Finite Group Databasebeta

← #13052879

Subgroups — #13052879

Order 1344.

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

normal subgroups only

This group has 324 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×#66677C211yesmaximal, characteristic
2C3×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#66557C211yesmaximal, characteristic, complemented
4#10620648C211yesmaximal, characteristic, complemented
5C7×#810C211yesmaximal, characteristic, complemented
6#10620507C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic, complemented
8#89575—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×C4×D7—21no—
12C2×C3×C4×D7C2211yescharacteristic
13C2×#4246—21no—
14C2×D21⋊C4—21no—
15#66677—22no—
16#66677C414yescomplemented
17C3×C7×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C7×C3⋊C8C2211yescharacteristic
20C3×C4⋊Dic7C2211yescharacteristic
21C3×Dic7⋊C4—21no—
22#66557—22no—
23C2×#1161—31no—
24C2×#1131—31no—
25D7×C4⋊C4D311yescharacteristic, complemented
26C7×#103—31no—
27#6409—31no—
28#6395—31no—
29#6383—31no—
30#4966—71nomaximal, Hall
31C22×D21—21no—
32C22×C3×D7D411yescharacteristic
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35D84—21no—
36D84C2×C412yescomplemented
37C2×C7×Dic3—21no—
38C7×Dic6—21no—
39C7×Dic6C2×C412yescomplemented
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7—42no—
43C3×C4×D7C2×C412yes—
44C3×C4×D7D412yes—
45#4246—24no—
46D21⋊C4—24no—
47C3⋊(C7⋊C8)—21no—
48C7×C3⋊C8—21no—
49C2×C7×Q8—31no—
50#1161—34no—
51#1161—62no—
52C2×D28—31no—
53C2×D28—61no—
54C2×C4×D7—21no—
55C2×C4×D7—61no—
56C2×C4×D7C2×D311yescharacteristic
57C2×C7×C8—31no—
58C7×C4⋊C4C2×D311yescharacteristic
59C4⋊Dic7C2×D311yescharacteristic
60Dic7⋊C4—21no—
61C2×C7⋊C8—31no—
62#1131—62no—
63C2×#235—71no—
64C2×C3×C4⋊C4—71no—
65C2×#208—71no—
66#810—71no—
67#810D711yescharacteristic, complemented
68#809—71no—
69#809—71no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—22no—
72C2×D21—41no—
73C2×D21—41no—
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7C22⋊C412yes—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80C3×Dic7—41no—
81C3×Dic7C22⋊C412yes—
82C7×Dic3—22no—
83#417—211noSylow
84C22×D7—61no—
85C22×D7#6711yescharacteristic
86C7×Q8—32no—
87C7×Q8—61no—
88C2×C4×C7—21no—
89C2×C4×C7—61no—
90C2×C4×C7C22×D311yescharacteristic
91C2×Dic7—21no—
92C2×Dic7#6711yescharacteristic
93D28—32no—
94D28—64no—
95D28—61no—
96C4×D7—42no—
97C4×D7—64no—
98C4×D7C4×D312yes—
99C4×D7D1212yes—
100C7×C8—61no—
101C7⋊C8—61no—
102C22×C3×C4—71no—
103C22×C3×C4—141no—
104C2×#67—141no—
105#235—74no—
106#235—142no—
107C2×D12—71no—
108C2×C4×D3—141no—
109C2×Dic6C2×D711yescharacteristic
110C3×C4⋊C4—71no—
111C3×C4⋊C4—141no—
112C3×C4⋊C4C2×D711yescharacteristic
113#208—142no—
114C2×C3⋊C8—71no—
115C2×C3⋊C8C2×D711yescharacteristic
116C2×C3×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119D21—42no—
120C3×D7—22no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—71no—
124#103—31no—
125#103—211no—
126#102—211no—
127#102—211no—
128C22×C7C2×#6711yescharacteristic
129C2×D7—21no—
130C2×D7—21no—
131C2×D7—62no—
132C2×D7—121no—
133C2×D7—121no—
134C2×D7(C2×D3)⋊C412yes—
135C4×C7—41no—
136C4×C7—62no—
137C4×C7C2×C4×D311yescharacteristic
138C4×C7C2×D1211yescharacteristic
139Dic7—41no—
140Dic7(C2×D3)⋊C412yes—
141C23×C3—71no—
142C22×D3—141no—
143C2×C3×C4—21no—
144C2×C3×C4—71no—
145C2×C3×C4—72no—
146C2×C3×C4—72no—
147C2×C3×C4—141no—
148C2×C3×C4—282no—
149C2×C3×C4C22×D711yescharacteristic
150#67—144no—
151C2×Dic3—21no—
152D12—72no—
153D12—141no—
154C4×D3—144no—
155Dic6—21no—
156Dic6C4×D712yescomplemented
157C3⋊C8—21no—
158C3⋊C8—141no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8—31no—
163C2×D4—211no—
164C2×D4—421no—
165C22×C4—71no—
166C22×C4—141no—
167C22×C4—421no—
168M16—422no—
169C2×C8—31no—
170C2×C8—211no—
171C4⋊C4—71no—
172C4⋊C4—141no—
173C4⋊C4D3×D711yescharacteristic
174C2×C7C2×(C2×D3)⋊C411yescharacteristic
175C2×C7#93211yescharacteristic
176C2×C7#94211yescharacteristic
177D7—22no—
178D7—122no—
179C22×C3—72no—
180C22×C3—141no—
181C22×C3—141no—
182C22×C3D4×D711yescharacteristic
183C2×D3—142no—
184C2×D3—281no—
185C2×D3—281no—
186C3×C4—41no—
187C3×C4—72no—
188C3×C4—281no—
189C3×C4C2×C4×D711yescharacteristic
190C3×C4D4×D711yescharacteristic
191Dic3—22no—
192C23—71no—
193C23—421no—
194Q8—32no—
195Q8—61no—
196D4—212no—
197D4—424no—
198D4—421no—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—71no—
202C2×C4—72no—
203C2×C4—72no—
204C2×C4—141no—
205C2×C4—282no—
206C2×C4—424no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—421no—
210C7#496611yescharacteristic, complemented, Sylow
211C2×C3—142no—
212C2×C3D7×C22⋊C411yescharacteristic
213C2×C3#647711yescharacteristic
214C2×C3#648111yescharacteristic
215D3—282no—
216C22—72no—
217C22—141no—
218C22—141no—
219C22—422no—
220C22—841no—
221C22—841no—
222C22D7×#6711yescharacteristic
223C4—41no—
224C4—62no—
225C4—72no—
226C4—281no—
227C4C4×D3×D711yescharacteristic
228C4D7×D1211yescharacteristic
229C3#8957511yescharacteristic, complemented, Sylow
230C2—142no—
231C2—842no—
232C2D7×(C2×D3)⋊C411yescharacteristic
233C2#1062105311yescharacteristic
234C2#1062107711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66677C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#417
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×D7×C4⋊C4C2semidirect
3. C2×#66557C2semidirect
4. #10620648C2semidirect
5. C7×#810C2semidirect
6. #10620507C2semidirect
7. #10620503C2semidirect
16. #66677C4semidirect
25. D7×C4⋊C4D3semidirect
36. D84C2×C4semidirect
39. C7×Dic6C2×C4semidirect
67. #810D7semidirect
156. Dic6C4×D7semidirect
159. C3×C7#417semidirect
210. C7#4966semidirect
229. C3#89575semidirect

Maximal subgroups

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

subgroupindex
C2×#666772
C3×D7×C4⋊C42
C2×#665572
#106206482
C7×#8102
#106205072
#106205032
#895753
#49667