Finite Group Databasebeta

← #13056293

Subgroups — #13056293

Order 1344.

364 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 121 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
2C2×C4⋊Dic21C212yesmaximal
3C2×#66629C211yesmaximal, characteristic
4C3×C7×#95C211yesmaximal, characteristic
5#10620670C212yesmaximal
6#89428—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×C3×C4×C7C2212yes—
9C22×Dic21C2211yescharacteristic
10C22×Dic21C2212yes—
11C22×Dic21C2211yescharacteristic
12C2×Dic42—22no—
13C2×Dic42—24no—
14C4⋊Dic21—24no—
15#66629—22no—
16C22×Dic14—31no—
17C2×C4⋊Dic7—32no—
18C2×Dic7⋊C4—31no—
19C7×#95D311yescharacteristic
20C7⋊#95—32no—
21#4656—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7Q814yescomplemented
28C2×Dic21—21no—
29C2×Dic21—22no—
30C2×Dic21—22no—
31C2×Dic21—21no—
32C2×Dic21—21no—
33C2×Dic21—22no—
34C2×Dic21D414yes—
35C2×Dic21Q812yescomplemented
36Dic42—28no—
37C22×C4×C7C2×D312yes—
38C22×C4×C7C2×D311yescharacteristic
39C22×Dic7—31no—
40C22×Dic7—32no—
41C22×Dic7—31no—
42C2×Dic14—64no—
43C2×Dic14—62no—
44C4⋊Dic7—64no—
45Dic7⋊C4—62no—
46C22×Dic6—71no—
47C2×C4⋊Dic3—72no—
48C2×Dic3⋊C4—71no—
49C3×#95D711yescharacteristic
50C3⋊#95—72no—
51C22×C3×C7C2×D411yescharacteristic
52C22×C3×C7C2×D412yes—
53C22×C3×C7C2×Q812yes—
54C22×C3×C7C2×Q811yescharacteristic
55C22×C3×C7#4111yescharacteristic
56C3×C4×C7—22no—
57C3×C4×C7—24no—
58Dic21—24no—
59Dic21—22no—
60Dic21—41no—
61#395—211noSylow
62C23×C7C22×D311yescharacteristic
63C2×C4×C7—22no—
64C2×C4×C7—22no—
65C2×C4×C7—22no—
66C2×C4×C7Dic614yescomplemented
67C2×C4×C7D1212yes—
68C2×Dic7—34no—
69C2×Dic7—32no—
70C2×Dic7—61no—
71C2×Dic7—62no—
72C2×Dic7—62no—
73C2×Dic7—61no—
74C2×Dic7—61no—
75C2×Dic7—62no—
76Dic14—68no—
77C22×C3×C4C2×D712yes—
78C22×C3×C4C2×D711yescharacteristic
79C22×Dic3—71no—
80C22×Dic3—72no—
81C22×Dic3—71no—
82C2×Dic6—144no—
83C2×Dic6—142no—
84C4⋊Dic3—144no—
85Dic3⋊C4—142no—
86C2×C3×C7#12011yescharacteristic
87C2×C3×C7C22⋊Q812yes—
88C2×C3×C7C22⋊Q811yescharacteristic
89C2×C3×C7C4⋊Q811yescharacteristic
90C2×C3×C7C4⋊Q812yes—
91C22×Q8—211no—
92C2×C4⋊C4—211no—
93C2×C4⋊C4—212no—
94#95—212no—
95#95D2111yescharacteristic
96C22×C7C2×Dic612yes—
97C22×C7C2×D1211yescharacteristic
98C22×C7D3×D412yes—
99C22×C7#23711yescharacteristic
100C22×C7D3×Q811yescharacteristic
101C4×C7—24no—
102C4×C7—22no—
103Dic7—64no—
104Dic7—62no—
105Dic7—121no—
106C23×C3C22×D711yescharacteristic
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4Dic1414yescomplemented
111C2×C3×C4D2812yes—
112C2×Dic3—74no—
113C2×Dic3—72no—
114C2×Dic3—141no—
115C2×Dic3—142no—
116C2×Dic3—142no—
117C2×Dic3—141no—
118C2×Dic3—141no—
119C2×Dic3—142no—
120Dic6—148no—
121C3×C7#39511yescharacteristic, complemented, Hall
122C2×Q8—424no—
123C2×Q8—422no—
124C22×C4—211no—
125C22×C4—212no—
126C22×C4—211no—
127C22×C4C2×D2112yes—
128C22×C4C2×D2111yescharacteristic
129C4⋊C4—424no—
130C4⋊C4—422no—
131C2×C7#86911yescharacteristic
132C2×C7#87812yes—
133C2×C7#88211yescharacteristic
134C2×C7#88812yes—
135C2×C7#89711yescharacteristic
136C22×C3C2×Dic1412yes—
137C22×C3C2×D2811yescharacteristic
138C22×C3D4×D712yes—
139C22×C3#115911yescharacteristic
140C22×C3Q8×D711yescharacteristic
141C3×C4—24no—
142C3×C4—22no—
143Dic3—144no—
144Dic3—142no—
145Dic3—281no—
146C23C22×D2111yescharacteristic
147Q8—428no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—214no—
152C2×C4—212no—
153C2×C4—421no—
154C2×C4—422no—
155C2×C4—422no—
156C2×C4—421no—
157C2×C4—421no—
158C2×C4—422no—
159C2×C4Dic4214yescomplemented
160C2×C4D8412yes—
161C7#465611yescharacteristic, complemented, Sylow
162C2×C3#643211yescharacteristic
163C2×C3#644112yes—
164C2×C3#644511yescharacteristic
165C2×C3#645112yes—
166C2×C3#646011yescharacteristic
167C22C2×Dic4212yes—
168C22C2×D8411yescharacteristic
169C22D4×D2112yes—
170C22#6673011yescharacteristic
171C22Q8×D2111yescharacteristic
172C4—24no—
173C4—22no—
174C4—424no—
175C4—422no—
176C4—841no—
177C3#8942811yescharacteristic, complemented, Sylow
178C2#1062135211yescharacteristic
179C2#1062136112yes—
180C2#1062136511yescharacteristic
181C2#1062137112yes—
182C2#1062138011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC3×C7×#95C2
hypercenter#95D21
hypocenterC3×C7#395
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
27. C2×C3×C4×C7Q8semidirect
35. C2×Dic21Q8semidirect
66. C2×C4×C7Dic6semidirect
110. C2×C3×C4Dic14semidirect
121. C3×C7#395semidirect
159. C2×C4Dic42semidirect
161. C7#4656semidirect
177. C3#89428semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C2×C4⋊Dic212
C2×#666292
C3×C7×#952
#106206702
#894283
#46567