Finite Group Databasebeta

← #13054397

Subgroups — #13054397

Order 1344.

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

normal subgroups only

This group has 260 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×#66632C211yesmaximal, characteristic, complemented
2C3×#6501C211yesmaximal, characteristic, complemented
3#10621012C211yesmaximal, characteristic, complemented
4C2×#66557C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620521C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic, complemented
8#89554—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C2×C3×C4×D7C2211yescharacteristic
13C2×#4246—21no—
14#66632—22no—
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C22⋊Dic7—21no—
19C3×C4⋊Dic7C2211yescharacteristic
20#66575—21no—
21#66557—22no—
22#6501D311yescharacteristic, complemented
23C2×#1141—31no—
24C2×#1131—31no—
25#6458—31no—
26C7×#102—31no—
27#6395—31no—
28#6382—31no—
29#5039—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7C2311yescharacteristic
36D84—21no—
37D84—21no—
38C2×C7×Dic3—21no—
39C3×C22⋊D7—42no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44#4246—42no—
45C3⋊(C7⋊C8)D412yescomplemented
46C7×C3⋊C8—21no—
47C2×C7×D4C2×D311yescharacteristic
48C2×C22⋊D7—21no—
49C2×D28—31no—
50C2×D28—61no—
51C2×C4×D7C2×D311yescharacteristic
52C2×C7×C8—31no—
53C7×C4⋊C4—31no—
54C22⋊Dic7—21no—
55#1141—62no—
56(C2×D7)⋊C4—61no—
57C4⋊Dic7C2×D311yescharacteristic
58C2×C7⋊C8—31no—
59#1131—62no—
60C3×#121—71no—
61C2×#213—71no—
62#930—71no—
63C2×#208—71no—
64#832D711yescharacteristic, complemented
65#809—71no—
66#808—71no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—41no—
72C2×D21—41no—
73C2×D21—41no—
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7—21no—
77C3×C4×C7C2×D411yescharacteristic
78C3×C4×C7#4111yescharacteristic
79C3×Dic7—21no—
80C3×Dic7—41no—
81C7×Dic3—41no—
82#469—211noSylow
83C23×C7—21no—
84C22×D7—61no—
85C22×D7#6711yescharacteristic
86C7×D4—21no—
87C7×D4—21no—
88C2×C4×C7—61no—
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—42no—
91C2×Dic7—21no—
92C2×Dic7#6711yescharacteristic
93D28—61no—
94D28—61no—
95D28—122no—
96C4×D7—21no—
97C4×D7—21no—
98C7×C8—61no—
99C7⋊C8—32no—
100C2×C3×D4—141no—
101C2×C3×D4C2×D711yescharacteristic
102C22×C3×C4—71no—
103C2×#67—141no—
104C2×D12—71no—
105C3×C4⋊C4—71no—
106C3×C22⋊C4—141no—
107#213—142no—
108(C2×D3)⋊C4—141no—
109C4⋊Dic3C2×D711yescharacteristic
110#208—142no—
111C2×C3⋊C8—71no—
112C2×C3⋊C8C2×D711yescharacteristic
113C2×C3×C7—81no—
114C2×C3×C7#12111yescharacteristic
115C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
116C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
117D21—81no—
118C3×D7—41no—
119C2×D8—211no—
120C2×M16—211no—
121#121—71no—
122#121—211no—
123#106—211no—
124#102—31no—
125#102—211no—
126C22×C7—41no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7C2×#6711yescharacteristic
130C2×D7—21no—
131C2×D7—21no—
132C2×D7—21no—
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C4×C7—121no—
137C4×C7D3×D411yescharacteristic
138C4×C7#23711yescharacteristic
139Dic7—21no—
140Dic7—41no—
141C23×C3—21no—
142C23×C3—71no—
143C22×D3—141no—
144C3×D4—21no—
145C3×D4—21no—
146C3×D4—282no—
147C2×C3×C4—71no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4—141no—
151C2×C3×C4C22×D711yescharacteristic
152#67—282no—
153C2×Dic3—21no—
154D12—141no—
155D12—141no—
156C3⋊C8—21no—
157C3⋊C8—72no—
158C3×C7#46911yescharacteristic, complemented, Hall
159C2×D4—141no—
160C2×D4—211no—
161C2×D4—421no—
162C2×D4D3×D711yescharacteristic
163C22×C4—71no—
164D8—422no—
165M16—422no—
166C2×C8—31no—
167C2×C8—211no—
168C4⋊C4—31no—
169C4⋊C4—71no—
170C22⋊C4—141no—
171C22⋊C4—421no—
172C2×C7—81no—
173C2×C7#93211yescharacteristic
174C2×C7#93911yescharacteristic
175C2×C7#94911yescharacteristic
176D7—41no—
177D7—241no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—141no—
182C22×C3—141no—
183C22×C3—141no—
184C22×C3D4×D711yescharacteristic
185C2×D3—281no—
186C2×D3—281no—
187C2×D3—281no—
188C3×C4—141no—
189C3×C4—281no—
190C3×C4#115711yescharacteristic
191C3×C4D4×D711yescharacteristic
192Dic3—41no—
193C23—21no—
194C23—71no—
195C23—421no—
196D4—21no—
197D4—21no—
198D4—282no—
199D4—421no—
200D4—421no—
201D4—842no—
202C2×C4—61no—
203C2×C4—71no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—212no—
210C7#503911yescharacteristic, complemented, Sylow
211C2×C3—81no—
212C2×C3—281no—
213C2×C3#644711yescharacteristic
214C2×C3#647411yescharacteristic
215C2×C3#647711yescharacteristic
216D3—561no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—141no—
221C22—141no—
222C22—141no—
223C22—841no—
224C22—841no—
225C22—841no—
226C22D7×#6711yescharacteristic
227C4—121no—
228C4—141no—
229C4—281no—
230C4#6668311yescharacteristic
231C4#6669411yescharacteristic
232C3#8955411yescharacteristic, complemented, Sylow
233C2—81no—
234C2—281no—
235C2—1681no—
236C2#1062104011yescharacteristic
237C2#1062105311yescharacteristic
238C2#1062111211yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#469
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66632C2semidirect
2. C3×#6501C2semidirect
3. #10621012C2semidirect
4. C2×#66557C2semidirect
5. C7×#832C2semidirect
6. #10620521C2semidirect
7. #10620503C2semidirect
22. #6501D3semidirect
45. C3⋊(C7⋊C8)D4semidirect
64. #832D7semidirect
158. C3×C7#469semidirect
210. C7#5039semidirect
232. C3#89554semidirect

Maximal subgroups

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

subgroupindex
C2×#666322
C3×#65012
#106210122
C2×#665572
C7×#8322
#106205212
#106205032
#895543
#50397