Finite Group Databasebeta

← #13052548

Subgroups — #13052548

Order 1344.

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

normal subgroups only

This group has 204 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
1C3×#6453C211yesmaximal, characteristic, complemented
2#10620980C211yesmaximal, characteristic
3#10620648C211yesmaximal, characteristic
4C7×#808C211yesmaximal, characteristic, complemented
5#10620524C211yesmaximal, characteristic, complemented
6Dic7⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic
8#89643—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×D21⋊C4—21no—
11C3×C7×C4⋊C4C2211yescharacteristic
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13C7×C4⋊Dic3C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×C4⋊Dic7—21no—
16C3×C4⋊Dic7C2211yescharacteristic
17C3×Dic7⋊C4—21no—
18C3×C4×Dic7C2211yescharacteristic
19#66575—21no—
20#6456—31no—
21#6453D311yescharacteristic, complemented
22C7×#106—31no—
23#6395—31no—
24Dic7⋊C8—31no—
25#6383—31no—
26#6381—31no—
27#5066—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31D84—21no—
32D84Q812yes—
33C2×C7×Dic3—21no—
34C2×C3×Dic7—21no—
35C2×C3×Dic7D411yescharacteristic
36C2×C3×Dic7D411yescharacteristic
37D21⋊C4—42no—
38C3⋊(C7⋊C8)—21no—
39C7×C3⋊C8—21no—
40C2×D28—31no—
41C2×C4×D7—61no—
42C2×C7×C8—31no—
43C7×C4⋊C4—31no—
44C7×C4⋊C4C2×D311yescharacteristic
45(C2×D7)⋊C4—61no—
46C4⋊Dic7—21no—
47C4⋊Dic7C2×D311yescharacteristic
48Dic7⋊C4—21no—
49C4×Dic7C2×D311yescharacteristic
50C2×C7⋊C8—31no—
51C3×#125—71no—
52C4×D12—71no—
53#809—71no—
54#809—71no—
55#808D711yescharacteristic, complemented
56#807—71no—
57C4⋊(C3⋊C8)—71no—
58C22×C3×C7C2×D411yescharacteristic
59C2×D21—22no—
60C2×D21—41no—
61C2×D21—41no—
62C3×C4×C7—41no—
63C3×C4×C7C2×Q811yescharacteristic
64C3×C4×C7#4111yescharacteristic
65C3×Dic7—21no—
66C3×Dic7—21no—
67C3×Dic7—41no—
68C7×Dic3—41no—
69#478—211noSylow
70C22×D7—61no—
71C2×C4×C7—21no—
72C2×C4×C7—61no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7#6711yescharacteristic
76C2×Dic7#6711yescharacteristic
77D28—32no—
78D28—61no—
79C4×D7—122no—
80C7×C8—61no—
81C7⋊C8—61no—
82C2×D12—71no—
83C2×C4×D3—141no—
84C3×C4⋊C4—71no—
85C3×C4⋊C4—141no—
86C3×C4⋊C4—141no—
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C42—71no—
89(C2×D3)⋊C4—141no—
90C4⋊Dic3C2×D711yescharacteristic
91C2×C3⋊C8—71no—
92C2×C3⋊C8C2×D711yescharacteristic
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#13511yescharacteristic
95C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
96D21—42no—
97#125—71no—
98C4×D4—211no—
99#107—211no—
100#106—31no—
101C4⋊C8—211no—
102#102—211no—
103#102—211no—
104C22×C7C2×#6711yescharacteristic
105C2×D7—62no—
106C2×D7—121no—
107C2×D7—121no—
108C4×C7—41no—
109C4×C7—121no—
110C4×C7D3×Q811yescharacteristic
111C4×C7#23911yescharacteristic
112Dic7—21no—
113Dic7—21no—
114Dic7—41no—
115C22×D3—141no—
116C2×C3×C4—21no—
117C2×C3×C4—71no—
118C2×C3×C4—71no—
119C2×C3×C4—141no—
120C2×C3×C4C22×D711yescharacteristic
121C2×Dic3—21no—
122D12—72no—
123D12—141no—
124C4×D3—282no—
125C3⋊C8—21no—
126C3⋊C8—141no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4—211no—
129C22×C4—421no—
130C2×C8—31no—
131C2×C8—211no—
132C4⋊C4—31no—
133C4⋊C4—71no—
134C4⋊C4—141no—
135C4⋊C4—141no—
136C4⋊C4D3×D711yescharacteristic
137C22⋊C4—421no—
138C42—71no—
139C2×C7#94611yescharacteristic
140C2×C7#94911yescharacteristic
141C2×C7#95011yescharacteristic
142D7—122no—
143C22×C3D4×D711yescharacteristic
144C2×D3—142no—
145C2×D3—281no—
146C2×D3—281no—
147C3×C4—41no—
148C3×C4—141no—
149C3×C4—141no—
150C3×C4—281no—
151C3×C4#115711yescharacteristic
152C3×C4Q8×D711yescharacteristic
153Dic3—41no—
154C23—421no—
155D4—212no—
156D4—421no—
157C2×C4—21no—
158C2×C4—61no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—141no—
162C2×C4—842no—
163C2×C4C2×D3×D711yescharacteristic
164C8—61no—
165C8—421no—
166C7#506611yescharacteristic, complemented, Sylow
167C2×C3#645911yescharacteristic
168C2×C3#647711yescharacteristic
169C2×C3#647911yescharacteristic
170D3—282no—
171C22—422no—
172C22—841no—
173C22—841no—
174C22D7×#6711yescharacteristic
175C4—41no—
176C4—121no—
177C4—141no—
178C4—141no—
179C4—281no—
180C4#6667311yescharacteristic
181C4D21⋊Q811yescharacteristic
182C3#8964311yescharacteristic, complemented, Sylow
183C2—842no—
184C2#1062097311yescharacteristic
185C2#1062105411yescharacteristic
186C2#1062106411yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#478
LCSG ▷ C2×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. C3×#6453C2semidirect
4. C7×#808C2semidirect
5. #10620524C2semidirect
6. Dic7⋊(C3⋊C8)C2semidirect
21. #6453D3semidirect
55. #808D7semidirect
127. C3×C7#478semidirect
166. C7#5066semidirect
182. C3#89643semidirect

Maximal subgroups

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

subgroupindex
C3×#64532
#106209802
#106206482
C7×#8082
#106205242
Dic7⋊(C3⋊C8)2
#106205032
#896433
#50667