Finite Group Databasebeta

← D7⋊(2−1+4⋊C3) #13062199

Subgroups — D7⋊(2−1+4⋊C3) #13062199

Order 1344.

408 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 62 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
12−1+4⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#10621541C211yesmaximal, characteristic, complemented
3D7⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
4#10621538C212yesmaximal, complemented
5C2×D7⋊SL(2,𝔽3)C212yesmaximal, complemented
6D7×2−1+4C311yesmaximal, characteristic, complemented, Hall
7#41⋊ΓL(1,𝔽8)C2211yescharacteristic, complemented
8C2×C7⋊SL(2,𝔽3)C2212yescomplemented
9D7⋊SL(2,𝔽3)C2211yescharacteristic
10D7⋊SL(2,𝔽3)C2212yescomplemented
11#66665C2211yescharacteristic, complemented
12D4×AGL(1,𝔽7)—41nomaximal
13C7×2−1+4C2×C311yescharacteristic, complemented
14#6554—32no—
15#6554—31no—
16#6554C2×C311yescharacteristic, complemented
17D7×#41—31no—
18D7×#41—32no—
19D7×#41C2×C311yescharacteristic, complemented
20#6551—31no—
21#6551C2×C312yescomplemented
22C2×Q8×D7—31no—
23C2×Q8×D7C2×C312yescomplemented
24C2×2−1+4⋊C3—71nomaximal, Hall
25C22×AGL(1,𝔽7)—42no—
26C7⋊SL(2,𝔽3)C2311yescharacteristic
27D4×ΓL(1,𝔽8)—41no—
28(C22⋊D7)⋊C3—42no—
29C4⋊AGL(1,𝔽7)—41no—
30C4×AGL(1,𝔽7)—41no—
31C7×#41—31no—
32C7×#41—32no—
33C7×#41C22×C311yescharacteristic, complemented
34C2×C7×Q8—31no—
35C2×C7×Q8C22×C312yescomplemented
36#1161—31no—
37#1161—32no—
38#1161C22×C311yescharacteristic, complemented
39Q8×D7—32no—
40Q8×D7—31no—
41Q8×D7—32no—
42Q8×D7—32no—
43Q8×D7—32no—
44Q8×D7C22×C311yescharacteristic
45Q8×D7C22×C312yescomplemented
46#1159—34no—
47#1159—31no—
48#1159—32no—
49#1159—32no—
50#1159—31no—
51D4×D7—32no—
52D4×D7—31no—
53D4×D7A411yescharacteristic
54#1157—34no—
55#1157—32no—
56#1157—32no—
57#1157—31no—
58#1157—31no—
59C2×C4×D7—32no—
60C2×C4×D7—32no—
61C2×C4×D7—31no—
62C2×Dic14—32no—
63C2×Dic14—32no—
64C2×Dic14—31no—
652−1+4⋊C3—72no—
662−1+4⋊C3—71no—
672−1+4⋊C3—71no—
68C2×#41⋊C3—71no—
69C22×SL(2,𝔽3)—72no—
70C22×ΓL(1,𝔽8)—42no—
71C2×AGL(1,𝔽7)—41no—
72C2×AGL(1,𝔽7)—42no—
73C2×AGL(1,𝔽7)—84no—
74C4×ΓL(1,𝔽8)—41no—
75ΓL(1,𝔽8)⋊C4—41no—
76C2×2−1+4—71noSylow
77C22×D7—31no—
78C22×D7C2×A412yes—
79C7×Q8—32no—
80C7×Q8—31no—
81C7×Q8C23×C311yescharacteristic
82C7×D4—31no—
83C7×D4—32no—
84C7×D4C2×A411yescharacteristic
85C2×C4×C7—32no—
86C2×C4×C7—32no—
87C2×C4×C7—31no—
88C22⋊D7—32no—
89C22⋊D7—32no—
90C22⋊D7—32no—
91C22⋊D7C2×A412yes—
92C2×Dic7—32no—
93C2×Dic7—32no—
94C2×Dic7—31no—
95D28—31no—
96D28—32no—
97D28C2×A411yescharacteristic
98C4×D7—34no—
99C4×D7—32no—
100C4×D7—31no—
101C4×D7—32no—
102C4×D7—32no—
103C4×D7—31no—
104C4×D7—31no—
105C4×D7C2×A411yescharacteristic
106Dic14—34no—
107Dic14—31no—
108Dic14—32no—
109Dic14—31no—
110Dic14—32no—
111C2×C3×D4—281no—
112#41⋊C3—71no—
113#41⋊C3—71no—
114C2×SL(2,𝔽3)—71no—
115C2×SL(2,𝔽3)—72no—
116C2×SL(2,𝔽3)—72no—
117C2×ΓL(1,𝔽8)—41no—
118C2×ΓL(1,𝔽8)—82no—
119AGL(1,𝔽7)—42no—
120AGL(1,𝔽7)—82no—
1212−1+4—71no—
1222−1+4—72no—
1232−1+4—211no—
1242−1+4—211no—
1252−1+4—212no—
1262−1+4AGL(1,𝔽7)11yescharacteristic, complemented
127C2×#41—71no—
128C2×#41—211no—
129C2×#41—212no—
130C22×Q8—72no—
131C22×Q8—211no—
132C22×C7—31no—
133C22×C7C22×A412yes—
134C2×D7—24no—
135C2×D7—31no—
136C2×D7—62no—
137C2×D7C22×A412yes—
138C2×D7C22×A411yescharacteristic
139C4×C7—32no—
140C4×C7—31no—
141C4×C7C22×A411yescharacteristic
142Dic7—31no—
143Dic7—32no—
144Dic7C22×A411yescharacteristic
145C23×C3—282no—
146C3×D4—281no—
147C3×D4—282no—
148C3×D4—281no—
149C2×C3×C4—281no—
150SL(2,𝔽3)—71no—
151ΓL(1,𝔽8)—41noHall
152#41—31no—
153#41—32no—
154#41—71no—
155#41—212no—
156#41—211no—
157#41—214no—
158#41—212no—
159#41—211no—
160#41—212no—
161#41—214no—
162#41—212no—
163#41—211no—
164#41—211no—
165#41—212no—
166#41—211no—
167#41C2×AGL(1,𝔽7)11yescharacteristic, complemented
168C2×Q8—31no—
169C2×Q8—71no—
170C2×Q8—72no—
171C2×Q8—211no—
172C2×Q8—212no—
173C2×Q8—212no—
174C2×Q8—211no—
175C2×Q8—212no—
176C2×Q8—212no—
177C2×Q8—212no—
178C2×Q8—212no—
179C2×Q8C2×AGL(1,𝔽7)12yescomplemented
180C2×D4—71no—
181C2×D4—212no—
182C2×D4—211no—
183C22×C4—211no—
184C22×C4—212no—
185C22×C4—212no—
186C2×C7—22no—
187C2×C7—61no—
188C2×C7C23×A411yescharacteristic
189D7—22no—
190D7—61no—
191D72−1+4⋊C312yescomplemented
192C22×C3—281no—
193C22×C3—282no—
194C22×C3—282no—
195C22×C3—564no—
196C3×C4—281no—
197C3×C4—281no—
198C23—72no—
199C23—211no—
200Q8—32no—
201Q8—31no—
202Q8—212no—
203Q8—211no—
204Q8—212no—
205Q8—214no—
206Q8—211no—
207Q8C22×AGL(1,𝔽7)11yescharacteristic
208D4—31no—
209D4—32no—
210D4—72no—
211D4—71no—
212D4—212no—
213D4—211no—
214D4—212no—
215D4—212no—
216D4—212no—
217D4D7⋊A411yescharacteristic
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—31no—
221C2×C4—71no—
222C2×C4—211no—
223C2×C4—211no—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—212no—
227C2×C4—214no—
228C2×C4—211no—
229C2×C4—212no—
230C2×C4—212no—
231C2×C4—211no—
232C7C2×2−1+4⋊C311yescharacteristic, complemented, Sylow
233C2×C3—282no—
234C2×C3—281no—
235C2×C3—562no—
236C2×C3—562no—
237C22—31no—
238C22—71no—
239C22—72no—
240C22—144no—
241C22—211no—
242C22—422no—
243C22C2×D7⋊A412yes—
244C4—31no—
245C4—32no—
246C4—71no—
247C4—211no—
248C4—212no—
249C4C2×D7⋊A411yescharacteristic
250C3—281noSylow
251C2—22no—
252C2—61no—
253C2—72no—
254C2—142no—
255C2—421no—
256C2C22×D7⋊A411yescharacteristic
subgroupG/H
center, FrattiniC2C22×D7⋊A4
commutator, cosocle, hypocenterC7×Q8C23×C3
FittingC7×2−1+4C2×C3
socleC2×C7C23×A4
hypercenterD4D7⋊A4
LCSG ▷ C7×Q8
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊ΓL(1,𝔽8)C2semidirect
2. #10621541C2semidirect
3. D7⋊(#41⋊C3)C2semidirect
4. #10621538C2semidirect
5. C2×D7⋊SL(2,𝔽3)C2semidirect
6. D7×2−1+4C3semidirect
7. #41⋊ΓL(1,𝔽8)C22semidirect
8. C2×C7⋊SL(2,𝔽3)C22semidirect
10. D7⋊SL(2,𝔽3)C22semidirect
11. #66665C22semidirect
13. C7×2−1+4C2×C3semidirect
16. #6554C2×C3semidirect
19. D7×#41C2×C3semidirect
21. #6551C2×C3semidirect
23. C2×Q8×D7C2×C3semidirect
33. C7×#41C22×C3semidirect
35. C2×C7×Q8C22×C3semidirect
38. #1161C22×C3semidirect
45. Q8×D7C22×C3semidirect
126. 2−1+4AGL(1,𝔽7)semidirect
167. #41C2×AGL(1,𝔽7)semidirect
179. C2×Q8C2×AGL(1,𝔽7)semidirect
191. D72−1+4⋊C3semidirect
232. C7C2×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊ΓL(1,𝔽8)2
#106215412
D7⋊(#41⋊C3)2
#106215382
C2×D7⋊SL(2,𝔽3)2
D7×2−1+43
D4×AGL(1,𝔽7)4
C2×2−1+4⋊C37