Finite Group Databasebeta

← #13056623

Subgroups — #13056623

Order 1344.

228 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2#10621413C211yesmaximal, characteristic
3#10620669C211yesmaximal, characteristic, complemented
4#10620648C212yesmaximal, complemented
5#10620646C212yesmaximal, complemented
6#89739—31nomaximal, Hall
7C22×C3×C4×C7—21no—
8C22×C3×C4×C7C2211yescharacteristic
9C2×#4268—21no—
10C2×D84C2211yescharacteristic
11C3×C7×C4⋊C4—21no—
12C3×C7×C4⋊C4C2212yes—
13#66631—21no—
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2212yes—
16C2×C7×C4⋊C4D311yescharacteristic, complemented
17#6493—31no—
18C22⋊(C7⋊C8)—31no—
19#6383—32no—
20#6381—32no—
21#4967—71nomaximal, Hall
22C23×C3×C7D411yescharacteristic
23C22×D21—21no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7—42no—
27C2×C3×C4×C7D411yescharacteristic
28C2×C3×C4×C7C2311yescharacteristic
29#4268—42no—
30C2×Dic21—21no—
31D84—22no—
32C3⋊(C7⋊C8)—22no—
33C22×C4×C7—21no—
34C22×C4×C7C2×D311yescharacteristic
35C2×C22⋊D7—61no—
36C2×D28—31no—
37C7×C4⋊C4—21no—
38C7×C4⋊C4C2×D312yes—
39(C2×D7)⋊C4—61no—
40C4⋊Dic7—31no—
41C2×C7⋊C8—32no—
42C2×C3×C4⋊C4D711yescharacteristic, complemented
43#930—71no—
44#830—71no—
45#809—72no—
46#807—72no—
47C22×C3×C7—21no—
48C22×C3×C7—21no—
49C22×C3×C7D812yes—
50C22×C3×C7C2×D411yescharacteristic
51C2×D21—41no—
52C2×D21—41no—
53C2×D21—41no—
54C3×C4×C7—21no—
55C3×C4×C7—42no—
56C3×C4×C7#4112yes—
57Dic21—41no—
58#480—211noSylow
59C23×C7#6711yescharacteristic
60C22×D7—61no—
61C2×C4×C7—22no—
62C2×C4×C7—22no—
63C2×C4×C7—42no—
64C2×C4×C7#6711yescharacteristic
65C2×C4×C7C22×D311yescharacteristic
66C22⋊D7—122no—
67C2×Dic7—61no—
68D28—62no—
69C7⋊C8—62no—
70C22×C3×C4—21no—
71C22×C3×C4C2×D711yescharacteristic
72C2×#67—141no—
73C2×D12—71no—
74C3×C4⋊C4—21no—
75C3×C4⋊C4C2×D712yes—
76(C2×D3)⋊C4—141no—
77C4⋊Dic3—71no—
78C2×C3⋊C8—72no—
79C2×C3×C7—22no—
80C2×C3×C7#12311yescharacteristic
81C2×C3×C7C2×D811yescharacteristic
82C2×C3×C7#13711yescharacteristic
83D21—81no—
84#121—211no—
85C2×C4⋊C4D2111yescharacteristic, complemented
86#107—212no—
87#102—212no—
88C22⋊C8—211no—
89C22×C7—21no—
90C22×C7—21no—
91C22×C7#21312yes—
92C22×C7C2×#6711yescharacteristic
93C2×D7—121no—
94C2×D7—121no—
95C2×D7—121no—
96C4×C7—21no—
97C4×C7—42no—
98C4×C7#23512yes—
99Dic7—121no—
100C23×C3C22⋊D711yescharacteristic
101C22×D3—141no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—42no—
105C2×C3×C4C22⋊D711yescharacteristic
106C2×C3×C4C22×D711yescharacteristic
107#67—282no—
108C2×Dic3—141no—
109D12—142no—
110C3⋊C8—142no—
111C3×C7#48011yescharacteristic, complemented, Hall
112C2×D4—211no—
113C2×D4—421no—
114C22×C4—21no—
115C22×C4C2×D2111yescharacteristic
116C2×C8—212no—
117C4⋊C4—21no—
118C4⋊C4—211no—
119C4⋊C4C2×D2112yes—
120C22⋊C4—421no—
121C2×C7—22no—
122C2×C7#92911yescharacteristic
123C2×C7C2×#21311yescharacteristic
124C2×C7#94211yescharacteristic
125D7—241no—
126C22×C3—21no—
127C22×C3—21no—
128C22×C3#114112yes—
129C22×C3C2×C22⋊D711yescharacteristic
130C2×D3—281no—
131C2×D3—281no—
132C2×D3—281no—
133C3×C4—21no—
134C3×C4—42no—
135C3×C4#115712yes—
136Dic3—281no—
137C23—421no—
138C23#426811yescharacteristic
139D4—422no—
140D4—842no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—421no—
145C2×C4#426811yescharacteristic
146C2×C4C22×D2111yescharacteristic
147C8—422no—
148C7#496711yescharacteristic, complemented, Sylow
149C2×C3—22no—
150C2×C3#649211yescharacteristic
151C2×C3C2×#114111yescharacteristic
152C2×C3#650511yescharacteristic
153D3—561no—
154C22—21no—
155C22—21no—
156C22—841no—
157C22—841no—
158C22—841no—
159C22#6663212yes—
160C22C2×#426811yescharacteristic
161C4—21no—
162C4—42no—
163C4—841no—
164C4#6672812yes—
165C3#8973911yescharacteristic, complemented, Sylow
166C2—22no—
167C2—1681no—
168C2#1062141211yescharacteristic
169C2C2×#6663211yescharacteristic
170C2#1062142511yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC2×C3×C7×C4⋊C4C2
socleC22×C3×C7C2×D4
prime radical#10621413C2
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#480
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×C4⋊C4C2semidirect
3. #10620669C2semidirect
4. #10620648C2semidirect
5. #10620646C2semidirect
16. C2×C7×C4⋊C4D3semidirect
42. C2×C3×C4⋊C4D7semidirect
85. C2×C4⋊C4D21semidirect
111. C3×C7#480semidirect
148. C7#4967semidirect
165. C3#89739semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
#106214132
#106206692
#106206482
#106206462
#897393
#49677