Finite Group Databasebeta

← #13056376

Subgroups — #13056376

Order 1344.

296 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×Dic42C211yesmaximal, characteristic
2#10621407C211yesmaximal, characteristic
3C2×Dic84C212yesmaximal, complemented
4C3×C7×C22⋊C8C211yesmaximal, characteristic
5#10620655C212yesmaximal, complemented
6#89511—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×Dic21—21no—
9C2×Dic42—21no—
10C2×Dic42—24no—
11C2×Dic42C2211yescharacteristic
12C2×Dic42C2212yes—
13C2×C3×C7×C8C2212yes—
14#66636—21no—
15C4⋊Dic21C2211yescharacteristic
16#66629—21no—
17Dic84—24no—
18C22×Dic14—31no—
19#6487—31no—
20C2×Dic28—32no—
21C7×C22⋊C8D311yescharacteristic
22#6390—32no—
23#4739—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7D411yescharacteristic
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—24no—
29C2×Dic21—22no—
30C2×Dic21—21no—
31C2×Dic21—21no—
32Dic42—24no—
33Dic42—22no—
34Dic42—22no—
35Dic42D414yes—
36C3×C7×C8—22no—
37C22×C4×C7C2×D311yescharacteristic
38C22×Dic7—61no—
39C2×Dic14—32no—
40C2×Dic14—31no—
41C2×Dic14—64no—
42C2×Dic14—61no—
43C2×C7×C8C2×D312yes—
44C22⋊Dic7—61no—
45C4⋊Dic7—31no—
46Dic7⋊C4—61no—
47Dic28—64no—
48C22×Dic6—71no—
49#924—71no—
50C2×Dic12—72no—
51C3×C22⋊C8D711yescharacteristic
52#816—72no—
53C22×C3×C7—21no—
54C22×C3×C7—21no—
55C22×C3×C7Q1612yescomplemented
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—21no—
58C3×C4×C7C2×D412yes—
59Dic21—24no—
60Dic21—41no—
61Dic21—41no—
62#451—211noSylow
63C23×C7D1211yescharacteristic
64C2×C4×C7—22no—
65C2×C4×C7D1211yescharacteristic
66C2×C4×C7C22×D311yescharacteristic
67C2×Dic7—64no—
68C2×Dic7—62no—
69C2×Dic7—61no—
70C2×Dic7—61no—
71Dic14—34no—
72Dic14—64no—
73Dic14—62no—
74Dic14—62no—
75C7×C8—22no—
76C22×C3×C4C2×D711yescharacteristic
77C22×Dic3—141no—
78C2×Dic6—72no—
79C2×Dic6—71no—
80C2×Dic6—144no—
81C2×Dic6—141no—
82C2×C3×C8C2×D712yes—
83#217—141no—
84C4⋊Dic3—71no—
85Dic3⋊C4—141no—
86Dic12—144no—
87C2×C3×C7—22no—
88C2×C3×C7#12011yescharacteristic
89C2×C3×C7C2×Q1611yescharacteristic
90C2×C3×C7#13711yescharacteristic
91C22×Q8—211no—
92C2×Q16—212no—
93C22⋊Q8—211no—
94#103—212no—
95C22⋊C8D2111yescharacteristic
96C22×C7—21no—
97C22×C7—21no—
98C22×C7Dic1212yescomplemented
99C22×C7C2×D1211yescharacteristic
100C4×C7—21no—
101C4×C7D3×D412yes—
102Dic7—64no—
103Dic7—121no—
104Dic7—121no—
105C23×C3D2811yescharacteristic
106C2×C3×C4—22no—
107C2×C3×C4D2811yescharacteristic
108C2×C3×C4C22×D711yescharacteristic
109C2×Dic3—144no—
110C2×Dic3—142no—
111C2×Dic3—141no—
112C2×Dic3—141no—
113Dic6—74no—
114Dic6—144no—
115Dic6—142no—
116Dic6—142no—
117C3×C8—22no—
118C3×C7#45111yescharacteristic, complemented, Hall
119C2×Q8—212no—
120C2×Q8—211no—
121C2×Q8—424no—
122C2×Q8—421no—
123C22×C4—421no—
124C22×C4C2×D2111yescharacteristic
125Q16—424no—
126C2×C8C2×D2112yes—
127C4⋊C4—211no—
128C4⋊C4—421no—
129C22⋊C4—421no—
130C2×C7—22no—
131C2×C7#88211yescharacteristic
132C2×C7C2×Dic1211yescharacteristic
133C2×C7#90911yescharacteristic
134C22×C3—21no—
135C22×C3—21no—
136C22×C3Dic2812yescomplemented
137C22×C3C2×D2811yescharacteristic
138C3×C4—21no—
139C3×C4D4×D712yes—
140Dic3—144no—
141Dic3—281no—
142Dic3—281no—
143C23D8411yescharacteristic
144Q8—214no—
145Q8—424no—
146Q8—422no—
147Q8—422no—
148C2×C4—22no—
149C2×C4—424no—
150C2×C4—422no—
151C2×C4—421no—
152C2×C4—421no—
153C2×C4D8411yescharacteristic
154C2×C4C22×D2111yescharacteristic
155C8—22no—
156C7#473911yescharacteristic, complemented, Sylow
157C2×C3—22no—
158C2×C3#644511yescharacteristic
159C2×C3C2×Dic2811yescharacteristic
160C2×C3#647211yescharacteristic
161C22—21no—
162C22—21no—
163C22Dic8412yescomplemented
164C22C2×D8411yescharacteristic
165C4—21no—
166C4—424no—
167C4—841no—
168C4—841no—
169C4D4×D2112yes—
170C3#8951111yescharacteristic, complemented, Sylow
171C2—22no—
172C2#1062136511yescharacteristic
173C2C2×Dic8411yescharacteristic
174C2#1062139211yescharacteristic
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#451
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
3. C2×Dic84C2semidirect
5. #10620655C2semidirect
55. C22×C3×C7Q16semidirect
98. C22×C7Dic12semidirect
118. C3×C7#451semidirect
136. C22×C3Dic28semidirect
156. C7#4739semidirect
163. C22Dic84semidirect
170. C3#89511semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
#106214072
C2×Dic842
C3×C7×C22⋊C82
#106206552
#895113
#47397