Finite Group Databasebeta

← C2×C7×#2984 #13237460

Subgroups — C2×C7×#2984 #13237460

Order 1792.

2022 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×Q8C211yesmaximal, characteristic
2C7×#2984C2116yesmaximal, direct product factor
3C2×C7×#446C216yesmaximal, complemented
4C2×C7×C8×Q8C212yesmaximal, complemented
5C2×C7×#432C213yesmaximal
6C2×C7×#431C213yesmaximal
7C23×C7×Q8C411yescharacteristic
8C2×C4×C7×Q8C418yes—
9C2×C4×C7×Q8C413yes—
10C2×C4×C7×Q8C411yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×Q8C2213yes—
13C2×C4×C7×Q8C2218yes—
14C22×C7×C4⋊C4C413yes—
15C22×C42×C7C2213yes—
16C7×#446C22148yescomplemented
17C7×C8×Q8C22116yescomplemented
18C7×#432C22124yes—
19C7×#431C22124yes—
20C2×C7×C4⋊C8C2216yes—
21C2×C7×C4⋊C8C2216yes—
22C2×C7×C22⋊C8C2211yescharacteristic
23C2×C7×C22⋊C8C2213yes—
24C2×C7×#97C2216yes—
25C2×C4×C7×C8C2216yes—
26C2×#2984C711yesmaximal, characteristic, direct product factor, Sylow
27C22×C7×Q8—28no—
28C22×C7×Q8C2×C418yes—
29C22×C7×Q8C2×C413yes—
30C22×C7×Q8C2×C411yescharacteristic
31C23×C4×C7C2×C413yes—
32C23×C4×C7C2×C413yes—
33C23×C4×C7C2311yescharacteristic
34C22×C7×C8C2312yes—
35C22×C7×C8C2316yes—
36C4×C7×Q8C2×C4112yes—
37C4×C7×Q8C2×C4124yes—
38C4×C7×Q8C2×C414yes—
39C4×C7×Q8C2×C418yes—
40C4×C7×Q8C2314yes—
41C4×C7×Q8C23112yes—
42C2×C7×C4⋊C4C2×C416yes—
43C2×C7×C4⋊C4C2×C4124yes—
44C2×C7×C4⋊C4C2×C413yes—
45C2×C7×C4⋊C4C2×C413yes—
46C2×C42×C7C2×C413yes—
47C2×C42×C7C2×C4112yes—
48C2×C42×C7C2×C413yes—
49C2×C42×C7C23112yes—
50C2×C42×C7C2313yes—
51C2×C42×C7C2313yes—
52C7×C4⋊C8C23124yes—
53C7×C4⋊C8C23124yes—
54C7×C22⋊C8C2314yes—
55C7×C22⋊C8C23112yes—
56C7×#97C23124yes—
57C4×C7×C8C23124yes—
58C22×C4×Q8C2×C711yescharacteristic
59#2984C2×C7116yesdirect product factor
60C2×#446C2×C716yescomplemented
61C2×C8×Q8C2×C712yescomplemented
62C2×#432C2×C713yes—
63C2×#431C2×C713yes—
64C24×C7C22×C411yescharacteristic
65C2×C7×Q8—232no—
66C2×C7×Q8—212no—
67C2×C7×Q8M1618yes—
68C2×C7×Q8C22×C4112yes—
69C2×C7×Q8C22×C414yes—
70C22×C4×C7—212no—
71C22×C4×C7—212no—
72C22×C4×C7—24no—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C22×C412yes—
75C22×C4×C7C22×C4112yes—
76C22×C4×C7C22×C413yes—
77C22×C4×C7C22×C413yes—
78C22×C4×C7C22×C413yes—
79C22×C4×C7C22×C4112yes—
80C22×C4×C7C22×C413yes—
81C22×C4×C7C2412yes—
82C22×C4×C7C2411yescharacteristic
83C2×C7×C8—24no—
84C2×C7×C8—212no—
85C2×C7×C8C2414yes—
86C2×C7×C8C24112yes—
87C7×C4⋊C4C22×C4124yes—
88C7×C4⋊C4C22×C4112yes—
89C7×C4⋊C4C22×C4112yes—
90C42×C7C22×C4112yes—
91C42×C7C22×C4112yes—
92C42×C7C22×C416yes—
93C42×C7C22×C416yes—
94C42×C7C2416yes—
95C42×C7C2416yes—
96C23×Q8C4×C711yescharacteristic
97C2×C4×Q8C4×C718yes—
98C2×C4×Q8C4×C713yes—
99C2×C4×Q8C4×C711yescharacteristic
100C2×C4×Q8C22×C711yescharacteristic
101C2×C4×Q8C22×C713yes—
102C2×C4×Q8C22×C718yes—
103C22×C4⋊C4C4×C713yes—
104C22×C42C22×C713yes—
105#446C22×C7148yescomplemented
106C8×Q8C22×C7116yescomplemented
107#432C22×C7124yes—
108#431C22×C7124yes—
109C2×C4⋊C8C22×C716yes—
110C2×C4⋊C8C22×C716yes—
111C2×C22⋊C8C22×C711yescharacteristic
112C2×C22⋊C8C22×C713yes—
113C2×#97C22×C716yes—
114C2×C4×C8C22×C716yes—
115C23×C7—22no—
116C23×C7—22no—
117C23×C7#13114yes—
118C23×C7C23×C412yes—
119C23×C7C23×C411yescharacteristic
120C7×Q8—224no—
121C7×Q8C2×M16116yes—
122C2×C4×C7—22no—
123C2×C4×C7—26no—
124C2×C4×C7—224no—
125C2×C4×C7—26no—
126C2×C4×C7—26no—
127C2×C4×C7—24no—
128C2×C4×C7—224no—
129C2×C4×C7—24no—
130C2×C4×C7C2×M16112yes—
131C2×C4×C7C23×C412yes—
132C2×C4×C7C23×C416yes—
133C2×C4×C7C23×C416yes—
134C2×C4×C7C23×C416yes—
135C2×C4×C7C23×C411yescharacteristic
136C2×C4×C7C23×C416yes—
137C2×C4×C72−1+414yes—
138C2×C4×C7C2511yescharacteristic
139C7×C8—24no—
140C7×C8—212no—
141C22×Q8—28no—
142C22×Q8C2×C4×C718yes—
143C22×Q8C2×C4×C713yes—
144C22×Q8C2×C4×C711yescharacteristic
145C23×C4C2×C4×C713yes—
146C23×C4C2×C4×C713yes—
147C23×C4C23×C711yescharacteristic
148C22×C8C23×C712yes—
149C22×C8C23×C716yes—
150C4×Q8C2×C4×C7112yes—
151C4×Q8C2×C4×C7124yes—
152C4×Q8C2×C4×C714yes—
153C4×Q8C2×C4×C718yes—
154C4×Q8C23×C714yes—
155C4×Q8C23×C7112yes—
156C2×C4⋊C4C2×C4×C716yes—
157C2×C4⋊C4C2×C4×C7124yes—
158C2×C4⋊C4C2×C4×C713yes—
159C2×C4⋊C4C2×C4×C713yes—
160C2×C42C2×C4×C713yes—
161C2×C42C2×C4×C7112yes—
162C2×C42C2×C4×C713yes—
163C2×C42C23×C713yes—
164C2×C42C23×C713yes—
165C2×C42C23×C7112yes—
166C4⋊C8C23×C7124yes—
167C4⋊C8C23×C7124yes—
168C22⋊C8C23×C714yes—
169C22⋊C8C23×C7112yes—
170#97C23×C7124yes—
171C4×C8C23×C7124yes—
172C22×C7—22no—
173C22×C7—28no—
174C22×C7—22no—
175C22×C7#51912yes—
176C22×C7C22×M1612yes—
177C22×C7C2×#13114yes—
178C22×C7C2×#13112yes—
179C22×C7C24×C411yescharacteristic
180C4×C7—22no—
181C4×C7—26no—
182C4×C7—26no—
183C4×C7—26no—
184C4×C7#51912yes—
185C4×C7C22×M16112yes—
186C4×C7C2×2−1+412yes—
187C24C22×C4×C711yescharacteristic
188C2×Q8—232no—
189C2×Q8—212no—
190C2×Q8C7×M1618yes—
191C2×Q8C22×C4×C7112yes—
192C2×Q8C22×C4×C714yes—
193C22×C4—212no—
194C22×C4—212no—
195C22×C4—24no—
196C22×C4C22×C4×C711yescharacteristic
197C22×C4C22×C4×C712yes—
198C22×C4C22×C4×C7112yes—
199C22×C4C22×C4×C713yes—
200C22×C4C22×C4×C713yes—
201C22×C4C22×C4×C713yes—
202C22×C4C22×C4×C7112yes—
203C22×C4C22×C4×C713yes—
204C22×C4C24×C712yes—
205C22×C4C24×C711yescharacteristic
206C2×C8—24no—
207C2×C8—212no—
208C2×C8C24×C714yes—
209C2×C8C24×C7112yes—
210C4⋊C4C22×C4×C7124yes—
211C4⋊C4C22×C4×C7112yes—
212C4⋊C4C22×C4×C7112yes—
213C42C22×C4×C7112yes—
214C42C22×C4×C7112yes—
215C42C22×C4×C716yes—
216C42C22×C4×C716yes—
217C42C24×C716yes—
218C42C24×C716yes—
219C2×C7—24no—
220C2×C7#298414yesdirect product factor
221C2×C7C2×#51911yescharacteristic
222C2×C7C23×M1611yescharacteristic
223C2×C7C22×#13111yescharacteristic
224C23—22no—
225C23—22no—
226C23C7×#13114yes—
227C23C23×C4×C712yes—
228C23C23×C4×C711yescharacteristic
229Q8—224no—
230Q8C2×C7×M16116yes—
231C2×C4—22no—
232C2×C4—26no—
233C2×C4—224no—
234C2×C4—26no—
235C2×C4—26no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—224no—
239C2×C4C2×C7×M16112yes—
240C2×C4C23×C4×C712yes—
241C2×C4C23×C4×C716yes—
242C2×C4C23×C4×C716yes—
243C2×C4C23×C4×C716yes—
244C2×C4C23×C4×C711yescharacteristic
245C2×C4C23×C4×C716yes—
246C2×C4C7×2−1+414yes—
247C2×C4C25×C711yescharacteristic
248C8—24no—
249C8—212no—
250C7C2×#298411yescharacteristic, direct product factor, Sylow
251C22—22no—
252C22—28no—
253C22—22no—
254C22C7×#51912yes—
255C22C22×C7×M1612yes—
256C22C2×C7×#13114yes—
257C22C2×C7×#13112yes—
258C22C24×C4×C711yescharacteristic
259C4—26no—
260C4—26no—
261C4—26no—
262C4—22no—
263C4C7×#51912yes—
264C4C22×C7×M16112yes—
265C4C2×C7×2−1+412yes—
266C2—24no—
267C2C7×#298414yesdirect product factor
268C2C2×C7×#51911yescharacteristic
269C2C23×C7×M1611yescharacteristic
270C2C22×C7×#13111yescharacteristic
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×#2984C2direct (N × H)
3. C2×C7×#446C2semidirect
4. C2×C7×C8×Q8C2semidirect
16. C7×#446C22semidirect
17. C7×C8×Q8C22semidirect
26. C2×#2984C7direct (N × H)
59. #2984C2×C7direct (N × H)
60. C2×#446C2×C7semidirect
61. C2×C8×Q8C2×C7semidirect
105. #446C22×C7semidirect
106. C8×Q8C22×C7semidirect
220. C2×C7#2984direct (N × H)
250. C7C2×#2984direct (N × H)
267. C2C7×#2984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×Q82
C7×#29842
C2×C7×#4462
C2×C7×C8×Q82
C2×C7×#4322
C2×C7×#4312
C2×#29847