Finite Group Databasebeta

← #13054475

Subgroups — #13054475

Order 1344.

316 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 96 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
2#10620920C211yesmaximal, characteristic, complemented
3C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
4#10620671C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620506C211yesmaximal, characteristic, complemented
7#10620505C211yesmaximal, characteristic, complemented
8#89544—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C22×C3×Dic7—21no—
12C3×#1159—22no—
13C3×#1159C414yescomplemented
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17C2×C3⋊(C7⋊C8)C2211yescharacteristic
18C7×C4⋊Dic3C2211yescharacteristic
19C2×C7×C3⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21Dic3×Dic7—21no—
22D7×C3⋊C8—22no—
23C2×#1159D311yescharacteristic
24C2×C8×D7—31no—
25#6455—31no—
26C7×#102—31no—
27#6406—31no—
28#6390—31no—
29#6384—31no—
30#5233—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4C2×C412yes—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×C7×Dic3—21no—
38C3×C22⋊D7—24no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7—24no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7C2×C412yes—
43C3×C4×D7D412yes—
44C3×Dic14—21no—
45C3×Dic14C2×C412yes—
46C3⋊(C7⋊C8)—21no—
47C7×C3⋊C8—21no—
48C2×C7×D4C2×D311yescharacteristic
49C2×C22⋊D7—21no—
50C22×Dic7—21no—
51#1159—22no—
52#1159Dic314yescomplemented
53C2×C4×D7C2×D311yescharacteristic
54C2×Dic14C2×D311yescharacteristic
55C2×C7×C8—31no—
56C7×C4⋊C4—31no—
57(C2×D7)⋊C4—61no—
58C4⋊Dic7—31no—
59C4×Dic7—61no—
60C2×C7⋊C8—31no—
61C8×D7—62no—
62C2×C3×#41—71no—
63#926—71no—
64C22×C3⋊C8—71no—
65#835—71no—
66#835—71no—
67#832—71no—
68#832D711yescharacteristic, complemented
69C22×C3×C7—22no—
70C22×C3×C7—41no—
71C22×C3×C7—41no—
72C22×C3×C7C2×D411yescharacteristic
73C2×C3×D7—21no—
74C2×C3×D7—21no—
75C2×C3×D7C22⋊C412yes—
76C3×C4×C7C22×C411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78Dic21—41no—
79C3×Dic7—22no—
80C3×Dic7C22⋊C412yes—
81C7×Dic3—41no—
82#416—211noSylow
83C23×C7—21no—
84C22×D7#6711yescharacteristic
85C7×D4—21no—
86C7×D4C2×Dic312yes—
87C2×C4×C7—61no—
88C2×C4×C7C22×D311yescharacteristic
89C22⋊D7—24no—
90C2×Dic7—21no—
91C2×Dic7—24no—
92C2×Dic7—61no—
93C2×Dic7#6711yescharacteristic
94C4×D7C2×Dic312yes—
95C4×D7#6712yes—
96Dic14—21no—
97Dic14C2×Dic312yes—
98C7×C8—61no—
99C7⋊C8—61no—
100C3×#41—74no—
101C3×#41—142no—
102C2×C3×Q8—71no—
103C2×C3×D4—141no—
104C2×C3×D4C2×D711yescharacteristic
105C22×C3×C4—71no—
106C22×C3×C4—141no—
107#217—141no—
108C4⋊Dic3—71no—
109C4⋊Dic3C2×D711yescharacteristic
110C4×Dic3—141no—
111C2×C3⋊C8—71no—
112C2×C3⋊C8—142no—
113C2×C3⋊C8C2×D711yescharacteristic
114C2×C3×C7—42no—
115C2×C3×C7C2×C22⋊C411yescharacteristic
116C2×C3×C7#13511yescharacteristic
117C2×C3×C7#13511yescharacteristic
118C3×D7—22no—
119C2×#41—71no—
120C22×C8—211no—
121#117—211no—
122#103—211no—
123#103—211no—
124#102—31no—
125#102—211no—
126C22×C7—22no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7C2×#6711yescharacteristic
130C2×D7—21no—
131C2×D7—21no—
132C2×D7#21712yes—
133C4×C7—121no—
134C4×C7C22×Dic311yescharacteristic
135C4×C7C2×#6711yescharacteristic
136Dic7—22no—
137Dic7—121no—
138Dic7#21712yes—
139C23×C3—21no—
140C23×C3—71no—
141C3×Q8—72no—
142C3×Q8—141no—
143C3×D4—21no—
144C3×D4—144no—
145C3×D4C4×D712yes—
146C2×C3×C4—71no—
147C2×C3×C4—72no—
148C2×C3×C4—72no—
149C2×C3×C4—144no—
150C2×C3×C4—141no—
151C2×C3×C4C22×D711yescharacteristic
152C2×Dic3—21no—
153C2×Dic3—141no—
154C3⋊C8—21no—
155C3⋊C8—141no—
156C3×C7#41611yescharacteristic, complemented, Hall
157#41—74no—
158#41—142no—
159C2×Q8—71no—
160C2×D4—141no—
161C2×D4D3×D711yescharacteristic
162C22×C4—71no—
163C22×C4—141no—
164C2×C8—31no—
165C2×C8—211no—
166C2×C8—422no—
167C4⋊C4—31no—
168C4⋊C4—211no—
169C22⋊C4—421no—
170C42—421no—
171C2×C7—42no—
172C2×C7#95011yescharacteristic
173C2×C7#95011yescharacteristic
174C2×C7C2×#21711yescharacteristic
175D7—22no—
176C22×C3—22no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—72no—
180C22×C3—141no—
181C22×C3—141no—
182C22×C3D4×D711yescharacteristic
183C3×C4—72no—
184C3×C4—142no—
185C3×C4C2×C4×D711yescharacteristic
186C3×C4D4×D711yescharacteristic
187Dic3—41no—
188Dic3—281no—
189C23—21no—
190C23—71no—
191Q8—72no—
192Q8—141no—
193D4—21no—
194D4—144no—
195D4Dic3×D712yes—
196C2×C4—61no—
197C2×C4—71no—
198C2×C4—72no—
199C2×C4—72no—
200C2×C4—144no—
201C2×C4—141no—
202C2×C4—421no—
203C2×C4C2×D3×D711yescharacteristic
204C8—61no—
205C8—421no—
206C7#523311yescharacteristic, complemented, Sylow
207C2×C3—42no—
208C2×C3—142no—
209C2×C3D7×C22⋊C411yescharacteristic
210C2×C3#647511yescharacteristic
211C2×C3#647911yescharacteristic
212C22—22no—
213C22—41no—
214C22—41no—
215C22—72no—
216C22—141no—
217C22—141no—
218C22D7×#6711yescharacteristic
219C4—72no—
220C4—121no—
221C4—142no—
222C4—841no—
223C4C2×Dic3×D711yescharacteristic
224C4D7×#6711yescharacteristic
225C3#8954411yescharacteristic, complemented, Sylow
226C2—42no—
227C2—142no—
228C2#1062105211yescharacteristic
229C2#1062105411yescharacteristic
230C2D7×#21711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1159C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#416
LCSG ▷ 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. #10620920C2semidirect
3. C2×D7×C3⋊C8C2semidirect
4. #10620671C2semidirect
5. C7×#832C2semidirect
6. #10620506C2semidirect
7. #10620505C2semidirect
13. C3×#1159C4semidirect
52. #1159Dic3semidirect
68. #832D7semidirect
156. C3×C7#416semidirect
206. C7#5233semidirect
225. C3#89544semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
#106209202
C2×D7×C3⋊C82
#106206712
C7×#8322
#106205062
#106205052
#895443
#52337