Finite Group Databasebeta

← #13054232

Subgroups — #13054232

Order 1344.

220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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#10621407C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic
3C2×C3×#1136C211yesmaximal, characteristic
4C2×#66571C211yesmaximal, characteristic, complemented
5#10620520C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620508C211yesmaximal, characteristic, complemented
8#89824—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12#66636—21no—
13C4⋊Dic21C2211yescharacteristic
14#66629—21no—
15C7×#217—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C7×Dic3⋊C4—21no—
18C3×#1136—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20C2×C3×C7⋊C8C2211yescharacteristic
21#66571—22no—
22C7×C22⋊Q8—31no—
23C2×#1144—31no—
24#6487—31no—
25C2×#1136D311yescharacteristic
26#6409—31no—
27#6384—31no—
28#6382—31no—
29#5137—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7D411yescharacteristic
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21—21no—
37Dic42—21no—
38Dic42—21no—
39C2×C7×Dic3—21no—
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6—21no—
43C3×C7⋊C8—21no—
44C3×C7⋊C8D412yes—
45C2×C7×Q8—31no—
46C22×C4×C7C2×D311yescharacteristic
47C2×Dic14—31no—
48C7×C4⋊C4—31no—
49C7×C4⋊C4—61no—
50C7×C22⋊C4—61no—
51C22⋊Dic7—61no—
52#1144—62no—
53C4⋊Dic7—31no—
54Dic7⋊C4—61no—
55#1136—22no—
56C2×C7⋊C8C2×D311yescharacteristic
57C2×C7⋊C8C2×D311yescharacteristic
58C2×C3×M16—71no—
59#924—71no—
60#924D711yescharacteristic
61C2×Dic12—71no—
62#817—71no—
63#816—71no—
64#816—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
72Dic21—41no—
73Dic21—41no—
74C7×Dic3—41no—
75C7×Dic3—41no—
76#470—211noSylow
77C23×C7D1211yescharacteristic
78C7×Q8—61no—
79C7×Q8—61no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—61no—
83C2×C4×C7—61no—
84C2×C4×C7D1211yescharacteristic
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—61no—
87C2×Dic7—61no—
88Dic14—61no—
89Dic14—61no—
90C7⋊C8—21no—
91C7⋊C8#6712yes—
92C22×C3×C4C2×D711yescharacteristic
93C2×Dic6—71no—
94C2×Dic6C2×D711yescharacteristic
95C3×M16—142no—
96C2×C3×C8—71no—
97C2×C3×C8—71no—
98#217—21no—
99#217—141no—
100C4⋊Dic3—71no—
101C4⋊Dic3C2×D711yescharacteristic
102Dic3⋊C4—21no—
103Dic3⋊C4—141no—
104Dic12—142no—
105C2×C3×C7—41no—
106C2×C3×C7#12111yescharacteristic
107C2×C3×C7#13711yescharacteristic
108C2×C3×C7#13711yescharacteristic
109C2×Q16—211no—
110C2×M16—71no—
111C22⋊Q8—31no—
112C22⋊Q8—211no—
113#106—211no—
114#103—211no—
115#103—211no—
116C22×C7—21no—
117C22×C7—21no—
118C22×C7—21no—
119C22×C7C2×D1211yescharacteristic
120C4×C7—21no—
121C4×C7—121no—
122C4×C7—121no—
123C4×C7#23511yescharacteristic
124C4×C7C2×#6711yescharacteristic
125Dic7—121no—
126Dic7—121no—
127C23×C3C22⋊D711yescharacteristic
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4C22⋊D711yescharacteristic
131C2×C3×C4C22×D711yescharacteristic
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—141no—
135C2×Dic3—141no—
136Dic6—21no—
137Dic6—21no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—72no—
141C3×C8—141no—
142C3×C7#47011yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8—211no—
145C22×C4D3×D711yescharacteristic
146Q16—422no—
147M16—142no—
148C2×C8—71no—
149C2×C8—71no—
150C4⋊C4—31no—
151C4⋊C4—61no—
152C4⋊C4—211no—
153C4⋊C4—421no—
154C22⋊C4—61no—
155C22⋊C4—421no—
156C2×C7—41no—
157C2×C7#90911yescharacteristic
158C2×C7#90911yescharacteristic
159C2×C7#93011yescharacteristic
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3C2×C22⋊D711yescharacteristic
164C3×C4—21no—
165C3×C4D4×D711yescharacteristic
166C3×C4#115911yescharacteristic
167Dic3—41no—
168Dic3—41no—
169Dic3—281no—
170Dic3—281no—
171C23#424711yescharacteristic
172Q8—61no—
173Q8—61no—
174Q8—421no—
175Q8—421no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4#424711yescharacteristic
183C2×C4C2×D3×D711yescharacteristic
184C8—72no—
185C8—141no—
186C7#513711yescharacteristic, complemented, Sylow
187C2×C3—41no—
188C2×C3#650211yescharacteristic
189C2×C3#650511yescharacteristic
190C2×C3#651411yescharacteristic
191C22—21no—
192C22—21no—
193C22—21no—
194C22C2×#424711yescharacteristic
195C4—21no—
196C4—121no—
197C4—121no—
198C4—841no—
199C4—841no—
200C4#6668611yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8982411yescharacteristic, complemented, Sylow
203C2—41no—
204C2#1062088311yescharacteristic
205C2#1062088411yescharacteristic
206C2#1062112811yescharacteristic
subgroupG/H
centerC22C2×#4247
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×#66571C2semidirect
5. #10620520C2semidirect
6. #10620510C2semidirect
7. #10620508C2semidirect
142. C3×C7#470semidirect
186. C7#5137semidirect
202. C3#89824semidirect

Maximal subgroups

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

subgroupindex
#106214072
C7×#9242
C2×C3×#11362
C2×#665712
#106205202
#106205102
#106205082
#898243
#51377