Finite Group Databasebeta

← C7×#63937 #72245833

Subgroups — C7×#63937 #72245833

Order 2016.

326 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 76 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×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C7×#3796C211yesmaximal, characteristic, complemented
3C3×C7×#214C211yesmaximal, characteristic, complemented
4C3×C7×#213C211yesmaximal, characteristic, complemented
5C7×#3717C211yesmaximal, characteristic, complemented
6C7×#3714C211yesmaximal, characteristic, complemented
7C7×#3712C211yesmaximal, characteristic, complemented
8C7×#914—31nomaximal
9C7×#911—31nomaximal
10C22×C7×C3⋊D3—21no—
11C32×C7×D4C2211yescharacteristic
12C7×#638—21no—
13C7×C4⋊(C3⋊D3)C2211yescharacteristic
14C4×C7×C3⋊D3C2211yescharacteristic
15C3×C7×D12C2211yescharacteristic, complemented
16C3×C7×Dic6C2211yescharacteristic, complemented
17C7×#626—21no—
18D3×C7×Dic3—21no—
19C3×C7×C3⋊C8C2211yescharacteristic, complemented
20C3×C7×C3⋊C8C2211yescharacteristic, complemented
21C7×#239—31no—
22C7×#237—31no—
23D3×C7×D4—31no—
24D3×C7×D4—31no—
25D3×C7×D4—61no—
26C3×C7×SD16—31no—
27C3×C7×D8—31no—
28C7×#215—31no—
29C7×#214D311yescharacteristic, complemented
30C7×#213—31no—
31C7×#213D311yescharacteristic, complemented
32C7×D24—31no—
33C7×#204—31no—
34C7×ΓL(1,𝔽25)—31no—
35C7×ΓL(1,𝔽25)—31no—
36#63937C711yesmaximal, characteristic, direct product factor, Hall
37C22×C32×C7—21no—
38C2×C7×C3⋊D3—21no—
39C2×C7×C3⋊D3—42no—
40C2×C7×C3⋊D3D411yescharacteristic
41C2×C3×D3×C7—21no—
42C32×C4×C7C2311yescharacteristic
43C7×C3⋊Dic3D411yescharacteristic
44C3×C7×Dic3—21no—
45C7×AGL(1,ℤ/8)—91noHall
46C22×D3×C7—61no—
47C22×D3×C7—61no—
48C22×D3×C7—61no—
49C22×D3×C7—61no—
50C3×C7×Q8—31no—
51C3×C7×D4—21no—
52C3×C7×D4—31no—
53C3×C7×D4C2×D311yescharacteristic
54C3×C7×D4C2×D311yescharacteristic
55C7×#67—61no—
56C7×#67—61no—
57C7×#67—61no—
58C7×#67—61no—
59C7×#67—61no—
60C2×C7×Dic3—61no—
61C7×D12—31no—
62C7×D12—31no—
63C7×D12—61no—
64C7×D12—61no—
65C7×D12C2×D311yescharacteristic, complemented
66C4×D3×C7—31no—
67C4×D3×C7—31no—
68C4×D3×C7—61no—
69C4×D3×C7—61no—
70C7×Dic6C2×D311yescharacteristic, complemented
71C3×C7×C8—31no—
72C3×C7×C8—31no—
73C7×C3⋊C8C2×D311yescharacteristic, complemented
74C7×C3⋊C8C2×D311yescharacteristic, complemented
75D4×C3⋊D3C2×C711yescharacteristic, complemented
76#3796C2×C711yescharacteristic, complemented
77C3×#214C2×C711yescharacteristic, complemented
78C3×#213C2×C711yescharacteristic, complemented
79#3717C2×C711yescharacteristic, complemented
80#3714C2×C711yescharacteristic, complemented
81#3712C2×C711yescharacteristic, complemented
82C2×C32×C7—41no—
83C2×C32×C7C2×D411yescharacteristic
84C7×C3⋊D3—21no—
85C7×C3⋊D3—41no—
86C3×D3×C7—41no—
87C7×#41—91no—
88C2×C7×D4—91no—
89C7×SD16—31no—
90C7×SD16—91no—
91C7×D8—31no—
92C7×D8—91no—
93C7×M16—91no—
94#914—31no—
95#911—31no—
96C22×C3×C7—21no—
97C22×C3×C7—21no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C22×C3×C7—61no—
101C2×D3×C7—21no—
102C2×D3×C7—31no—
103C2×D3×C7—31no—
104C2×D3×C7—61no—
105C2×D3×C7—61no—
106C2×D3×C7—61no—
107C2×D3×C7—61no—
108C2×D3×C7—61no—
109C2×D3×C7—122no—
110C2×D3×C7—122no—
111C2×D3×C7—122no—
112C2×D3×C7—122no—
113C3×C4×C7—21no—
114C3×C4×C7—61no—
115C3×C4×C7C22×D311yescharacteristic
116C3×C4×C7C22×D311yescharacteristic
117C7×Dic3—21no—
118C7×Dic3—31no—
119C7×Dic3—31no—
120C7×Dic3—61no—
121C22×C3⋊D3—21no—
122C32×D4C22×C711yescharacteristic
123#638—21no—
124C4⋊(C3⋊D3)C22×C711yescharacteristic
125C4×C3⋊D3C22×C711yescharacteristic
126C3×D12C22×C711yescharacteristic, complemented
127C3×Dic6C22×C711yescharacteristic, complemented
128#626—21no—
129D3×Dic3—21no—
130C3×C3⋊C8C22×C711yescharacteristic, complemented
131C3×C3⋊C8C22×C711yescharacteristic, complemented
132C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
133C23×C7—181no—
134C7×Q8—31no—
135C7×D4—31no—
136C7×D4—91no—
137C7×D4—181no—
138C7×D4—181no—
139C7×D4D3211yescharacteristic
140C2×C4×C7—91no—
141C2×C4×C7—181no—
142C7×C8—31no—
143C7×C8—31no—
144#239—31no—
145#237—31no—
146D3×D4—31no—
147D3×D4—31no—
148D3×D4—61no—
149C3×SD16—31no—
150C3×D8—31no—
151#215—31no—
152#214D3×C711yescharacteristic, complemented
153#213—31no—
154#213D3×C711yescharacteristic, complemented
155D24—31no—
156#204—31no—
157ΓL(1,𝔽25)—31no—
158ΓL(1,𝔽25)—31no—
159C2×C3×C7—21no—
160C2×C3×C7—41no—
161C2×C3×C7—41no—
162C2×C3×C7—41no—
163C2×C3×C7—41no—
164C2×C3×C7—121no—
165C2×C3×C7D3×D411yescharacteristic
166C2×C3×C7D3×D411yescharacteristic
167D3×C7—41no—
168D3×C7—61no—
169D3×C7—61no—
170D3×C7—61no—
171D3×C7—61no—
172D3×C7—121no—
173D3×C7—121no—
174D3×C7—121no—
175D3×C7—121no—
176C22×C32—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—42no—
179C2×C3⋊D3C7×D411yescharacteristic
180C2×C3×D3—21no—
181C32×C4C23×C711yescharacteristic
182C3⋊Dic3C7×D411yescharacteristic
183C3×Dic3—21no—
184AGL(1,ℤ/8)—91noSylow
185C22×C7—21no—
186C22×C7—61no—
187C22×C7—91no—
188C22×C7—181no—
189C22×C7—362no—
190C4×C7—61no—
191C4×C7—91no—
192C4×C7C2×D3211yescharacteristic
193C22×D3—61no—
194C22×D3—61no—
195C22×D3—61no—
196C22×D3—61no—
197C3×Q8—31no—
198C3×D4—21no—
199C3×D4—31no—
200C3×D4C2×D3×C711yescharacteristic
201C3×D4C2×D3×C711yescharacteristic
202#67—61no—
203#67—61no—
204#67—61no—
205#67—61no—
206#67—61no—
207C2×Dic3—61no—
208D12—31no—
209D12—31no—
210D12—61no—
211D12—61no—
212D12C2×D3×C711yescharacteristic, complemented
213C4×D3—31no—
214C4×D3—31no—
215C4×D3—61no—
216C4×D3—61no—
217Dic6C2×D3×C711yescharacteristic, complemented
218C3×C8—31no—
219C3×C8—31no—
220C3⋊C8C2×D3×C711yescharacteristic, complemented
221C3⋊C8C2×D3×C711yescharacteristic, complemented
222C3×C7—21no—
223C3×C7#91111yescharacteristic, complemented
224C3×C7#91411yescharacteristic, complemented
225C2×C32—41no—
226C2×C32C2×C7×D411yescharacteristic
227C3⋊D3—21no—
228C3⋊D3—41no—
229C3×D3—41no—
230#41—91no—
231C2×D4—91no—
232SD16—31no—
233SD16—91no—
234D8—31no—
235D8—91no—
236M16—91no—
237C2×C7—41no—
238C2×C7—121no—
239C2×C7—181no—
240C2×C7—361no—
241C2×C7#381111yescharacteristic
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—61no—
247C2×D3—21no—
248C2×D3—31no—
249C2×D3—31no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—61no—
253C2×D3—61no—
254C2×D3—61no—
255C2×D3—122no—
256C2×D3—122no—
257C2×D3—122no—
258C2×D3—122no—
259C3×C4—21no—
260C3×C4—61no—
261C3×C4C22×D3×C711yescharacteristic
262C3×C4C22×D3×C711yescharacteristic
263Dic3—21no—
264Dic3—31no—
265Dic3—31no—
266Dic3—61no—
267C32C7×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
268C23—181no—
269Q8—31no—
270D4—31no—
271D4—91no—
272D4—181no—
273D4—181no—
274D4D32×C711yescharacteristic
275C2×C4—91no—
276C2×C4—181no—
277C8—31no—
278C8—31no—
279C7#6393711yescharacteristic, direct product factor, Sylow
280C2×C3—21no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—121no—
286C2×C3D3×C7×D411yescharacteristic
287C2×C3D3×C7×D411yescharacteristic
288D3—41no—
289D3—61no—
290D3—61no—
291D3—61no—
292D3—61no—
293D3—121no—
294D3—121no—
295D3—121no—
296D3—121no—
297C22—21no—
298C22—61no—
299C22—91no—
300C22—181no—
301C22—362no—
302C4—61no—
303C4—91no—
304C4C2×D32×C711yescharacteristic
305C3—21no—
306C3C7×#91111yescharacteristic, complemented
307C3C7×#91411yescharacteristic, complemented
308C2—41no—
309C2—121no—
310C2—181no—
311C2—361no—
312C2C7×#381111yescharacteristic
subgroupG/H
centerC2×C7#3811
commutator, cosocleC32×C4C23×C7
FrattiniC4C2×D32×C7
FittingC32×C7×D4C22
socleC2×C32×C7C2×D4
hypercenterC7×D4D32
hypocenterC32C7×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×D4×C3⋊D3C2semidirect
2. C7×#3796C2semidirect
3. C3×C7×#214C2semidirect
4. C3×C7×#213C2semidirect
5. C7×#3717C2semidirect
6. C7×#3714C2semidirect
7. C7×#3712C2semidirect
15. C3×C7×D12C22semidirect
16. C3×C7×Dic6C22semidirect
19. C3×C7×C3⋊C8C22semidirect
20. C3×C7×C3⋊C8C22semidirect
29. C7×#214D3semidirect
31. C7×#213D3semidirect
36. #63937C7direct (N × H)
65. C7×D12C2×D3semidirect
70. C7×Dic6C2×D3semidirect
73. C7×C3⋊C8C2×D3semidirect
74. C7×C3⋊C8C2×D3semidirect
75. D4×C3⋊D3C2×C7semidirect
76. #3796C2×C7semidirect
77. C3×#214C2×C7semidirect
78. C3×#213C2×C7semidirect
79. #3717C2×C7semidirect
80. #3714C2×C7semidirect
81. #3712C2×C7semidirect
126. C3×D12C22×C7semidirect
127. C3×Dic6C22×C7semidirect
130. C3×C3⋊C8C22×C7semidirect
131. C3×C3⋊C8C22×C7semidirect
132. C32×C7AGL(1,ℤ/8)semidirect
152. #214D3×C7semidirect
154. #213D3×C7semidirect
212. D12C2×D3×C7semidirect
217. Dic6C2×D3×C7semidirect
220. C3⋊C8C2×D3×C7semidirect
221. C3⋊C8C2×D3×C7semidirect
223. C3×C7#911semidirect
224. C3×C7#914semidirect
267. C32C7×AGL(1,ℤ/8)semidirect
279. C7#63937direct (N × H)
306. C3C7×#911semidirect
307. C3C7×#914semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C3⋊D32
C7×#37962
C3×C7×#2142
C3×C7×#2132
C7×#37172
C7×#37142
C7×#37122
C7×#9143
C7×#9113
#639377