Finite Group Databasebeta

← #13054214

Subgroups — #13054214

Order 1344.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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#10621407C211yesmaximal, characteristic, complemented
2C2×C7×#208C211yesmaximal, characteristic, complemented
3C3×#6493C211yesmaximal, characteristic
4C2×#66565C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8#89906—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C2×C3×D28C2211yescharacteristic
13#66636—21no—
14C4⋊Dic21C2211yescharacteristic
15#66629—21no—
16C7×#208—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×(C2×D7)⋊C4—21no—
20C3×C4⋊Dic7C2211yescharacteristic
21#66565—22no—
22C2×C7×M16—31no—
23#6493D311yescharacteristic
24#6487—31no—
25C2×#1132—31no—
26#6395—31no—
27#6392—31no—
28#6390—31no—
29#5042—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C22×C3×D7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7D411yescharacteristic
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21—21no—
38Dic42—21no—
39Dic42—21no—
40C3×C22⋊D7—42no—
41C2×C3×Dic7—21no—
42C3×D28—21no—
43C3×D28—21no—
44C7×C3⋊C8—21no—
45C7×C3⋊C8D412yescomplemented
46C22×C4×C7C2×D311yescharacteristic
47C2×C22⋊D7—21no—
48C2×D28C2×D311yescharacteristic
49C2×Dic14—31no—
50C7×M16—62no—
51C2×C7×C8—31no—
52C2×C7×C8—31no—
53C22⋊Dic7—61no—
54(C2×D7)⋊C4—21no—
55C4⋊Dic7—31no—
56C4⋊Dic7C2×D311yescharacteristic
57Dic7⋊C4—61no—
58#1132—62no—
59C3×#121—71no—
60C2×#214—71no—
61#924—71no—
62C2×#208D711yescharacteristic, complemented
63#832—71no—
64#810—71no—
65#807—71no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—41no—
71C2×C3×D7—41no—
72C2×C3×D7—41no—
73C3×C4×C7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—41no—
77Dic21—41no—
78C3×Dic7—41no—
79#468—211noSylow
80C23×C7#6711yescharacteristic
81C22×D7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7#6711yescharacteristic
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—42no—
87C2×Dic7—21no—
88C2×Dic7—61no—
89C2×Dic7—61no—
90D28—21no—
91D28—21no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—32no—
95C7×C8—61no—
96C2×C3×D4—71no—
97C2×C3×D4—141no—
98C22×C3×C4C2×D711yescharacteristic
99C2×Dic6—71no—
100C3×C4⋊C4—71no—
101C3×C22⋊C4—141no—
102#217—141no—
103#214—142no—
104C4⋊Dic3—71no—
105Dic3⋊C4—141no—
106#208—22no—
107C2×C3⋊C8C2×D711yescharacteristic
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7—41no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113C3×D7—81no—
114C2×SD16—211no—
115C2×M16—31no—
116C22⋊Q8—211no—
117#121—71no—
118#107—211no—
119#103—211no—
120#102—211no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7—21no—
124C22×C7C2×#6711yescharacteristic
125C2×D7—41no—
126C2×D7—41no—
127C2×D7—41no—
128C4×C7—21no—
129C4×C7D3×D411yescharacteristic
130C4×C7#23711yescharacteristic
131Dic7—41no—
132Dic7—121no—
133Dic7—121no—
134C23×C3—141no—
135C23×C3D2811yescharacteristic
136C3×D4—141no—
137C3×D4—141no—
138C3×D4—282no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—141no—
142C2×C3×C4D2811yescharacteristic
143C2×C3×C4C22×D711yescharacteristic
144C2×Dic3—141no—
145C2×Dic3—141no—
146Dic6—141no—
147Dic6—141no—
148C3⋊C8—21no—
149C3⋊C8C22⋊D712yescomplemented
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—71no—
153C2×D4—141no—
154C22×C4D3×D711yescharacteristic
155SD16—422no—
156M16—62no—
157C2×C8—31no—
158C2×C8—31no—
159C4⋊C4—71no—
160C4⋊C4—211no—
161C4⋊C4—421no—
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7—41no—
165C2×C7#93211yescharacteristic
166C2×C7#93911yescharacteristic
167C2×C7#95111yescharacteristic
168D7—81no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—281no—
173C22×C3—281no—
174C22×C3—281no—
175C22×C3C2×D2811yescharacteristic
176C3×C4—21no—
177C3×C4—281no—
178C3×C4#115711yescharacteristic
179C3×C4C2×C22⋊D711yescharacteristic
180Dic3—281no—
181Dic3—281no—
182C23—141no—
183C23#424611yescharacteristic
184Q8—421no—
185Q8—421no—
186D4—141no—
187D4—141no—
188D4—282no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—141no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4#424611yescharacteristic
195C2×C4C2×D3×D711yescharacteristic
196C8—32no—
197C8—61no—
198C7#504211yescharacteristic, complemented, Sylow
199C2×C3—41no—
200C2×C3—561no—
201C2×C3#647111yescharacteristic
202C2×C3#647211yescharacteristic
203C2×C3#649311yescharacteristic
204C22—21no—
205C22—21no—
206C22—21no—
207C22—281no—
208C22—281no—
209C22—281no—
210C22C2×#424611yescharacteristic
211C4—21no—
212C4—281no—
213C4—841no—
214C4—841no—
215C4#6668311yescharacteristic
216C4D3×C22⋊D711yescharacteristic
217C3#8990611yescharacteristic, complemented, Sylow
218C2—41no—
219C2—561no—
220C2#1062087411yescharacteristic
221C2#1062087511yescharacteristic
222C2#1062113111yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC3×#6493C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#468
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #10621407C2semidirect
2. C2×C7×#208C2semidirect
4. C2×#66565C2semidirect
5. #10620525C2semidirect
6. #10620509C2semidirect
7. #10620500C2semidirect
45. C7×C3⋊C8D4semidirect
62. C2×#208D7semidirect
149. C3⋊C8C22⋊D7semidirect
150. C3×C7#468semidirect
198. C7#5042semidirect
217. C3#89906semidirect

Maximal subgroups

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

subgroupindex
#106214072
C2×C7×#2082
C3×#64932
C2×#665652
#106205252
#106205092
#106205002
#899063
#50427