Finite Group Databasebeta

← #13054430

Subgroups — #13054430

Order 1344.

320 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#10621657C211yesmaximal, characteristic
2C7×#935C211yesmaximal, characteristic, complemented
3#10621104C212yesmaximal, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5#10620532C212yesmaximal, complemented
6#89825—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268—21no—
9C2×#4268C2212yes—
10#66730—22no—
11D4×D21—22no—
12#66728—21no—
13C22×C7×Dic3C2211yescharacteristic, complemented
14C7×#217—21no—
15C7×#217C2211yescharacteristic, complemented
16C7×Dic3⋊C4—21no—
17C3×C22⋊Dic7C2212yescomplemented
18Dic7⋊Dic3—22no—
19#66575—22no—
20#6548—31no—
21C7×#123—31no—
22#6492—32no—
23#6408D311yescharacteristic, complemented
24#6380—32no—
25#4742—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×D21—21no—
29C22×D21D412yes—
30C3×C7×D4—22no—
31C2×C3×C4×C7D411yescharacteristic
32#4268—22no—
33#4268—22no—
34#4268—22no—
35C2×Dic21—21no—
36C2×Dic21D412yes—
37D84—21no—
38C4×D21—22no—
39Dic42—21no—
40C2×C7×Dic3—22no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3—21no—
43C2×C3×Dic7—22no—
44C2×C7×D4C2×D311yescharacteristic
45C22×C4×C7—31no—
46C2×C22⋊D7—32no—
47C2×C22⋊D7—61no—
48#1159—62no—
49D4×D7—62no—
50#1157—61no—
51C7×C4⋊C4—61no—
52C7×C22⋊C4—31no—
53C7×C22⋊C4—61no—
54C22⋊Dic7C2×D312yescomplemented
55(C2×D7)⋊C4—62no—
56Dic7⋊C4—62no—
57#1004—71no—
58#935D711yescharacteristic, complemented
59#886—72no—
60C3×#99—71no—
61#806—72no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D412yes—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—22no—
68C2×D21—42no—
69C2×D21—81no—
70C3×C4×C7—21no—
71Dic21—22no—
72C3×Dic7—42no—
73C7×Dic3—41no—
74C7×Dic3—41no—
75#458—211noSylow
76C23×C7D1211yescharacteristic
77C23×C7C22×D311yescharacteristic
78C22×D7—32no—
79C22×D7—61no—
80C7×D4—22no—
81C2×C4×C7—61no—
82C2×C4×C7—61no—
83C2×C4×C7—62no—
84C2×C4×C7D1211yescharacteristic
85C22⋊D7—62no—
86C22⋊D7—62no—
87C22⋊D7—62no—
88C2×Dic7—22no—
89C2×Dic7—32no—
90C2×Dic7—61no—
91D28—61no—
92C4×D7—62no—
93Dic14—61no—
94C2×C3×D4C2×D711yescharacteristic
95C2×#67—72no—
96C2×#67—141no—
97C22×Dic3C2×D711yescharacteristic, complemented
98#237—142no—
99D3×D4—142no—
100#235—141no—
101C3×C22⋊C4—72no—
102#217—21no—
103#217C2×D711yescharacteristic, complemented
104(C2×D3)⋊C4—142no—
105C4⋊Dic3—142no—
106Dic3⋊C4—21no—
107C2×C3×C7—21no—
108C2×C3×C7—22no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111D21—42no—
112H(5,𝔽2)—211no—
113#123—31no—
114#123—212no—
115#99—71no—
116#99—212no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7C2×D1211yescharacteristic
121C22×C7D3×D412yes—
122C2×D7—62no—
123C2×D7—122no—
124C2×D7—241no—
125C4×C7—21no—
126C4×C7—121no—
127C4×C7—121no—
128Dic7—42no—
129Dic7—62no—
130C23×C3C22⋊D711yescharacteristic
131C23×C3C22×D711yescharacteristic
132C22×D3—72no—
133C22×D3—141no—
134C3×D4—22no—
135C2×C3×C4—142no—
136C2×C3×C4C22⋊D711yescharacteristic
137#67—142no—
138#67—142no—
139#67—142no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—72no—
144C2×Dic3—141no—
145D12—141no—
146C4×D3—142no—
147Dic6—141no—
148C3×C7#45811yescharacteristic, complemented, Hall
149#41—422no—
150#41—421no—
151C2×D4—212no—
152C2×D4—422no—
153C2×D4—421no—
154C2×D4D3×D711yescharacteristic
155C22×C4—31no—
156C4⋊C4—61no—
157C4⋊C4—422no—
158C22⋊C4—31no—
159C22⋊C4—61no—
160C22⋊C4—72no—
161C22⋊C4—422no—
162C2×C7—22no—
163C2×C7—21no—
164C2×C7—41no—
165C2×C7#88211yescharacteristic
166D7—122no—
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3D4×D712yes—
171C22×C3C2×C22⋊D711yescharacteristic
172C2×D3—142no—
173C2×D3—282no—
174C2×D3—561no—
175C3×C4—21no—
176C3×C4—282no—
177Dic3—41no—
178Dic3—41no—
179Dic3—142no—
180C23—212no—
181C23—421no—
182C23#424711yescharacteristic
183C23C2×D3×D711yescharacteristic
184Q8—421no—
185D4—22no—
186D4—421no—
187D4—422no—
188D4—422no—
189D4—422no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—62no—
193C2×C4—142no—
194C2×C4—212no—
195C2×C4—422no—
196C2×C4—421no—
197C2×C4#424711yescharacteristic
198C7#474211yescharacteristic, complemented, Sylow
199C2×C3—21no—
200C2×C3—22no—
201C2×C3—41no—
202C2×C3#650011yescharacteristic
203D3—282no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—422no—
208C22—842no—
209C22—1681no—
210C22C2×#424711yescharacteristic
211C22#6669412yes—
212C4—21no—
213C4—121no—
214C4—121no—
215C4—282no—
216C4—422no—
217C3#8982511yescharacteristic, complemented, Sylow
218C2—21no—
219C2—22no—
220C2—41no—
221C2—842no—
222C2#1062113511yescharacteristic
subgroupG/H
centerC2#10621135
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radical#10621657C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#458
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C7×#935C2semidirect
3. #10621104C2semidirect
4. C3×#6408C2semidirect
5. #10620532C2semidirect
13. C22×C7×Dic3C22semidirect
15. C7×#217C22semidirect
17. C3×C22⋊Dic7C22semidirect
23. #6408D3semidirect
54. C22⋊Dic7C2×D3semidirect
58. #935D7semidirect
97. C22×Dic3C2×D7semidirect
103. #217C2×D7semidirect
148. C3×C7#458semidirect
198. C7#4742semidirect
217. C3#89825semidirect

Maximal subgroups

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

subgroupindex
#106216572
C7×#9352
#106211042
C3×#64082
#106205322
#898253
#47427