Finite Group Databasebeta

← #13054411

Subgroups — #13054411

Order 1344.

304 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 72 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×C3×#1159C211yesmaximal, characteristic
2C2×C7×#214C211yesmaximal, characteristic, complemented
3#10620921C211yesmaximal, characteristic, complemented
4C2×#66569C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620506C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89945—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C22×C3×Dic7—21no—
13C3×#1159—22no—
14C3×#1159—22no—
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C4⋊Dic21C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C7×#214—22no—
20C2×C7×C3⋊C8C2211yescharacteristic
21#66577—21no—
22(C2×D7)⋊Dic3—21no—
23#66569—22no—
24C2×#1159D311yescharacteristic
25C2×C7×SD16—31no—
26#6509—31no—
27C2×#1144—31no—
28#6406—31no—
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#5239—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×C3×D7D411yescharacteristic
34C3×C7×D4—21no—
35C3×C7×D4D412yes—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×C7×Dic3—21no—
39C7×Dic6—21no—
40C7×Dic6—21no—
41C3×C22⋊D7—24no—
42C2×C3×Dic7—24no—
43C2×C3×Dic7—21no—
44C2×C3×Dic7D411yescharacteristic
45C3×C4×D7—21no—
46C3×C4×D7—21no—
47C3×Dic14—21no—
48C3×Dic14D412yes—
49C3⋊(C7⋊C8)—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8—31no—
52C2×C7×D4C2×D311yescharacteristic
53C2×C22⋊D7—21no—
54C22×Dic7—21no—
55#1159—22no—
56#1159—22no—
57C2×C4×D7C2×D311yescharacteristic
58C2×Dic14C2×D311yescharacteristic
59C7×SD16—62no—
60C2×C7×C8—31no—
61#1144—62no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—31no—
64Dic7⋊C4—61no—
65C2×C7⋊C8—31no—
66C2×C3×#41—71no—
67C2×#216—71no—
68C2×#214D711yescharacteristic, complemented
69#924—71no—
70#835—71no—
71#832—71no—
72#830—71no—
73C22×C3×C7—22no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7C2×D411yescharacteristic
77C2×C3×D7—21no—
78C2×C3×D7—21no—
79C2×C3×D7—21no—
80C3×C4×C7C2×D411yescharacteristic
81C3×C4×C7C2×D411yescharacteristic
82Dic21—41no—
83C3×Dic7—22no—
84C3×Dic7—21no—
85C7×Dic3—41no—
86#452—211noSylow
87C23×C7—21no—
88C22×D7#6711yescharacteristic
89C7×Q8—61no—
90C7×Q8—61no—
91C7×D4—21no—
92C7×D4#6712yes—
93C2×C4×C7—61no—
94C2×C4×C7C22×D311yescharacteristic
95C22⋊D7—24no—
96C2×Dic7—24no—
97C2×Dic7—21no—
98C2×Dic7—61no—
99C2×Dic7#6711yescharacteristic
100C4×D7—21no—
101C4×D7—21no—
102Dic14—21no—
103Dic14#6712yes—
104C7×C8—61no—
105C7⋊C8—61no—
106C3×#41—142no—
107C3×#41—142no—
108C2×C3×Q8—71no—
109C2×C3×D4—141no—
110C2×C3×D4C2×D711yescharacteristic
111C22×C3×C4—71no—
112C22×C3×C4—141no—
113C2×Dic6C2×D711yescharacteristic
114#217—141no—
115#216—142no—
116#214—22no—
117C4⋊Dic3—71no—
118Dic3⋊C4—141no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7—42no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125C3×D7—41no—
126C2×#41—71no—
127C2×Q16—211no—
128C2×SD16—31no—
129C22⋊Q8—211no—
130#103—211no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7—22no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7C2×#6711yescharacteristic
137C2×D7—21no—
138C2×D7—21no—
139C2×D7—21no—
140C4×C7—121no—
141C4×C7C2×#6711yescharacteristic
142C4×C7C2×#6711yescharacteristic
143Dic7—22no—
144Dic7—21no—
145Dic7—121no—
146C23×C3—21no—
147C23×C3—71no—
148C3×Q8—72no—
149C3×Q8—141no—
150C3×D4—21no—
151C3×D4—144no—
152C3×D4C22⋊D712yes—
153C2×C3×C4—71no—
154C2×C3×C4—144no—
155C2×C3×C4—141no—
156C2×C3×C4—141no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159C2×Dic3—21no—
160C2×Dic3—141no—
161Dic6—21no—
162Dic6—21no—
163C3⋊C8—21no—
164C3⋊C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—142no—
167#41—142no—
168C2×Q8—31no—
169C2×Q8—71no—
170C2×D4—141no—
171C2×D4D3×D711yescharacteristic
172C22×C4—71no—
173C22×C4—141no—
174Q16—422no—
175SD16—62no—
176C2×C8—31no—
177C2×C8—211no—
178C4⋊C4—211no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7—42no—
182C2×C7#95011yescharacteristic
183C2×C7#95111yescharacteristic
184C2×C7#95311yescharacteristic
185D7—41no—
186C22×C3—22no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—141no—
190C22×C3—141no—
191C22×C3—141no—
192C22×C3D4×D711yescharacteristic
193C3×C4—141no—
194C3×C4—142no—
195C3×C4D4×D711yescharacteristic
196C3×C4C2×C22⋊D711yescharacteristic
197Dic3—41no—
198Dic3—281no—
199C23—21no—
200C23—71no—
201Q8—61no—
202Q8—61no—
203Q8—72no—
204Q8—141no—
205D4—21no—
206D4—144no—
207D4#424512yes—
208C2×C4—61no—
209C2×C4—71no—
210C2×C4—144no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—421no—
218C7#523911yescharacteristic, complemented, Sylow
219C2×C3—42no—
220C2×C3—281no—
221C2×C3#647811yescharacteristic
222C2×C3#647911yescharacteristic
223C2×C3#650011yescharacteristic
224C22—22no—
225C22—41no—
226C22—41no—
227C22—141no—
228C22—141no—
229C22—141no—
230C22D7×#6711yescharacteristic
231C4—121no—
232C4—141no—
233C4—142no—
234C4—841no—
235C4C2×#424511yescharacteristic
236C4D7×#6711yescharacteristic
237C3#8994511yescharacteristic, complemented, Sylow
238C2—42no—
239C2—281no—
240C2#1062104611yescharacteristic
241C2#1062105411yescharacteristic
242C2#1062112511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1159C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C7×#214C2semidirect
3. #10620921C2semidirect
4. C2×#66569C2semidirect
5. #10620671C2semidirect
6. #10620506C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
68. C2×#214D7semidirect
165. C3×C7#452semidirect
218. C7#5239semidirect
237. C3#89945semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
C2×C7×#2142
#106209212
C2×#665692
#106206712
#106205062
(C2×D7)⋊(C3⋊C8)2
#899453
#52397