Finite Group Databasebeta

← #13054702

Subgroups — #13054702

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×#1161C211yesmaximal, characteristic
2#10620920C211yesmaximal, characteristic, complemented
3C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
4#10620674C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89576—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×#1161—22no—
11C3×#1161C414yescomplemented
12C2×C3×D28—21no—
13C2×C3×D28C2211yescharacteristic
14C2×C3×C4×D7—21no—
15C2×C3×C4×D7C2211yescharacteristic
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×#1161D311yescharacteristic
24C2×C8×D7—31no—
25#6455—31no—
26C7×#103—31no—
27#6409—31no—
28#6395—31no—
29#6383—31no—
30#5233—71nomaximal, Hall
31C22×C3×D7—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8C2×C412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×C7×Dic3—21no—
39C2×C3×Dic7D411yescharacteristic
40C3×D28—24no—
41C3×D28—21no—
42C3×D28C2×C412yes—
43C3×C4×D7—24no—
44C3×C4×D7C2×C412yes—
45C3×C4×D7D412yes—
46C3⋊(C7⋊C8)—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8C2×D311yescharacteristic
49#1161—22no—
50#1161Dic314yescomplemented
51C2×D28—21no—
52C2×D28C2×D311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7C2×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#835D711yescharacteristic, complemented
67#832—71no—
68#832—71no—
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—21no—
71C2×C3×D7—21no—
72C2×C3×D7—22no—
73C2×C3×D7—41no—
74C2×C3×D7—41no—
75C2×C3×D7C22⋊C412yes—
76C3×C4×C7—22no—
77C3×C4×C7C22×C411yescharacteristic
78C3×C4×C7C2×D411yescharacteristic
79Dic21—41no—
80C3×Dic7C22⋊C412yes—
81C7×Dic3—41no—
82#416—211noSylow
83C22×D7—21no—
84C22×D7#6711yescharacteristic
85C7×Q8—21no—
86C7×Q8C2×Dic312yes—
87C2×C4×C7—21no—
88C2×C4×C7—61no—
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—61no—
91C2×Dic7#6711yescharacteristic
92D28—24no—
93D28—21no—
94D28C2×Dic312yes—
95C4×D7—24no—
96C4×D7C2×Dic312yes—
97C4×D7#6712yes—
98C7×C8—61no—
99C7⋊C8—61no—
100C3×#41—74no—
101C3×#41—142no—
102C2×C3×Q8C2×D711yescharacteristic
103C2×C3×D4—71no—
104C2×C3×D4—141no—
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×C7C2×C22⋊C411yescharacteristic
115C2×C3×C7#13511yescharacteristic
116C2×C3×C7#13511yescharacteristic
117C3×D7—22no—
118C3×D7—42no—
119C2×#41—71no—
120C22×C8—211no—
121#117—211no—
122#103—31no—
123#103—211no—
124#102—211no—
125#102—211no—
126C22×C7C2×#6711yescharacteristic
127C2×D7—21no—
128C2×D7—21no—
129C2×D7—22no—
130C2×D7—41no—
131C2×D7—41no—
132C2×D7#21712yes—
133C4×C7—22no—
134C4×C7—121no—
135C4×C7C22×Dic311yescharacteristic
136C4×C7C2×#6711yescharacteristic
137Dic7—121no—
138Dic7#21712yes—
139C23×C3—71no—
140C23×C3—141no—
141C3×Q8—21no—
142C3×Q8C4×D712yes—
143C3×D4—72no—
144C3×D4—144no—
145C3×D4—141no—
146C2×C3×C4—21no—
147C2×C3×C4—71no—
148C2×C3×C4—72no—
149C2×C3×C4—72no—
150C2×C3×C4—144no—
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×Q8D3×D711yescharacteristic
160C2×D4—71no—
161C2×D4—141no—
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#95011yescharacteristic
172C2×C7#95011yescharacteristic
173C2×C7C2×#21711yescharacteristic
174D7—22no—
175D7—42no—
176C22×C3—72no—
177C22×C3—141no—
178C22×C3—141no—
179C22×C3—142no—
180C22×C3—281no—
181C22×C3—281no—
182C22×C3D4×D711yescharacteristic
183C3×C4—22no—
184C3×C4—72no—
185C3×C4C2×C4×D711yescharacteristic
186C3×C4D4×D711yescharacteristic
187Dic3—41no—
188Dic3—281no—
189C23—71no—
190C23—141no—
191Q8—21no—
192Q8Dic3×D712yes—
193D4—72no—
194D4—144no—
195D4—141no—
196C2×C4—21no—
197C2×C4—61no—
198C2×C4—71no—
199C2×C4—72no—
200C2×C4—72no—
201C2×C4—144no—
202C2×C4—421no—
203C2×C4C2×D3×D711yescharacteristic
204C8—61no—
205C8—421no—
206C7#523311yescharacteristic, complemented, Sylow
207C2×C3—142no—
208C2×C3—282no—
209C2×C3D7×C22⋊C411yescharacteristic
210C2×C3#647911yescharacteristic
211C2×C3#648211yescharacteristic
212C22—72no—
213C22—141no—
214C22—141no—
215C22—142no—
216C22—281no—
217C22—281no—
218C22D7×#6711yescharacteristic
219C4—22no—
220C4—72no—
221C4—121no—
222C4—841no—
223C4C2×Dic3×D711yescharacteristic
224C4D7×#6711yescharacteristic
225C3#8957611yescharacteristic, complemented, Sylow
226C2—142no—
227C2—282no—
228C2#1062107611yescharacteristic
229C2#1062107811yescharacteristic
230C2D7×#21711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1161C2
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7#416
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10620920C2semidirect
3. C2×D7×C3⋊C8C2semidirect
4. #10620674C2semidirect
5. C7×#835C2semidirect
6. #10620500C2semidirect
7. #10620499C2semidirect
11. C3×#1161C4semidirect
50. #1161Dic3semidirect
66. #835D7semidirect
156. C3×C7#416semidirect
206. C7#5233semidirect
225. C3#89576semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11612
#106209202
C2×D7×C3⋊C82
#106206742
C7×#8352
#106205002
#106204992
#895763
#52337