Finite Group Databasebeta

← C2×#10621538 #13062192

Subgroups — C2×#10621538 #13062192

Order 1344.

408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C22×C7⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10621538C214yesmaximal, direct product factor
3C2×D7⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#66665C211yesmaximal, characteristic, complemented
5C2×#6551C311yesmaximal, characteristic, complemented, Hall
6C2×C7⋊SL(2,𝔽3)C2211yescharacteristic
7C2×C7⋊SL(2,𝔽3)C2212yescomplemented
8D7⋊SL(2,𝔽3)C2212yescomplemented
9#66665C2212yescomplemented
10C2×(C22⋊D7)⋊C3—41nomaximal
11C22×C7×Q8C2×C311yescharacteristic, complemented
12#6551—34no—
13#6551C2×C314yescomplemented
14C2×#1161—31no—
15C2×#1161C2×C311yescharacteristic, complemented
16C2×Q8×D7—31no—
17C2×Q8×D7C2×C311yescharacteristic, complemented
18C2×#1157—31no—
19C2×#1157—31no—
20C2×2−1+4⋊C3—71nomaximal, Hall
21C23×ΓL(1,𝔽8)—41no—
22C22×AGL(1,𝔽7)—41no—
23C7⋊SL(2,𝔽3)C2311yescharacteristic
24(C22⋊D7)⋊C3—44no—
25C2×ΓL(1,𝔽8)⋊C4—41no—
26C2×C7×Q8—31no—
27C2×C7×Q8—32no—
28C2×C7×Q8C22×C311yescharacteristic
29C2×C7×Q8C22×C312yescomplemented
30C22×C4×C7—31no—
31C2×C22⋊D7—31no—
32C2×C22⋊D7A411yescharacteristic
33#1161—34no—
34#1161—34no—
35#1161—32no—
36#1161C22×C312yescomplemented
37Q8×D7—34no—
38Q8×D7—32no—
39Q8×D7—34no—
40Q8×D7C22×C312yescomplemented
41#1157—34no—
42#1157—34no—
43#1157—34no—
44#1157—34no—
45C2×D28—31no—
46C2×D28—31no—
47C2×C4×D7—32no—
48C2×C4×D7—31no—
49C2×C4×D7—31no—
50C2×Dic14—31no—
51C2×Dic14—31no—
522−1+4⋊C3—74no—
53C2×#41⋊C3—71no—
54C22×SL(2,𝔽3)—71no—
55C22×SL(2,𝔽3)—71no—
56C22×ΓL(1,𝔽8)—41no—
57C22×ΓL(1,𝔽8)—42no—
58C22×ΓL(1,𝔽8)—82no—
59C2×AGL(1,𝔽7)—42no—
60C2×AGL(1,𝔽7)—82no—
61ΓL(1,𝔽8)⋊C4—42no—
62C2×2−1+4—71noSylow
63C23×C7C2×A411yescharacteristic
64C22×D7—31no—
65C22×D7C2×A411yescharacteristic
66C7×Q8—31no—
67C7×Q8—32no—
68C7×Q8—32no—
69C7×Q8C23×C311yescharacteristic
70C2×C4×C7—31no—
71C2×C4×C7—31no—
72C2×C4×C7—32no—
73C2×C4×C7—32no—
74C22⋊D7—34no—
75C22⋊D7C2×A414yes—
76C2×Dic7—31no—
77C2×Dic7C2×A411yescharacteristic
78D28—32no—
79D28—32no—
80D28—34no—
81C4×D7—38no—
82C4×D7—32no—
83C4×D7—32no—
84C4×D7—32no—
85C4×D7—32no—
86Dic14—32no—
87Dic14—34no—
88Dic14—32no—
89C2×C3×D4—281no—
90#41⋊C3—72no—
91C2×SL(2,𝔽3)—72no—
92C2×SL(2,𝔽3)—71no—
93C2×SL(2,𝔽3)—72no—
94C2×ΓL(1,𝔽8)—42no—
95C2×ΓL(1,𝔽8)—41no—
96C2×ΓL(1,𝔽8)—82no—
97AGL(1,𝔽7)—82no—
982−1+4—74no—
992−1+4—214no—
100C2×#41—71no—
101C2×#41—211no—
102C2×#41—211no—
103C2×#41—211no—
104C22×Q8—71no—
105C22×Q8—211no—
106C22×Q8AGL(1,𝔽7)11yescharacteristic, complemented
107C22×C7—22no—
108C22×C7C22×A412yes—
109C22×C7C22×A411yescharacteristic
110C2×D7—22no—
111C2×D7—32no—
112C2×D7—62no—
113C2×D7C22×A412yes—
114C4×C7—31no—
115C4×C7—32no—
116C4×C7—31no—
117Dic7—32no—
118Dic7C22×A412yes—
119C23×C3—281no—
120C23×C3—281no—
121C3×D4—284no—
122C2×C3×C4—281no—
123SL(2,𝔽3)—71no—
124ΓL(1,𝔽8)—41noHall
125#41—72no—
126#41—214no—
127#41—214no—
128#41—214no—
129#41—214no—
130#41—214no—
131#41—214no—
132#41—212no—
133C2×Q8—31no—
134C2×Q8—32no—
135C2×Q8—72no—
136C2×Q8—211no—
137C2×Q8—211no—
138C2×Q8—214no—
139C2×Q8—214no—
140C2×Q8—212no—
141C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
142C2×Q8C2×AGL(1,𝔽7)12yescomplemented
143C2×D4—71no—
144C2×D4—211no—
145C2×D4—211no—
146C2×D4—211no—
147C22×C4—31no—
148C22×C4—211no—
149C22×C4—211no—
150C22×C4—212no—
151C2×C7—22no—
152C2×C72−1+4⋊C312yescomplemented
153C2×C7C23×A411yescharacteristic
154D7—22no—
155D7—62no—
156C22×C3—282no—
157C22×C3—282no—
158C22×C3—281no—
159C22×C3—562no—
160C22×C3—562no—
161C3×C4—282no—
162C23—71no—
163C23—211no—
164C23D7⋊A411yescharacteristic
165Q8—31no—
166Q8—32no—
167Q8—32no—
168Q8—214no—
169Q8—212no—
170Q8—212no—
171Q8C22×AGL(1,𝔽7)11yescharacteristic
172D4—74no—
173D4—214no—
174D4—212no—
175D4—212no—
176D4—214no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—71no—
182C2×C4—212no—
183C2×C4—212no—
184C2×C4—212no—
185C2×C4—212no—
186C2×C4—218no—
187C2×C4—211no—
188C7C2×2−1+4⋊C311yescharacteristic, complemented, Sylow
189C2×C3—282no—
190C2×C3—281no—
191C2×C3—562no—
192C2×C3—562no—
193C22—22no—
194C22—72no—
195C22—142no—
196C22—212no—
197C22—422no—
198C22C2×D7⋊A412yes—
199C22C2×D7⋊A411yescharacteristic
200C4—31no—
201C4—32no—
202C4—31no—
203C4—72no—
204C4—212no—
205C3—281noSylow
206C2—22no—
207C2—142no—
208C2—422no—
209C2#1062153812yesdirect product factor
210C2C22×D7⋊A411yescharacteristic
subgroupG/H
centerC22C2×D7⋊A4
commutator, cosocle, hypocenterC7×Q8C23×C3
FrattiniC2C22×D7⋊A4
FittingC22×C7×Q8C2×C3
socleC22×C7C22×A4
hypercenterC23D7⋊A4
LCSG ▷ C7×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C7⋊SL(2,𝔽3)C2semidirect
2. #10621538C2direct (N × H)
3. C2×D7⋊SL(2,𝔽3)C2semidirect
4. C2×#66665C2semidirect
5. C2×#6551C3semidirect
7. C2×C7⋊SL(2,𝔽3)C22semidirect
8. D7⋊SL(2,𝔽3)C22semidirect
9. #66665C22semidirect
11. C22×C7×Q8C2×C3semidirect
13. #6551C2×C3semidirect
15. C2×#1161C2×C3semidirect
17. C2×Q8×D7C2×C3semidirect
29. C2×C7×Q8C22×C3semidirect
36. #1161C22×C3semidirect
40. Q8×D7C22×C3semidirect
106. C22×Q8AGL(1,𝔽7)semidirect
142. C2×Q8C2×AGL(1,𝔽7)semidirect
152. C2×C72−1+4⋊C3semidirect
188. C7C2×2−1+4⋊C3semidirect
209. C2#10621538direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7⋊SL(2,𝔽3)2
#106215382
C2×D7⋊SL(2,𝔽3)2
C2×#666652
C2×#65513
C2×(C22⋊D7)⋊C34
C2×2−1+4⋊C37