Finite Group Databasebeta

← #13056430

Subgroups — #13056430

Order 1344.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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#10621430C211yesmaximal, characteristic
2#10621371C211yesmaximal, characteristic, complemented
3C3×C7×#103C211yesmaximal, characteristic, complemented
4#10620674C211yesmaximal, characteristic, complemented
5#10620660C211yesmaximal, characteristic
6#10620654C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic
8#89567—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×Dic42—21no—
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66631—21no—
15#66631—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6510—31no—
21#6451—31no—
22C7×#103D311yescharacteristic, complemented
23#6409—31no—
24#6395—31no—
25#6389—31no—
26#6383—31no—
27#4794—71nomaximal, Hall
28C22×D21—21no—
29C3×C7×Q8—21no—
30C3×C7×Q8—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37D84—21no—
38D84—21no—
39Dic42—42no—
40C3×C7×C8—21no—
41C3⋊(C7⋊C8)—21no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×D28—31no—
44C2×Dic14—61no—
45C2×C7×C8C2×D311yescharacteristic
46C7×C4⋊C4C2×D311yescharacteristic
47(C2×D7)⋊C4—61no—
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50Dic7⋊C4—61no—
51C4×Dic7—31no—
52C2×C7⋊C8—31no—
53#947—71no—
54#888—71no—
55C3×#103D711yescharacteristic, complemented
56#835—71no—
57#821—71no—
58#815—71no—
59#809—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—41no—
62C2×D21—41no—
63C2×D21—41no—
64C3×C4×C7—41no—
65C3×C4×C7—41no—
66C3×C4×C7#4111yescharacteristic
67C3×C4×C7#4111yescharacteristic
68Dic21—21no—
69Dic21—21no—
70Dic21—41no—
71#489—211noSylow
72C22×D7—61no—
73C7×Q8—21no—
74C7×Q8—21no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—31no—
79C2×Dic7—31no—
80C2×Dic7—61no—
81D28—61no—
82D28—61no—
83Dic14—122no—
84C7×C8—21no—
85C7⋊C8—61no—
86C2×C3×Q8C2×D711yescharacteristic
87C2×D12—71no—
88C2×Dic6—141no—
89C2×C3×C8C2×D711yescharacteristic
90C3×C4⋊C4C2×D711yescharacteristic
91(C2×D3)⋊C4—141no—
92(C2×D3)⋊C4—141no—
93C4⋊Dic3—71no—
94Dic3⋊C4—141no—
95C4×Dic3—71no—
96C2×C3⋊C8—71no—
97C2×C3×C7#12411yescharacteristic
98C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
99C2×C3×C7#13711yescharacteristic
100D21—81no—
101C4⋊Q8—211no—
102#124—211no—
103#103—211no—
104#103D2111yescharacteristic, complemented
105#102—211no—
106#102—211no—
107#97—211no—
108C22×C7D3×D411yescharacteristic
109C2×D7—121no—
110C2×D7—121no—
111C2×D7—121no—
112C4×C7—41no—
113C4×C7—41no—
114C4×C7#23511yescharacteristic
115C4×C7#23711yescharacteristic
116Dic7—61no—
117Dic7—61no—
118Dic7—121no—
119C22×D3—141no—
120C3×Q8—21no—
121C3×Q8—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4C22×D711yescharacteristic
125C2×Dic3—71no—
126C2×Dic3—71no—
127C2×Dic3—141no—
128D12—141no—
129D12—141no—
130Dic6—282no—
131C3×C8—21no—
132C3⋊C8—141no—
133C3×C7#48911yescharacteristic, complemented, Hall
134C2×Q8—421no—
135C2×Q8C2×D2111yescharacteristic
136C2×D4—211no—
137C2×C8—211no—
138C2×C8C2×D2111yescharacteristic
139C4⋊C4—211no—
140C4⋊C4—421no—
141C4⋊C4C2×D2111yescharacteristic
142C22⋊C4—421no—
143C22⋊C4—421no—
144C42—211no—
145C2×C7#88511yescharacteristic
146C2×C7#91411yescharacteristic
147C2×C7#91811yescharacteristic
148D7—241no—
149C22×C3D4×D711yescharacteristic
150C2×D3—281no—
151C2×D3—281no—
152C2×D3—281no—
153C3×C4—41no—
154C3×C4—41no—
155C3×C4#115711yescharacteristic
156C3×C4#115911yescharacteristic
157Dic3—141no—
158Dic3—141no—
159Dic3—281no—
160C23—421no—
161Q8—21no—
162Q8—21no—
163Q8—842no—
164D4—421no—
165D4—421no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—211no—
169C2×C4—211no—
170C2×C4—421no—
171C2×C4C22×D2111yescharacteristic
172C8—21no—
173C8—421no—
174C7#479411yescharacteristic, complemented, Sylow
175C2×C3#644811yescharacteristic
176C2×C3#647711yescharacteristic
177C2×C3#648111yescharacteristic
178D3—561no—
179C22—841no—
180C22—841no—
181C22—841no—
182C22D4×D2111yescharacteristic
183C4—41no—
184C4—41no—
185C4—421no—
186C4—421no—
187C4—841no—
188C4#6672811yescharacteristic
189C4#6673011yescharacteristic
190C3#8956711yescharacteristic, complemented, Sylow
191C2—1681no—
192C2#1062136811yescharacteristic
193C2#1062139711yescharacteristic
194C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#103D21
hypocenterC3×C7#489
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. #10621371C2semidirect
3. C3×C7×#103C2semidirect
4. #10620674C2semidirect
6. #10620654C2semidirect
22. C7×#103D3semidirect
55. C3×#103D7semidirect
104. #103D21semidirect
133. C3×C7#489semidirect
174. C7#4794semidirect
190. C3#89567semidirect

Maximal subgroups

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

subgroupindex
#106214302
#106213712
C3×C7×#1032
#106206742
#106206602
#106206542
#106206482
#895673
#47947