Finite Group Databasebeta

← C7×#12859 #13190075

Subgroups — C7×#12859 #13190075

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×#2311C211yesmaximal, characteristic
2C7×#1782C212yesmaximal
3C7×#1393C214yesmaximal, complemented
4C22×C7×#41C2211yescharacteristic
5C2×C7×#117C411yescharacteristic, complemented
6C22×C7×C4⋊C4C411yescharacteristic, complemented
7C2×C7×#99—22no—
8C2×C7×#99C2214yes—
9C7×#391C412yescomplemented
10C7×#390C2212yes—
11C7×#386—22no—
12C7×#381—22no—
13C7×#380—22no—
14#12859C711yesmaximal, characteristic, direct product factor, Sylow
15C2×C7×#41—24no—
16C2×C7×#41—42no—
17C2×C7×#41—42no—
18C22×C7×Q8C2×C411yescharacteristic
19C22×C7×D4D412yes—
20C22×C7×D4C2311yescharacteristic
21C23×C4×C7—21no—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D412yes—
24C7×#117—42no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4—42no—
28C2×C7×C4⋊C4C2×C412yes—
29C2×C7×C22⋊C4—21no—
30C2×C7×C22⋊C4—24no—
31C2×C7×C22⋊C4—24no—
32C2×C42×C7C2×C412yes—
33C7×#99—28no—
34C7×#99—44no—
35C7×#95—24no—
36C7×#95—44no—
37#2311C2×C711yescharacteristic
38#1782C2×C712yes—
39#1393C2×C714yescomplemented
40C24×C7C22⋊C411yescharacteristic
41C24×C7C2×D412yes—
42C7×#41—416no—
43C2×C7×Q8—22no—
44C2×C7×Q8—41no—
45C2×C7×Q8—41no—
46C2×C7×D4—24no—
47C2×C7×D4—24no—
48C2×C7×D4—42no—
49C2×C7×D4—41no—
50C2×C7×D4—41no—
51C2×C7×D4#4114yes—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—24no—
56C22×C4×C7—24no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—42no—
60C22×C4×C7—42no—
61C22×C4×C7—44no—
62C22×C4×C7—82no—
63C22×C4×C7C22⋊C412yes—
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22×C411yescharacteristic
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C22⋊C4—42no—
70C7×C22⋊C4—44no—
71C7×C22⋊C4—44no—
72C7×C22⋊C4—84no—
73C42×C7—42no—
74C22×#41C22×C711yescharacteristic
75C2×#117C4×C711yescharacteristic, complemented
76C22×C4⋊C4C4×C711yescharacteristic, complemented
77C2×#99—22no—
78C2×#99C22×C714yes—
79#391C4×C712yescomplemented
80#390C22×C712yes—
81#386—22no—
82#381—22no—
83#380—22no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—41no—
89C23×C7—41no—
90C23×C7—42no—
91C23×C7—42no—
92C23×C7#9912yes—
93C23×C7C2×C22⋊C411yescharacteristic
94C23×C7#12314yes—
95C7×Q8—44no—
96C7×D4—44no—
97C7×D4—48no—
98C2×C4×C7—28no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—48no—
107C2×C4×C7—42no—
108C2×C4×C7—44no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—84no—
113C2×C4×C7—84no—
114C2×C4×C7—84no—
115C2×C4×C7—84no—
116C2×C4×C7#9912yes—
117C2×C4×C7#11712yes—
118C2×#41—24no—
119C2×#41—42no—
120C2×#41—42no—
121C22×Q8C2×C4×C711yescharacteristic
122C22×D4C7×D412yes—
123C22×D4C23×C711yescharacteristic
124C23×C4—21no—
125C23×C4C2×C4×C711yescharacteristic
126C23×C4C7×D412yes—
127#117—42no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×C4×C712yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C42C2×C4×C712yes—
136#99—28no—
137#99—44no—
138#95—24no—
139#95—44no—
140C22×C7—21no—
141C22×C7—21no—
142C22×C7—22no—
143C22×C7—24no—
144C22×C7—42no—
145C22×C7—44no—
146C22×C7—42no—
147C22×C7—41no—
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—41no—
151C22×C7—82no—
152C22×C7#38111yescharacteristic
153C22×C7C2×#9911yescharacteristic
154C22×C7C2×#9911yescharacteristic
155C4×C7—22no—
156C4×C7—22no—
157C4×C7—42no—
158C4×C7—82no—
159C4×C7—82no—
160C4×C7—164no—
161C24C7×C22⋊C411yescharacteristic
162C24C2×C7×D412yes—
163#41—416no—
164C2×Q8—22no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4C7×#4114yes—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—82no—
184C22×C4C7×C22⋊C412yes—
185C22×C4C7×C22⋊C411yescharacteristic
186C22×C4C22×C4×C711yescharacteristic
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—84no—
194C42—42no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7—44no—
198C2×C7—42no—
199C2×C7#181911yescharacteristic
200C2×C7#212811yescharacteristic
201C2×C7#212911yescharacteristic
202C23—21no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—42no—
209C23—42no—
210C23C7×#9912yes—
211C23C2×C7×C22⋊C411yescharacteristic
212C23C7×#12314yes—
213Q8—44no—
214D4—44no—
215D4—48no—
216C2×C4—28no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—48no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4C7×#9912yes—
235C2×C4C7×#11712yes—
236C7#1285911yescharacteristic, direct product factor, Sylow
237C22—21no—
238C22—21no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—82no—
249C22C7×#38111yescharacteristic
250C22C2×C7×#9911yescharacteristic
251C22C2×C7×#9911yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—164no—
258C2—21no—
259C2—21no—
260C2—44no—
261C2—42no—
262C2C7×#181911yescharacteristic
263C2C7×#212811yescharacteristic
264C2C7×#212911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×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×#1393C2semidirect
5. C2×C7×#117C4semidirect
6. C22×C7×C4⋊C4C4semidirect
9. C7×#391C4semidirect
14. #12859C7direct (N × H)
39. #1393C2×C7semidirect
75. C2×#117C4×C7semidirect
76. C22×C4⋊C4C4×C7semidirect
79. #391C4×C7semidirect
236. C7#12859direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23112
C7×#17822
C7×#13932
#128597