Finite Group Databasebeta

← #13056690

Subgroups — #13056690

Order 1344.

320 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#10621657C211yesmaximal, characteristic
2C3×C7×#123C211yesmaximal, characteristic, complemented
3#10621412C212yesmaximal, complemented
4#10620673C211yesmaximal, characteristic, complemented
5#10620645C212yesmaximal, complemented
6#89825—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic, complemented
9C2×#4268—21no—
10C2×#4268C2212yes—
11#66730—22no—
12D4×D21—22no—
13#66728—21no—
14C3×C7×C4⋊C4—21no—
15C3×C7×C22⋊C4—21no—
16C3×C7×C22⋊C4C2211yescharacteristic, complemented
17#66636C2212yescomplemented
18#66631—22no—
19#66629—22no—
20#6548—31no—
21C7×#123D311yescharacteristic, complemented
22#6492—32no—
23#6408—31no—
24#6380—32no—
25#5053—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×D21—21no—
29C22×D21D412yes—
30C3×C7×D4—22no—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7D411yescharacteristic
35#4268—22no—
36#4268—22no—
37#4268—22no—
38C2×Dic21—21no—
39C2×Dic21—22no—
40C2×Dic21D412yes—
41D84—21no—
42C4×D21—22no—
43Dic42—21no—
44C2×C7×D4C2×D311yescharacteristic
45C22×C4×C7C2×D311yescharacteristic, complemented
46C2×C22⋊D7—32no—
47C2×C22⋊D7—61no—
48#1159—62no—
49D4×D7—62no—
50#1157—61no—
51C7×C4⋊C4—21no—
52C7×C22⋊C4—21no—
53C7×C22⋊C4C2×D311yescharacteristic, complemented
54C22⋊Dic7—32no—
55(C2×D7)⋊C4—62no—
56Dic7⋊C4—62no—
57#1004—71no—
58C3×#123D711yescharacteristic, complemented
59#929—72no—
60#834—71no—
61#806—72no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D412yes—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—22no—
68C2×D21—42no—
69C2×D21—81no—
70C3×C4×C7—21no—
71C3×C4×C7—41no—
72C3×C4×C7—41no—
73Dic21—22no—
74Dic21—42no—
75#458—211noSylow
76C23×C7#6711yescharacteristic
77C23×C7C22×D311yescharacteristic
78C22×D7—32no—
79C22×D7—61no—
80C7×D4—22no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7#6711yescharacteristic
85C22⋊D7—62no—
86C22⋊D7—62no—
87C22⋊D7—62no—
88C2×Dic7—32no—
89C2×Dic7—61no—
90C2×Dic7—62no—
91D28—61no—
92C4×D7—62no—
93Dic14—61no—
94C2×C3×D4C2×D711yescharacteristic
95C22×C3×C4C2×D711yescharacteristic, complemented
96C2×#67—72no—
97C2×#67—141no—
98#237—142no—
99D3×D4—142no—
100#235—141no—
101C3×C4⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4C2×D711yescharacteristic, complemented
104#217—72no—
105(C2×D3)⋊C4—142no—
106Dic3⋊C4—142no—
107C2×C3×C7—22no—
108C2×C3×C7—21no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111D21—42no—
112H(5,𝔽2)—211no—
113#123—212no—
114#123D2111yescharacteristic, complemented
115#99—211no—
116#99—212no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7D3×D412yes—
121C22×C7C2×#6711yescharacteristic
122C2×D7—62no—
123C2×D7—122no—
124C2×D7—241no—
125C4×C7—21no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—62no—
129Dic7—122no—
130C23×C3C22⋊D711yescharacteristic
131C23×C3C22×D711yescharacteristic
132C22×D3—72no—
133C22×D3—141no—
134C3×D4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4C22⋊D711yescharacteristic
139#67—142no—
140#67—142no—
141#67—142no—
142C2×Dic3—72no—
143C2×Dic3—141no—
144C2×Dic3—142no—
145D12—141no—
146C4×D3—142no—
147Dic6—141no—
148C3×C7#45811yescharacteristic, complemented, Hall
149#41—422no—
150#41—421no—
151C2×D4—212no—
152C2×D4—422no—
153C2×D4—421no—
154C2×D4C2×D2111yescharacteristic
155C22×C4C2×D2111yescharacteristic, complemented
156C4⋊C4—21no—
157C4⋊C4—422no—
158C22⋊C4—21no—
159C22⋊C4—212no—
160C22⋊C4—422no—
161C22⋊C4C2×D2111yescharacteristic, complemented
162C2×C7—22no—
163C2×C7—21no—
164C2×C7—41no—
165C2×C7#93711yescharacteristic
166D7—122no—
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3D4×D712yes—
171C22×C3C2×C22⋊D711yescharacteristic
172C2×D3—142no—
173C2×D3—282no—
174C2×D3—561no—
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—41no—
178Dic3—142no—
179Dic3—282no—
180C23—212no—
181C23—421no—
182C23#426811yescharacteristic
183C23C22×D2111yescharacteristic
184Q8—421no—
185D4—22no—
186D4—421no—
187D4—422no—
188D4—422no—
189D4—422no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—212no—
194C2×C4—422no—
195C2×C4—421no—
196C2×C4—422no—
197C2×C4#426811yescharacteristic
198C7#505311yescharacteristic, complemented, Sylow
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—41no—
202C2×C3#650011yescharacteristic
203D3—282no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—422no—
208C22—842no—
209C22—1681no—
210C22D4×D2112yes—
211C22C2×#426811yescharacteristic
212C4—21no—
213C4—41no—
214C4—41no—
215C4—422no—
216C4—842no—
217C3#8982511yescharacteristic, complemented, Sylow
218C2—21no—
219C2—22no—
220C2—41no—
221C2—842no—
222C2#1062142011yescharacteristic
subgroupG/H
centerC2#10621420
commutator, cosocleC23×C3×C7C23
FrattiniC23C22×D21
FittingC3×C7×#123C2
socleC2×C3×C7#120
prime radical#10621657C2
hypercenter#123D21
hypocenterC3×C7#458
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C3×C7×#123C2semidirect
3. #10621412C2semidirect
4. #10620673C2semidirect
5. #10620645C2semidirect
8. C22×C3×C4×C7C22semidirect
16. C3×C7×C22⋊C4C22semidirect
17. #66636C22semidirect
21. C7×#123D3semidirect
45. C22×C4×C7C2×D3semidirect
53. C7×C22⋊C4C2×D3semidirect
58. C3×#123D7semidirect
95. C22×C3×C4C2×D7semidirect
103. C3×C22⋊C4C2×D7semidirect
114. #123D21semidirect
148. C3×C7#458semidirect
155. C22×C4C2×D21semidirect
161. C22⋊C4C2×D21semidirect
198. C7#5053semidirect
217. C3#89825semidirect

Maximal subgroups

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

subgroupindex
#106216572
C3×C7×#1232
#106214122
#106206732
#106206452
#898253
#50537