Finite Group Databasebeta

← #13052831

Subgroups — #13052831

Order 1344.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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#10621370C211yesmaximal, characteristic, complemented
2#10620997C211yesmaximal, characteristic
3C2×#66567C211yesmaximal, characteristic
4C7×#809C211yesmaximal, characteristic
5C3×#6381C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620489C211yesmaximal, characteristic, complemented
8#89529—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×D3×Dic7—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13#66629—21no—
14C4×Dic21—21no—
15C4×Dic21C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic7—21no—
20#66567C414yescomplemented
21D4×Dic7—31no—
22C2×#1142—31no—
23#6450—31no—
24C7×#102—31no—
25#6390—31no—
26#6389—31no—
27#6381D311yescharacteristic, complemented
28#4889—71nomaximal, Hall
29C22×D3×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35Dic42—21no—
36Dic42C2×C412yescomplemented
37C7×D12—21no—
38C7×D12C2×C412yes—
39C2×C3×Dic7—21no—
40D3×Dic7—42no—
41C3×C7⋊C8C2×C412yescomplemented
42C7×C3⋊C8—21no—
43C2×C7×D4—31no—
44C22×Dic7—61no—
45C2×Dic14—31no—
46C2×C7×C8—31no—
47C7×C4⋊C4C2×D311yescharacteristic
48C22⋊Dic7—61no—
49#1142—34no—
50C4⋊Dic7C2×D311yescharacteristic
51Dic7⋊C4—61no—
52C4×Dic7—31no—
53C4×Dic7—61no—
54C2×C7⋊C8C2×D311yescharacteristic
55C2×#204—71no—
56#893—71no—
57#887—71no—
58C3×#107—71no—
59#815—71no—
60#810—71no—
61#809D711yescharacteristic
62C22×C3×C7C2×D411yescharacteristic
63C2×D3×C7—22no—
64C2×D3×C7—41no—
65C2×D3×C7—41no—
66C3×C4×C7—41no—
67C3×C4×C7C22×C411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—22no—
70Dic21—21no—
71Dic21—21no—
72C3×Dic7—41no—
73#440—211noSylow
74C23×C7—61no—
75C7×D4—32no—
76C7×D4—61no—
77C2×C4×C7—21no—
78C2×C4×C7C22×D311yescharacteristic
79C2×Dic7—21no—
80C2×Dic7—31no—
81C2×Dic7—31no—
82C2×Dic7—61no—
83C2×Dic7—122no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—61no—
87C7⋊C8C4×D312yescomplemented
88C2×D12C2×D711yescharacteristic
89C2×C4×D3—141no—
90C2×Dic6—71no—
91C2×C3×C8—71no—
92C3×C4⋊C4—71no—
93C3×C4⋊C4C2×D711yescharacteristic
94(C2×D3)⋊C4—141no—
95Dic3⋊C4—141no—
96C4×Dic3—71no—
97C4×Dic3—141no—
98C2×C3⋊C8C2×D711yescharacteristic
99#204—74no—
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103D3×C7—42no—
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—211no—
107#107—71no—
108#103—211no—
109#102—31no—
110#97—211no—
111C22×C7—62no—
112C22×C7—121no—
113C22×C7—121no—
114C22×C7D3×D411yescharacteristic
115C4×C7—41no—
116C4×C7C2×C4×D311yescharacteristic
117C4×C7#23911yescharacteristic
118Dic7—41no—
119Dic7—62no—
120Dic7—61no—
121Dic7—61no—
122C22×D3—21no—
123C2×C3×C4—21no—
124C2×C3×C4—141no—
125C2×C3×C4C22×D711yescharacteristic
126C2×Dic3—71no—
127C2×Dic3—71no—
128C2×Dic3—141no—
129D12—21no—
130D12C4×D712yes—
131C4×D3—282no—
132Dic6—72no—
133Dic6—141no—
134C3×C8—72no—
135C3⋊C8—21no—
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4—31no—
139C22×C4—421no—
140SD16—214no—
141C2×C8—31no—
142C2×C8—71no—
143C4⋊C4—71no—
144C4⋊C4—421no—
145C4⋊C4D3×D711yescharacteristic
146C22⋊C4—421no—
147C42—211no—
148C42—421no—
149C2×C7—122no—
150C2×C7#89311yescharacteristic
151C2×C7#91111yescharacteristic
152C2×C7#91811yescharacteristic
153C22×C3D4×D711yescharacteristic
154C2×D3—22no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—41no—
158C3×C4—281no—
159C3×C4C2×C4×D711yescharacteristic
160C3×C4#115911yescharacteristic
161Dic3—142no—
162Dic3—141no—
163Dic3—141no—
164C23—61no—
165Q8—212no—
166Q8—421no—
167D4—32no—
168D4—61no—
169C2×C4—21no—
170C2×C4—141no—
171C2×C4—211no—
172C2×C4—211no—
173C2×C4—421no—
174C2×C4—842no—
175C2×C4C2×D3×D711yescharacteristic
176C8—61no—
177C8—72no—
178C7#488911yescharacteristic, complemented, Sylow
179C2×C3(C22⋊D7)⋊C411yescharacteristic
180C2×C3#647411yescharacteristic
181C2×C3#647811yescharacteristic
182D3—42no—
183C22—62no—
184C22—121no—
185C22—121no—
186C22#6669411yescharacteristic
187C4—41no—
188C4—281no—
189C4—422no—
190C4—421no—
191C4—421no—
192C4#6667211yescharacteristic
193C4C4×D3×D711yescharacteristic
194C3#8952911yescharacteristic, complemented, Sylow
195C2—122no—
196C2#1062096811yescharacteristic
197C2#1062104411yescharacteristic
198C2#1062108611yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#440
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
1. #10621370C2semidirect
5. C3×#6381C2semidirect
6. #10620509C2semidirect
7. #10620489C2semidirect
20. #66567C4semidirect
27. #6381D3semidirect
36. Dic42C2×C4semidirect
41. C3×C7⋊C8C2×C4semidirect
87. C7⋊C8C4×D3semidirect
136. C3×C7#440semidirect
178. C7#4889semidirect
194. C3#89529semidirect

Maximal subgroups

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

subgroupindex
#106213702
#106209972
C2×#665672
C7×#8092
C3×#63812
#106205092
#106204892
#895293
#48897