Finite Group Databasebeta

← #13054239

Subgroups — #13054239

Order 1344.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66627C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic
3C3×#6487C211yesmaximal, characteristic
4C2×#66569C211yesmaximal, characteristic, complemented
5#10620521C211yesmaximal, characteristic, complemented
6#10620507C211yesmaximal, characteristic, complemented
7#10620505C211yesmaximal, characteristic, complemented
8#89824—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66627—22no—
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×#217—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C7×Dic3⋊C4—21no—
18C3×C22⋊Dic7—21no—
19C3×C4⋊Dic7C2211yescharacteristic
20C3×Dic7⋊C4—21no—
21#66569—22no—
22C7×C22⋊Q8—31no—
23C2×#1144—31no—
24#6487D311yescharacteristic
25C2×#1136—31no—
26#6409—31no—
27#6384—31no—
28#6382—31no—
29#5052—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7D411yescharacteristic
34C2×C3×C4×C7C2311yescharacteristic
35C2×C7×Dic3—21no—
36C2×C7×Dic3—21no—
37C7×Dic6—21no—
38C7×Dic6—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C3⋊(C7⋊C8)D412yes—
45C2×C7×Q8—31no—
46C22×C4×C7C2×D311yescharacteristic
47C2×Dic14C2×D311yescharacteristic
48C7×C4⋊C4—31no—
49C7×C4⋊C4—61no—
50C7×C22⋊C4—61no—
51C22⋊Dic7—21no—
52#1144—62no—
53C4⋊Dic7C2×D311yescharacteristic
54Dic7⋊C4—21no—
55#1136—62no—
56C2×C7⋊C8—31no—
57C2×C7⋊C8—31no—
58C3×C22⋊Q8—71no—
59C2×#216—71no—
60#924D711yescharacteristic
61C2×#208—71no—
62#835—71no—
63#810—71no—
64#808—71no—
65C22×C3×C7—21no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7C2×D411yescharacteristic
69C3×C4×C7—21no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72C3×Dic7—41no—
73C3×Dic7—41no—
74C7×Dic3—41no—
75C7×Dic3—41no—
76#470—211noSylow
77C23×C7#6711yescharacteristic
78C7×Q8—61no—
79C7×Q8—61no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—61no—
83C2×C4×C7—61no—
84C2×C4×C7#6711yescharacteristic
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—21no—
87C2×Dic7—21no—
88Dic14—21no—
89Dic14—21no—
90C7⋊C8—32no—
91C7⋊C8—61no—
92C2×C3×Q8—71no—
93C22×C3×C4C2×D711yescharacteristic
94C2×Dic6C2×D711yescharacteristic
95C3×C4⋊C4—71no—
96C3×C4⋊C4—141no—
97C3×C22⋊C4—141no—
98#217—21no—
99#216—142no—
100C4⋊Dic3C2×D711yescharacteristic
101Dic3⋊C4—21no—
102#208—142no—
103C2×C3⋊C8—71no—
104C2×C3⋊C8—71no—
105C2×C3×C7—41no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7#13711yescharacteristic
108C2×C3×C7#13711yescharacteristic
109C2×Q16—211no—
110C2×M16—211no—
111C22⋊Q8—31no—
112C22⋊Q8—71no—
113#106—211no—
114#103—211no—
115#103—211no—
116C22×C7—21no—
117C22×C7—21no—
118C22×C7—21no—
119C22×C7C2×#6711yescharacteristic
120C4×C7—21no—
121C4×C7—121no—
122C4×C7—121no—
123C4×C7D3×D411yescharacteristic
124C4×C7#23711yescharacteristic
125Dic7—41no—
126Dic7—41no—
127C23×C3C22⋊D711yescharacteristic
128C3×Q8—141no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4C22⋊D711yescharacteristic
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—21no—
138Dic6—21no—
139Dic6—21no—
140C3⋊C8—72no—
141C3⋊C8—141no—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8—71no—
145C22×C4D3×D711yescharacteristic
146Q16—422no—
147M16—422no—
148C2×C8—211no—
149C2×C8—211no—
150C4⋊C4—31no—
151C4⋊C4—61no—
152C4⋊C4—71no—
153C4⋊C4—141no—
154C22⋊C4—61no—
155C22⋊C4—141no—
156C2×C7—41no—
157C2×C7#93911yescharacteristic
158C2×C7#94211yescharacteristic
159C2×C7#95111yescharacteristic
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3C2×C22⋊D711yescharacteristic
164C3×C4—21no—
165C3×C4—281no—
166C3×C4—281no—
167C3×C4D4×D711yescharacteristic
168C3×C4#115911yescharacteristic
169Dic3—41no—
170Dic3—41no—
171C23#424511yescharacteristic
172Q8—61no—
173Q8—61no—
174Q8—141no—
175Q8—141no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4#424511yescharacteristic
183C2×C4C2×D3×D711yescharacteristic
184C8—212no—
185C8—421no—
186C7#505211yescharacteristic, complemented, Sylow
187C2×C3—41no—
188C2×C3#650211yescharacteristic
189C2×C3#650511yescharacteristic
190C2×C3#651411yescharacteristic
191C22—21no—
192C22—21no—
193C22—21no—
194C22C2×#424511yescharacteristic
195C4—21no—
196C4—121no—
197C4—121no—
198C4—281no—
199C4—281no—
200C4#6668411yescharacteristic
201C4#6669411yescharacteristic
202C3#8982411yescharacteristic, complemented, Sylow
203C2—41no—
204C2#1062086911yescharacteristic
205C2#1062087111yescharacteristic
206C2#1062113311yescharacteristic
subgroupG/H
centerC22C2×#4245
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#470
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#66569C2semidirect
5. #10620521C2semidirect
6. #10620507C2semidirect
7. #10620505C2semidirect
142. C3×C7#470semidirect
186. C7#5052semidirect
202. C3#89824semidirect

Maximal subgroups

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

subgroupindex
C2×#666272
C7×#9242
C3×#64872
C2×#665692
#106205212
#106205072
#106205052
#898243
#50527