Finite Group Databasebeta

← #13056401

Subgroups — #13056401

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#10621419C211yesmaximal, characteristic
2#10621371C211yesmaximal, characteristic, complemented
3C3×C7×#102C211yesmaximal, characteristic
4#10620671C211yesmaximal, characteristic
5#10620655C211yesmaximal, characteristic, complemented
6#10620654C211yesmaximal, characteristic, complemented
7#10620649C211yesmaximal, characteristic, complemented
8#89537—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42—21no—
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66636—21no—
15#66636—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6499—31no—
21#6451—31no—
22C7×#102D311yescharacteristic
23#6406—31no—
24#6390—31no—
25#6389—31no—
26#6384—31no—
27#4765—71nomaximal, Hall
28C23×C3×C7—21no—
29C3×C7×D4—21no—
30C3×C7×D4—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37Dic42—21no—
38Dic42—21no—
39Dic42—42no—
40C3×C7×C8—21no—
41C3⋊(C7⋊C8)—21no—
42C2×C7×D4C2×D311yescharacteristic
43C2×Dic14—31no—
44C2×Dic14—61no—
45C2×C7×C8C2×D311yescharacteristic
46C7×C4⋊C4C2×D311yescharacteristic
47C22⋊Dic7—61no—
48C22⋊Dic7—61no—
49C4⋊Dic7—31no—
50Dic7⋊C4—61no—
51C4×Dic7—31no—
52C2×C7⋊C8—31no—
53#936—71no—
54#888—71no—
55C3×#102D711yescharacteristic
56#832—71no—
57#816—71no—
58#815—71no—
59#810—71no—
60C22×C3×C7—41no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7C2×D411yescharacteristic
64C3×C4×C7—41no—
65C3×C4×C7#4111yescharacteristic
66C3×C4×C7#4111yescharacteristic
67Dic21—21no—
68Dic21—21no—
69Dic21—41no—
70Dic21—41no—
71#489—211noSylow
72C23×C7—21no—
73C7×D4—21no—
74C7×D4—21no—
75C2×C4×C7—21no—
76C2×C4×C7C22×D311yescharacteristic
77C2×Dic7—31no—
78C2×Dic7—31no—
79C2×Dic7—61no—
80C2×Dic7—61no—
81Dic14—61no—
82Dic14—61no—
83Dic14—122no—
84C7×C8—21no—
85C7⋊C8—61no—
86C2×C3×D4C2×D711yescharacteristic
87C2×Dic6—71no—
88C2×Dic6—141no—
89C2×C3×C8C2×D711yescharacteristic
90C3×C4⋊C4C2×D711yescharacteristic
91#217—141no—
92#217—141no—
93C4⋊Dic3—71no—
94Dic3⋊C4—141no—
95C4×Dic3—71no—
96C2×C3⋊C8—71no—
97C2×C3×C7—81no—
98C2×C3×C7#12411yescharacteristic
99C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
100C2×C3×C7#13711yescharacteristic
101C4⋊Q8—211no—
102#124—211no—
103#103—211no—
104#103—211no—
105#102—211no—
106#102D2111yescharacteristic
107#97—211no—
108C22×C7—41no—
109C22×C7—41no—
110C22×C7—41no—
111C22×C7D3×D411yescharacteristic
112C4×C7—41no—
113C4×C7#23511yescharacteristic
114C4×C7#23711yescharacteristic
115Dic7—61no—
116Dic7—61no—
117Dic7—121no—
118Dic7—121no—
119C23×C3—21no—
120C3×D4—21no—
121C3×D4—21no—
122C2×C3×C4—21no—
123C2×C3×C4C22×D711yescharacteristic
124C2×Dic3—71no—
125C2×Dic3—71no—
126C2×Dic3—141no—
127C2×Dic3—141no—
128Dic6—141no—
129Dic6—141no—
130Dic6—282no—
131C3×C8—21no—
132C3⋊C8—141no—
133C3×C7#48911yescharacteristic, complemented, Hall
134C2×Q8—211no—
135C2×Q8—421no—
136C2×D4C2×D2111yescharacteristic
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—81no—
146C2×C7#88511yescharacteristic
147C2×C7#91111yescharacteristic
148C2×C7#91511yescharacteristic
149C22×C3—41no—
150C22×C3—41no—
151C22×C3—41no—
152C22×C3D4×D711yescharacteristic
153C3×C4—41no—
154C3×C4#115711yescharacteristic
155C3×C4#115911yescharacteristic
156Dic3—141no—
157Dic3—141no—
158Dic3—281no—
159Dic3—281no—
160C23—21no—
161Q8—421no—
162Q8—421no—
163Q8—842no—
164D4—21no—
165D4—21no—
166C2×C4—21no—
167C2×C4—211no—
168C2×C4—211no—
169C2×C4—421no—
170C2×C4—421no—
171C2×C4C22×D2111yescharacteristic
172C8—21no—
173C8—421no—
174C7#476511yescharacteristic, complemented, Sylow
175C2×C3—81no—
176C2×C3#644811yescharacteristic
177C2×C3#647411yescharacteristic
178C2×C3#647811yescharacteristic
179C22—41no—
180C22—41no—
181C22—41no—
182C22D4×D2111yescharacteristic
183C4—41no—
184C4—421no—
185C4—421no—
186C4—841no—
187C4—841no—
188C4#6672811yescharacteristic
189C4#6673011yescharacteristic
190C3#8953711yescharacteristic, complemented, Sylow
191C2—81no—
192C2#1062136811yescharacteristic
193C2#1062139411yescharacteristic
194C2#1062139811yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×C7×D4C22
hypercenter#102D21
hypocenterC3×C7#489
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #10621371C2semidirect
5. #10620655C2semidirect
6. #10620654C2semidirect
7. #10620649C2semidirect
133. C3×C7#489semidirect
174. C7#4765semidirect
190. C3#89537semidirect

Maximal subgroups

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

subgroupindex
#106214192
#106213712
C3×C7×#1022
#106206712
#106206552
#106206542
#106206492
#895373
#47657