Finite Group Databasebeta

← #13054726

Subgroups — #13054726

Order 1344.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#10621425C211yesmaximal, characteristic
2C2×C7×#216C211yesmaximal, characteristic, complemented
3#10620883C211yesmaximal, characteristic
4#10620853C211yesmaximal, characteristic
5C3×#6410C211yesmaximal, characteristic, complemented
6#10620528C211yesmaximal, characteristic, complemented
7#10620497C211yesmaximal, characteristic, complemented
8#89964—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12#66635—21no—
13#66634—21no—
14#66627C2211yescharacteristic
15C7×#216—21no—
16C7×#216—21no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1136C2211yescharacteristic
19C3×#1136C2211yescharacteristic
20#66571—21no—
21#66568—21no—
22#66559—21no—
23D21⋊C8—21no—
24C2×C7×Q16—31no—
25#6505—31no—
26#6505—31no—
27#6464—31no—
28#6410—31no—
29#6410D311yescharacteristic, complemented
30#6393—31no—
31#4755—71nomaximal, Hall
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37D84D411yescharacteristic
38C4×D21—21no—
39Dic42D411yescharacteristic
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8—21no—
45C3×C7⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C2×C7×Q8—31no—
49C2×C7×Q8C2×D311yescharacteristic
50#1157—31no—
51C7×Q16—61no—
52C7×Q16—61no—
53C2×C7×C8—31no—
54#1144—61no—
55#1144—61no—
56#1143—61no—
57#1143—61no—
58#1136—31no—
59#1136C2×D311yescharacteristic
60#1136C2×D311yescharacteristic
61#1131—61no—
62C8×D7—61no—
63C2×#216D711yescharacteristic, complemented
64#942—71no—
65#909—71no—
66#907—71no—
67C3×#101—71no—
68#824—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75Dic21—21no—
76C7×Dic3—41no—
77#473—211noSylow
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—61no—
81C7×Q8—61no—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C22⋊D7—61no—
86D28—31no—
87C4×D7—61no—
88Dic14—31no—
89C7×C8—31no—
90C7×C8—31no—
91C7⋊C8—21no—
92C7⋊C8—21no—
93C7⋊C8—61no—
94C2×C3×Q8C2×D711yescharacteristic
95#235—71no—
96C2×Dic6C2×D711yescharacteristic
97C3×M16—71no—
98C3×M16—71no—
99#216—21no—
100#216—21no—
101#216—141no—
102#215—141no—
103#208—71no—
104C2×C3⋊C8C2×D711yescharacteristic
105Dic12—141no—
106#204—141no—
107ΓL(1,𝔽25)—141no—
108D3×C8—141no—
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111D21—41no—
112#137—211no—
113#137—211no—
114C2×Q16—31no—
115#131—211no—
116#108—211no—
117#101—71no—
118#101—211no—
119C22×C7#23511yescharacteristic
120C2×D7—61no—
121C4×C7—41no—
122C4×C7—121no—
123C4×C7D3×D411yescharacteristic
124C4×C7D3×D411yescharacteristic
125Dic7—61no—
126C3×Q8—21no—
127C3×Q8—21no—
128C2×C3×C4—21no—
129C2×C3×C4C22×D711yescharacteristic
130#67—141no—
131C2×Dic3—21no—
132D12—71no—
133C4×D3—141no—
134Dic6—21no—
135Dic6—21no—
136Dic6—71no—
137C3×C8—141no—
138C3×C8—141no—
139C3⋊C8—141no—
140C3⋊C8C22⋊D711yescharacteristic
141C3⋊C8C22⋊D711yescharacteristic
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—211no—
144C2×Q8—31no—
145C2×Q8D3×D711yescharacteristic
146Q16—61no—
147Q16—61no—
148Q16—421no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152M16—71no—
153M16—71no—
154M16—211no—
155M16—421no—
156C2×C8—31no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7#88411yescharacteristic
160D7—121no—
161C22×C3#115911yescharacteristic
162C2×D3—141no—
163C3×C4—41no—
164C3×C4D4×D711yescharacteristic
165C3×C4C2×C22⋊D711yescharacteristic
166Dic3—41no—
167Dic3—141no—
168Q8—21no—
169Q8—21no—
170Q8—61no—
171Q8—61no—
172Q8—211no—
173D4—211no—
174D4—421no—
175C2×C4—21no—
176C2×C4—61no—
177C2×C4—421no—
178C2×C4C2×D3×D711yescharacteristic
179C8—31no—
180C8—31no—
181C8—141no—
182C8—141no—
183C8—421no—
184C7#475511yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3#650111yescharacteristic
187D3—281no—
188C22—421no—
189C22#6668611yescharacteristic
190C4—41no—
191C4—121no—
192C4—421no—
193C4D3×C22⋊D711yescharacteristic
194C4#6669411yescharacteristic
195C3#8996411yescharacteristic, complemented, Sylow
196C2—21no—
197C2—841no—
198C2#1062111911yescharacteristic
subgroupG/H
centerC2#10621119
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radical#66728C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#473
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C7×#216C2semidirect
5. C3×#6410C2semidirect
6. #10620528C2semidirect
7. #10620497C2semidirect
29. #6410D3semidirect
63. C2×#216D7semidirect
142. C3×C7#473semidirect
184. C7#4755semidirect
195. C3#89964semidirect

Maximal subgroups

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

subgroupindex
#106214252
C2×C7×#2162
#106208832
#106208532
C3×#64102
#106205282
#106204972
#899643
#47557