Finite Group Databasebeta

← #13052795

Subgroups — #13052795

Order 1344.

236 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 78 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#10621375C211yesmaximal, characteristic
2#10620999C211yesmaximal, characteristic
3C2×#66559C211yesmaximal, characteristic
4C7×#807C211yesmaximal, characteristic, complemented
5C3×#6381C211yesmaximal, characteristic, complemented
6#10620522C211yesmaximal, characteristic, complemented
7#10620520C211yesmaximal, characteristic, complemented
8#89653—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×D21⋊C4—21no—
11C3×C7×C4⋊C4C2211yescharacteristic
12#66631—21no—
13C4⋊Dic21C2211yescharacteristic
14C4×Dic21—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19#66577—21no—
20#66559C414yescomplemented
21C2×#1131—31no—
22#6455—31no—
23D7×C4⋊C4—31no—
24C7×#107—31no—
25#6391—31no—
26#6382—31no—
27#6381D311yescharacteristic, complemented
28#4881—71nomaximal, Hall
29C22×D21D411yescharacteristic
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C4×D21C2×C412yes—
35C4×D21Q812yes—
36C2×C7×Dic3—21no—
37C2×C3×Dic7—21no—
38D21⋊C4—42no—
39C3×C7⋊C8C2×C412yescomplemented
40C7×C3⋊C8C2×C412yescomplemented
41C2×C4×D7—31no—
42C2×C4×D7—61no—
43C2×C7×C8—31no—
44C7×C4⋊C4—31no—
45C7×C4⋊C4C2×D311yescharacteristic
46(C2×D7)⋊C4—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50C4×Dic7—61no—
51C2×C7⋊C8C2×D311yescharacteristic
52#1131—34no—
53C2×ΓL(1,𝔽25)—71no—
54#892—71no—
55D3×C4⋊C4—71no—
56C3×#107—71no—
57#817—71no—
58#808—71no—
59#807D711yescharacteristic, complemented
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21C4⋊C412yes—
64C3×C4×C7—41no—
65C3×C4×C7C22×C411yescharacteristic
66C3×C4×C7C2×Q811yescharacteristic
67Dic21—41no—
68Dic21C4⋊C412yes—
69C3×Dic7—41no—
70C7×Dic3—41no—
71#428—211noSylow
72C22×D7—31no—
73C2×C4×C7—21no—
74C2×C4×C7—61no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—31no—
78C2×Dic7—61no—
79C4×D7—32no—
80C4×D7—32no—
81C4×D7—122no—
82C7×C8—32no—
83C7⋊C8C4×D312yescomplemented
84C2×C4×D3—71no—
85C2×C4×D3—141no—
86C2×C3×C8—71no—
87C3×C4⋊C4—71no—
88C3×C4⋊C4C2×D711yescharacteristic
89(C2×D3)⋊C4—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3C2×D711yescharacteristic
92Dic3⋊C4—141no—
93C4×Dic3—141no—
94C2×C3⋊C8C2×D711yescharacteristic
95ΓL(1,𝔽25)—74no—
96C2×C3×C7C2×C4⋊C411yescharacteristic
97C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
98C2×C3×C7#13711yescharacteristic
99D21—22no—
100C2×M16—211no—
101#117—211no—
102C2×C4⋊C4—211no—
103#107—31no—
104#107—71no—
105#106—211no—
106#106—211no—
107C22×C7D3×D411yescharacteristic
108C2×D7—32no—
109C2×D7—61no—
110C2×D7—61no—
111C4×C7—41no—
112C4×C7—121no—
113C4×C7C2×C4×D311yescharacteristic
114C4×C7D3×Q811yescharacteristic
115Dic7—32no—
116Dic7—41no—
117Dic7—121no—
118C22×D3—71no—
119C2×C3×C4—21no—
120C2×C3×C4—141no—
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—71no—
124C2×Dic3—141no—
125C4×D3—72no—
126C4×D3—72no—
127C4×D3—282no—
128C3×C8—72no—
129C3⋊C8C4×D712yescomplemented
130C3×C7#42811yescharacteristic, complemented, Hall
131C22×C4—211no—
132C22×C4—421no—
133M16—214no—
134C2×C8—31no—
135C2×C8—71no—
136C4⋊C4—31no—
137C4⋊C4—71no—
138C4⋊C4—211no—
139C4⋊C4—421no—
140C4⋊C4D3×D711yescharacteristic
141C22⋊C4—421no—
142C42—421no—
143C2×C7D3×C4⋊C411yescharacteristic
144C2×C7#91111yescharacteristic
145C2×C7#91811yescharacteristic
146D7—62no—
147C22×C3D4×D711yescharacteristic
148C2×D3—72no—
149C2×D3—141no—
150C2×D3—141no—
151C3×C4—41no—
152C3×C4—281no—
153C3×C4C2×C4×D711yescharacteristic
154C3×C4Q8×D711yescharacteristic
155Dic3—41no—
156Dic3—72no—
157Dic3—281no—
158C23—211no—
159C2×C4—21no—
160C2×C4—61no—
161C2×C4—141no—
162C2×C4—211no—
163C2×C4—212no—
164C2×C4—212no—
165C2×C4—421no—
166C2×C4—842no—
167C2×C4C2×D3×D711yescharacteristic
168C8—32no—
169C8—72no—
170C7#488111yescharacteristic, complemented, Sylow
171C2×C3D7×C4⋊C411yescharacteristic
172C2×C3#647411yescharacteristic
173C2×C3#648111yescharacteristic
174D3—142no—
175C22—212no—
176C22—421no—
177C22—421no—
178C22#6669411yescharacteristic
179C4—41no—
180C4—121no—
181C4—212no—
182C4—281no—
183C4—841no—
184C4D21⋊Q811yescharacteristic
185C4C4×D3×D711yescharacteristic
186C3#8965311yescharacteristic, complemented, Sylow
187C2—422no—
188C2#1062096611yescharacteristic
189C2#1062104411yescharacteristic
190C2#1062107411yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#428
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C7×#807C2semidirect
5. C3×#6381C2semidirect
6. #10620522C2semidirect
7. #10620520C2semidirect
20. #66559C4semidirect
27. #6381D3semidirect
39. C3×C7⋊C8C2×C4semidirect
40. C7×C3⋊C8C2×C4semidirect
59. #807D7semidirect
83. C7⋊C8C4×D3semidirect
129. C3⋊C8C4×D7semidirect
130. C3×C7#428semidirect
170. C7#4881semidirect
186. C3#89653semidirect

Maximal subgroups

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

subgroupindex
#106213752
#106209992
C2×#665592
C7×#8072
C3×#63812
#106205222
#106205202
#896533
#48817