Finite Group Databasebeta

← #13062120

Subgroups — #13062120

Order 1344.

288 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C2×C7×#41⋊C3C211yesmaximal, characteristic, complemented
2#10621504C211yesmaximal, characteristic, complemented
3#10621503C212yesmaximal, complemented
4#10621502C211yesmaximal, characteristic, complemented
5#10621501C212yesmaximal, complemented
6#90513—31nomaximal, Hall
7#66728—41nomaximal
8C7×#41⋊C3C2211yescharacteristic
9C7×#41⋊C3C2211yescharacteristic, complemented
10C2×C7×SL(2,𝔽3)C2211yescharacteristic, complemented
11C7⋊GL(2,𝔽3)C2212yescomplemented
12C7⋊#226C2212yescomplemented
13C2×C7×#41D311yescharacteristic, complemented
14C2×#1157—31no—
15#6514—31no—
16#6514—31no—
17#6513—32no—
18#6513—32no—
19#6512—31no—
20#6512—31no—
21#6505—32no—
22#6495—32no—
23C2×#1136—31no—
24#5923—71nomaximal, Hall
25C2×C3×C4×C7—41no—
26#4268—42no—
27D84—41no—
28C4×D21—42no—
29Dic42—41no—
30C7×SL(2,𝔽3)C2311yescharacteristic
31C7×#41—31no—
32C7×#41—31no—
33C7×#41C2×D311yescharacteristic
34C7×#41C2×D311yescharacteristic
35C2×C7×Q8C2×D311yescharacteristic
36C2×C7×D4—31no—
37C22×C4×C7—31no—
38C2×C22⋊D7—61no—
39#1157—32no—
40#1157—32no—
41#1157—61no—
42#1157—61no—
43C2×D28—31no—
44C2×C4×D7—61no—
45C2×Dic14—31no—
46#1144—32no—
47#1144—32no—
48#1143—32no—
49#1143—32no—
50#1142—32no—
51#1142—32no—
52#1141—32no—
53#1141—32no—
54#1136—32no—
55#1136—31no—
56#1136—31no—
57C2×C7⋊C8—32no—
58C2×#41⋊C3D711yescharacteristic, complemented
59#986—71no—
60#985—72no—
61#984—71no—
62#983—72no—
63C22×C3×C7—41no—
64C2×D21—42no—
65C3×C4×C7—41no—
66C3×C4×C7—41no—
67Dic21—42no—
68#575—211noSylow
69C23×C7—31no—
70C22×D7—61no—
71C7×Q8—31no—
72C7×Q8C22×D311yescharacteristic
73C7×D4—31no—
74C7×D4—31no—
75C7×D4—61no—
76C2×C4×C7—31no—
77C2×C4×C7—31no—
78C2×C4×C7—31no—
79C2×C4×C7—31no—
80C2×C4×C7—31no—
81C2×C4×C7S411yescharacteristic
82C22⋊D7—62no—
83C22⋊D7—62no—
84C2×Dic7—61no—
85D28—32no—
86D28—61no—
87C4×D7—62no—
88C4×D7—62no—
89Dic14—32no—
90Dic14—61no—
91C7⋊C8—32no—
92C7⋊C8—32no—
93#235—281no—
94#41⋊C3C2×D711yescharacteristic
95#41⋊C3C2×D711yescharacteristic, complemented
96C2×SL(2,𝔽3)C2×D711yescharacteristic, complemented
97GL(2,𝔽3)—72no—
98#226—72no—
99C2×C3×C7—41no—
100C2×C3×C7—81no—
101D21—82no—
102C2×#41—211no—
103C2×#41D2111yescharacteristic, complemented
104#137—211no—
105#137—211no—
106#137—212no—
107AGL(1,ℤ/8)—212no—
108AGL(1,ℤ/8)—211no—
109AGL(1,ℤ/8)—211no—
110#135—212no—
111#135—212no—
112C2×M16—211no—
113C22×C7—31no—
114C22×C7—31no—
115C22×C7—121no—
116C22×C7C2×S411yescharacteristic
117C2×D7—62no—
118C2×D7—122no—
119C4×C7—31no—
120C4×C7—31no—
121C4×C7C2×S411yescharacteristic
122C4×C7C2×S411yescharacteristic
123Dic7—62no—
124C2×C3×C4—41no—
125#67—282no—
126D12—281no—
127C4×D3—282no—
128Dic6—281no—
129SL(2,𝔽3)C22×D711yescharacteristic
130C3×C7—41noHall
131#41—31no—
132#41—31no—
133#41—212no—
134#41—212no—
135#41—421no—
136#41—421no—
137#41C2×D2111yescharacteristic
138#41C2×D2111yescharacteristic
139C2×Q8—211no—
140C2×Q8C2×D2111yescharacteristic
141C2×D4—31no—
142C2×D4—211no—
143C2×D4—421no—
144C22×C4—31no—
145C22×C4—421no—
146Q16—212no—
147Q16—212no—
148SD16—212no—
149SD16—212no—
150SD16—212no—
151SD16—212no—
152D8—212no—
153D8—212no—
154M16—212no—
155M16—211no—
156M16—211no—
157C2×C8—212no—
158C2×C7—21no—
159C2×C7—61no—
160C2×C7—61no—
161C2×C7C22×S411yescharacteristic
162D7—122no—
163C22×C3—41no—
164C2×D3—282no—
165C3×C4—41no—
166C3×C4—41no—
167Dic3—282no—
168C23—31no—
169C23—421no—
170Q8—31no—
171Q8—212no—
172Q8—421no—
173Q8C22×D2111yescharacteristic
174D4—31no—
175D4—31no—
176D4—61no—
177D4—212no—
178D4—422no—
179D4—422no—
180D4—421no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—422no—
187C2×C4—421no—
188C2×C4—422no—
189C2×C4C7⋊S411yescharacteristic
190C8—212no—
191C8—212no—
192C7#592311yescharacteristic, complemented, Sylow
193C2×C3—41no—
194C2×C3—81no—
195D3—562no—
196C22—31no—
197C22—31no—
198C22—121no—
199C22—422no—
200C22—842no—
201C22C2×C7⋊S411yescharacteristic
202C4—31no—
203C4—31no—
204C4—422no—
205C4C2×C7⋊S411yescharacteristic
206C4C2×C7⋊S411yescharacteristic
207C3—41noSylow
208C2—21no—
209C2—61no—
210C2—61no—
211C2—842no—
212C2C22×C7⋊S411yescharacteristic
subgroupG/H
centerC4C2×C7⋊S4
commutator, cosocle, hypocenterC7×SL(2,𝔽3)C23
FrattiniC2C22×C7⋊S4
FittingC2×C7×#41D3
socleC2×C7C22×S4
hypercenterC2×C4C7⋊S4
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C7×#41⋊C3C2semidirect
2. #10621504C2semidirect
3. #10621503C2semidirect
4. #10621502C2semidirect
5. #10621501C2semidirect
9. C7×#41⋊C3C22semidirect
10. C2×C7×SL(2,𝔽3)C22semidirect
11. C7⋊GL(2,𝔽3)C22semidirect
12. C7⋊#226C22semidirect
13. C2×C7×#41D3semidirect
58. C2×#41⋊C3D7semidirect
95. #41⋊C3C2×D7semidirect
96. C2×SL(2,𝔽3)C2×D7semidirect
103. C2×#41D21semidirect
192. C7#5923semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#41⋊C32
#106215042
#106215032
#106215022
#106215012
#905133
#667284
#59237