Finite Group Databasebeta

← #13054426

Subgroups — #13054426

Order 1344.

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

normal subgroups only

This group has 320 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
1C3×#6548C211yesmaximal, characteristic
2#10621418C211yesmaximal, characteristic, complemented
3#10621099C212yesmaximal, complemented
4C7×#834C211yesmaximal, characteristic, complemented
5#10620530C212yesmaximal, complemented
6#89514—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C22×Dic21C2211yescharacteristic, complemented
9C2×C3×C22⋊D7—21no—
10C2×C3×C22⋊D7C2212yes—
11C3×#1159—22no—
12C3×D4×D7—22no—
13C3×#1157—21no—
14#66636—21no—
15#66636C2211yescharacteristic, complemented
16#66629—21no—
17C7×#217C2212yescomplemented
18Dic3⋊Dic7—22no—
19(C2×D7)⋊Dic3—22no—
20#6548D311yescharacteristic
21#6498—31no—
22#6449—32no—
23C7×#99—31no—
24#6380—32no—
25#5243—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×C3×D7—21no—
29C22×C3×D7D412yes—
30C3×C7×D4—22no—
31C2×C3×C4×C7D411yescharacteristic
32C2×Dic21—22no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×C7×Dic3—22no—
36C3×C22⋊D7—22no—
37C3×C22⋊D7—22no—
38C3×C22⋊D7—22no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7D412yes—
41C3×D28—21no—
42C3×C4×D7—22no—
43C3×Dic14—21no—
44C2×C7×D4C2×D311yescharacteristic
45C2×C22⋊D7—21no—
46C2×C22⋊D7C2×D312yes—
47C22×Dic7—31no—
48#1159—22no—
49D4×D7—22no—
50#1157—21no—
51C7×C22⋊C4—32no—
52C22⋊Dic7—31no—
53C22⋊Dic7—61no—
54(C2×D7)⋊C4—62no—
55C4⋊Dic7—62no—
56Dic7⋊C4—61no—
57C3×H(5,𝔽2)—71no—
58#935—71no—
59#935—72no—
60#834—72no—
61#834D711yescharacteristic, complemented
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D411yescharacteristic
66C22×C3×C7C2×D412yes—
67C2×C3×D7—22no—
68C2×C3×D7—42no—
69C2×C3×D7—81no—
70C3×C4×C7—21no—
71Dic21—41no—
72Dic21—41no—
73C3×Dic7—22no—
74C7×Dic3—42no—
75#458—211noSylow
76C23×C7#6711yescharacteristic
77C23×C7C22×D311yescharacteristic
78C22×D7—21no—
79C22×D7#6712yes—
80C7×D4—22no—
81C2×C4×C7—62no—
82C2×C4×C7#6711yescharacteristic
83C22⋊D7—22no—
84C22⋊D7—22no—
85C22⋊D7—22no—
86C2×Dic7—21no—
87C2×Dic7—61no—
88C2×Dic7—61no—
89C2×Dic7—62no—
90C2×Dic7#6712yes—
91D28—21no—
92C4×D7—22no—
93Dic14—21no—
94C3×#41—142no—
95C3×#41—141no—
96C2×C3×D4—72no—
97C2×C3×D4—142no—
98C2×C3×D4—141no—
99C2×C3×D4C2×D711yescharacteristic
100C22×Dic3—71no—
101#217—71no—
102#217—142no—
103#217—141no—
104#217C2×D712yescomplemented
105Dic3⋊C4—142no—
106Dic3⋊C4—141no—
107C2×C3×C7—22no—
108C2×C3×C7—21no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111C3×D7—42no—
112H(5,𝔽2)—71no—
113#123—212no—
114#123—211no—
115#99—31no—
116#99—212no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7C2×#6711yescharacteristic
121C22×C7C2×#6712yes—
122C2×D7—22no—
123C2×D7—42no—
124C2×D7—81no—
125C4×C7—21no—
126C4×C7—122no—
127Dic7—22no—
128Dic7—121no—
129Dic7—121no—
130C23×C3—72no—
131C23×C3—141no—
132C23×C3D2811yescharacteristic
133C23×C3C22×D711yescharacteristic
134C3×Q8—141no—
135C3×D4—22no—
136C3×D4—141no—
137C3×D4—142no—
138C3×D4—142no—
139C3×D4—142no—
140C2×C3×C4—72no—
141C2×C3×C4—142no—
142C2×C3×C4—141no—
143C2×C3×C4D2811yescharacteristic
144C2×Dic3—22no—
145C2×Dic3—141no—
146C2×Dic3—141no—
147C2×Dic3—142no—
148C3×C7#45811yescharacteristic, complemented, Hall
149#41—142no—
150#41—141no—
151C2×D4—72no—
152C2×D4—142no—
153C2×D4—141no—
154C2×D4D3×D711yescharacteristic
155C22×C4—211no—
156C4⋊C4—422no—
157C4⋊C4—421no—
158C22⋊C4—32no—
159C22⋊C4—211no—
160C22⋊C4—422no—
161C22⋊C4—421no—
162C2×C7—21no—
163C2×C7—22no—
164C2×C7—41no—
165C2×C7#95311yescharacteristic
166D7—42no—
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—142no—
171C22×C3—282no—
172C22×C3—561no—
173C22×C3C2×D2811yescharacteristic
174C22×C3D4×D712yes—
175C3×C4—21no—
176C3×C4—142no—
177Dic3—42no—
178Dic3—281no—
179Dic3—281no—
180C23—72no—
181C23—141no—
182C23#424611yescharacteristic
183C23C2×D3×D711yescharacteristic
184Q8—141no—
185D4—22no—
186D4—141no—
187D4—142no—
188D4—142no—
189D4—142no—
190C2×C4—62no—
191C2×C4—72no—
192C2×C4—142no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4—421no—
196C2×C4—422no—
197C2×C4#424611yescharacteristic
198C7#524311yescharacteristic, complemented, Sylow
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—41no—
202C2×C3—282no—
203C2×C3#644511yescharacteristic
204C22—21no—
205C22—41no—
206C22—41no—
207C22—142no—
208C22—282no—
209C22—561no—
210C22C2×#424611yescharacteristic
211C22D7×#6712yes—
212C4—21no—
213C4—122no—
214C4—142no—
215C4—841no—
216C4—841no—
217C3#8951411yescharacteristic, complemented, Sylow
218C2—22no—
219C2—21no—
220C2—41no—
221C2—282no—
222C2#1062112611yescharacteristic
subgroupG/H
centerC2#10621126
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radicalC3×#6548C2
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. #10621418C2semidirect
3. #10621099C2semidirect
4. C7×#834C2semidirect
5. #10620530C2semidirect
8. C22×Dic21C22semidirect
15. #66636C22semidirect
17. C7×#217C22semidirect
61. #834D7semidirect
104. #217C2×D7semidirect
148. C3×C7#458semidirect
198. C7#5243semidirect
217. C3#89514semidirect

Maximal subgroups

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

subgroupindex
C3×#65482
#106214182
#106210992
C7×#8342
#106205302
#895143
#52437