Finite Group Databasebeta

← #13056614

Subgroups — #13056614

Order 1344.

372 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 105 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×C3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#66631C211yesmaximal, characteristic
3C2×#66631C212yesmaximal
4C2×#66629C212yesmaximal, complemented
5#10620670C211yesmaximal, characteristic, complemented
6#89764—31nomaximal, Hall
7C23×D21C2211yescharacteristic
8C22×C3×C4×C7C2212yes—
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2212yes—
12C3×C7×C4⋊C4—22no—
13#66631—22no—
14#66631—24no—
15#66629—24no—
16C2×C7×C4⋊C4D311yescharacteristic, complemented
17C2×(C2×D7)⋊C4—32no—
18C2×(C2×D7)⋊C4—31no—
19C2×Dic7⋊C4—32no—
20C7⋊#95—31no—
21#4993—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C22×D21—22no—
24C22×D21—21no—
25C22×D21—22no—
26C22×D21—21no—
27C22×D21Q812yes—
28C2×C3×C4×C7—22no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7D412yes—
33C2×Dic21—22no—
34C2×Dic21—22no—
35C2×Dic21—22no—
36C2×Dic21—21no—
37C2×Dic21D414yes—
38C23×D7—31no—
39C22×C4×C7C2×D312yes—
40C22×C4×C7C2×D311yescharacteristic
41C22×Dic7—32no—
42C22×Dic7—31no—
43C7×C4⋊C4—22no—
44(C2×D7)⋊C4—64no—
45(C2×D7)⋊C4—62no—
46Dic7⋊C4—64no—
47C2×C3×C4⋊C4D711yescharacteristic, complemented
48C2×(C2×D3)⋊C4—72no—
49C2×(C2×D3)⋊C4—71no—
50C2×Dic3⋊C4—72no—
51C3⋊#95—71no—
52C22×C3×C7C2×D412yes—
53C22×C3×C7C2×D411yescharacteristic
54C22×C3×C7C2×Q811yescharacteristic
55C22×C3×C7#4112yes—
56C22×C3×C7#4111yescharacteristic
57C2×D21—24no—
58C2×D21—22no—
59C2×D21—42no—
60C2×D21—41no—
61C2×D21—41no—
62C3×C4×C7—22no—
63C3×C4×C7—42no—
64Dic21—24no—
65Dic21—41no—
66#399—211noSylow
67C23×C7C22×D311yescharacteristic
68C22×D7—32no—
69C22×D7—62no—
70C22×D7—62no—
71C22×D7—61no—
72C22×D7—61no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7#6712yes—
78C2×Dic7—34no—
79C2×Dic7—62no—
80C2×Dic7—62no—
81C2×Dic7—62no—
82C2×Dic7—61no—
83C23×D3—71no—
84C22×C3×C4C2×D712yes—
85C22×C3×C4C2×D711yescharacteristic
86C22×Dic3—72no—
87C22×Dic3—71no—
88C3×C4⋊C4—22no—
89(C2×D3)⋊C4—144no—
90(C2×D3)⋊C4—142no—
91Dic3⋊C4—144no—
92C2×C3×C7C22⋊Q812yes—
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#12312yes—
95C2×C3×C7#12311yescharacteristic
96C2×C3×C7#12711yescharacteristic
97D21—42no—
98C2×C4⋊C4—212no—
99C2×C4⋊C4D2111yescharacteristic, complemented
100C2×C22⋊C4—212no—
101C2×C22⋊C4—211no—
102#95—211no—
103C22×C7#23512yes—
104C22×C7D3×D412yes—
105C22×C7D3×Q811yescharacteristic
106C22×C7#23911yescharacteristic
107C22×C7C2×#6711yescharacteristic
108C2×D7—64no—
109C2×D7—62no—
110C2×D7—122no—
111C2×D7—121no—
112C2×D7—121no—
113C4×C7—22no—
114C4×C7—42no—
115Dic7—64no—
116Dic7—121no—
117C23×C3C22×D711yescharacteristic
118C22×D3—72no—
119C22×D3—142no—
120C22×D3—142no—
121C22×D3—141no—
122C22×D3—141no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4C22⋊D712yes—
128C2×Dic3—74no—
129C2×Dic3—142no—
130C2×Dic3—142no—
131C2×Dic3—142no—
132C2×Dic3—141no—
133C3×C7#39911yescharacteristic, complemented, Hall
134C24—211no—
135C22×C4—212no—
136C22×C4—211no—
137C22×C4C2×D2112yes—
138C22×C4C2×D2111yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—424no—
141C22⋊C4—424no—
142C22⋊C4—422no—
143C2×C7#89412yes—
144C2×C7#89612yes—
145C2×C7#92911yescharacteristic
146C2×C7#94011yescharacteristic
147C2×C7#94611yescharacteristic
148D7—122no—
149C22×C3#115712yes—
150C22×C3D4×D712yes—
151C22×C3Q8×D711yescharacteristic
152C22×C3#116111yescharacteristic
153C22×C3C2×C22⋊D711yescharacteristic
154C2×D3—144no—
155C2×D3—142no—
156C2×D3—282no—
157C2×D3—281no—
158C2×D3—281no—
159C3×C4—22no—
160C3×C4—42no—
161Dic3—144no—
162Dic3—281no—
163C23—212no—
164C23—422no—
165C23—422no—
166C23—421no—
167C23—421no—
168C23C22×D2111yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—214no—
174C2×C4—422no—
175C2×C4—422no—
176C2×C4—422no—
177C2×C4—421no—
178C2×C4#426812yes—
179C7#499311yescharacteristic, complemented, Sylow
180C2×C3#645712yes—
181C2×C3#645912yes—
182C2×C3#649211yescharacteristic
183C2×C3#650311yescharacteristic
184C2×C3#650911yescharacteristic
185D3—282no—
186C22—424no—
187C22—422no—
188C22—842no—
189C22—841no—
190C22—841no—
191C22#6672812yes—
192C22D4×D2112yes—
193C22Q8×D2111yescharacteristic
194C22#6673211yescharacteristic
195C22C2×#426811yescharacteristic
196C4—22no—
197C4—42no—
198C4—424no—
199C4—841no—
200C3#8976411yescharacteristic, complemented, Sylow
201C2—842no—
202C2#1062137712yes—
203C2#1062137912yes—
204C2#1062141211yescharacteristic
205C2#1062142311yescharacteristic
206C2#1062142911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, cosocleC23×C3×C7C23
FittingC2×C3×C7×C4⋊C4C2
prime radicalC23×D21C22
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#399
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
1. C2×C3×C7×C4⋊C4C2semidirect
4. C2×#66629C2semidirect
5. #10620670C2semidirect
16. C2×C7×C4⋊C4D3semidirect
47. C2×C3×C4⋊C4D7semidirect
99. C2×C4⋊C4D21semidirect
133. C3×C7#399semidirect
179. C7#4993semidirect
200. C3#89764semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
C2×#666312
C2×#666292
#106206702
#897643
#49937