Finite Group Databasebeta

← C2×C7×#2986 #13237462

Subgroups — C2×C7×#2986 #13237462

Order 1792.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic
2C7×#2986C2116yesmaximal, direct product factor
3C2×C7×#446C214yesmaximal, complemented
4C2×C7×C8×Q8C214yesmaximal, complemented
5C2×C7×#433C214yesmaximal
6C2×C7×#431C212yesmaximal
7C7×#533C418yes—
8C7×#533C2218yes—
9C2×C7×C4⋊Q8C412yes—
10C2×C7×#125C412yes—
11C2×C7×C22⋊Q8C414yes—
12C2×C4×C7×Q8C2214yes—
13C2×C7×#117C2212yes—
14C22×C42×C7C2211yescharacteristic
15C7×#446C22132yescomplemented
16C7×C8×Q8C22132yescomplemented
17C7×#433C22132yes—
18C7×#431C22116yes—
19C2×C7×C4⋊C8C2212yes—
20C2×C7×C4⋊C8C2212yes—
21C2×C7×C4⋊C8C2218yes—
22C2×C7×C22⋊C8C2214yes—
23C2×C7×#97C2214yes—
24C2×C4×C7×C8C2214yes—
25C2×C4×C7×C8C2214yes—
26C2×#2986C711yesmaximal, characteristic, direct product factor, Sylow
27C22×C7×Q8C2×C414yes—
28C23×C4×C7C2×C412yes—
29C23×C4×C7C2311yescharacteristic
30C22×C7×C8C2314yes—
31C22×C7×C8C2314yes—
32C7×C4⋊Q8C2×C4116yes—
33C7×#125C2×C4116yes—
34C7×C22⋊Q8C2×C4132yes—
35C4×C7×Q8C2×C4116yes—
36C4×C7×Q8C23116yes—
37C7×#117C2×C418yes—
38C7×#117C2318yes—
39C2×C7×C4⋊C4C2×C414yes—
40C2×C7×C4⋊C4C2×C418yes—
41C2×C7×C4⋊C4C2×C414yes—
42C2×C7×C22⋊C4C2×C414yes—
43C2×C42×C7C2×C411yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7C2×C414yes—
46C2×C42×C7C2314yes—
47C2×C42×C7C2312yes—
48C2×C42×C7C2314yes—
49C7×C4⋊C8C2318yes—
50C7×C4⋊C8C2318yes—
51C7×C4⋊C8C23132yes—
52C7×C22⋊C8C23116yes—
53C7×#97C23116yes—
54C4×C7×C8C23116yes—
55C4×C7×C8C23116yes—
56C2×#533C2×C711yescharacteristic
57#2986C2×C7116yesdirect product factor
58C2×#446C2×C714yescomplemented
59C2×C8×Q8C2×C714yescomplemented
60C2×#433C2×C714yes—
61C2×#431C2×C712yes—
62C24×C7C22×C411yescharacteristic
63C2×C7×Q8—216no—
64C2×C7×Q8C22×C4116yes—
65C22×C4×C7—28no—
66C22×C4×C7—24no—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C22×C412yes—
69C22×C4×C7C22×C412yes—
70C22×C4×C7C22×C418yes—
71C22×C4×C7C22×C418yes—
72C22×C4×C7C22×C412yes—
73C22×C4×C7C2412yes—
74C22×C4×C7C2411yescharacteristic
75C2×C7×C8—28no—
76C2×C7×C8—28no—
77C2×C7×C8C2418yes—
78C2×C7×C8C2418yes—
79C7×C4⋊C4C22×C4116yes—
80C7×C4⋊C4C22×C4132yes—
81C7×C4⋊C4C22×C4116yes—
82C7×C22⋊C4C22×C4116yes—
83C42×C7C22×C414yes—
84C42×C7C22×C414yes—
85C42×C7C22×C414yes—
86C42×C7C22×C418yes—
87C42×C7C2414yes—
88C42×C7C2418yes—
89#533C4×C718yes—
90#533C22×C718yes—
91C2×C4⋊Q8C4×C712yes—
92C2×#125C4×C712yes—
93C2×C22⋊Q8C4×C714yes—
94C2×C4×Q8C22×C714yes—
95C2×#117C22×C712yes—
96C22×C42C22×C711yescharacteristic
97#446C22×C7132yescomplemented
98C8×Q8C22×C7132yescomplemented
99#433C22×C7132yes—
100#431C22×C7116yes—
101C2×C4⋊C8C22×C712yes—
102C2×C4⋊C8C22×C712yes—
103C2×C4⋊C8C22×C718yes—
104C2×C22⋊C8C22×C714yes—
105C2×#97C22×C714yes—
106C2×C4×C8C22×C714yes—
107C2×C4×C8C22×C714yes—
108C23×C7—22no—
109C23×C7—24no—
110C23×C7C23×C412yes—
111C23×C7C23×C411yescharacteristic
112C7×Q8—232no—
113C2×C4×C7—22no—
114C2×C4×C7—24no—
115C2×C4×C7—216no—
116C2×C4×C7—24no—
117C2×C4×C7—216no—
118C2×C4×C7—24no—
119C2×C4×C7#13118yes—
120C2×C4×C7C23×C412yes—
121C2×C4×C7C23×C414yes—
122C2×C4×C7C23×C4116yes—
123C2×C4×C7C23×C411yescharacteristic
124C2×C4×C7C23×C414yes—
125C2×C4×C72−1+414yes—
126C2×C4×C7C2511yescharacteristic
127C7×C8—28no—
128C7×C8—28no—
129C22×Q8C2×C4×C714yes—
130C23×C4C2×C4×C712yes—
131C23×C4C23×C711yescharacteristic
132C22×C8C23×C714yes—
133C22×C8C23×C714yes—
134C4⋊Q8C2×C4×C7116yes—
135#125C2×C4×C7116yes—
136C22⋊Q8C2×C4×C7132yes—
137C4×Q8C2×C4×C7116yes—
138C4×Q8C23×C7116yes—
139#117C2×C4×C718yes—
140#117C23×C718yes—
141C2×C4⋊C4C2×C4×C714yes—
142C2×C4⋊C4C2×C4×C718yes—
143C2×C4⋊C4C2×C4×C714yes—
144C2×C22⋊C4C2×C4×C714yes—
145C2×C42C2×C4×C711yescharacteristic
146C2×C42C2×C4×C711yescharacteristic
147C2×C42C2×C4×C714yes—
148C2×C42C23×C714yes—
149C2×C42C23×C712yes—
150C2×C42C23×C714yes—
151C4⋊C8C23×C718yes—
152C4⋊C8C23×C718yes—
153C4⋊C8C23×C7132yes—
154C22⋊C8C23×C7116yes—
155#97C23×C7116yes—
156C4×C8C23×C7116yes—
157C4×C8C23×C7116yes—
158C22×C7—22no—
159C22×C7—24no—
160C22×C7—44no—
161C22×C7#51912yes—
162C22×C7C2×#13114yes—
163C22×C7C24×C411yescharacteristic
164C4×C7—22no—
165C4×C7—24no—
166C4×C7—216no—
167C4×C7#51912yes—
168C4×C7C2×#13118yes—
169C4×C7C2×2−1+412yes—
170C24C22×C4×C711yescharacteristic
171C2×Q8—216no—
172C2×Q8C22×C4×C7116yes—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4C22×C4×C711yescharacteristic
176C22×C4C22×C4×C712yes—
177C22×C4C22×C4×C712yes—
178C22×C4C22×C4×C718yes—
179C22×C4C22×C4×C718yes—
180C22×C4C22×C4×C712yes—
181C22×C4C24×C712yes—
182C22×C4C24×C711yescharacteristic
183C2×C8—28no—
184C2×C8—28no—
185C2×C8C24×C718yes—
186C2×C8C24×C718yes—
187C4⋊C4C22×C4×C7116yes—
188C4⋊C4C22×C4×C7132yes—
189C4⋊C4C22×C4×C7116yes—
190C22⋊C4C22×C4×C7116yes—
191C42C22×C4×C714yes—
192C42C22×C4×C714yes—
193C42C22×C4×C714yes—
194C42C22×C4×C718yes—
195C42C24×C714yes—
196C42C24×C718yes—
197C2×C7—42no—
198C2×C7#298614yesdirect product factor
199C2×C7C2×#51911yescharacteristic
200C2×C7C22×#13112yes—
201C23—22no—
202C23—24no—
203C23C23×C4×C712yes—
204C23C23×C4×C711yescharacteristic
205Q8—232no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—216no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—216no—
212C2×C4C7×#13118yes—
213C2×C4C23×C4×C712yes—
214C2×C4C23×C4×C714yes—
215C2×C4C23×C4×C7116yes—
216C2×C4C23×C4×C711yescharacteristic
217C2×C4C23×C4×C714yes—
218C2×C4C7×2−1+414yes—
219C2×C4C25×C711yescharacteristic
220C8—28no—
221C8—28no—
222C7C2×#298611yescharacteristic, direct product factor, Sylow
223C22—24no—
224C22—22no—
225C22—44no—
226C22C7×#51912yes—
227C22C2×C7×#13114yes—
228C22C24×C4×C711yescharacteristic
229C4—24no—
230C4—216no—
231C4—22no—
232C4C7×#51912yes—
233C4C2×C7×#13118yes—
234C4C2×C7×2−1+412yes—
235C2—42no—
236C2C7×#298614yesdirect product factor
237C2C2×C7×#51911yescharacteristic
238C2C22×C7×#13112yes—
subgroupG/H
centerC22×C4×C7C24
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC23×C7C23×C4
prime radicalC24×C7C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2986C2direct (N × H)
3. C2×C7×#446C2semidirect
4. C2×C7×C8×Q8C2semidirect
15. C7×#446C22semidirect
16. C7×C8×Q8C22semidirect
26. C2×#2986C7direct (N × H)
57. #2986C2×C7direct (N × H)
58. C2×#446C2×C7semidirect
59. C2×C8×Q8C2×C7semidirect
97. #446C22×C7semidirect
98. C8×Q8C22×C7semidirect
198. C2×C7#2986direct (N × H)
222. C7C2×#2986direct (N × H)
236. C2C7×#2986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C7×#29862
C2×C7×#4462
C2×C7×C8×Q82
C2×C7×#4332
C2×C7×#4312
C2×#29867