Finite Group Databasebeta

← #13056518

Subgroups — #13056518

Order 1344.

236 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 83 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×#66622C211yesmaximal, characteristic
2#10621375C211yesmaximal, characteristic
3C4⋊C4×D21C211yesmaximal, characteristic
4C3×C7×#107C211yesmaximal, characteristic, complemented
5#10620656C211yesmaximal, characteristic, complemented
6#10620647C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89653—31nomaximal, Hall
9C2×C4×D21—21no—
10C2×C4×D21C2211yescharacteristic
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66631—21no—
15C4⋊Dic21C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21—21no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#66622C414yescomplemented
21C2×#1131—31no—
22#6455—31no—
23D7×C4⋊C4—31no—
24C7×#107D311yescharacteristic, complemented
25#6391—31no—
26#6382—31no—
27#6381—31no—
28#4881—71nomaximal, Hall
29C22×D21D411yescharacteristic
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C4×D21—42no—
37C4×D21C2×C412yes—
38C4×D21Q812yes—
39C3×C7×C8C2×C412yescomplemented
40C3⋊(C7⋊C8)C2×C412yescomplemented
41C2×C4×D7—31no—
42C2×C4×D7—61no—
43C2×C7×C8C2×D311yescharacteristic
44C7×C4⋊C4C2×D311yescharacteristic
45C7×C4⋊C4C2×D311yescharacteristic
46(C2×D7)⋊C4—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7—31no—
49Dic7⋊C4—61no—
50C4×Dic7—61no—
51C2×C7⋊C8—31no—
52#1131—34no—
53C2×ΓL(1,𝔽25)—71no—
54#892—71no—
55D3×C4⋊C4—71no—
56C3×#107D711yescharacteristic, complemented
57#817—71no—
58#808—71no—
59#807—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21C4⋊C412yes—
64C3×C4×C7—41no—
65C3×C4×C7—41no—
66C3×C4×C7C22×C411yescharacteristic
67C3×C4×C7C2×Q811yescharacteristic
68Dic21—41no—
69Dic21—41no—
70Dic21C4⋊C412yes—
71#428—211noSylow
72C22×D7—31no—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—31no—
77C2×Dic7—61no—
78C2×Dic7—61no—
79C4×D7—32no—
80C4×D7—32no—
81C4×D7—122no—
82C7×C8C4×D312yescomplemented
83C7⋊C8—32no—
84C2×C4×D3—71no—
85C2×C4×D3—141no—
86C2×C3×C8C2×D711yescharacteristic
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C4⋊C4C2×D711yescharacteristic
89(C2×D3)⋊C4—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3—71no—
92Dic3⋊C4—141no—
93C4×Dic3—141no—
94C2×C3⋊C8—71no—
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—211no—
104#107D2111yescharacteristic, complemented
105#106—211no—
106#106—211no—
107C22×C7D3×D411yescharacteristic
108C2×D7—32no—
109C2×D7—61no—
110C2×D7—61no—
111C4×C7—41no—
112C4×C7—41no—
113C4×C7C2×C4×D311yescharacteristic
114C4×C7D3×Q811yescharacteristic
115Dic7—32no—
116Dic7—121no—
117Dic7—121no—
118C22×D3—71no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—71no—
123C2×Dic3—141no—
124C2×Dic3—141no—
125C4×D3—72no—
126C4×D3—72no—
127C4×D3—282no—
128C3×C8C4×D712yescomplemented
129C3⋊C8—72no—
130C3×C7#42811yescharacteristic, complemented, Hall
131C22×C4—211no—
132C22×C4—421no—
133M16—214no—
134C2×C8—211no—
135C2×C8C2×D2111yescharacteristic
136C4⋊C4—211no—
137C4⋊C4—211no—
138C4⋊C4—421no—
139C4⋊C4C2×D2111yescharacteristic
140C4⋊C4C2×D2111yescharacteristic
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—41no—
153C3×C4C2×C4×D711yescharacteristic
154C3×C4Q8×D711yescharacteristic
155Dic3—72no—
156Dic3—281no—
157Dic3—281no—
158C23—211no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—211no—
162C2×C4—212no—
163C2×C4—212no—
164C2×C4—421no—
165C2×C4—421no—
166C2×C4—842no—
167C2×C4C22×D2111yescharacteristic
168C8—212no—
169C8C4×D2112yescomplemented
170C7#488111yescharacteristic, complemented, Sylow
171C2×C3D7×C4⋊C411yescharacteristic
172C2×C3#647411yescharacteristic
173C2×C3#648111yescharacteristic
174D3—142no—
175C22—212no—
176C22—421no—
177C22—421no—
178C22D4×D2111yescharacteristic
179C4—41no—
180C4—41no—
181C4—212no—
182C4—841no—
183C4—841no—
184C4C2×C4×D2111yescharacteristic
185C4Q8×D2111yescharacteristic
186C3#8965311yescharacteristic, complemented, Sylow
187C2—422no—
188C2C4⋊C4×D2111yescharacteristic
189C2#1062139411yescharacteristic
190C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
cosocleC2×C3×C4×C7C23
hypercenter#107D21
hypocenterC3×C7#428
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
4. C3×C7×#107C2semidirect
5. #10620656C2semidirect
6. #10620647C2semidirect
7. #10620646C2semidirect
20. #66622C4semidirect
24. C7×#107D3semidirect
39. C3×C7×C8C2×C4semidirect
40. C3⋊(C7⋊C8)C2×C4semidirect
56. C3×#107D7semidirect
82. C7×C8C4×D3semidirect
104. #107D21semidirect
128. C3×C8C4×D7semidirect
130. C3×C7#428semidirect
169. C8C4×D21semidirect
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
C2×#666222
#106213752
C4⋊C4×D212
C3×C7×#1072
#106206562
#106206472
#106206462
#896533
#48817