Finite Group Databasebeta

← #13056671

Subgroups — #13056671

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
1C3×C7×#121C211yesmaximal, characteristic
2C2×C4⋊Dic21C211yesmaximal, characteristic, complemented
3#10620671C212yesmaximal, complemented
4#10620669C211yesmaximal, characteristic, complemented
5#10620647C212yesmaximal, complemented
6#89806—31nomaximal, Hall
7C2×C3×C7×D4—21no—
8C2×C3×C7×D4C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21—21no—
11C3×C7×C4⋊C4C2211yescharacteristic
12C3×C7×C22⋊C4—21no—
13C4⋊Dic21—21no—
14C4⋊Dic21C2212yes—
15C2×C3⋊(C7⋊C8)C2212yes—
16C7×#121D311yescharacteristic
17C2×C4⋊Dic7—31no—
18#6406—32no—
19C22⋊(C7⋊C8)—31no—
20#6382—32no—
21#5034—71nomaximal, Hall
22C23×C3×C7—21no—
23C23×C3×C7D411yescharacteristic
24C3×C7×D4—22no—
25C3×C7×D4—42no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7D411yescharacteristic
29C2×C3×C4×C7C2311yescharacteristic
30C2×Dic21—22no—
31C2×Dic21—42no—
32C3⋊(C7⋊C8)—22no—
33C2×C7×D4—21no—
34C2×C7×D4C2×D311yescharacteristic
35C22×C4×C7C2×D311yescharacteristic
36C22×Dic7—61no—
37C7×C4⋊C4C2×D311yescharacteristic
38C7×C22⋊C4—21no—
39C4⋊Dic7—32no—
40C4⋊Dic7—61no—
41C2×C7⋊C8—32no—
42C3×#121D711yescharacteristic
43C2×C4⋊Dic3—71no—
44#832—72no—
45#830—71no—
46#808—72no—
47C22×C3×C7—21no—
48C22×C3×C7—21no—
49C22×C3×C7—41no—
50C22×C3×C7—41no—
51C22×C3×C7—41no—
52C22×C3×C7SD1612yes—
53C22×C3×C7C2×D411yescharacteristic
54C3×C4×C7—21no—
55C3×C4×C7—41no—
56C3×C4×C7#4112yes—
57Dic21—42no—
58#481—211noSylow
59C23×C7—21no—
60C23×C7#6711yescharacteristic
61C7×D4—22no—
62C7×D4—42no—
63C2×C4×C7—22no—
64C2×C4×C7—21no—
65C2×C4×C7#6711yescharacteristic
66C2×C4×C7C22×D311yescharacteristic
67C2×Dic7—62no—
68C2×Dic7—122no—
69C7⋊C8—62no—
70C2×C3×D4—21no—
71C2×C3×D4C2×D711yescharacteristic
72C22×C3×C4C2×D711yescharacteristic
73C22×Dic3—141no—
74C3×C4⋊C4C2×D711yescharacteristic
75C3×C22⋊C4—21no—
76C4⋊Dic3—72no—
77C4⋊Dic3—141no—
78C2×C3⋊C8—72no—
79C2×C3×C7—22no—
80C2×C3×C7—81no—
81C2×C3×C7#12311yescharacteristic
82C2×C3×C7C2×SD1611yescharacteristic
83C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
84#121D2111yescharacteristic
85C2×C4⋊C4—211no—
86#106—212no—
87#102—212no—
88C22⋊C8—211no—
89C22×C7—21no—
90C22×C7—21no—
91C22×C7—41no—
92C22×C7—41no—
93C22×C7—41no—
94C22×C7#21412yes—
95C22×C7C2×#6711yescharacteristic
96C4×C7—21no—
97C4×C7—41no—
98C4×C7#23712yes—
99Dic7—122no—
100C23×C3—21no—
101C23×C3C22⋊D711yescharacteristic
102C3×D4—22no—
103C3×D4—42no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4C22⋊D711yescharacteristic
107C2×C3×C4C22×D711yescharacteristic
108C2×Dic3—142no—
109C2×Dic3—282no—
110C3⋊C8—142no—
111C3×C7#48111yescharacteristic, complemented, Hall
112C2×D4—21no—
113C2×D4C2×D2111yescharacteristic
114C22×C4—421no—
115C22×C4C2×D2111yescharacteristic
116C2×C8—212no—
117C4⋊C4—212no—
118C4⋊C4—421no—
119C4⋊C4C2×D2111yescharacteristic
120C22⋊C4—21no—
121C2×C7—22no—
122C2×C7—81no—
123C2×C7C2×#21411yescharacteristic
124C2×C7#93511yescharacteristic
125C2×C7#94911yescharacteristic
126C22×C3—21no—
127C22×C3—21no—
128C22×C3—41no—
129C22×C3—41no—
130C22×C3—41no—
131C22×C3#114212yes—
132C22×C3C2×C22⋊D711yescharacteristic
133C3×C4—21no—
134C3×C4—41no—
135C3×C4#115912yes—
136Dic3—282no—
137C23—21no—
138C23#426811yescharacteristic
139D4—22no—
140D4—42no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—422no—
144C2×C4—842no—
145C2×C4#426811yescharacteristic
146C2×C4C22×D2111yescharacteristic
147C8—422no—
148C7#503411yescharacteristic, complemented, Sylow
149C2×C3—22no—
150C2×C3—81no—
151C2×C3C2×#114211yescharacteristic
152C2×C3#649811yescharacteristic
153C2×C3#651211yescharacteristic
154C22—21no—
155C22—21no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22#6663312yes—
160C22C2×#426811yescharacteristic
161C4—21no—
162C4—41no—
163C4—842no—
164C4#6673012yes—
165C3#8980611yescharacteristic, complemented, Sylow
166C2—22no—
167C2—81no—
168C2C2×#6663311yescharacteristic
169C2#1062141811yescharacteristic
170C2#1062143211yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
Fitting, prime radicalC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#481
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊Dic21C2semidirect
3. #10620671C2semidirect
4. #10620669C2semidirect
5. #10620647C2semidirect
111. C3×C7#481semidirect
148. C7#5034semidirect
165. C3#89806semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
C2×C4⋊Dic212
#106206712
#106206692
#106206472
#898063
#50347