Finite Group Databasebeta

← C7×#21334 #13198550

Subgroups — C7×#21334 #13198550

Order 1792.

1142 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic
2C7×#2316C211yesmaximal, characteristic
3C7×#2274C211yesmaximal, characteristic
4C7×#1996C218yesmaximal
5C7×#1740C214yesmaximal
6C7×#544—24no—
7C7×#544—24no—
8C2×C7×C4⋊Q8C414yescomplemented
9C2×C7×#125C414yescomplemented
10C2×C7×C22⋊Q8C2214yes—
11C2×C7×#117C2212yes—
12C2×C7×#117C2212yes—
13C2×C7×#117C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C7×#389—24no—
16C7×#380C2214yes—
17C4×C7×C4⋊C4—24no—
18C4×C7×C22⋊C4C2214yes—
19C7×#342C22116yes—
20#21334C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8C2×C414yes—
22C23×C4×C7C2312yes—
23C23×C4×C7C2311yescharacteristic
24C7×C4⋊Q8—216no—
25C7×#125—216no—
26C7×C22⋊Q8—28no—
27C7×C22⋊Q8—28no—
28C7×#117—24no—
29C7×#117—24no—
30C7×#117—24no—
31C7×#117—28no—
32C7×#117Q818yescomplemented
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—24no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4C2×C412yes—
38C2×C7×C4⋊C4C2×C414yes—
39C2×C7×C4⋊C4C2×C418yes—
40C2×C7×C4⋊C4C2×C414yes—
41C2×C7×C4⋊C4D414yes—
42C2×C7×C4⋊C4D414yes—
43C2×C7×C22⋊C4C2314yes—
44C2×C7×C22⋊C4C2314yes—
45C2×C7×C22⋊C4C2314yes—
46C2×C42×C7—22no—
47C2×C42×C7—22no—
48C2×C42×C7—24no—
49C2×C42×C7C2×C414yes—
50C7×#95—24no—
51C7×#95—24no—
52C2×#544C2×C711yescharacteristic
53#2316C2×C711yescharacteristic
54#2274C2×C711yescharacteristic
55#1996C2×C718yes—
56#1740C2×C714yes—
57C24×C7C2411yescharacteristic
58C2×C7×Q8—28no—
59C2×C7×Q8—84no—
60C22×C4×C7—24no—
61C22×C4×C7—24no—
62C22×C4×C7—82no—
63C22×C4×C7C22×C414yes—
64C22×C4×C7C22×C414yes—
65C22×C4×C7C22×C414yes—
66C22×C4×C7C22×C412yes—
67C22×C4×C7C2×D414yes—
68C22×C4×C7C2×D414yes—
69C22×C4×C7C2×D412yes—
70C22×C4×C7C2×D412yes—
71C22×C4×C7C2×Q814yes—
72C22×C4×C7#4114yes—
73C7×C4⋊C4—24no—
74C7×C4⋊C4—24no—
75C7×C4⋊C4—28no—
76C7×C4⋊C4—216no—
77C7×C4⋊C4—28no—
78C7×C4⋊C4—44no—
79C7×C4⋊C4—84no—
80C7×C4⋊C4C22⋊C4116yes—
81C7×C22⋊C4—24no—
82C7×C22⋊C4—24no—
83C7×C22⋊C4—24no—
84C7×C22⋊C4—28no—
85C7×C22⋊C4C2×Q818yes—
86C42×C7—28no—
87C42×C7—28no—
88C42×C7—44no—
89C42×C7—84no—
90#544—24no—
91#544—24no—
92C2×C4⋊Q8C4×C714yescomplemented
93C2×#125C4×C714yescomplemented
94C2×C22⋊Q8C22×C714yes—
95C2×#117C22×C712yes—
96C2×#117C22×C712yes—
97C2×#117C22×C712yes—
98C22×C4⋊C4C22×C711yescharacteristic
99#389—24no—
100#380C22×C714yes—
101C4×C4⋊C4—24no—
102C4×C22⋊C4C22×C714yes—
103#342C22×C7116yes—
104C23×C7—41no—
105C23×C7—41no—
106C23×C7C23×C411yescharacteristic
107C23×C7C22×D411yescharacteristic
108C23×C7C22×D411yescharacteristic
109C23×C7C22×Q812yes—
110C23×C7C2×#4112yes—
111C7×Q8—88no—
112C2×C4×C7—24no—
113C2×C4×C7—24no—
114C2×C4×C7—28no—
115C2×C4×C7—44no—
116C2×C4×C7—44no—
117C2×C4×C7—44no—
118C2×C4×C7—44no—
119C2×C4×C7—44no—
120C2×C4×C7—44no—
121C2×C4×C7—44no—
122C2×C4×C7—44no—
123C2×C4×C7—44no—
124C2×C4×C7—44no—
125C2×C4×C7C2×C22⋊C418yes—
126C2×C4×C7C2×C22⋊C414yes—
127C2×C4×C7C4×Q818yes—
128C2×C4×C7C22⋊Q814yes—
129C2×C4×C7C22⋊Q814yes—
130C2×C4×C7#12414yes—
131C2×C4×C7C4⋊Q814yes—
132C22×Q8C2×C4×C714yes—
133C23×C4C23×C712yes—
134C23×C4C23×C711yescharacteristic
135C4⋊Q8—216no—
136#125—216no—
137C22⋊Q8—28no—
138C22⋊Q8—28no—
139#117—24no—
140#117—24no—
141#117—24no—
142#117—28no—
143#117C7×Q818yescomplemented
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4C2×C4×C712yes—
148C2×C4⋊C4C2×C4×C712yes—
149C2×C4⋊C4C2×C4×C714yes—
150C2×C4⋊C4C2×C4×C718yes—
151C2×C4⋊C4C2×C4×C714yes—
152C2×C4⋊C4C7×D414yes—
153C2×C4⋊C4C7×D414yes—
154C2×C22⋊C4C23×C714yes—
155C2×C22⋊C4C23×C714yes—
156C2×C22⋊C4C23×C714yes—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—24no—
160C2×C42C2×C4×C714yes—
161#95—24no—
162#95—24no—
163C22×C7—21no—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7C22×C22⋊C411yescharacteristic
172C22×C7C2×C4×Q812yes—
173C22×C7C2×C22⋊Q812yes—
174C22×C7C2×#12411yescharacteristic
175C22×C7C2×C4⋊Q811yescharacteristic
176C4×C7—28no—
177C4×C7—48no—
178C4×C7—44no—
179C4×C7—44no—
180C4×C7—84no—
181C4×C7#39118yes—
182C24C24×C711yescharacteristic
183C2×Q8—28no—
184C2×Q8—84no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—82no—
188C22×C4C22×C4×C714yes—
189C22×C4C22×C4×C714yes—
190C22×C4C22×C4×C714yes—
191C22×C4C22×C4×C712yes—
192C22×C4C2×C7×D414yes—
193C22×C4C2×C7×D414yes—
194C22×C4C2×C7×D412yes—
195C22×C4C2×C7×D412yes—
196C22×C4C2×C7×Q814yes—
197C22×C4C7×#4114yes—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—28no—
201C4⋊C4—216no—
202C4⋊C4—28no—
203C4⋊C4—44no—
204C4⋊C4—84no—
205C4⋊C4C7×C22⋊C4116yes—
206C22⋊C4—24no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—28no—
210C22⋊C4C2×C7×Q818yes—
211C42—28no—
212C42—28no—
213C42—44no—
214C42—84no—
215C2×C7—22no—
216C2×C7—21no—
217C2×C7—22no—
218C2×C7—21no—
219C2×C7C2×#39111yescharacteristic
220C2×C7#288311yescharacteristic
221C2×C7#288411yescharacteristic
222C23—41no—
223C23—41no—
224C23C23×C4×C711yescharacteristic
225C23C22×C7×D411yescharacteristic
226C23C22×C7×D411yescharacteristic
227C23C22×C7×Q812yes—
228C23C2×C7×#4112yes—
229Q8—88no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—28no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4C2×C7×C22⋊C418yes—
244C2×C4C2×C7×C22⋊C414yes—
245C2×C4C4×C7×Q818yes—
246C2×C4C7×C22⋊Q814yes—
247C2×C4C7×C22⋊Q814yes—
248C2×C4C7×#12414yes—
249C2×C4C7×C4⋊Q814yes—
250C7#2133411yescharacteristic, direct product factor, Sylow
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22C22×C7×C22⋊C411yescharacteristic
260C22C2×C4×C7×Q812yes—
261C22C2×C7×C22⋊Q812yes—
262C22C2×C7×#12411yescharacteristic
263C22C2×C7×C4⋊Q811yescharacteristic
264C4—28no—
265C4—48no—
266C4—44no—
267C4—44no—
268C4—84no—
269C4C7×#39118yes—
270C2—22no—
271C2—21no—
272C2—22no—
273C2—21no—
274C2C2×C7×#39111yescharacteristic
275C2C7×#288311yescharacteristic
276C2C7×#288411yescharacteristic
subgroupG/H
center, socleC22×C7C22×C22⋊C4
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC24×C7C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
8. C2×C7×C4⋊Q8C4semidirect
9. C2×C7×#125C4semidirect
20. #21334C7direct (N × H)
32. C7×#117Q8semidirect
92. C2×C4⋊Q8C4×C7semidirect
93. C2×#125C4×C7semidirect
143. #117C7×Q8semidirect
250. C7#21334direct (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×#23162
C7×#22742
C7×#19962
C7×#17402
#213347