Finite Group Databasebeta

← C7×#21359 #13198575

Subgroups — C7×#21359 #13198575

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#431C211yesmaximal, characteristic
2C7×#2285C211yesmaximal, characteristic
3C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
4C7×#1948C214yesmaximal, complemented
5C7×#1853C214yesmaximal
6C4×C7×#106C214yesmaximal, complemented
7C2×C7×#117—21no—
8C22×C7×C4⋊C4C2212yes—
9C22×C42×C7—21no—
10C22×C42×C7C2211yescharacteristic
11C7×#431C418yescomplemented
12C2×C7×#106C2218yes—
13C2×C7×C4⋊C8C2212yes—
14C2×C7×C22⋊C8C2212yes—
15C2×C4×C7×C8C2212yes—
16C7×#384C2212yes—
17C7×#382—21no—
18C7×#380—21no—
19C7×#380C2212yes—
20C4×C7×C4⋊C4—21no—
21C4×C7×C4⋊C4—21no—
22C4×C7×C4⋊C4—21no—
23C4×C7×C4⋊C4C2212yes—
24C4×C7×C4⋊C4C2212yes—
25C4×C7×C4⋊C4C2212yes—
26C2×C7×#95—21no—
27C7×#340C2218yes—
28#21359C711yesmaximal, characteristic, direct product factor, Sylow
29C23×C4×C7—22no—
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C22×C7×C8C2314yes—
34C7×#117—44no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—28no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4C2314yes—
40C2×C7×C4⋊C4C2314yes—
41C2×C7×C22⋊C4—22no—
42C2×C42×C7—22no—
43C2×C42×C7—22no—
44C2×C42×C7—22no—
45C2×C42×C7—44no—
46C2×C42×C7C2×C414yes—
47C2×C42×C7D411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C2×C42×C7Q814yes—
50C2×C42×C7C2311yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C7×#106—216no—
53C7×C4⋊C8C2×C418yescomplemented
54C7×C22⋊C8C2×C418yes—
55C4×C7×C8C2×C418yescomplemented
56C7×#95—22no—
57C7×#95—22no—
58C7×#95—22no—
59C7×#95—22no—
60C2×#431C2×C711yescharacteristic
61#2285C2×C711yescharacteristic
62C2×C4×C4⋊C4C2×C711yescharacteristic
63#1948C2×C714yescomplemented
64#1853C2×C714yes—
65C4×#106C2×C714yescomplemented
66C24×C7C2×D411yescharacteristic
67C22×C4×C7—24no—
68C22×C4×C7—28no—
69C22×C4×C7—24no—
70C22×C4×C7—24no—
71C22×C4×C7—42no—
72C22×C4×C7—44no—
73C22×C4×C7—44no—
74C22×C4×C7C4⋊C414yes—
75C22×C4×C7C4⋊C414yes—
76C22×C4×C7SD1614yes—
77C22×C4×C7SD1614yes—
78C22×C4×C7C22×C412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×Q812yes—
85C22×C4×C7C2411yescharacteristic
86C2×C7×C8—28no—
87C2×C7×C8C22×C418yes—
88C7×C4⋊C4—28no—
89C7×C4⋊C4—24no—
90C7×C4⋊C4—28no—
91C7×C4⋊C4—44no—
92C7×C4⋊C4—44no—
93C7×C4⋊C4#4118yes—
94C7×C22⋊C4—44no—
95C42×C7—44no—
96C42×C7—44no—
97C42×C7—44no—
98C42×C7C4⋊C414yes—
99C42×C7C4⋊C414yes—
100C42×C7C22×C412yes—
101C42×C7C22×C412yes—
102C42×C7C2×Q812yes—
103C42×C7C2×Q812yes—
104C2×#117—21no—
105C22×C4⋊C4C22×C712yes—
106C22×C42—21no—
107C22×C42C22×C711yescharacteristic
108#431C4×C718yescomplemented
109C2×#106C22×C718yes—
110C2×C4⋊C8C22×C712yes—
111C2×C22⋊C8C22×C712yes—
112C2×C4×C8C22×C712yes—
113#384C22×C712yes—
114#382—21no—
115#380—21no—
116#380C22×C712yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C22×C712yes—
121C4×C4⋊C4C22×C712yes—
122C4×C4⋊C4C22×C712yes—
123C2×#95—21no—
124#340C22×C718yes—
125C23×C7—22no—
126C23×C7—22no—
127C23×C7C2×C4⋊C412yes—
128C23×C7C2×SD1612yes—
129C23×C7C2×SD1612yes—
130C23×C7C22×D411yescharacteristic
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—24no—
135C2×C4×C7—24no—
136C2×C4×C7—28no—
137C2×C4×C7—44no—
138C2×C4×C7—416no—
139C2×C4×C7—44no—
140C2×C4×C7—44no—
141C2×C4×C7—44no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7#106116yescomplemented
145C2×C4×C7C2×C4⋊C412yes—
146C2×C4×C7C2×C4⋊C412yes—
147C2×C4×C7C2×C4⋊C412yes—
148C2×C4×C7C2×C4⋊C412yes—
149C2×C4×C7C2×C4⋊C412yes—
150C2×C4×C7C2×SD1612yes—
151C2×C4×C7C2×SD1612yes—
152C2×C4×C7C2×SD1612yes—
153C2×C4×C7C2×SD1612yes—
154C2×C4×C7C23×C411yescharacteristic
155C2×C4×C7C22×Q811yescharacteristic
156C2×C4×C7C2×#4114yes—
157C7×C8—28no—
158C23×C4—22no—
159C23×C4C7×D411yescharacteristic
160C23×C4C7×D411yescharacteristic
161C23×C4C23×C711yescharacteristic
162C22×C8C23×C714yes—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C23×C714yes—
169C2×C4⋊C4C23×C714yes—
170C2×C22⋊C4—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—44no—
175C2×C42C2×C4×C714yes—
176C2×C42C7×D411yescharacteristic
177C2×C42C7×D411yescharacteristic
178C2×C42C7×Q814yes—
179C2×C42C23×C711yescharacteristic
180C2×C42C23×C711yescharacteristic
181#106—216no—
182C4⋊C8C2×C4×C718yescomplemented
183C22⋊C8C2×C4×C718yes—
184C4×C8C2×C4×C718yescomplemented
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C22×C7—24no—
190C22×C7—24no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7C2×#10614yes—
194C22×C7C22×C4⋊C411yescharacteristic
195C22×C7#52512yes—
196C22×C7C22×SD1611yescharacteristic
197C22×C7C22×SD1611yescharacteristic
198C22×C7#57512yes—
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—22no—
202C4×C7—48no—
203C4×C7—84no—
204C4×C7C2×#10614yes—
205C4×C7C2×#10614yes—
206C4×C7C4×#4112yes—
207C4×C7#53312yes—
208C24C2×C7×D411yescharacteristic
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4C7×C4⋊C414yes—
217C22×C4C7×C4⋊C414yes—
218C22×C4C7×SD1614yes—
219C22×C4C7×SD1614yes—
220C22×C4C22×C4×C712yes—
221C22×C4C2×C7×D411yescharacteristic
222C22×C4C2×C7×D411yescharacteristic
223C22×C4C2×C7×D411yescharacteristic
224C22×C4C2×C7×D411yescharacteristic
225C22×C4C2×C7×D411yescharacteristic
226C22×C4C2×C7×Q812yes—
227C22×C4C24×C711yescharacteristic
228C2×C8—28no—
229C2×C8C22×C4×C718yes—
230C4⋊C4—28no—
231C4⋊C4—24no—
232C4⋊C4—28no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4C7×#4118yes—
236C22⋊C4—44no—
237C42—44no—
238C42—44no—
239C42—44no—
240C42C7×C4⋊C414yes—
241C42C7×C4⋊C414yes—
242C42C22×C4×C712yes—
243C42C22×C4×C712yes—
244C42C2×C7×Q812yes—
245C42C2×C7×Q812yes—
246C2×C7—24no—
247C2×C7#229611yescharacteristic
248C2×C7C22×#10611yescharacteristic
249C2×C7#291111yescharacteristic
250C2×C7#310112yes—
251C2×C7#310312yes—
252C23—22no—
253C23—22no—
254C23C2×C7×C4⋊C412yes—
255C23C2×C7×SD1612yes—
256C23C2×C7×SD1612yes—
257C23C22×C7×D411yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—28no—
264C2×C4—44no—
265C2×C4—416no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4C7×#106116yescomplemented
272C2×C4C2×C7×C4⋊C412yes—
273C2×C4C2×C7×C4⋊C412yes—
274C2×C4C2×C7×C4⋊C412yes—
275C2×C4C2×C7×C4⋊C412yes—
276C2×C4C2×C7×C4⋊C412yes—
277C2×C4C2×C7×SD1612yes—
278C2×C4C2×C7×SD1612yes—
279C2×C4C2×C7×SD1612yes—
280C2×C4C2×C7×SD1612yes—
281C2×C4C23×C4×C711yescharacteristic
282C2×C4C22×C7×Q811yescharacteristic
283C2×C4C2×C7×#4114yes—
284C8—28no—
285C7#2135911yescharacteristic, direct product factor, Sylow
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22C2×C7×#10614yes—
291C22C22×C7×C4⋊C411yescharacteristic
292C22C7×#52512yes—
293C22C22×C7×SD1611yescharacteristic
294C22C22×C7×SD1611yescharacteristic
295C22C7×#57512yes—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—48no—
300C4—84no—
301C4C2×C7×#10614yes—
302C4C2×C7×#10614yes—
303C4C4×C7×#4112yes—
304C4C7×#53312yes—
305C2—24no—
306C2C7×#229611yescharacteristic
307C2C22×C7×#10611yescharacteristic
308C2C7×#291111yescharacteristic
309C2C7×#310112yes—
310C2C7×#310312yes—
subgroupG/H
centerC22×C4×C7C2×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1948C2semidirect
6. C4×C7×#106C2semidirect
11. C7×#431C4semidirect
28. #21359C7direct (N × H)
53. C7×C4⋊C8C2×C4semidirect
55. C4×C7×C8C2×C4semidirect
63. #1948C2×C7semidirect
65. C4×#106C2×C7semidirect
108. #431C4×C7semidirect
144. C2×C4×C7#106semidirect
182. C4⋊C8C2×C4×C7semidirect
184. C4×C8C2×C4×C7semidirect
271. C2×C4C7×#106semidirect
285. C7#21359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4312
C7×#22852
C2×C4×C7×C4⋊C42
C7×#19482
C7×#18532
C4×C7×#1062
#213597