Finite Group Databasebeta

← #13054389

Subgroups — #13054389

Order 1344.

396 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 60 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, complemented
2C7×#937C211yesmaximal, characteristic, complemented
3#10621132C212yesmaximal, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5#10620532C212yesmaximal, complemented
6#89804—31nomaximal, Hall
7C23×D3×C7C2211yescharacteristic, complemented
8C2×C3×C7×D4C2211yescharacteristic
9C2×#4268—21no—
10C2×#4268C2212yes—
11#66730—22no—
12D4×D21—22no—
13#66728—21no—
14C2×C7×#67—21no—
15C2×#4247—22no—
16C7×#217C2211yescharacteristic, complemented
17C7×(C2×D3)⋊C4—21no—
18C3×C22⋊Dic7C2212yescomplemented
19(C2×D3)⋊Dic7—22no—
20#6548—31no—
21C7×#120—31no—
22#6516—32no—
23#6408D311yescharacteristic, complemented
24#6380—32no—
25#4741—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic, complemented
27C23×C3×C7C2311yescharacteristic
28C22×D21—21no—
29C22×D21D412yescomplemented
30C22×D3×C7—22no—
31C22×D3×C7—21no—
32C22×D3×C7—42no—
33C3×C7×D4—22no—
34C2×C3×C4×C7D411yescharacteristic, complemented
35#4268—22no—
36#4268—22no—
37#4268—22no—
38C2×Dic21—21no—
39C2×Dic21D412yescomplemented
40D84—21no—
41C4×D21—22no—
42Dic42—21no—
43C7×#67—42no—
44C2×C7×Dic3—21no—
45C2×C3×Dic7—22no—
46#4247—44no—
47C24×C7—31no—
48C2×C7×D4—61no—
49C2×C7×D4C2×D311yescharacteristic
50C2×C22⋊D7—32no—
51C2×C22⋊D7—62no—
52C2×C22⋊D7—61no—
53#1159—62no—
54D4×D7—62no—
55#1157—61no—
56C7×C22⋊C4—31no—
57C7×C22⋊C4—61no—
58C22⋊Dic7—62no—
59C22⋊Dic7C2×D312yescomplemented
60#1004—71no—
61#937D711yescharacteristic, complemented
62#882—72no—
63C3×#99—71no—
64#806—72no—
65C22×C3×C7—21no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7C2×D412yes—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—22no—
71C2×D21—42no—
72C2×D21—81no—
73C2×D3×C7—24no—
74C2×D3×C7—22no—
75C2×D3×C7—41no—
76C2×D3×C7—42no—
77C2×D3×C7—41no—
78C3×C4×C7—21no—
79Dic21—22no—
80C3×Dic7—42no—
81C7×Dic3—41no—
82B(4,𝔽2)—211noSylow
83C23×C7—61no—
84C23×C7—62no—
85C23×C7—122no—
86C23×C7D1211yescharacteristic, complemented
87C23×C7C22×D311yescharacteristic
88C22×D7—32no—
89C22×D7—61no—
90C7×D4—22no—
91C7×D4—122no—
92C2×C4×C7—61no—
93C2×C4×C7D1211yescharacteristic, complemented
94C22⋊D7—62no—
95C22⋊D7—62no—
96C22⋊D7—62no—
97C22⋊D7—124no—
98C2×Dic7—22no—
99C2×Dic7—32no—
100C2×Dic7—61no—
101D28—61no—
102C4×D7—62no—
103Dic14—61no—
104C23×D3C2×D711yescharacteristic, complemented
105C2×C3×D4C2×D711yescharacteristic
106C2×#67—21no—
107C2×#67—72no—
108C2×#67—141no—
109#237—142no—
110D3×D4—142no—
111#235—141no—
112C2×D12—142no—
113C3×C22⋊C4—72no—
114#217C2×D711yescharacteristic, complemented
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—142no—
117C2×C3×C7—21no—
118C2×C3×C7—22no—
119C2×C3×C7—41no—
120C2×C3×C7#12011yescharacteristic
121D21—42no—
122D3×C7—42no—
123H(5,𝔽2)—211no—
124#120—31no—
125#120—212no—
126#99—71no—
127#99—212no—
128C22×C7—21no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7—64no—
132C22×C7—62no—
133C22×C7—122no—
134C22×C7—121no—
135C22×C7—121no—
136C22×C7C2×D1211yescharacteristic
137C22×C7D3×D412yes—
138C2×D7—62no—
139C2×D7—122no—
140C2×D7—241no—
141C4×C7—21no—
142C4×C7—121no—
143Dic7—42no—
144Dic7—62no—
145C23×C3C22⋊D711yescharacteristic, complemented
146C23×C3C22×D711yescharacteristic
147C22×D3—22no—
148C22×D3—21no—
149C22×D3—42no—
150C22×D3—72no—
151C22×D3—141no—
152C3×D4—22no—
153C2×C3×C4—142no—
154C2×C3×C4C22⋊D711yescharacteristic, complemented
155#67—42no—
156#67—142no—
157#67—142no—
158#67—142no—
159C2×Dic3—21no—
160C2×Dic3—72no—
161C2×Dic3—141no—
162D12—141no—
163D12—284no—
164C4×D3—142no—
165Dic6—141no—
166C3×C7B(4,𝔽2)11yescharacteristic, complemented, Hall
167C24—31no—
168#41—422no—
169#41—421no—
170C2×D4—61no—
171C2×D4—212no—
172C2×D4—422no—
173C2×D4—422no—
174C2×D4—421no—
175C2×D4D3×D711yescharacteristic
176C22⋊C4—31no—
177C22⋊C4—61no—
178C22⋊C4—72no—
179C22⋊C4—422no—
180C2×C7—22no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7—122no—
184C2×C7#88211yescharacteristic
185D7—122no—
186C22×C3—21no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3D4×D712yes—
190C22×C3C2×C22⋊D711yescharacteristic
191C2×D3—24no—
192C2×D3—22no—
193C2×D3—41no—
194C2×D3—42no—
195C2×D3—41no—
196C2×D3—142no—
197C2×D3—282no—
198C2×D3—561no—
199C3×C4—21no—
200C3×C4—282no—
201Dic3—41no—
202Dic3—142no—
203C23—61no—
204C23—62no—
205C23—122no—
206C23—212no—
207C23—421no—
208C23#424711yescharacteristic, complemented
209C23C2×D3×D711yescharacteristic
210Q8—421no—
211D4—22no—
212D4—122no—
213D4—421no—
214D4—422no—
215D4—422no—
216D4—422no—
217D4—844no—
218C2×C4—61no—
219C2×C4—142no—
220C2×C4—212no—
221C2×C4—422no—
222C2×C4—421no—
223C2×C4#424711yescharacteristic, complemented
224C7#474111yescharacteristic, complemented, Sylow
225C2×C3—21no—
226C2×C3—22no—
227C2×C3—41no—
228C2×C3#650011yescharacteristic
229D3—42no—
230D3—282no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—62no—
235C22—64no—
236C22—121no—
237C22—122no—
238C22—121no—
239C22—422no—
240C22—842no—
241C22—1681no—
242C22C2×#424711yescharacteristic
243C22#6669412yes—
244C4—21no—
245C4—121no—
246C4—282no—
247C4—422no—
248C3#8980411yescharacteristic, complemented, Sylow
249C2—21no—
250C2—22no—
251C2—41no—
252C2—122no—
253C2—842no—
254C2#1062113511yescharacteristic
subgroupG/H
centerC2#10621135
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
hypercenterC2×D4D3×D7
hypocenterC3×C7B(4,𝔽2)
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621657C2semidirect
2. C7×#937C2semidirect
3. #10621132C2semidirect
4. C3×#6408C2semidirect
5. #10620532C2semidirect
7. C23×D3×C7C22semidirect
16. C7×#217C22semidirect
18. C3×C22⋊Dic7C22semidirect
23. #6408D3semidirect
26. C23×C3×C7D4semidirect
29. C22×D21D4semidirect
34. C2×C3×C4×C7D4semidirect
39. C2×Dic21D4semidirect
59. C22⋊Dic7C2×D3semidirect
61. #937D7semidirect
86. C23×C7D12semidirect
93. C2×C4×C7D12semidirect
104. C23×D3C2×D7semidirect
114. #217C2×D7semidirect
145. C23×C3C22⋊D7semidirect
154. C2×C3×C4C22⋊D7semidirect
166. C3×C7B(4,𝔽2)semidirect
208. C23#4247semidirect
223. C2×C4#4247semidirect
224. C7#4741semidirect
248. C3#89804semidirect

Maximal subgroups

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

subgroupindex
#106216572
C7×#9372
#106211322
C3×#64082
#106205322
#898043
#47417