Finite Group Databasebeta

← #13056526

Subgroups — #13056526

Order 1344.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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×#66623C211yesmaximal, characteristic
2#10621376C211yesmaximal, characteristic
3#10621370C211yesmaximal, characteristic, complemented
4C3×C7×#107C211yesmaximal, characteristic, complemented
5#10620654C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic
8#89661—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C4×D21—21no—
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66631—21no—
16#66629—21no—
17C4×Dic21—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#66623C414yescomplemented
21C2×#1132—31no—
22#6456—31no—
23#6450—31no—
24C7×#107D311yescharacteristic, complemented
25#6389—31no—
26#6384—31no—
27#6383—31no—
28#4889—71nomaximal, Hall
29C22×D21—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C2×Dic21D411yescharacteristic
36D84—21no—
37D84C2×C412yes—
38C4×D21—42no—
39Dic42—21no—
40Dic42C2×C412yescomplemented
41C3×C7×C8C2×C412yescomplemented
42C3⋊(C7⋊C8)—21no—
43C2×D28—31no—
44C2×C4×D7—61no—
45C2×Dic14—31no—
46C2×C7×C8C2×D311yescharacteristic
47C7×C4⋊C4C2×D311yescharacteristic
48C7×C4⋊C4C2×D311yescharacteristic
49(C2×D7)⋊C4—61no—
50Dic7⋊C4—61no—
51C4×Dic7—31no—
52C4×Dic7—61no—
53C2×C7⋊C8—31no—
54#1132—34no—
55C2×#204—71no—
56#893—71no—
57#887—71no—
58C3×#107D711yescharacteristic, complemented
59#815—71no—
60#810—71no—
61#809—71no—
62C22×C3×C7C2×D411yescharacteristic
63C2×D21—22no—
64C2×D21—41no—
65C2×D21—41no—
66C3×C4×C7—41no—
67C3×C4×C7—41no—
68C3×C4×C7C22×C411yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—22no—
71Dic21—21no—
72Dic21—21no—
73#440—211noSylow
74C22×D7—61no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—31no—
79C2×Dic7—31no—
80C2×Dic7—61no—
81D28—32no—
82D28—61no—
83C4×D7—122no—
84Dic14—32no—
85Dic14—61no—
86C7×C8C4×D312yescomplemented
87C7⋊C8—61no—
88C2×D12—71no—
89C2×C4×D3—141no—
90C2×Dic6—71no—
91C2×C3×C8C2×D711yescharacteristic
92C3×C4⋊C4C2×D711yescharacteristic
93C3×C4⋊C4C2×D711yescharacteristic
94(C2×D3)⋊C4—141no—
95Dic3⋊C4—141no—
96C4×Dic3—71no—
97C4×Dic3—141no—
98C2×C3⋊C8—71no—
99#204—74no—
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103D21—42no—
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—211no—
107#107D2111yescharacteristic, complemented
108#103—211no—
109#102—211no—
110#97—211no—
111C22×C7D3×D411yescharacteristic
112C2×D7—62no—
113C2×D7—121no—
114C2×D7—121no—
115C4×C7—41no—
116C4×C7—41no—
117C4×C7C2×C4×D311yescharacteristic
118C4×C7#23911yescharacteristic
119Dic7—62no—
120Dic7—61no—
121Dic7—61no—
122C22×D3—141no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4C22×D711yescharacteristic
126C2×Dic3—71no—
127C2×Dic3—71no—
128C2×Dic3—141no—
129D12—72no—
130D12—141no—
131C4×D3—282no—
132Dic6—72no—
133Dic6—141no—
134C3×C8C4×D712yescomplemented
135C3⋊C8—141no—
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4—211no—
139C22×C4—421no—
140SD16—214no—
141C2×C8—211no—
142C2×C8C2×D2111yescharacteristic
143C4⋊C4—421no—
144C4⋊C4C2×D2111yescharacteristic
145C4⋊C4C2×D2111yescharacteristic
146C22⋊C4—421no—
147C42—211no—
148C42—421no—
149C2×C7#89311yescharacteristic
150C2×C7#91111yescharacteristic
151C2×C7#91811yescharacteristic
152D7—122no—
153C22×C3D4×D711yescharacteristic
154C2×D3—142no—
155C2×D3—281no—
156C2×D3—281no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4C2×C4×D711yescharacteristic
160C3×C4#116111yescharacteristic
161Dic3—142no—
162Dic3—141no—
163Dic3—141no—
164C23—421no—
165Q8—212no—
166Q8—421no—
167D4—212no—
168D4—421no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—211no—
172C2×C4—211no—
173C2×C4—421no—
174C2×C4—842no—
175C2×C4C22×D2111yescharacteristic
176C8—421no—
177C8C4×D2112yescomplemented
178C7#488911yescharacteristic, complemented, Sylow
179C2×C3#645611yescharacteristic
180C2×C3#647411yescharacteristic
181C2×C3#648111yescharacteristic
182D3—282no—
183C22—422no—
184C22—841no—
185C22—841no—
186C22D4×D2111yescharacteristic
187C4—41no—
188C4—41no—
189C4—422no—
190C4—421no—
191C4—421no—
192C4C2×C4×D2111yescharacteristic
193C4#6673211yescharacteristic
194C3#8966111yescharacteristic, complemented, Sylow
195C2—842no—
196C2#1062137611yescharacteristic
197C2#1062139411yescharacteristic
198C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
cosocleC2×C3×C4×C7C23
hypercenter#107D21
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. #10621370C2semidirect
4. C3×C7×#107C2semidirect
5. #10620654C2semidirect
6. #10620649C2semidirect
20. #66623C4semidirect
24. C7×#107D3semidirect
40. Dic42C2×C4semidirect
41. C3×C7×C8C2×C4semidirect
58. C3×#107D7semidirect
86. C7×C8C4×D3semidirect
107. #107D21semidirect
134. C3×C8C4×D7semidirect
136. C3×C7#440semidirect
177. C8C4×D21semidirect
178. C7#4889semidirect
194. C3#89661semidirect

Maximal subgroups

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

subgroupindex
C2×#666232
#106213762
#106213702
C3×C7×#1072
#106206542
#106206492
#106206482
#896613
#48897