Finite Group Databasebeta

← #13052878

Subgroups — #13052878

Order 1344.

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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×#66676C211yesmaximal, characteristic
2C3×#6455C211yesmaximal, characteristic, complemented
3C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
4#10620649C211yesmaximal, characteristic, complemented
5C7×#809C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620501C211yesmaximal, characteristic, complemented
8#89544—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C2×#4245—21no—
13C2×D3×Dic7—21no—
14#66676—22no—
15#66676C414yescomplemented
16C3×C7×C4⋊C4C2211yescharacteristic
17C2×C3⋊(C7⋊C8)C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×(C2×D7)⋊C4—21no—
20C3×C4⋊Dic7C2211yescharacteristic
21C3×C4×Dic7—21no—
22D7×C3⋊C8—22no—
23C2×#1159—31no—
24C2×C8×D7—31no—
25#6455D311yescharacteristic, complemented
26C7×#102—31no—
27#6406—31no—
28#6390—31no—
29#6384—31no—
30#5002—71nomaximal, Hall
31C22×D3×C7—21no—
32C22×C3×D7D411yescharacteristic
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36Dic42—21no—
37Dic42C2×C412yes—
38C7×D12—21no—
39C7×D12C2×C412yes—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7C2×C412yes—
43C3×C4×D7D412yes—
44#4245—24no—
45D3×Dic7—24no—
46C3⋊(C7⋊C8)—21no—
47C7×C3⋊C8—21no—
48C2×C7×D4—31no—
49C2×C22⋊D7—61no—
50C22×Dic7—61no—
51#1159—34no—
52#1159—62no—
53C2×C4×D7C2×D311yescharacteristic
54C2×Dic14—31no—
55C2×C7×C8—31no—
56C7×C4⋊C4C2×D311yescharacteristic
57(C2×D7)⋊C4—21no—
58C4⋊Dic7C2×D311yescharacteristic
59C4×Dic7—21no—
60C2×C7⋊C8—31no—
61C8×D7—62no—
62C2×#235—71no—
63C3×#117—71no—
64C22×C3⋊C8—71no—
65#810—71no—
66#810—71no—
67#809—71no—
68#809D711yescharacteristic, complemented
69C22×C3×C7C2×D411yescharacteristic
70C2×D3×C7—22no—
71C2×D3×C7—41no—
72C2×D3×C7—41no—
73C2×C3×D7—21no—
74C2×C3×D7—21no—
75C2×C3×D7C22⋊C412yes—
76C3×C4×C7—41no—
77C3×C4×C7C22×C411yescharacteristic
78C3×C4×C7C2×D411yescharacteristic
79Dic21—22no—
80C3×Dic7—41no—
81C3×Dic7C22⋊C412yes—
82#416—211noSylow
83C23×C7—61no—
84C22×D7#6711yescharacteristic
85C7×D4—32no—
86C7×D4—61no—
87C2×C4×C7—21no—
88C2×C4×C7C22×D311yescharacteristic
89C22⋊D7—64no—
90C2×Dic7—21no—
91C2×Dic7—61no—
92C2×Dic7—64no—
93C2×Dic7#6711yescharacteristic
94C4×D7C4×D312yes—
95C4×D7D1212yes—
96Dic14—32no—
97Dic14—61no—
98C7×C8—61no—
99C7⋊C8—61no—
100C22×C3×C4—71no—
101C2×#67—141no—
102#235—74no—
103#235—142no—
104C2×D12C2×D711yescharacteristic
105C2×C4×D3—141no—
106C2×Dic6—71no—
107C3×C4⋊C4—71no—
108C3×C4⋊C4C2×D711yescharacteristic
109C3×C22⋊C4—141no—
110C3×C42—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—
118D3×C7—42no—
119C2×#41—211no—
120C22×C8—211no—
121#117—71no—
122#103—211no—
123#103—211no—
124#102—31no—
125#102—211no—
126C22×C7—62no—
127C22×C7—121no—
128C22×C7—121no—
129C22×C7C2×#6711yescharacteristic
130C2×D7—21no—
131C2×D7—21no—
132C2×D7(C2×D3)⋊C412yes—
133C4×C7—41no—
134C4×C7C2×C4×D311yescharacteristic
135C4×C7C2×D1211yescharacteristic
136Dic7—41no—
137Dic7—62no—
138Dic7(C2×D3)⋊C412yes—
139C23×C3—71no—
140C22×D3—21no—
141C2×C3×C4—21no—
142C2×C3×C4—71no—
143C2×C3×C4—72no—
144C2×C3×C4—72no—
145C2×C3×C4—141no—
146C2×C3×C4C22×D711yescharacteristic
147#67—144no—
148C2×Dic3—141no—
149D12—21no—
150D12C4×D712yes—
151C4×D3—144no—
152Dic6—72no—
153Dic6—141no—
154C3⋊C8—21no—
155C3⋊C8—141no—
156C3×C7#41611yescharacteristic, complemented, Hall
157#41—214no—
158#41—422no—
159C2×Q8—211no—
160C2×D4—31no—
161C2×D4—421no—
162C22×C4—71no—
163C22×C4—421no—
164C2×C8—31no—
165C2×C8—211no—
166C2×C8—422no—
167C4⋊C4—71no—
168C4⋊C4D3×D711yescharacteristic
169C22⋊C4—141no—
170C42—141no—
171C2×C7—122no—
172C2×C7C2×(C2×D3)⋊C411yescharacteristic
173C2×C7#95011yescharacteristic
174C2×C7#95011yescharacteristic
175D7—22no—
176C22×C3—72no—
177C22×C3—141no—
178C22×C3—141no—
179C22×C3D4×D711yescharacteristic
180C2×D3—22no—
181C2×D3—41no—
182C2×D3—41no—
183C3×C4—41no—
184C3×C4—72no—
185C3×C4—281no—
186C3×C4C2×C4×D711yescharacteristic
187C3×C4D4×D711yescharacteristic
188Dic3—142no—
189C23—61no—
190C23—71no—
191Q8—212no—
192Q8—421no—
193D4—32no—
194D4—61no—
195D4—424no—
196C2×C4—21no—
197C2×C4—71no—
198C2×C4—72no—
199C2×C4—72no—
200C2×C4—141no—
201C2×C4—424no—
202C2×C4—421no—
203C2×C4C2×D3×D711yescharacteristic
204C8—61no—
205C8—421no—
206C7#500211yescharacteristic, complemented, Sylow
207C2×C3—142no—
208C2×C3D7×C22⋊C411yescharacteristic
209C2×C3#647511yescharacteristic
210C2×C3#647911yescharacteristic
211D3—42no—
212C22—62no—
213C22—72no—
214C22—121no—
215C22—121no—
216C22—141no—
217C22—141no—
218C22D7×#6711yescharacteristic
219C4—41no—
220C4—72no—
221C4—281no—
222C4—422no—
223C4C4×D3×D711yescharacteristic
224C4D7×D1211yescharacteristic
225C3#8954411yescharacteristic, complemented, Sylow
226C2—122no—
227C2—142no—
228C2D7×(C2×D3)⋊C411yescharacteristic
229C2#1062105211yescharacteristic
230C2#1062107611yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66676C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#416
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#6455C2semidirect
3. C2×D7×C3⋊C8C2semidirect
4. #10620649C2semidirect
5. C7×#809C2semidirect
6. #10620509C2semidirect
7. #10620501C2semidirect
15. #66676C4semidirect
25. #6455D3semidirect
68. #809D7semidirect
156. C3×C7#416semidirect
206. C7#5002semidirect
225. C3#89544semidirect

Maximal subgroups

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

subgroupindex
C2×#666762
C3×#64552
C2×D7×C3⋊C82
#106206492
C7×#8092
#106205092
#106205012
#895443
#50027