Finite Group Databasebeta

← #13054385

Subgroups — #13054385

Order 1344.

396 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#6548C211yesmaximal, characteristic, complemented
2#10621420C211yesmaximal, characteristic, complemented
3#10621134C212yesmaximal, complemented
4C7×#834C211yesmaximal, characteristic, complemented
5#10620530C212yesmaximal, complemented
6#89513—31nomaximal, Hall
7C23×D21C2211yescharacteristic, complemented
8C2×C3×C7×D4C2211yescharacteristic
9C2×#4268—21no—
10C2×C3×C22⋊D7—21no—
11C2×C3×C22⋊D7C2212yes—
12C3×#1159—22no—
13C3×D4×D7—22no—
14C3×#1157—21no—
15C2×#4246—22no—
16#66636C2211yescharacteristic, complemented
17#66631—21no—
18C7×#217C2212yescomplemented
19#66575—22no—
20#6548D311yescharacteristic, complemented
21#6500—31no—
22#6445—32no—
23C7×#99—31no—
24#6380—32no—
25#5244—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic, complemented
27C23×C3×C7C2311yescharacteristic
28C22×D21—22no—
29C22×D21—21no—
30C22×D21—42no—
31C22×C3×D7—21no—
32C22×C3×D7D412yescomplemented
33C3×C7×D4—22no—
34C2×C3×C4×C7D411yescharacteristic, complemented
35#4268—42no—
36C2×Dic21—21no—
37C2×C7×Dic3—22no—
38C3×C22⋊D7—22no—
39C3×C22⋊D7—22no—
40C3×C22⋊D7—22no—
41C2×C3×Dic7—21no—
42C2×C3×Dic7D412yescomplemented
43C3×D28—21no—
44C3×C4×D7—22no—
45C3×Dic14—21no—
46#4246—44no—
47C23×D7—31no—
48C2×C7×D4C2×D311yescharacteristic
49C2×C22⋊D7—21no—
50C2×C22⋊D7—61no—
51C2×C22⋊D7C2×D312yes—
52#1159—22no—
53D4×D7—22no—
54#1157—21no—
55C2×D28—62no—
56C7×C22⋊C4—32no—
57C22⋊Dic7—31no—
58(C2×D7)⋊C4—61no—
59(C2×D7)⋊C4—62no—
60C3×H(5,𝔽2)—71no—
61#937—71no—
62#937—72no—
63#834—72no—
64#834D711yescharacteristic, complemented
65C22×C3×C7—21no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7C2×D411yescharacteristic
69C22×C3×C7C2×D412yes—
70C2×D21—24no—
71C2×D21—22no—
72C2×D21—42no—
73C2×D21—41no—
74C2×D21—41no—
75C2×C3×D7—22no—
76C2×C3×D7—42no—
77C2×C3×D7—81no—
78C3×C4×C7—21no—
79Dic21—41no—
80C3×Dic7—22no—
81C7×Dic3—42no—
82B(4,𝔽2)—211noSylow
83C23×C7#6711yescharacteristic, complemented
84C23×C7C22×D311yescharacteristic
85C22×D7—21no—
86C22×D7—62no—
87C22×D7—61no—
88C22×D7—122no—
89C22×D7#6712yescomplemented
90C7×D4—22no—
91C2×C4×C7—62no—
92C2×C4×C7#6711yescharacteristic, complemented
93C22⋊D7—22no—
94C22⋊D7—22no—
95C22⋊D7—22no—
96C22⋊D7—122no—
97C2×Dic7—21no—
98C2×Dic7—61no—
99C2×Dic7#6712yescomplemented
100D28—21no—
101D28—124no—
102C4×D7—22no—
103Dic14—21no—
104C23×D3—71no—
105C3×#41—142no—
106C3×#41—141no—
107C2×C3×D4—72no—
108C2×C3×D4—142no—
109C2×C3×D4—141no—
110C2×C3×D4C2×D711yescharacteristic
111C2×#67—142no—
112C2×#67—141no—
113#217—71no—
114#217C2×D712yescomplemented
115(C2×D3)⋊C4—142no—
116(C2×D3)⋊C4—141no—
117C2×C3×C7—22no—
118C2×C3×C7—21no—
119C2×C3×C7—41no—
120C2×C3×C7#12011yescharacteristic
121D21—42no—
122C3×D7—42no—
123H(5,𝔽2)—71no—
124#120—212no—
125#120—211no—
126#99—31no—
127#99—212no—
128C22×C7—21no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7C2×#6711yescharacteristic
132C22×C7C2×#6712yes—
133C2×D7—22no—
134C2×D7—42no—
135C2×D7—64no—
136C2×D7—62no—
137C2×D7—81no—
138C2×D7—121no—
139C2×D7—122no—
140C2×D7—121no—
141C4×C7—21no—
142C4×C7—122no—
143Dic7—22no—
144Dic7—121no—
145C23×C3—72no—
146C23×C3—141no—
147C23×C3D2811yescharacteristic, complemented
148C23×C3C22×D711yescharacteristic
149C22×D3—141no—
150C22×D3—142no—
151C22×D3—282no—
152C3×Q8—141no—
153C3×D4—22no—
154C3×D4—141no—
155C3×D4—142no—
156C3×D4—142no—
157C3×D4—142no—
158C2×C3×C4—72no—
159C2×C3×C4—142no—
160C2×C3×C4—141no—
161C2×C3×C4D2811yescharacteristic, complemented
162#67—284no—
163#67—282no—
164C2×Dic3—22no—
165C2×Dic3—141no—
166C3×C7B(4,𝔽2)11yescharacteristic, complemented, Hall
167C24—211no—
168#41—142no—
169#41—141no—
170C2×D4—72no—
171C2×D4—142no—
172C2×D4—141no—
173C2×D4—422no—
174C2×D4—421no—
175C2×D4D3×D711yescharacteristic
176C22⋊C4—32no—
177C22⋊C4—211no—
178C22⋊C4—422no—
179C22⋊C4—421no—
180C2×C7—21no—
181C2×C7—22no—
182C2×C7—41no—
183C2×C7#95311yescharacteristic
184D7—42no—
185D7—122no—
186C22×C3—21no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—142no—
190C22×C3—282no—
191C22×C3—561no—
192C22×C3C2×D2811yescharacteristic
193C22×C3D4×D712yes—
194C2×D3—144no—
195C2×D3—142no—
196C2×D3—282no—
197C2×D3—281no—
198C2×D3—281no—
199C3×C4—21no—
200C3×C4—142no—
201Dic3—42no—
202Dic3—281no—
203C23—72no—
204C23—141no—
205C23—421no—
206C23—422no—
207C23—842no—
208C23#424611yescharacteristic, complemented
209C23C2×D3×D711yescharacteristic
210Q8—141no—
211D4—22no—
212D4—141no—
213D4—142no—
214D4—142no—
215D4—142no—
216D4—844no—
217D4—842no—
218C2×C4—62no—
219C2×C4—72no—
220C2×C4—142no—
221C2×C4—141no—
222C2×C4—421no—
223C2×C4#424611yescharacteristic, complemented
224C7#524411yescharacteristic, complemented, Sylow
225C2×C3—22no—
226C2×C3—21no—
227C2×C3—41no—
228C2×C3—282no—
229C2×C3#644511yescharacteristic
230D3—282no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—142no—
235C22—282no—
236C22—424no—
237C22—422no—
238C22—561no—
239C22—842no—
240C22—841no—
241C22—841no—
242C22C2×#424611yescharacteristic
243C22D7×#6712yes—
244C4—21no—
245C4—122no—
246C4—142no—
247C4—841no—
248C3#8951311yescharacteristic, complemented, Sylow
249C2—22no—
250C2—21no—
251C2—41no—
252C2—282no—
253C2—842no—
254C2#1062112611yescharacteristic
subgroupG/H
centerC2#10621126
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. C3×#6548C2semidirect
2. #10621420C2semidirect
3. #10621134C2semidirect
4. C7×#834C2semidirect
5. #10620530C2semidirect
7. C23×D21C22semidirect
16. #66636C22semidirect
18. C7×#217C22semidirect
20. #6548D3semidirect
26. C23×C3×C7D4semidirect
32. C22×C3×D7D4semidirect
34. C2×C3×C4×C7D4semidirect
42. C2×C3×Dic7D4semidirect
64. #834D7semidirect
83. C23×C7#67semidirect
89. C22×D7#67semidirect
92. C2×C4×C7#67semidirect
99. C2×Dic7#67semidirect
114. #217C2×D7semidirect
147. C23×C3D28semidirect
161. C2×C3×C4D28semidirect
166. C3×C7B(4,𝔽2)semidirect
208. C23#4246semidirect
223. C2×C4#4246semidirect
224. C7#5244semidirect
248. C3#89513semidirect

Maximal subgroups

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

subgroupindex
C3×#65482
#106214202
#106211342
C7×#8342
#106205302
#895133
#52447