
43C7×#117—28no—
44C2×C7×C4⋊C4—28no—
45C2×C7×C4⋊C4C2×C418yes—
46C2×C7×C22⋊C4—28no—
47C2×C7×C22⋊C4C2×C418yes—
48C2×C42×C7—28no—
49C2×C42×C7C2×C414yes—
50C2×C42×C7C2314yes—
51C23×M16C2×C712yes—
52C22×#117C2×C711yescharacteristic
53#1729C2×C718yescomplemented
54#1728C2×C714yescomplemented
55C24×C7—44no—
56C24×C7C4214yes—
57C24×C7C22⋊C414yes—
58C24×C7C4⋊C414yes—
59C24×C7C22×C412yes—
60C24×C7C22×C411yescharacteristic
61C22×C4×C7—28no—
62C22×C4×C7—48no—
63C22×C4×C7C4214yes—
64C22×C4×C7C4218yes—
65C22×C4×C7C22⋊C414yes—
66C22×C4×C7C22⋊C418yes—
67C22×C4×C7C22⋊C4132yes—
68C22×C4×C7C4⋊C414yes—
69C22×C4×C7C4⋊C418yes—
70C22×C4×C7C4⋊C4132yes—
71C22×C4×C7C22×C418yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C22×C412yes—
74C22×C4×C7C22×C414yes—
75C22×C4×C7C22×C414yes—
76C22×C4×C7C22×C414yes—
77C22×C4×C7C2×D414yes—
78C22×C4×C7C2×D412yes—
79C22×C4×C7C2×D414yes—
80C22×C4×C7C2×D414yes—
81C22×C4×C7C2×D414yes—
82C22×C4×C7C2×Q812yes—
83C22×C4×C7C2×Q814yes—
84C22×C4×C7C2411yescharacteristic
85C7×M16—232no—
86C7×M16—232no—
87C2×C7×C8—232no—
88C2×C7×C8—28no—
89C2×C7×C8—28no—
90C2×C7×C8C22×C4116yes—
91C7×C4⋊C4—216no—
92C7×C22⋊C4—216no—
93C42×C7—28no—
94C42×C7—28no—
95C24×C4C22×C711yescharacteristic
96C22×M16—28no—
97C22×M16—28no—
98C22×M16C4×C7116yescomplemented
99C22×M16C22×C714yes—
100C22×M16C22×C714yes—
101C23×C8C22×C714yes—
102C2×#117—28no—
103C2×#117C4×C714yes—
104C2×#117C22×C714yes—
105C22×C4⋊C4C4×C712yes—
106C22×C22⋊C4C4×C712yes—
107C22×C42C22×C712yes—
108#336C22×C7116yescomplemented
109C23×C7—24no—
110C23×C7—28no—
111C23×C7—48no—
112C23×C7—48no—
113C23×C7—48no—
114C23×C7#95116yes—
115C23×C7C2×C4214yes—
116C23×C7C2×C4212yes—
117C23×C7C2×C22⋊C414yes—
118C23×C7C2×C22⋊C412yes—
119C23×C7C2×C4⋊C414yes—
120C23×C7C2×C4⋊C412yes—
121C23×C7C23×C411yescharacteristic
122C2×C4×C7—28no—
123C2×C4×C7—216no—
124C2×C4×C7—416no—
125C2×C4×C7#95116yes—
126C2×C4×C7#95132yes—
127C2×C4×C7C2×C4212yes—
128C2×C4×C7C2×C4214yes—
129C2×C4×C7C2×C22⋊C418yes—
130C2×C4×C7C2×C22⋊C418yes—
131C2×C4×C7C2×C22⋊C418yes—
132C2×C4×C7C2×C22⋊C412yes—
133C2×C4×C7C2×C22⋊C414yes—
134C2×C4×C7C2×C4⋊C418yes—
135C2×C4×C7C2×C4⋊C418yes—
136C2×C4×C7C2×C4⋊C418yes—
137C2×C4×C7C2×C4⋊C412yes—
138C2×C4×C7C2×C4⋊C414yes—
139C2×C4×C7C23×C412yes—
140C2×C4×C7C22×D411yescharacteristic
141C2×C4×C7C22×D412yes—
142C2×C4×C7C22×Q811yescharacteristic
143C7×C8—216no—
144C25C2×C4×C711yescharacteristic
145C23×C4C2×C4×C714yes—
146C23×C4C2×C4×C711yescharacteristic
147C23×C4C2×C4×C712yes—
148C23×C4C2×C4×C718yes—
149C23×C4C7×D414yes—
150C23×C4C7×D418yes—
151C23×C4C7×Q814yes—
152C23×C4C23×C712yes—
153C23×C4C23×C711yescharacteristic
154C2×M16—264no—
155C2×M16—28no—
156C2×M16—28no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16C2×C4×C7116yes—
160C2×M16C2×C4×C7116yescomplemented
161C22×C8—28no—
162C22×C8—28no—
163C22×C8C2×C4×C7116yescomplemented
164C22×C8C23×C718yes—
165#117—216no—
166#117—28no—
167#117—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4C2×C4×C718yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4C2×C4×C718yes—
172C2×C42—28no—
173C2×C42C2×C4×C714yes—
174C2×C42C23×C714yes—
175C22×C7—24no—
176C22×C7—28no—
177C22×C7—24no—
178C22×C7—28no—
179C22×C7—24no—
180C22×C7—28no—
181C22×C7—416no—
182C22×C7C2×#9518yes—
183C22×C7C2×#9514yes—
184C22×C7C22×C4211yescharacteristic
185C22×C7C22×C22⋊C411yescharacteristic
186C22×C7C22×C4⋊C411yescharacteristic
187C22×C7#56814yes—
188C4×C7—48no—
189C4×C7C2×#9518yes—
190C4×C7C2×#9514yes—
191C4×C7C22×C22⋊C412yes—
192C4×C7C22×C4⋊C412yes—
193C24—44no—
194C24C42×C714yes—
195C24C7×C22⋊C414yes—
196C24C7×C4⋊C414yes—
197C24C22×C4×C712yes—
198C24C22×C4×C711yescharacteristic
199C22×C4—28no—
200C22×C4—48no—
201C22×C4C42×C714yes—
202C22×C4C42×C718yes—
203C22×C4C7×C22⋊C414yes—
204C22×C4C7×C22⋊C418yes—
205C22×C4C7×C22⋊C4132yes—
206C22×C4C7×C4⋊C414yes—
207C22×C4C7×C4⋊C418yes—
208C22×C4C7×C4⋊C4132yes—
209C22×C4C22×C4×C718yes—
210C22×C4C22×C4×C711yescharacteristic
211C22×C4C22×C4×C712yes—
212C22×C4C22×C4×C714yes—
213C22×C4C22×C4×C714yes—
214C22×C4C22×C4×C714yes—
215C22×C4C2×C7×D414yes—
216C22×C4C2×C7×D412yes—
217C22×C4C2×C7×D414yes—
218C22×C4C2×C7×D414yes—
219C22×C4C2×C7×D414yes—
220C22×C4C2×C7×Q814yes—
221C22×C4C2×C7×Q812yes—
222C22×C4C24×C711yescharacteristic
223M16—232no—
224M16—232no—
225C2×C8—232no—
226C2×C8—28no—
227C2×C8—28no—
228C2×C8C22×C4×C7116yes—
229C4⋊C4—216no—
230C22⋊C4—216no—
231C42—28no—
232C42—28no—
233C2×C7—24no—
234C2×C7—28no—
235C2×C7C22×#9511yescharacteristic
236C2×C7#287312yes—
237C2×C7#287912yes—
238C2×C7#290512yes—
239C23—24no—
240C23—28no—
241C23—48no—
242C23—48no—
243C23—48no—
244C23C7×#95116yes—
245C23C2×C42×C714yes—
246C23C2×C42×C712yes—
247C23C2×C7×C22⋊C414yes—
248C23C2×C7×C22⋊C412yes—
249C23C2×C7×C4⋊C414yes—
250C23C2×C7×C4⋊C412yes—
251C23C23×C4×C711yescharacteristic
252C2×C4—28no—
253C2×C4—216no—
254C2×C4—416no—
255C2×C4C7×#95116yes—
256C2×C4C7×#95132yes—
257C2×C4C2×C42×C712yes—
258C2×C4C2×C42×C714yes—
259C2×C4C2×C7×C22⋊C412yes—
260C2×C4C2×C7×C22⋊C414yes—
261C2×C4C2×C7×C22⋊C418yes—
262C2×C4C2×C7×C22⋊C418yes—
263C2×C4C2×C7×C22⋊C418yes—
264C2×C4C2×C7×C4⋊C412yes—
265C2×C4C2×C7×C4⋊C414yes—
266C2×C4C2×C7×C4⋊C418yes—
267C2×C4C2×C7×C4⋊C418yes—
268C2×C4C2×C7×C4⋊C418yes—
269C2×C4C23×C4×C712yes—
270C2×C4C22×C7×D411yescharacteristic
271C2×C4C22×C7×D412yes—
272C2×C4C22×C7×Q811yescharacteristic
273C8—216no—
274C7#2033511yescharacteristic, direct product factor, Sylow
275C22—24no—
276C22—28no—
277C22—24no—
278C22—28no—
279C22—24no—
280C22—28no—
281C22—416no—
282C22C2×C7×#9518yes—
283C22C2×C7×#9514yes—
284C22C22×C42×C711yescharacteristic
285C22C22×C7×C22⋊C411yescharacteristic
286C22C22×C7×C4⋊C411yescharacteristic
287C22C7×#56814yes—
288C4—48no—
289C4C2×C7×#9514yes—
290C4C2×C7×#9518yes—
291C4C22×C7×C22⋊C412yes—
292C4C22×C7×C4⋊C412yes—
293C2—24no—
294C2—28no—
295C2C22×C7×#9511yescharacteristic
296C2C7×#287312yes—
297C2C7×#287912yes—
298C2C7×#290512yes—
subgroupG/H
centerC22×C4×C7C24
commutatorC22C22×C42×C7
Frattini, cosocleC22×C4C24×C7
socleC23×C7C23×C4
prime radicalC25×C7C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1729C2semidirect
4. C7×#1728C2semidirect
8. C22×C7×M16C4semidirect
18. C7×#336C22semidirect
19. #20335C7direct (N × H)
36. C2×C7×M16C2×C4semidirect
39. C22×C7×C8C2×C4semidirect
53. #1729C2×C7semidirect
54. #1728C2×C7semidirect
98. C22×M16C4×C7semidirect
108. #336C22×C7semidirect
160. C2×M16C2×C4×C7semidirect
163. C22×C8C2×C4×C7semidirect
274. C7#20335direct (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×#17292
C7×#17282
#203357