Finite Group Databasebeta

← #13054459

Subgroups — #13054459

Order 1344.

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

normal subgroups only

This group has 236 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
1C3×#6503C211yesmaximal, characteristic
2#10621030C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5#10620500C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7C3⋊C8×Dic7C211yesmaximal, characteristic, complemented
8#89532—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×C22⋊D7—21no—
12C2×C3×D28C2211yescharacteristic
13C2×#4248—21no—
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4×Dic7C2211yescharacteristic
19Dic7⋊Dic3—21no—
20#6503D311yescharacteristic
21#6451—31no—
22C7×#102—31no—
23#6406—31no—
24#6395—31no—
25C8×Dic7—31no—
26#6383—31no—
27#5070—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×C3×D7—21no—
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C3×C22⋊D7—42no—
36C3×C22⋊D7—42no—
37C2×C3×Dic7D411yescharacteristic
38C2×C3×Dic7D411yescharacteristic
39C3×D28—21no—
40C3×D28—21no—
41#4248—42no—
42C3⋊(C7⋊C8)—21no—
43C7×C3⋊C8—21no—
44C2×C7×D4C2×D311yescharacteristic
45C2×C22⋊D7—21no—
46C2×C22⋊D7—21no—
47C2×D28C2×D311yescharacteristic
48C2×Dic14—61no—
49C2×C7×C8—31no—
50C7×C4⋊C4—31no—
51C4⋊Dic7—31no—
52Dic7⋊C4—61no—
53C4×Dic7C2×D311yescharacteristic
54C2×C7⋊C8—31no—
55C3×#127—71no—
56#869—71no—
57#832—71no—
58#832—71no—
59#832—71no—
60#832D711yescharacteristic, complemented
61C4×C3⋊C8—71no—
62C22×C3×C7—41no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D411yescharacteristic
66C2×C3×D7—41no—
67C2×C3×D7—41no—
68C2×C3×D7—41no—
69C3×C4×C7#4111yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—41no—
72C3×Dic7D812yes—
73C3×Dic7SD1612yes—
74C7×Dic3—41no—
75#486—211noSylow
76C23×C7—21no—
77C22×D7—21no—
78C7×D4—21no—
79C7×D4—21no—
80C2×C4×C7—61no—
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—42no—
83C22⋊D7—42no—
84C2×Dic7—61no—
85C2×Dic7#6711yescharacteristic
86C2×Dic7#6711yescharacteristic
87D28—21no—
88D28—21no—
89Dic14—122no—
90C7×C8—61no—
91C7⋊C8—61no—
92C2×C3×D4—71no—
93C2×C3×D4—141no—
94C2×C3×D4—141no—
95C2×C3×D4C2×D711yescharacteristic
96C2×Dic6—141no—
97C3×C42—71no—
98C4⋊Dic3—71no—
99C4⋊Dic3—141no—
100C4⋊Dic3C2×D711yescharacteristic
101C2×C3⋊C8—71no—
102C2×C3⋊C8C2×D711yescharacteristic
103C2×C3×C7—81no—
104C2×C3×C7#12411yescharacteristic
105C2×C3×C7C2×D811yescharacteristic
106C2×C3×C7C2×SD1611yescharacteristic
107C3×D7—81no—
108C4⋊Q8—211no—
109#127—71no—
110#102—31no—
111#102—211no—
112#102—211no—
113#102—211no—
114C4×C8—211no—
115C22×C7—41no—
116C22×C7—41no—
117C22×C7—41no—
118C22×C7C2×#6711yescharacteristic
119C2×D7—41no—
120C2×D7—41no—
121C2×D7—41no—
122C4×C7—121no—
123C4×C7#23711yescharacteristic
124C4×C7#23711yescharacteristic
125Dic7—121no—
126Dic7#21312yes—
127Dic7#21412yes—
128C23×C3—21no—
129C23×C3—141no—
130C3×D4—21no—
131C3×D4—21no—
132C3×D4—141no—
133C3×D4—141no—
134C3×D4—282no—
135C3×D4—282no—
136C2×C3×C4—71no—
137C2×C3×C4—71no—
138C2×C3×C4C22×D711yescharacteristic
139C2×Dic3—21no—
140C2×Dic3—141no—
141Dic6—282no—
142C3⋊C8—21no—
143C3⋊C8—141no—
144C3×C7#48611yescharacteristic, complemented, Hall
145C2×Q8—421no—
146C2×D4—71no—
147C2×D4—141no—
148C2×D4—141no—
149C2×D4D3×D711yescharacteristic
150C2×C8—31no—
151C2×C8—211no—
152C4⋊C4—31no—
153C4⋊C4—211no—
154C4⋊C4—421no—
155C42—71no—
156C2×C7—81no—
157C2×C7C2×#21311yescharacteristic
158C2×C7C2×#21411yescharacteristic
159C2×C7#93611yescharacteristic
160D7—81no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—281no—
165C22×C3—281no—
166C22×C3—281no—
167C22×C3D4×D711yescharacteristic
168C3×C4—72no—
169C3×C4—72no—
170C3×C4#115711yescharacteristic
171C3×C4#115911yescharacteristic
172Dic3—41no—
173Dic3—281no—
174C23—21no—
175C23—141no—
176Q8—842no—
177D4—21no—
178D4—21no—
179D4—141no—
180D4—141no—
181D4—282no—
182D4—282no—
183C2×C4—61no—
184C2×C4—71no—
185C2×C4—71no—
186C2×C4—421no—
187C2×C4C2×D3×D711yescharacteristic
188C8—61no—
189C8—421no—
190C7#507011yescharacteristic, complemented, Sylow
191C2×C3—81no—
192C2×C3—561no—
193C2×C3#644811yescharacteristic
194C2×C3D7×D811yescharacteristic
195C2×C3D7×SD1611yescharacteristic
196C22—41no—
197C22—41no—
198C22—41no—
199C22—281no—
200C22—281no—
201C22—281no—
202C22D7×#6711yescharacteristic
203C4—72no—
204C4—72no—
205C4—121no—
206C4—841no—
207C4#6668311yescharacteristic
208C4#6668411yescharacteristic
209C3#8953211yescharacteristic, complemented, Sylow
210C2—81no—
211C2—561no—
212C2D7×#21311yescharacteristic
213C2D7×#21411yescharacteristic
214C2#1062109611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6503C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#486
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
2. #10621030C2semidirect
3. #10620671C2semidirect
4. C7×#832C2semidirect
5. #10620500C2semidirect
6. #10620499C2semidirect
7. C3⋊C8×Dic7C2semidirect
60. #832D7semidirect
144. C3×C7#486semidirect
190. C7#5070semidirect
209. C3#89532semidirect

Maximal subgroups

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

subgroupindex
C3×#65032
#106210302
#106206712
C7×#8322
#106205002
#106204992
C3⋊C8×Dic72
#895323
#50707