Finite Group Databasebeta

← #13062179

Subgroups — #13062179

Order 1344.

390 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C7×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2D7×#41⋊C3C211yesmaximal, characteristic, complemented
3#10621528C211yesmaximal, characteristic, complemented
4#10621527C212yesmaximal, complemented
5C2×#66662C212yesmaximal, complemented
6#90620C311yesmaximal, characteristic, complemented, Hall
7C3×#1159—41nomaximal
8C7×#41⋊C3C2211yescharacteristic, complemented
9C2×C7×SL(2,𝔽3)C2212yescomplemented
10D7×SL(2,𝔽3)C2211yescharacteristic, complemented
11#66662C2211yescharacteristic, complemented
12#66662C2212yescomplemented
13C7×2−1+4C2×C311yescharacteristic, complemented
14#6553—32no—
15#6553—31no—
16#6553C2×C311yescharacteristic, complemented
17D7×#41—31no—
18D7×#41—32no—
19D7×#41C2×C311yescharacteristic, complemented
20#6551—31no—
21#6551C2×C312yescomplemented
22C2×#1161—31no—
23C2×#1161C2×C312yescomplemented
24#5945—71nomaximal, Hall
25C3×C7×D4—41no—
26C3×C22⋊D7—42no—
27C2×C3×Dic7—42no—
28C3×C4×D7—41no—
29C3×Dic14—41no—
30C7×SL(2,𝔽3)C2311yescharacteristic
31C7×#41—31no—
32C7×#41—32no—
33C7×#41C22×C311yescharacteristic
34C2×C7×Q8—31no—
35C2×C7×Q8C22×C312yes—
36#1161—32no—
37#1161—32no—
38#1161—31no—
39#1161—32no—
40#1161—32no—
41#1161C22×C311yescharacteristic
42#1161C22×C312yes—
43Q8×D7—31no—
44Q8×D7—32no—
45Q8×D7C22×C311yescharacteristic
46#1159—32no—
47#1159—31no—
48#1159A411yescharacteristic
49D4×D7—34no—
50D4×D7—31no—
51D4×D7—31no—
52D4×D7—32no—
53D4×D7—32no—
54#1157—34no—
55#1157—32no—
56#1157—32no—
57#1157—31no—
58#1157—31no—
59C2×D28—32no—
60C2×D28—32no—
61C2×D28—31no—
62C2×C4×D7—32no—
63C2×C4×D7—32no—
64C2×C4×D7—31no—
652−1+4⋊C3—72no—
662−1+4⋊C3D711yescharacteristic, complemented
67#994—71no—
68C2×#41⋊C3—71no—
69C2×#41⋊C3—72no—
70C22×C3×C7—42no—
71C2×C3×D7—41no—
72C3×C4×C7—41no—
73C3×Dic7—42no—
74C3×Dic7—41no—
75#585—71noSylow
76C22×D7—32no—
77C22×D7—32no—
78C22×D7—31no—
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—31no—
93C2×Dic7C2×A412yes—
94D28—31no—
95D28—32no—
96D28—34no—
97D28—32no—
98D28—31no—
99C4×D7—34no—
100C4×D7—31no—
101C4×D7—31no—
102C4×D7—32no—
103C4×D7—31no—
104C4×D7—32no—
105C4×D7—32no—
106C4×D7C2×A411yescharacteristic
107Dic14—32no—
108Dic14—31no—
109Dic14C2×A411yescharacteristic
110C3×#41—281no—
111#41⋊C3—71no—
112#41⋊C3—72no—
113#41⋊C3C2×D711yescharacteristic, complemented
114C2×SL(2,𝔽3)—71no—
115C2×SL(2,𝔽3)C2×D712yescomplemented
116C2×C3×C7—41no—
117C2×C3×C7—82no—
118C3×D7—81no—
1192−1+4—72no—
1202−1+4—211no—
1212−1+4C3×D711yescharacteristic, complemented
122H(5,𝔽2)—71no—
123H(5,𝔽2)—211no—
124H(5,𝔽2)—212no—
125C2×#41—71no—
126C2×#41—72no—
127C2×#41—211no—
128C2×#41—211no—
129C2×#41—212no—
130C22×C7—31no—
131C22×C7C22×A412yes—
132C2×D7—32no—
133C2×D7—31no—
134C2×D7—121no—
135C2×D7—122no—
136C2×D7—122no—
137C2×D7C22×A411yescharacteristic
138C4×C7—32no—
139C4×C7—31no—
140C4×C7C22×A411yescharacteristic
141Dic7—31no—
142Dic7C22×A412yes—
143Dic7C22×A411yescharacteristic
144C3×Q8—281no—
145C3×D4—41no—
146C3×D4—282no—
147C2×C3×C4—281no—
148C2×C3×C4—282no—
149SL(2,𝔽3)C22×D711yescharacteristic
150C3×C7—41noHall
151#41—31no—
152#41—32no—
153#41—71no—
154#41—71no—
155#41—72no—
156#41—212no—
157#41—211no—
158#41—214no—
159#41—212no—
160#41—211no—
161#41—212no—
162#41—211no—
163#41—212no—
164#41—212no—
165#41—211no—
166#41—212no—
167#41—212no—
168#41C2×C3×D711yescharacteristic
169C2×Q8—31no—
170C2×Q8—71no—
171C2×Q8—211no—
172C2×Q8—212no—
173C2×Q8C2×C3×D712yes—
174C2×D4—212no—
175C2×D4—212no—
176C2×D4—211no—
177C2×D4—212no—
178C2×D4—211no—
179C2×D4—214no—
180C2×D4—212no—
181C2×D4—211no—
182C22×C4—212no—
183C22×C4—211no—
184C22×C4—212no—
185C2×C7—22no—
186C2×C7—61no—
187C2×C7C23×A411yescharacteristic
188D7—21no—
189D7—61no—
190D7—62no—
191C22×C3—42no—
192C22×C3—281no—
193C3×C4—41no—
194C3×C4—281no—
195C3×C4—282no—
196C23—211no—
197C23—212no—
198C23—212no—
199Q8—32no—
200Q8—31no—
201Q8—71no—
202Q8—212no—
203Q8—211no—
204Q8C22×C3×D711yescharacteristic
205D4—31no—
206D4—32no—
207D4—72no—
208D4—212no—
209D4—211no—
210D4—212no—
211D4—214no—
212D4—212no—
213D4—212no—
214D4—212no—
215D4—211no—
216D4A4×D711yescharacteristic
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—31no—
220C2×C4—72no—
221C2×C4—71no—
222C2×C4—211no—
223C2×C4—212no—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—211no—
227C2×C4—214no—
228C2×C4—211no—
229C2×C4—211no—
230C7#594511yescharacteristic, complemented, Sylow
231C2×C3—41no—
232C2×C3—82no—
233C2×C3—561no—
234C22—31no—
235C22—71no—
236C22—211no—
237C22—212no—
238C22—842no—
239C22—841no—
240C22—842no—
241C22C2×A4×D712yes—
242C4—31no—
243C4—32no—
244C4—71no—
245C4—72no—
246C4—211no—
247C4C2×A4×D711yescharacteristic
248C3—41noSylow
249C2—22no—
250C2—61no—
251C2—141no—
252C2—421no—
253C2—422no—
254C2C22×A4×D711yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×D7
commutator, cosocle, hypocenterC7×Q8C23×C3
FittingC7×2−1+4C2×C3
socleC2×C7C23×A4
hypercenterD4A4×D7
LCSG ▷ C7×Q8
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×2−1+4⋊C3C2semidirect
2. D7×#41⋊C3C2semidirect
3. #10621528C2semidirect
4. #10621527C2semidirect
5. C2×#66662C2semidirect
6. #90620C3semidirect
8. C7×#41⋊C3C22semidirect
9. C2×C7×SL(2,𝔽3)C22semidirect
10. D7×SL(2,𝔽3)C22semidirect
11. #66662C22semidirect
12. #66662C22semidirect
13. C7×2−1+4C2×C3semidirect
16. #6553C2×C3semidirect
19. D7×#41C2×C3semidirect
21. #6551C2×C3semidirect
23. C2×#1161C2×C3semidirect
66. 2−1+4⋊C3D7semidirect
113. #41⋊C3C2×D7semidirect
115. C2×SL(2,𝔽3)C2×D7semidirect
121. 2−1+4C3×D7semidirect
230. C7#5945semidirect

Maximal subgroups

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

subgroupindex
C7×2−1+4⋊C32
D7×#41⋊C32
#106215282
#106215272
C2×#666622
#906203
C3×#11594
#59457