Finite Group Databasebeta

← #13053975

Subgroups — #13053975

Order 1344.

404 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 108 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×C7×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
3C2×(C2×D7)⋊Dic3C212yesmaximal
4#10620670C211yesmaximal, characteristic, complemented
5#10620529C212yesmaximal, complemented
6#90003—31nomaximal, Hall
7C23×C3×D7C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×Dic21C2212yes—
10C22×C7×Dic3C2212yes—
11C2×C7×Dic6—22no—
12C2×C7×Dic6—24no—
13C22×C3×Dic7C2211yescharacteristic
14C3×(C2×D7)⋊C4—22no—
15(C2×D7)⋊Dic3—24no—
16C22×C7×Q8—31no—
17C2×(C2×D7)⋊C4—32no—
18C2×(C2×D7)⋊C4D311yescharacteristic
19C7⋊#95—31no—
20C7⋊#95—32no—
21#4947—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C22×C3×D7—22no—
24C22×C3×D7—21no—
25C22×C3×D7—21no—
26C22×C3×D7—22no—
27C22×C3×D7Q812yescomplemented
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7D412yes—
30C2×Dic21—22no—
31C2×Dic21—22no—
32C2×Dic21—22no—
33C2×C7×Dic3—22no—
34C2×C7×Dic3—22no—
35C2×C7×Dic3D414yes—
36C7×Dic6—28no—
37C2×C3×Dic7—21no—
38C2×C3×Dic7—22no—
39C23×D7C2×D311yescharacteristic
40C2×C7×Q8—64no—
41C2×C7×Q8—62no—
42C22×C4×C7—32no—
43C22×C4×C7C2×D311yescharacteristic
44C22×Dic7—32no—
45C22×Dic7C2×D311yescharacteristic
46(C2×D7)⋊C4—22no—
47(C2×D7)⋊C4—64no—
48C22×Dic6D711yescharacteristic, complemented
49C2×C3×C22⋊C4—71no—
50C2×#217—72no—
51C3⋊#95—71no—
52C3⋊#95—72no—
53C22×C3×C7C2×D412yes—
54C22×C3×C7C2×D411yescharacteristic
55C22×C3×C7C2×Q811yescharacteristic
56C22×C3×C7#4111yescharacteristic
57C22×C3×C7#4112yes—
58C2×C3×D7—24no—
59C2×C3×D7—22no—
60C2×C3×D7—41no—
61C2×C3×D7—42no—
62C2×C3×D7—41no—
63C3×C4×C7—22no—
64Dic21—42no—
65C3×Dic7—41no—
66C7×Dic3—24no—
67#393—211noSylow
68C23×C7C22×D311yescharacteristic
69C22×D7—21no—
70C22×D7—22no—
71C22×D7—22no—
72C22×D7—21no—
73C22×D7Dic612yescomplemented
74C7×Q8—68no—
75C2×C4×C7—22no—
76C2×C4×C7—34no—
77C2×C4×C7—62no—
78C2×C4×C7—62no—
79C2×C4×C7#6712yes—
80C2×Dic7—22no—
81C2×Dic7—21no—
82C2×Dic7—62no—
83C2×Dic7—62no—
84C2×Dic7—62no—
85C24×C3—71no—
86C22×C3×C4—71no—
87C22×C3×C4C2×D711yescharacteristic
88C22×Dic3—72no—
89C22×Dic3C2×D712yes—
90C2×Dic6—22no—
91C2×Dic6—24no—
92C3×C22⋊C4—142no—
93#217—144no—
94C2×C3×C7#12011yescharacteristic
95C2×C3×C7C22⋊Q812yes—
96C2×C3×C7C22⋊Q811yescharacteristic
97C2×C3×C7#12412yes—
98C2×C3×C7#12411yescharacteristic
99C3×D7—42no—
100C22×Q8—31no—
101C2×C22⋊C4—71no—
102C2×C22⋊C4—212no—
103#95—212no—
104#95—211no—
105C22×C7C2×Dic611yescharacteristic
106C22×C7#23511yescharacteristic
107C22×C7D3×D412yes—
108C22×C7#23712yes—
109C22×C7C2×#6711yescharacteristic
110C2×D7—24no—
111C2×D7—22no—
112C2×D7—42no—
113C2×D7—41no—
114C2×D7—41no—
115C4×C7—22no—
116C4×C7—64no—
117Dic7—41no—
118Dic7—122no—
119C23×C3—72no—
120C23×C3—142no—
121C23×C3—141no—
122C23×C3—142no—
123C23×C3—141no—
124C23×C3C22×D711yescharacteristic
125C2×C3×C4—22no—
126C2×C3×C4—142no—
127C2×C3×C4—141no—
128C2×C3×C4C22⋊D712yes—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—142no—
132C2×Dic3—142no—
133C2×Dic3—142no—
134C2×Dic3C22⋊D714yes—
135Dic6—28no—
136C3×C7#39311yescharacteristic, complemented, Hall
137C24—71no—
138C2×Q8—64no—
139C2×Q8—62no—
140C22×C4—32no—
141C22×C4—71no—
142C22×C4—212no—
143C22×C4D3×D711yescharacteristic
144C22⋊C4—142no—
145C22⋊C4—424no—
146C2×C7#87812yes—
147C2×C7#88512yes—
148C2×C7#92411yescharacteristic
149C2×C7#93611yescharacteristic
150C2×C7#93711yescharacteristic
151D7—42no—
152C22×C3—144no—
153C22×C3—142no—
154C22×C3—281no—
155C22×C3—282no—
156C22×C3—281no—
157C22×C3Q8×D711yescharacteristic
158C22×C3#116112yes—
159C22×C3#116111yescharacteristic
160C22×C3C2×C22⋊D711yescharacteristic
161C22×C3C2×C22⋊D712yes—
162C3×C4—22no—
163C3×C4—281no—
164Dic3—24no—
165Dic3—282no—
166C23—72no—
167C23—142no—
168C23—141no—
169C23—141no—
170C23—142no—
171C23C2×D3×D711yescharacteristic
172Q8—68no—
173C2×C4—22no—
174C2×C4—34no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—142no—
178C2×C4—141no—
179C2×C4—422no—
180C2×C4—422no—
181C2×C4—422no—
182C2×C4#424512yes—
183C7#494711yescharacteristic, complemented, Sylow
184C2×C3—282no—
185C2×C3#650912yes—
186C2×C3#650911yescharacteristic
187C2×C3#651011yescharacteristic
188C2×C3#651012yes—
189C2×C3#651611yescharacteristic
190C22—144no—
191C22—142no—
192C22—281no—
193C22—282no—
194C22—281no—
195C22D7×Dic611yescharacteristic
196C22#6667012yes—
197C22#6667711yescharacteristic
198C22C2×#424511yescharacteristic
199C22D3×C22⋊D712yes—
200C4—22no—
201C4—64no—
202C4—281no—
203C4—842no—
204C3#9000311yescharacteristic, complemented, Sylow
205C2—282no—
206C2#1062092111yescharacteristic
207C2#1062093011yescharacteristic
208C2#1062098312yes—
209C2#1062101712yes—
210C2#1062113011yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
prime radicalC23×C3×D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#393
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C7×Dic6C2semidirect
4. #10620670C2semidirect
5. #10620529C2semidirect
27. C22×C3×D7Q8semidirect
48. C22×Dic6D7semidirect
73. C22×D7Dic6semidirect
136. C3×C7#393semidirect
183. C7#4947semidirect
204. C3#90003semidirect

Maximal subgroups

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

subgroupindex
C22×C7×Dic62
C2×C3×(C2×D7)⋊C42
C2×(C2×D7)⋊Dic32
#106206702
#106205292
#900033
#49477