Finite Group Databasebeta

← #13054472

Subgroups — #13054472

Order 1344.

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

normal subgroups only

This group has 216 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
1D4×Dic21C211yesmaximal, characteristic
2#10621032C211yesmaximal, characteristic, complemented
3C7×#832C211yesmaximal, characteristic
4C3×#6406C211yesmaximal, characteristic
5#10620522C211yesmaximal, characteristic, complemented
6#10620520C211yesmaximal, characteristic, complemented
7#10620518C211yesmaximal, characteristic, complemented
8#89533—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C22×Dic21—21no—
11C2×#4248—21no—
12#66636—21no—
13C4⋊Dic21C2211yescharacteristic
14C4×Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19#66577—21no—
20D4×Dic7—31no—
21#6451—31no—
22C7×#102—31no—
23#6406D311yescharacteristic
24#6391—31no—
25Dic7⋊C8—31no—
26#6382—31no—
27#4761—71nomaximal, Hall
28C23×C3×C7—21no—
29C3×C7×D4—21no—
30C3×C7×D4Q812yescomplemented
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21—42no—
34C2×Dic21D411yescharacteristic
35C2×Dic21D411yescharacteristic
36C2×C7×Dic3—21no—
37C2×C3×Dic7—21no—
38#4248—42no—
39C3×C7⋊C8—21no—
40C7×C3⋊C8—21no—
41C2×C7×D4C2×D311yescharacteristic
42C22×Dic7—61no—
43C2×Dic14—61no—
44C2×C7×C8—31no—
45C7×C4⋊C4—31no—
46C22⋊Dic7—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C2×C7⋊C8C2×D311yescharacteristic
52D4×Dic3—71no—
53#888—71no—
54C3×#102—71no—
55#832D711yescharacteristic
56#817—71no—
57Dic3⋊C8—71no—
58#808—71no—
59C22×C3×C7—22no—
60C22×C3×C7—41no—
61C22×C3×C7—41no—
62C22×C3×C7C2×D411yescharacteristic
63C3×C4×C7C2×Q811yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—21no—
66Dic21—41no—
67Dic21SD1612yes—
68C3×Dic7—41no—
69C7×Dic3—41no—
70#476—211noSylow
71C23×C7—21no—
72C7×D4—21no—
73C7×D4Dic612yescomplemented
74C2×C4×C7—61no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—31no—
78C2×Dic7—31no—
79C2×Dic7—61no—
80C2×Dic7—122no—
81Dic14—122no—
82C7×C8—61no—
83C7⋊C8—21no—
84C2×C3×D4C2×D711yescharacteristic
85C22×Dic3—141no—
86C2×Dic6—141no—
87C2×C3×C8—71no—
88C3×C4⋊C4—71no—
89#217—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3C2×D711yescharacteristic
92Dic3⋊C4—141no—
93C4×Dic3—71no—
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3×C7—42no—
96C2×C3×C7C22⋊Q811yescharacteristic
97C2×C3×C7C2×SD1611yescharacteristic
98C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
99C4⋊Q8—211no—
100C4×D4—211no—
101#106—211no—
102#106—211no—
103C4⋊C8—211no—
104#102—31no—
105#102—71no—
106C22×C7—22no—
107C22×C7—41no—
108C22×C7—41no—
109C22×C7D3×D411yescharacteristic
110C4×C7—121no—
111C4×C7C2×Dic611yescharacteristic
112C4×C7#23711yescharacteristic
113Dic7—32no—
114Dic7—41no—
115Dic7—61no—
116Dic7—121no—
117C23×C3—21no—
118C3×D4—21no—
119C3×D4Dic1412yescomplemented
120C2×C3×C4—141no—
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—71no—
124C2×Dic3—71no—
125C2×Dic3—141no—
126C2×Dic3—282no—
127Dic6—282no—
128C3×C8—141no—
129C3⋊C8—21no—
130C3×C7#47611yescharacteristic, complemented, Hall
131C2×Q8—421no—
132C2×D4D3×D711yescharacteristic
133C22×C4—421no—
134C2×C8—31no—
135C2×C8—71no—
136C4⋊C4—31no—
137C4⋊C4—71no—
138C4⋊C4—211no—
139C4⋊C4—421no—
140C22⋊C4—421no—
141C42—211no—
142C2×C7—42no—
143C2×C7#87811yescharacteristic
144C2×C7#91111yescharacteristic
145C2×C7D3×SD1611yescharacteristic
146C22×C3—22no—
147C22×C3—41no—
148C22×C3—41no—
149C22×C3D4×D711yescharacteristic
150C3×C4—281no—
151C3×C4C2×Dic1411yescharacteristic
152C3×C4#115911yescharacteristic
153Dic3—41no—
154Dic3—72no—
155Dic3—141no—
156Dic3—281no—
157C23—21no—
158Q8—842no—
159D4—21no—
160D4#424812yescomplemented
161C2×C4—61no—
162C2×C4—141no—
163C2×C4—211no—
164C2×C4—211no—
165C2×C4—421no—
166C2×C4—842no—
167C2×C4C2×D3×D711yescharacteristic
168C8—61no—
169C8—141no—
170C7#476111yescharacteristic, complemented, Sylow
171C2×C3—42no—
172C2×C3#644111yescharacteristic
173C2×C3#647411yescharacteristic
174C2×C3D7×SD1611yescharacteristic
175C22—22no—
176C22—41no—
177C22—41no—
178C22#6669411yescharacteristic
179C4—121no—
180C4—212no—
181C4—281no—
182C4—421no—
183C4—841no—
184C4#6668411yescharacteristic
185C4C2×#424811yescharacteristic
186C3#8953311yescharacteristic, complemented, Sylow
187C2—42no—
188C2#1062104411yescharacteristic
189C2#1062104911yescharacteristic
190C2#1062113811yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
Fitting, prime radicalC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#476
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10621032C2semidirect
5. #10620522C2semidirect
6. #10620520C2semidirect
7. #10620518C2semidirect
30. C3×C7×D4Q8semidirect
73. C7×D4Dic6semidirect
119. C3×D4Dic14semidirect
130. C3×C7#476semidirect
160. D4#4248semidirect
170. C7#4761semidirect
186. C3#89533semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
#106210322
C7×#8322
C3×#64062
#106205222
#106205202
#106205182
#895333
#47617