Finite Group Databasebeta

← C7×#12861 #13190077

Subgroups — C7×#12861 #13190077

Order 1792.

642 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2297C211yesmaximal, characteristic
2C7×#1785C212yesmaximal
3C7×#1394C214yesmaximal, complemented
4C22×C7×#41C2211yescharacteristic
5C2×C7×#117C411yescharacteristic
6C22×C42×C7C411yescharacteristic
7C2×C7×#101C2214yes—
8C2×C7×#100—22no—
9C7×#407—22no—
10C7×#391C2212yes—
11C7×#390C412yes—
12C7×#386—22no—
13C4×C7×C22⋊C4—22no—
14#12861C711yesmaximal, characteristic, direct product factor, Sylow
15C2×C7×#41—24no—
16C2×C7×#41—42no—
17C2×C7×#41—42no—
18C22×C7×Q8C2311yescharacteristic
19C22×C7×D4C2×C411yescharacteristic
20C22×C7×D4D412yes—
21C23×C4×C7—21no—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D412yes—
24C2×C7×M16—24no—
25C7×#117—42no—
26C2×C7×C4⋊C4C2×C412yes—
27C2×C7×C22⋊C4—24no—
28C2×C7×C22⋊C4—21no—
29C2×C42×C7—21no—
30C2×C42×C7—21no—
31C2×C42×C7—42no—
32C2×C42×C7C2×C412yes—
33C7×#101—28no—
34C7×#100—44no—
35C7×C22⋊C8—44no—
36C7×#95—24no—
37#2297C2×C711yescharacteristic
38#1785C2×C712yes—
39#1394C2×C714yescomplemented
40C24×C7C22⋊C412yes—
41C24×C7C22⋊C411yescharacteristic
42C7×#41—416no—
43C2×C7×Q8—41no—
44C2×C7×Q8—41no—
45C2×C7×Q8#4114yes—
46C2×C7×D4—24no—
47C2×C7×D4—22no—
48C2×C7×D4—24no—
49C2×C7×D4—41no—
50C2×C7×D4—41no—
51C2×C7×D4—42no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—24no—
57C22×C4×C7—24no—
58C22×C4×C7—22no—
59C22×C4×C7—42no—
60C22×C4×C7—44no—
61C22×C4×C7—42no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22×C411yescharacteristic
64C22×C4×C7C2×D412yes—
65C7×M16—44no—
66C7×M16—44no—
67C2×C7×C8—44no—
68C7×C4⋊C4—42no—
69C7×C22⋊C4—48no—
70C7×C22⋊C4—42no—
71C42×C7—42no—
72C42×C7—42no—
73C42×C7—42no—
74C22×#41C22×C711yescharacteristic
75C2×#117C4×C711yescharacteristic
76C22×C42C4×C711yescharacteristic
77C2×#101C22×C714yes—
78C2×#100—22no—
79#407—22no—
80#391C22×C712yes—
81#390C4×C712yes—
82#386—22no—
83C4×C22⋊C4—22no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—42no—
90C23×C7—44no—
91C23×C7#9912yes—
92C23×C7C2×C22⋊C411yescharacteristic
93C7×Q8—44no—
94C7×D4—48no—
95C7×D4—44no—
96C2×C4×C7—28no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—22no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—24no—
104C2×C4×C7—48no—
105C2×C4×C7—42no—
106C2×C4×C7—48no—
107C2×C4×C7—42no—
108C2×C4×C7—44no—
109C2×C4×C7—42no—
110C2×C4×C7#9912yes—
111C2×C4×C7#11712yes—
112C2×C4×C7#12314yes—
113C7×C8—84no—
114C2×#41—24no—
115C2×#41—42no—
116C2×#41—42no—
117C22×Q8C23×C711yescharacteristic
118C22×D4C2×C4×C711yescharacteristic
119C22×D4C7×D412yes—
120C23×C4—21no—
121C23×C4C2×C4×C711yescharacteristic
122C23×C4C7×D412yes—
123C2×M16—24no—
124#117—42no—
125C2×C4⋊C4C2×C4×C712yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—24no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—42no—
131C2×C42C2×C4×C712yes—
132#101—28no—
133#100—44no—
134C22⋊C8—44no—
135#95—24no—
136C22×C7—22no—
137C22×C7—22no—
138C22×C7—42no—
139C22×C7—42no—
140C22×C7—44no—
141C22×C7—42no—
142C22×C7—42no—
143C22×C7—44no—
144C22×C7—82no—
145C22×C7#38111yescharacteristic
146C22×C7C2×#9911yescharacteristic
147C22×C7C2×#9911yescharacteristic
148C4×C7—24no—
149C4×C7—22no—
150C4×C7—22no—
151C4×C7—44no—
152C4×C7—82no—
153C24C7×C22⋊C412yes—
154C24C7×C22⋊C411yescharacteristic
155#41—416no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C7×#4114yes—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4C7×C22⋊C411yescharacteristic
176C22×C4C22×C4×C711yescharacteristic
177C22×C4C2×C7×D412yes—
178M16—44no—
179M16—44no—
180C2×C8—44no—
181C4⋊C4—42no—
182C22⋊C4—48no—
183C22⋊C4—42no—
184C42—42no—
185C42—42no—
186C42—42no—
187C2×C7—21no—
188C2×C7—21no—
189C2×C7—42no—
190C2×C7—82no—
191C2×C7#181911yescharacteristic
192C2×C7#213212yes—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—42no—
199C23—44no—
200C23C7×#9912yes—
201C23C2×C7×C22⋊C411yescharacteristic
202Q8—44no—
203D4—48no—
204D4—44no—
205C2×C4—28no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—48no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4C7×#9912yes—
220C2×C4C7×#11712yes—
221C2×C4C7×#12314yes—
222C8—84no—
223C7#1286111yescharacteristic, direct product factor, Sylow
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22C7×#38111yescharacteristic
234C22C2×C7×#9911yescharacteristic
235C22C2×C7×#9911yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2C7×#181911yescharacteristic
246C2C7×#213212yes—
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×Q8C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1394C2semidirect
14. #12861C7direct (N × H)
39. #1394C2×C7semidirect
223. C7#12861direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#22972
C7×#17852
C7×#13942
#128617