Finite Group Databasebeta

← C7×#11782 #13188998

Subgroups — C7×#11782 #13188998

Order 1792.

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

normal subgroups only

This group has 414 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×#2546C211yesmaximal, characteristic
2C4×C7×#101C211yesmaximal, characteristic
3C7×#1467C211yesmaximal, characteristic
4C7×#1346C214yesmaximal
5C2×C7×C4⋊Q8—21no—
6C2×C7×C4⋊Q8C411yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#432—21no—
9C2×C7×#101—21no—
10C4×C7×M16—21no—
11C7×#398C412yes—
12C7×#398C2212yes—
13C7×#391—21no—
14C7×#389—21no—
15C7×#389—21no—
16C4×C7×C4⋊C4C411yescharacteristic
17C7×#343C2212yes—
18C7×#324—21no—
19C7×#324C2212yes—
20#11782C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8D411yescharacteristic
23C2×C7×M16—22no—
24C7×C4⋊Q8—41no—
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—44no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4C2311yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C7×C4⋊C8—41no—
42C7×#101—24no—
43C7×C22⋊C8—22no—
44C7×#97—41no—
45C7×#97—41no—
46C4×C7×C8—41no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95C2×C412yes—
50#2546C2×C711yescharacteristic
51C4×#101C2×C711yescharacteristic
52#1467C2×C711yescharacteristic
53#1346C2×C714yes—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—41no—
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C2×C7×Q8—42no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22×C411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C7×M16—24no—
70C7×M16—42no—
71C2×C7×C8—42no—
72C2×C7×C8—42no—
73C7×C4⋊C4—24no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—44no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—44no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—42no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8C4×C711yescharacteristic
91C2×C4×Q8C22×C711yescharacteristic
92#432—21no—
93C2×#101—21no—
94C4×M16—21no—
95#398C4×C712yes—
96#398C22×C712yes—
97#391—21no—
98#389—21no—
99#389—21no—
100C4×C4⋊C4C4×C711yescharacteristic
101#343C22×C712yes—
102#324—21no—
103#324C22×C712yes—
104C23×C7C2×C22⋊C411yescharacteristic
105C7×Q8—44no—
106C7×Q8—84no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—24no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7#9911yescharacteristic
128C2×C4×C7#9911yescharacteristic
129C2×C4×C7#10412yes—
130C2×C4×C7AGL(1,ℤ/8)12yes—
131C7×C8—44no—
132C7×C8—82no—
133C22×Q8—21no—
134C22×Q8C7×D411yescharacteristic
135C2×M16—22no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—44no—
139C4×Q8—41no—
140C4×Q8—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×C4×C712yes—
146C2×C4⋊C4C2×C4×C711yescharacteristic
147C2×C4⋊C4C7×D411yescharacteristic
148C2×C4⋊C4C23×C711yescharacteristic
149C2×C42—21no—
150C2×C42C2×C4×C711yescharacteristic
151C2×C42C7×D411yescharacteristic
152C2×C42C7×D411yescharacteristic
153C4⋊C8—41no—
154#101—24no—
155C22⋊C8—22no—
156#97—41no—
157#97—41no—
158C4×C8—41no—
159#95—22no—
160#95—22no—
161#95C2×C4×C712yes—
162C22×C7—21no—
163C22×C7—21no—
164C22×C7C2×#9911yescharacteristic
165C22×C7#41811yescharacteristic
166C22×C7C2×#10411yescharacteristic
167C4×C7—21no—
168C4×C7—22no—
169C4×C7—22no—
170C4×C7—44no—
171C4×C7—41no—
172C4×C7—41no—
173C4×C7—41no—
174C4×C7—41no—
175C4×C7—82no—
176C4×C7#35612yes—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4C7×C22⋊C411yescharacteristic
186C22×C4C7×C22⋊C411yescharacteristic
187C22×C4C7×C22⋊C411yescharacteristic
188C22×C4C7×C22⋊C411yescharacteristic
189C22×C4C22×C4×C711yescharacteristic
190C22×C4C2×C7×D411yescharacteristic
191C22×C4C2×C7×D411yescharacteristic
192M16—24no—
193M16—42no—
194C2×C8—42no—
195C2×C8—42no—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—42no—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#150711yescharacteristic
215C2×C7#212311yescharacteristic
216C2×C7C2×#35611yescharacteristic
217C23C2×C7×C22⋊C411yescharacteristic
218Q8—44no—
219Q8—84no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4C7×#9911yescharacteristic
241C2×C4C7×#9911yescharacteristic
242C2×C4C7×#10412yes—
243C2×C4C7×AGL(1,ℤ/8)12yes—
244C8—44no—
245C8—82no—
246C7#1178211yescharacteristic, direct product factor, Sylow
247C22—21no—
248C22—21no—
249C22C2×C7×#9911yescharacteristic
250C22C7×#41811yescharacteristic
251C22C2×C7×#10411yescharacteristic
252C4—22no—
253C4—22no—
254C4—21no—
255C4—44no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—82no—
261C4C7×#35612yes—
262C2—21no—
263C2—21no—
264C2C7×#150711yescharacteristic
265C2C7×#212311yescharacteristic
266C2C2×C7×#35611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7C2×C22⋊C4
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
20. #11782C7direct (N × H)
246. C7#11782direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25462
C4×C7×#1012
C7×#14672
C7×#13462
#117827