Finite Group Databasebeta

← #13056360

Subgroups — #13056360

Order 1344.

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

normal subgroups only

This group has 208 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×C4⋊Dic21C211yesmaximal, characteristic
2#10621407C211yesmaximal, characteristic
3C3×C7×C22⋊C8C211yesmaximal, characteristic
4#10620657C212yesmaximal, complemented
5#10620655C212yesmaximal, complemented
6#89494—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×Dic21—21no—
9C2×Dic42C2211yescharacteristic
10C2×C3×C7×C8C2212yes—
11#66636—21no—
12C4⋊Dic21—21no—
13C4⋊Dic21C2211yescharacteristic
14C4⋊Dic21C2212yes—
15#66629—21no—
16C2×C4⋊Dic7—31no—
17#6487—31no—
18C7×C22⋊C8D311yescharacteristic
19#6392—32no—
20#6390—32no—
21#4722—71nomaximal, Hall
22C23×C3×C7D411yescharacteristic
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7D411yescharacteristic
25C2×C3×C4×C7C2311yescharacteristic
26C2×Dic21—21no—
27C2×Dic21—21no—
28C2×Dic21—22no—
29C2×Dic21—42no—
30Dic42—22no—
31C3×C7×C8—22no—
32C22×C4×C7C2×D311yescharacteristic
33C22×Dic7—61no—
34C2×Dic14—31no—
35C2×C7×C8C2×D312yes—
36C22⋊Dic7—61no—
37C4⋊Dic7—31no—
38C4⋊Dic7—32no—
39C4⋊Dic7—61no—
40Dic7⋊C4—61no—
41C2×C4⋊Dic3—71no—
42#924—71no—
43C3×C22⋊C8D711yescharacteristic
44#818—72no—
45#816—72no—
46C22×C3×C7—21no—
47C22×C3×C7—21no—
48C22×C3×C7Q1612yes—
49C22×C3×C7C2×D411yescharacteristic
50C3×C4×C7—21no—
51C3×C4×C7#4112yes—
52Dic21—41no—
53Dic21—41no—
54Dic21—42no—
55#484—211noSylow
56C23×C7D1211yescharacteristic
57C2×C4×C7—22no—
58C2×C4×C7D1211yescharacteristic
59C2×C4×C7C22×D311yescharacteristic
60C2×Dic7—61no—
61C2×Dic7—61no—
62C2×Dic7—62no—
63C2×Dic7—122no—
64Dic14—62no—
65C7×C8—22no—
66C22×C3×C4C2×D711yescharacteristic
67C22×Dic3—141no—
68C2×Dic6—71no—
69C2×C3×C8C2×D712yes—
70#217—141no—
71C4⋊Dic3—72no—
72C4⋊Dic3—71no—
73C4⋊Dic3—141no—
74Dic3⋊C4—141no—
75C2×C3×C7—22no—
76C2×C3×C7#12311yescharacteristic
77C2×C3×C7C2×Q1611yescharacteristic
78C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
79C22⋊Q8—211no—
80C2×C4⋊C4—211no—
81#107—212no—
82#103—212no—
83C22⋊C8D2111yescharacteristic
84C22×C7—21no—
85C22×C7—21no—
86C22×C7Dic1212yes—
87C22×C7C2×D1211yescharacteristic
88C4×C7—21no—
89C4×C7#23712yes—
90Dic7—121no—
91Dic7—122no—
92Dic7—121no—
93C23×C3D2811yescharacteristic
94C2×C3×C4—22no—
95C2×C3×C4D2811yescharacteristic
96C2×C3×C4C22×D711yescharacteristic
97C2×Dic3—141no—
98C2×Dic3—141no—
99C2×Dic3—142no—
100C2×Dic3—282no—
101Dic6—142no—
102C3×C8—22no—
103C3×C7#48411yescharacteristic, complemented, Hall
104C2×Q8—211no—
105C22×C4—421no—
106C22×C4C2×D2111yescharacteristic
107C2×C8C2×D2112yes—
108C4⋊C4—212no—
109C4⋊C4—211no—
110C4⋊C4—421no—
111C4⋊C4—421no—
112C22⋊C4—421no—
113C2×C7—22no—
114C2×C7#88611yescharacteristic
115C2×C7C2×Dic1211yescharacteristic
116C2×C7#90811yescharacteristic
117C22×C3—21no—
118C22×C3—21no—
119C22×C3Dic2812yes—
120C22×C3C2×D2811yescharacteristic
121C3×C4—21no—
122C3×C4#115912yes—
123Dic3—281no—
124Dic3—282no—
125Dic3—281no—
126C23D8411yescharacteristic
127Q8—422no—
128C2×C4—22no—
129C2×C4—421no—
130C2×C4—421no—
131C2×C4—422no—
132C2×C4—842no—
133C2×C4D8411yescharacteristic
134C2×C4C22×D2111yescharacteristic
135C8—22no—
136C7#472211yescharacteristic, complemented, Sylow
137C2×C3—22no—
138C2×C3#644911yescharacteristic
139C2×C3C2×Dic2811yescharacteristic
140C2×C3#647111yescharacteristic
141C22—21no—
142C22—21no—
143C22Dic8412yes—
144C22C2×D8411yescharacteristic
145C4—21no—
146C4—841no—
147C4—842no—
148C4—841no—
149C4#6673012yes—
150C3#8949411yescharacteristic, complemented, Sylow
151C2—22no—
152C2#1062136911yescharacteristic
153C2C2×Dic8411yescharacteristic
154C2#1062139111yescharacteristic
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22⋊C8D21
hypocenterC3×C7#484
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #10620657C2semidirect
5. #10620655C2semidirect
103. C3×C7#484semidirect
136. C7#4722semidirect
150. C3#89494semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Dic212
#106214072
C3×C7×C22⋊C82
#106206572
#106206552
#894943
#47227