Finite Group Databasebeta

← #13054790

Subgroups — #13054790

Order 1344.

304 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 58 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#10621564C211yesmaximal, characteristic
2#10621425C211yesmaximal, characteristic, complemented
3C7×#942C211yesmaximal, characteristic, complemented
4C3×#6510C211yesmaximal, characteristic, complemented
5#10620516C211yesmaximal, characteristic, complemented
6#10620514C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89974—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic, complemented
10#66728C2211yescharacteristic
11C7×#235C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13D7×#67—22no—
14#66681—21no—
15D3×D28—21no—
16#66670—21no—
17#66669—21no—
18#66635—21no—
19#66634—21no—
20#66627C2211yescharacteristic, complemented
21C7×#216—21no—
22C7×#215—21no—
23C7×#208C2211yescharacteristic, complemented
24C3×(C2×D7)⋊C4—22no—
25C3×C4×Dic7C2211yescharacteristic, complemented
26#6553—31no—
27C7×#137—31no—
28#6510D311yescharacteristic, complemented
29#6505—31no—
30#6411—31no—
31#6399—31no—
32#6397—31no—
33#5061—71nomaximal, Hall
34C22×C3×D7D412yes—
35C2×D3×D7—22no—
36C3×C7×Q8—21no—
37C3×C7×Q8—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2311yescharacteristic
40#4268—21no—
41D84D411yescharacteristic
42C4×D21—21no—
43Dic42D411yescharacteristic
44C7×#67—21no—
45C7×D12D411yescharacteristic
46C4×D3×C7—21no—
47C7×Dic6D411yescharacteristic
48C2×C3×Dic7—21no—
49C3×D28—21no—
50C3×D28—21no—
51#4246—22no—
52#4245—22no—
53Dic3×D7—22no—
54C3⋊(C7⋊C8)—21no—
55C7×C3⋊C8—21no—
56C7×#41—31no—
57C2×C7×Q8C2×D311yescharacteristic, complemented
58#1161—61no—
59D4×D7—61no—
60D4×D7—62no—
61#1157—31no—
62#1157—61no—
63C2×D28—61no—
64C2×D28C2×D311yescharacteristic
65C7×Q16—61no—
66C7×SD16—61no—
67C7×M16—31no—
68#1144—61no—
69#1143—61no—
70(C2×D7)⋊C4—22no—
71C4×Dic7C2×D311yescharacteristic, complemented
72#1136—31no—
73#1004—71no—
74C3×#124—71no—
75#942—71no—
76#942D711yescharacteristic, complemented
77#833—71no—
78#805—71no—
79#805—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D3×C7—21no—
83C2×C3×D7—22no—
84C2×C3×D7—42no—
85D3×D7—82no—
86C3×C4×C7—41no—
87C3×C4×C7C2×D411yescharacteristic
88C3×C4×C7C2×D411yescharacteristic
89Dic21—21no—
90C3×Dic7—41no—
91C7×Dic3—21no—
92#455—211noSylow
93C22×D7—62no—
94C22×D7#6712yes—
95C7×Q8—21no—
96C7×Q8—21no—
97C7×Q8—31no—
98C7×D4—31no—
99C7×D4—61no—
100C2×C4×C7—21no—
101C2×C4×C7—61no—
102C2×C4×C7C22×D311yescharacteristic
103C22⋊D7—62no—
104C22⋊D7—61no—
105C2×Dic7—21no—
106D28—21no—
107D28—21no—
108D28—31no—
109D28—62no—
110C4×D7—62no—
111C4×D7—61no—
112Dic14—31no—
113C7×C8—61no—
114C7⋊C8—61no—
115C2×C3×Q8C2×D711yescharacteristic, complemented
116C2×C3×D4—71no—
117C2×#67—142no—
118#237—141no—
119#237—141no—
120D3×D4—141no—
121D3×D4—141no—
122#235—71no—
123#235C2×D711yescharacteristic
124C3×C22⋊C4—142no—
125C3×C42—71no—
126#216—21no—
127#216—141no—
128#215—21no—
129#215—141no—
130#208—71no—
131#208C2×D711yescharacteristic, complemented
132C2×C3×C7—21no—
133C2×C3×C7#12011yescharacteristic
134D21—41no—
135C3×D7—42no—
136D3×C7—41no—
137H(5,𝔽2)—211no—
138#137—31no—
139#137—211no—
140#124—71no—
141#104—211no—
142#104—211no—
143#100—211no—
144C22×C7—61no—
145C22×C7C2×#6711yescharacteristic
146C2×D7—22no—
147C2×D7—42no—
148C2×D7—61no—
149C2×D7—242no—
150C4×C7—41no—
151C4×C7—61no—
152C4×C7D3×D411yescharacteristic
153C4×C7D3×D411yescharacteristic
154Dic7—41no—
155Dic7—61no—
156C23×C3—72no—
157C22×D3—142no—
158C3×Q8—21no—
159C3×Q8—21no—
160C3×D4—141no—
161C3×D4—141no—
162C2×C3×C4—21no—
163C2×C3×C4—141no—
164C2×C3×C4C22×D711yescharacteristic
165#67—21no—
166#67—142no—
167#67—142no—
168#67—141no—
169C2×Dic3—142no—
170D12—71no—
171D12C22⋊D711yescharacteristic
172C4×D3—21no—
173C4×D3—141no—
174Dic6—71no—
175Dic6C22⋊D711yescharacteristic
176C3⋊C8—21no—
177C3⋊C8—141no—
178C3×C7#45511yescharacteristic, complemented, Hall
179#41—31no—
180#41—211no—
181#41—421no—
182#41—421no—
183C2×Q8D3×D711yescharacteristic, complemented
184C2×D4—71no—
185C2×D4—421no—
186C2×D4—421no—
187C2×D4—422no—
188Q16—61no—
189Q16—421no—
190SD16—61no—
191SD16—421no—
192M16—31no—
193M16—211no—
194C22⋊C4—142no—
195C42—71no—
196C2×C7—21no—
197C2×C7—121no—
198C2×C7#93711yescharacteristic
199D7—42no—
200D7—121no—
201C22×C3—142no—
202C22×C3—282no—
203C22×C3D4×D711yescharacteristic
204C2×D3—21no—
205C2×D3—141no—
206C2×D3—562no—
207C3×C4—41no—
208C3×C4—281no—
209C3×C4D4×D711yescharacteristic
210C3×C4C2×C22⋊D711yescharacteristic
211Dic3—21no—
212Dic3—141no—
213C23—72no—
214C23—422no—
215Q8—21no—
216Q8—21no—
217Q8—31no—
218Q8—211no—
219D4—31no—
220D4—61no—
221D4—141no—
222D4—141no—
223D4—211no—
224D4—422no—
225D4—422no—
226D4—421no—
227C2×C4—21no—
228C2×C4—61no—
229C2×C4—141no—
230C2×C4—422no—
231C2×C4—421no—
232C2×C4C2×D3×D711yescharacteristic
233C8—61no—
234C8—421no—
235C7#506111yescharacteristic, complemented, Sylow
236C2×C3—21no—
237C2×C3—282no—
238C2×C3#650011yescharacteristic
239D3—41no—
240D3—281no—
241C22—61no—
242C22—142no—
243C22—282no—
244C22—421no—
245C22—1682no—
246C22D7×#6711yescharacteristic
247C4—41no—
248C4—61no—
249C4—281no—
250C4—421no—
251C4D3×C22⋊D711yescharacteristic
252C4#6669411yescharacteristic
253C3#8997411yescharacteristic, complemented, Sylow
254C2—21no—
255C2—121no—
256C2—282no—
257C2—841no—
258C2#1062113911yescharacteristic
subgroupG/H
centerC2#10621139
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621564C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10621425C2semidirect
3. C7×#942C2semidirect
4. C3×#6510C2semidirect
5. #10620516C2semidirect
6. #10620514C2semidirect
7. #10620493C2semidirect
9. C2×C3×C7×Q8C22semidirect
20. #66627C22semidirect
23. C7×#208C22semidirect
25. C3×C4×Dic7C22semidirect
28. #6510D3semidirect
57. C2×C7×Q8C2×D3semidirect
71. C4×Dic7C2×D3semidirect
76. #942D7semidirect
115. C2×C3×Q8C2×D7semidirect
131. #208C2×D7semidirect
178. C3×C7#455semidirect
183. C2×Q8D3×D7semidirect
235. C7#5061semidirect
253. C3#89974semidirect

Maximal subgroups

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

subgroupindex
#106215642
#106214252
C7×#9422
C3×#65102
#106205162
#106205142
#106204932
#899743
#50617