Finite Group Databasebeta

← C7×#11780 #13188996

Subgroups — C7×#11780 #13188996

Order 1792.

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

normal subgroups only

This group has 464 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×#2486C211yesmaximal, characteristic
2C4×C7×#100C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1346C214yesmaximal, complemented
5C2×C7×#123—21no—
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C411yescharacteristic
8C2×C7×#100—21no—
9C7×#407—21no—
10C4×C7×M16—21no—
11C7×#398C2212yes—
12C7×#397C412yes—
13C7×#389—21no—
14C7×#381—21no—
15C7×#380—21no—
16C7×#380C411yescharacteristic
17C7×#343C2212yes—
18C7×#324C2212yes—
19C7×#323—21no—
20#11780C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—22no—
26C2×C7×M16—41no—
27C7×#123—44no—
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4C2311yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C7×#100—24no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—41no—
45C7×#97—41no—
46C4×C7×C8—41no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95C2×C412yes—
50#2486C2×C711yescharacteristic
51C4×#100C2×C711yescharacteristic
52#1465C2×C711yescharacteristic
53#1346C2×C714yescomplemented
54C24×C7C22⋊C411yescharacteristic
55C24×C7C22⋊C411yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—41no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
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—
77C7×M16—82no—
78C2×C7×C8—42no—
79C2×C7×C8—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—44no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—44no—
90C42×C7—21no—
91C42×C7—21no—
92C2×#123—21no—
93C2×C4×D4C22×C711yescharacteristic
94C22×C4⋊C4C4×C711yescharacteristic
95C2×#100—21no—
96#407—21no—
97C4×M16—21no—
98#398C22×C712yes—
99#397C4×C712yes—
100#389—21no—
101#381—21no—
102#380—21no—
103#380C4×C711yescharacteristic
104#343C22×C712yes—
105#324C22×C712yes—
106#323—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7#10412yes—
117C23×C7C2×C22⋊C411yescharacteristic
118C7×D4—44no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—81no—
138C2×C4×C7—84no—
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#9911yescharacteristic
141C2×C4×C7AGL(1,ℤ/8)12yes—
142C7×C8—44no—
143C7×C8—82no—
144C22×D4C7×D411yescharacteristic
145C23×C4—21no—
146C23×C4C2×C4×C711yescharacteristic
147C23×C4C7×D411yescharacteristic
148C2×M16—22no—
149C2×M16—41no—
150#123—44no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×C4×C712yes—
159C2×C4⋊C4C23×C711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4C2×C4×C711yescharacteristic
163C2×C22⋊C4C7×D411yescharacteristic
164C2×C42C7×D411yescharacteristic
165#100—24no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168#97—41no—
169C4×C8—41no—
170#95—22no—
171#95—22no—
172#95C2×C4×C712yes—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—81no—
185C22×C7C2×#9911yescharacteristic
186C22×C7#41811yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—21no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7—82no—
194C4×C7—82no—
195C4×C7#35212yes—
196C24C7×C22⋊C411yescharacteristic
197C24C7×C22⋊C411yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C7×C22⋊C411yescharacteristic
213C22×C4C7×C22⋊C411yescharacteristic
214C22×C4C22×C4×C711yescharacteristic
215C22×C4C2×C7×D411yescharacteristic
216C22×C4C2×C7×D411yescharacteristic
217M16—24no—
218M16—42no—
219M16—82no—
220C2×C8—42no—
221C2×C8—42no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C42—21no—
233C42—21no—
234C2×C7—21no—
235C2×C7—21no—
236C2×C7—42no—
237C2×C7—81no—
238C2×C7#150711yescharacteristic
239C2×C7C2×#35211yescharacteristic
240C2×C7#213511yescharacteristic
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23C7×#10412yes—
251C23C2×C7×C22⋊C411yescharacteristic
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—84no—
273C2×C4C7×#9911yescharacteristic
274C2×C4C7×#9911yescharacteristic
275C2×C4C7×AGL(1,ℤ/8)12yes—
276C8—44no—
277C8—82no—
278C7#1178011yescharacteristic, direct product factor, Sylow
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×C7×#9911yescharacteristic
292C22C7×#41811yescharacteristic
293C22C2×C7×#10411yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4C7×#35212yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2C7×#150711yescharacteristic
307C2C2×C7×#35211yescharacteristic
308C2C7×#213511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1346C2semidirect
20. #11780C7direct (N × H)
53. #1346C2×C7semidirect
278. C7#11780direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24862
C4×C7×#1002
C7×#14652
C7×#13462
#117807