Finite Group Databasebeta

← C7×#11781 #13188997

Subgroups — C7×#11781 #13188997

Order 1792.

482 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2548C211yesmaximal, characteristic
2C4×C7×#101C211yesmaximal, characteristic, complemented
3C7×#1467C211yesmaximal, characteristic, complemented
4C7×#1345C214yesmaximal, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#124—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#432—21no—
9C2×C7×#101—21no—
10C4×C7×M16—21no—
11C7×#394C412yes—
12C7×#394C2212yes—
13C7×#390—21no—
14C7×#390—21no—
15C7×#390—21no—
16C4×C7×C4⋊C4C411yescharacteristic
17C7×#343C2212yes—
18C7×#324—21no—
19C7×#324C2212yes—
20#11781C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×C7×D4—21no—
23C22×C7×D4—21no—
24C22×C7×D4C2×C412yes—
25C2×C7×M16—22no—
26C7×#127—41no—
27C7×#127—41no—
28C7×#124—44no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C42×C7—21no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C7×C4⋊C8—41no—
44C7×#101—24no—
45C7×C22⋊C8—22no—
46C7×#97—41no—
47C7×#97—41no—
48C4×C7×C8—41no—
49C7×#95C2×C412yes—
50#2548C2×C711yescharacteristic
51C4×#101C2×C711yescharacteristic, complemented
52#1467C2×C711yescharacteristic, complemented
53#1345C2×C714yescomplemented
54C24×C7—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—41no—
58C2×C7×D4—42no—
59C2×C7×D4—42no—
60C2×C7×D4—42no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C2×C7×D4—42no—
66C2×C7×D4—82no—
67C22×C4×C7—22no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C7×M16—24no—
76C7×M16—42no—
77C2×C7×C8—42no—
78C2×C7×C8—42no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—42no—
83C7×C22⋊C4—44no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—82no—
86C7×C22⋊C4—82no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—42no—
94C2×#127C4×C711yescharacteristic
95C2×#124—21no—
96C2×C4×Q8C22×C711yescharacteristic
97#432—21no—
98C2×#101—21no—
99C4×M16—21no—
100#394C4×C712yes—
101#394C22×C712yes—
102#390—21no—
103#390—21no—
104#390—21no—
105C4×C4⋊C4C4×C711yescharacteristic
106#343C22×C712yes—
107#324—21no—
108#324C22×C712yes—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7—42no—
112C23×C7—42no—
113C23×C7—42no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7C2×C22⋊C411yescharacteristic
117C7×Q8—44no—
118C7×D4—82no—
119C7×D4—84no—
120C7×D4—84no—
121C7×D4—82no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—22no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#9911yescharacteristic
141C2×C4×C7#10412yes—
142C2×C4×C7AGL(1,ℤ/8)12yes—
143C7×C8—44no—
144C7×C8—82no—
145C22×Q8C7×D411yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×C4×C712yes—
149C2×M16—22no—
150#127—41no—
151#127—41no—
152#124—44no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C2×C4×C711yescharacteristic
157C2×C4⋊C4C7×D411yescharacteristic
158C2×C4⋊C4C23×C711yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—21no—
164C2×C42C2×C4×C711yescharacteristic
165C2×C42C7×D411yescharacteristic
166C2×C42C7×D411yescharacteristic
167C4⋊C8—41no—
168#101—24no—
169C22⋊C8—22no—
170#97—41no—
171#97—41no—
172C4×C8—41no—
173#95C2×C4×C712yes—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—82no—
177C22×C7—82no—
178C22×C7—82no—
179C22×C7—82no—
180C22×C7—82no—
181C22×C7—82no—
182C22×C7—82no—
183C22×C7C2×#9911yescharacteristic
184C22×C7#41811yescharacteristic
185C22×C7C2×#10411yescharacteristic
186C4×C7—21no—
187C4×C7—22no—
188C4×C7—22no—
189C4×C7—41no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7—82no—
194C4×C7#35512yes—
195C24—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—82no—
208C22×C4—22no—
209C22×C4C7×C22⋊C411yescharacteristic
210C22×C4C7×C22⋊C411yescharacteristic
211C22×C4C7×C22⋊C411yescharacteristic
212C22×C4C7×C22⋊C411yescharacteristic
213C22×C4C22×C4×C711yescharacteristic
214C22×C4C2×C7×D411yescharacteristic
215C22×C4C2×C7×D411yescharacteristic
216M16—24no—
217M16—42no—
218C2×C8—42no—
219C2×C8—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C2×C7—21no—
236C2×C7—21no—
237C2×C7—162no—
238C2×C7#150711yescharacteristic
239C2×C7#212211yescharacteristic
240C2×C7C2×#35511yescharacteristic
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23C2×C7×C22⋊C411yescharacteristic
249Q8—44no—
250D4—84no—
251D4—82no—
252D4—84no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4C7×#9911yescharacteristic
272C2×C4C7×#9911yescharacteristic
273C2×C4C7×#10412yes—
274C2×C4C7×AGL(1,ℤ/8)12yes—
275C8—44no—
276C8—82no—
277C7#1178111yescharacteristic, direct product factor, Sylow
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×C7×#9911yescharacteristic
288C22C7×#41811yescharacteristic
289C22C2×C7×#10411yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4C7×#35512yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2C7×#150711yescharacteristic
303C2C7×#212211yescharacteristic
304C2C2×C7×#35511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#127C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C7×#101C2semidirect
3. C7×#1467C2semidirect
4. C7×#1345C2semidirect
20. #11781C7direct (N × H)
51. C4×#101C2×C7semidirect
52. #1467C2×C7semidirect
53. #1345C2×C7semidirect
277. C7#11781direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25482
C4×C7×#1012
C7×#14672
C7×#13452
#117817