Finite Group Databasebeta

← #13056611

Subgroups — #13056611

Order 1344.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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×C3×C7×C4⋊C4C211yesmaximal, characteristic
3C2×#66629C212yesmaximal
4#10620670C212yesmaximal
5#10620670C211yesmaximal, characteristic
6#89751—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2212yes—
11C22×Dic21C2212yes—
12C2×Dic42—22no—
13C2×Dic42—24no—
14C3×C7×C4⋊C4—22no—
15#66629—24no—
16C22×Dic14—31no—
17C2×C7×C4⋊C4D311yescharacteristic
18C2×Dic7⋊C4—32no—
19C7⋊#95—31no—
20C7⋊#95—32no—
21#4979—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C2×C3×C4×C7—21no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—21no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7D412yes—
29C2×C3×C4×C7Q812yescomplemented
30C2×Dic21—22no—
31C2×Dic21—22no—
32C2×Dic21—22no—
33C2×Dic21—22no—
34C2×Dic21D414yes—
35C2×Dic21Q814yescomplemented
36Dic42—28no—
37C22×C4×C7C2×D311yescharacteristic
38C22×C4×C7C2×D311yescharacteristic
39C22×C4×C7C2×D311yescharacteristic
40C22×Dic7—32no—
41C22×Dic7—32no—
42C2×Dic14—64no—
43C2×Dic14—62no—
44C7×C4⋊C4—22no—
45Dic7⋊C4—64no—
46C22×Dic6—71no—
47C2×C3×C4⋊C4D711yescharacteristic
48C2×Dic3⋊C4—72no—
49C3⋊#95—71no—
50C3⋊#95—72no—
51C22×C3×C7C2×D412yes—
52C22×C3×C7C2×D411yescharacteristic
53C22×C3×C7C2×Q811yescharacteristic
54C22×C3×C7C2×Q812yes—
55C22×C3×C7#4111yescharacteristic
56C3×C4×C7—22no—
57C3×C4×C7—22no—
58C3×C4×C7—41no—
59Dic21—24no—
60Dic21—24no—
61#395—211noSylow
62C23×C7C22×D311yescharacteristic
63C2×C4×C7—21no—
64C2×C4×C7—22no—
65C2×C4×C7—21no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7Dic612yescomplemented
69C2×C4×C7#6712yes—
70C2×Dic7—34no—
71C2×Dic7—34no—
72C2×Dic7—62no—
73C2×Dic7—62no—
74C2×Dic7—62no—
75C2×Dic7—62no—
76Dic14—68no—
77C22×C3×C4C2×D711yescharacteristic
78C22×C3×C4C2×D711yescharacteristic
79C22×C3×C4C2×D711yescharacteristic
80C22×Dic3—72no—
81C22×Dic3—72no—
82C2×Dic6—144no—
83C2×Dic6—142no—
84C3×C4⋊C4—22no—
85Dic3⋊C4—144no—
86C2×C3×C7#12011yescharacteristic
87C2×C3×C7C22⋊Q812yes—
88C2×C3×C7C22⋊Q811yescharacteristic
89C2×C3×C7C4⋊Q812yes—
90C2×C3×C7C4⋊Q811yescharacteristic
91C22×Q8—211no—
92C2×C4⋊C4—212no—
93C2×C4⋊C4D2111yescharacteristic
94#95—211no—
95#95—212no—
96C22×C7C2×Dic611yescharacteristic
97C22×C7#23511yescharacteristic
98C22×C7D3×D412yes—
99C22×C7D3×Q812yes—
100C22×C7C2×#6711yescharacteristic
101C4×C7—22no—
102C4×C7—22no—
103C4×C7—41no—
104Dic7—64no—
105Dic7—64no—
106C23×C3C22×D711yescharacteristic
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4Dic1412yescomplemented
113C2×C3×C4C22⋊D712yes—
114C2×Dic3—74no—
115C2×Dic3—74no—
116C2×Dic3—142no—
117C2×Dic3—142no—
118C2×Dic3—142no—
119C2×Dic3—142no—
120Dic6—148no—
121C3×C7#39511yescharacteristic, complemented, Hall
122C2×Q8—424no—
123C2×Q8—422no—
124C22×C4—212no—
125C22×C4—212no—
126C22×C4C2×D2111yescharacteristic
127C22×C4C2×D2111yescharacteristic
128C22×C4C2×D2111yescharacteristic
129C4⋊C4—22no—
130C4⋊C4—424no—
131C2×C7#88812yes—
132C2×C7#89612yes—
133C2×C7#92411yescharacteristic
134C2×C7#93711yescharacteristic
135C2×C7#94411yescharacteristic
136C22×C3C2×Dic1411yescharacteristic
137C22×C3#115711yescharacteristic
138C22×C3D4×D712yes—
139C22×C3Q8×D712yes—
140C22×C3C2×C22⋊D711yescharacteristic
141C3×C4—22no—
142C3×C4—22no—
143C3×C4—41no—
144Dic3—144no—
145Dic3—144no—
146C23C22×D2111yescharacteristic
147Q8—428no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—214no—
154C2×C4—214no—
155C2×C4—422no—
156C2×C4—422no—
157C2×C4—422no—
158C2×C4—422no—
159C2×C4Dic4212yescomplemented
160C2×C4#426812yes—
161C7#497911yescharacteristic, complemented, Sylow
162C2×C3#645112yes—
163C2×C3#645912yes—
164C2×C3#648711yescharacteristic
165C2×C3#650011yescharacteristic
166C2×C3#650711yescharacteristic
167C22C2×Dic4211yescharacteristic
168C22#6672811yescharacteristic
169C22D4×D2112yes—
170C22Q8×D2112yes—
171C22C2×#426811yescharacteristic
172C4—22no—
173C4—22no—
174C4—41no—
175C4—424no—
176C4—424no—
177C3#8975111yescharacteristic, complemented, Sylow
178C2#1062137112yes—
179C2#1062137912yes—
180C2#1062140711yescharacteristic
181C2#1062142011yescharacteristic
182C2#1062142711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC2×C3×C7×C4⋊C4C2
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#395
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
29. C2×C3×C4×C7Q8semidirect
35. C2×Dic21Q8semidirect
68. C2×C4×C7Dic6semidirect
112. C2×C3×C4Dic14semidirect
121. C3×C7#395semidirect
159. C2×C4Dic42semidirect
161. C7#4979semidirect
177. C3#89751semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C2×C3×C7×C4⋊C42
C2×#666292
#106206702
#897513
#49797