Finite Group Databasebeta

← #13054631

Subgroups — #13054631

Order 1344.

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 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×#66634C211yesmaximal, characteristic, complemented
2C3×#6509C211yesmaximal, characteristic, complemented
3#10621012C211yesmaximal, characteristic
4C2×#66557C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620524C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic, complemented
8#89584—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C2×#4246—21no—
13#66634—22no—
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18C3×C4⋊Dic7C2211yescharacteristic
19C3×Dic7⋊C4—21no—
20#66575—21no—
21#66557—22no—
22#6509D311yescharacteristic, complemented
23C2×#1143—31no—
24C2×#1131—31no—
25#6458—31no—
26C7×#103—31no—
27#6395—31no—
28#6381—31no—
29#5049—71nomaximal, Hall
30C22×D21—21no—
31C22×C3×D7D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36D84—21no—
37D84—21no—
38C2×C7×Dic3—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7D411yescharacteristic
41C3×C4×D7—21no—
42C3×C4×D7—21no—
43#4246—42no—
44C3⋊(C7⋊C8)D412yescomplemented
45C7×C3⋊C8—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C2×D28—61no—
49C2×C4×D7C2×D311yescharacteristic
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52#1143—62no—
53(C2×D7)⋊C4—21no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7C2×D311yescharacteristic
56Dic7⋊C4—21no—
57C2×C7⋊C8—31no—
58#1131—62no—
59C3×C22⋊Q8—71no—
60C2×#215—71no—
61#930—71no—
62C2×#208—71no—
63#835D711yescharacteristic, complemented
64#809—71no—
65#807—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—41no—
68C2×D21—41no—
69C2×D21—41no—
70C2×C3×D7—21no—
71C2×C3×D7—21no—
72C2×C3×D7—21no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76C3×Dic7—21no—
77C3×Dic7—41no—
78C7×Dic3—41no—
79#468—211noSylow
80C22×D7—61no—
81C22×D7#6711yescharacteristic
82C7×Q8—21no—
83C7×Q8—21no—
84C2×C4×C7—21no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—21no—
88C2×Dic7#6711yescharacteristic
89D28—61no—
90D28—61no—
91D28—122no—
92C4×D7—21no—
93C4×D7—21no—
94C7×C8—61no—
95C7⋊C8—32no—
96C2×C3×Q8C2×D711yescharacteristic
97C22×C3×C4—71no—
98C2×#67—141no—
99C2×D12—71no—
100C3×C4⋊C4—71no—
101C3×C4⋊C4—141no—
102C3×C22⋊C4—141no—
103#215—142no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3C2×D711yescharacteristic
106#208—142no—
107C2×C3⋊C8—71no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C3×C7#13711yescharacteristic
112D21—81no—
113C3×D7—41no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—71no—
117#121—211no—
118#107—211no—
119#103—31no—
120#102—211no—
121C22×C7C2×#6711yescharacteristic
122C2×D7—21no—
123C2×D7—21no—
124C2×D7—21no—
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7—121no—
130C4×C7D3×D411yescharacteristic
131C4×C7#23711yescharacteristic
132Dic7—21no—
133Dic7—41no—
134C23×C3—71no—
135C22×D3—141no—
136C3×Q8—21no—
137C3×Q8—21no—
138C2×C3×C4—21no—
139C2×C3×C4—71no—
140C2×C3×C4—141no—
141C2×C3×C4—141no—
142C2×C3×C4—141no—
143C2×C3×C4C22×D711yescharacteristic
144#67—282no—
145C2×Dic3—21no—
146D12—141no—
147D12—141no—
148C3⋊C8—21no—
149C3⋊C8—72no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8D3×D711yescharacteristic
152C2×D4—211no—
153C2×D4—421no—
154C22×C4—71no—
155SD16—422no—
156M16—422no—
157C2×C8—31no—
158C2×C8—211no—
159C4⋊C4—31no—
160C4⋊C4—71no—
161C4⋊C4—141no—
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7#93911yescharacteristic
165C2×C7#94211yescharacteristic
166C2×C7#94911yescharacteristic
167D7—41no—
168D7—241no—
169C22×C3—141no—
170C22×C3—141no—
171C22×C3—141no—
172C22×C3D4×D711yescharacteristic
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4—141no—
178C3×C4—281no—
179C3×C4#115711yescharacteristic
180C3×C4D4×D711yescharacteristic
181Dic3—41no—
182C23—71no—
183C23—421no—
184Q8—21no—
185Q8—21no—
186D4—421no—
187D4—421no—
188D4—842no—
189C2×C4—21no—
190C2×C4—61no—
191C2×C4—71no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—212no—
198C7#504911yescharacteristic, complemented, Sylow
199C2×C3—281no—
200C2×C3#644711yescharacteristic
201C2×C3#647711yescharacteristic
202C2×C3#648111yescharacteristic
203D3—561no—
204C22—141no—
205C22—141no—
206C22—141no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22D7×#6711yescharacteristic
211C4—41no—
212C4—121no—
213C4—141no—
214C4—281no—
215C4#6668311yescharacteristic
216C4#6669411yescharacteristic
217C3#8958411yescharacteristic, complemented, Sylow
218C2—281no—
219C2—1681no—
220C2#1062106411yescharacteristic
221C2#1062107711yescharacteristic
222C2#1062111211yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radical#10621012C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#468
LCSG ▷ C2×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
1. C2×#66634C2semidirect
2. C3×#6509C2semidirect
4. C2×#66557C2semidirect
5. C7×#835C2semidirect
6. #10620524C2semidirect
7. #10620503C2semidirect
22. #6509D3semidirect
44. C3⋊(C7⋊C8)D4semidirect
63. #835D7semidirect
150. C3×C7#468semidirect
198. C7#5049semidirect
217. C3#89584semidirect

Maximal subgroups

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

subgroupindex
C2×#666342
C3×#65092
#106210122
C2×#665572
C7×#8352
#106205242
#106205032
#895843
#50497