Finite Group Databasebeta

← #13054192

Subgroups — #13054192

Order 1344.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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
1#10621413C211yesmaximal, characteristic, complemented
2C2×C7×#208C211yesmaximal, characteristic, complemented
3C3×#6493C211yesmaximal, characteristic, complemented
4C2×#66561C211yesmaximal, characteristic, complemented
5#10620522C211yesmaximal, characteristic, complemented
6#10620503C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8#89907—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×#4268—21no—
11C2×D84C2211yescharacteristic
12C2×C3×C22⋊D7—21no—
13C2×C3×D28C2211yescharacteristic
14#66631—21no—
15C4⋊Dic21C2211yescharacteristic
16C7×#208—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×(C2×D7)⋊C4—21no—
20C3×C4⋊Dic7C2211yescharacteristic
21#66561—22no—
22C2×C7×M16—31no—
23#6493—31no—
24#6493D311yescharacteristic, complemented
25C2×D56—31no—
26#6395—31no—
27#6395—31no—
28#6391—31no—
29#5039—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C22×D21—21no—
32C22×C3×D7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7D411yescharacteristic
36C2×C3×C4×C7C2311yescharacteristic
37#4268—42no—
38C2×Dic21—21no—
39D84—21no—
40D84—21no—
41C3×C22⋊D7—42no—
42C2×C3×Dic7—21no—
43C3×D28—21no—
44C3×D28—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D412yescomplemented
47C22×C4×C7C2×D311yescharacteristic
48C2×C22⋊D7—21no—
49C2×C22⋊D7—61no—
50C2×D28—31no—
51C2×D28C2×D311yescharacteristic
52C7×M16—62no—
53C2×C7×C8—31no—
54C2×C7×C8—31no—
55(C2×D7)⋊C4—21no—
56(C2×D7)⋊C4—61no—
57C4⋊Dic7—31no—
58C4⋊Dic7C2×D311yescharacteristic
59D56—62no—
60C3×#121—71no—
61C2×#213—71no—
62#930—71no—
63C2×#208D711yescharacteristic, complemented
64#832—71no—
65#809—71no—
66#808—71no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7—21no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—41no—
72C2×D21—41no—
73C2×D21—41no—
74C2×C3×D7—41no—
75C2×C3×D7—41no—
76C2×C3×D7—41no—
77C3×C4×C7—21no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7#4111yescharacteristic
80Dic21—41no—
81C3×Dic7—41no—
82#469—211noSylow
83C23×C7#6711yescharacteristic
84C22×D7—21no—
85C22×D7—61no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7#6711yescharacteristic
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—42no—
91C22⋊D7—122no—
92C2×Dic7—21no—
93C2×Dic7—61no—
94D28—21no—
95D28—21no—
96D28—61no—
97D28—61no—
98C7×C8—32no—
99C7×C8—61no—
100C2×C3×D4—71no—
101C2×C3×D4—141no—
102C22×C3×C4C2×D711yescharacteristic
103C2×#67—141no—
104C2×D12—71no—
105C3×C4⋊C4—71no—
106C3×C22⋊C4—141no—
107#213—142no—
108(C2×D3)⋊C4—141no—
109C4⋊Dic3—71no—
110#208—22no—
111C2×C3⋊C8C2×D711yescharacteristic
112C2×C3⋊C8C2×D711yescharacteristic
113C2×C3×C7—41no—
114C2×C3×C7#12111yescharacteristic
115C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
116C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
117D21—81no—
118C3×D7—81no—
119C2×D8—211no—
120C2×M16—31no—
121#121—71no—
122#121—211no—
123#106—211no—
124#102—211no—
125#102—211no—
126C22×C7—21no—
127C22×C7—21no—
128C22×C7—21no—
129C22×C7C2×#6711yescharacteristic
130C2×D7—41no—
131C2×D7—41no—
132C2×D7—41no—
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C4×C7—21no—
137C4×C7D3×D411yescharacteristic
138C4×C7#23711yescharacteristic
139Dic7—41no—
140Dic7—121no—
141C23×C3—141no—
142C23×C3D2811yescharacteristic
143C22×D3—141no—
144C3×D4—141no—
145C3×D4—141no—
146C3×D4—282no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—141no—
150C2×C3×C4D2811yescharacteristic
151C2×C3×C4C22×D711yescharacteristic
152#67—282no—
153C2×Dic3—141no—
154D12—141no—
155D12—141no—
156C3⋊C8—21no—
157C3⋊C8C22⋊D712yescomplemented
158C3×C7#46911yescharacteristic, complemented, Hall
159C2×D4—71no—
160C2×D4—141no—
161C2×D4—211no—
162C2×D4—421no—
163C22×C4D3×D711yescharacteristic
164D8—422no—
165M16—62no—
166C2×C8—31no—
167C2×C8—31no—
168C4⋊C4—71no—
169C4⋊C4—211no—
170C22⋊C4—141no—
171C22⋊C4—421no—
172C2×C7—41no—
173C2×C7#93211yescharacteristic
174C2×C7#93911yescharacteristic
175C2×C7#94911yescharacteristic
176D7—81no—
177D7—241no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—281no—
182C22×C3—281no—
183C22×C3—281no—
184C22×C3C2×D2811yescharacteristic
185C2×D3—281no—
186C2×D3—281no—
187C2×D3—281no—
188C3×C4—21no—
189C3×C4—281no—
190C3×C4#115711yescharacteristic
191C3×C4C2×C22⋊D711yescharacteristic
192Dic3—281no—
193C23—141no—
194C23—421no—
195C23#424611yescharacteristic
196D4—141no—
197D4—141no—
198D4—282no—
199D4—421no—
200D4—421no—
201D4—842no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—141no—
205C2×C4—421no—
206C2×C4#424611yescharacteristic
207C2×C4C2×D3×D711yescharacteristic
208C8—32no—
209C8—61no—
210C7#503911yescharacteristic, complemented, Sylow
211C2×C3—41no—
212C2×C3—561no—
213C2×C3#647111yescharacteristic
214C2×C3#647111yescharacteristic
215C2×C3#649311yescharacteristic
216D3—561no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—281no—
221C22—281no—
222C22—281no—
223C22—841no—
224C22—841no—
225C22—841no—
226C22C2×#424611yescharacteristic
227C4—21no—
228C4—281no—
229C4—841no—
230C4#6668311yescharacteristic
231C4D3×C22⋊D711yescharacteristic
232C3#8990711yescharacteristic, complemented, Sylow
233C2—41no—
234C2—561no—
235C2—1681no—
236C2#1062086311yescharacteristic
237C2#1062087411yescharacteristic
238C2#1062113111yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#469
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #10621413C2semidirect
2. C2×C7×#208C2semidirect
3. C3×#6493C2semidirect
4. C2×#66561C2semidirect
5. #10620522C2semidirect
6. #10620503C2semidirect
7. #10620500C2semidirect
24. #6493D3semidirect
46. C7×C3⋊C8D4semidirect
63. C2×#208D7semidirect
157. C3⋊C8C22⋊D7semidirect
158. C3×C7#469semidirect
210. C7#5039semidirect
232. C3#89907semidirect

Maximal subgroups

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

subgroupindex
#106214132
C2×C7×#2082
C3×#64932
C2×#665612
#106205222
#106205032
#106205002
#899073
#50397