Finite Group Databasebeta

← #13053624

Subgroups — #13053624

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
2C7×#909C211yesmaximal, characteristic, complemented
3#10620930C211yesmaximal, characteristic, complemented
4#10620883C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C3×#6397C211yesmaximal, characteristic, complemented
7#10620513C211yesmaximal, characteristic, complemented
8#89974—31nomaximal, Hall
9#66728C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic, complemented
12C2×C3×D28C2211yescharacteristic
13D7×#67—22no—
14#66681—21no—
15D3×D28—21no—
16#66670—21no—
17#66669—21no—
18C3×C7×M16C2211yescharacteristic, complemented
19C4×Dic21C2211yescharacteristic, complemented
20C7×Dic12—21no—
21C7×#204—21no—
22C3×#1136C2211yescharacteristic, complemented
23(C2×D7)⋊Dic3—22no—
24#66571—21no—
25#66568—21no—
26#6553—31no—
27C7×#137—31no—
28#6510—31no—
29#6505—31no—
30#6411—31no—
31#6399—31no—
32#6397D311yescharacteristic, complemented
33#4746—71nomaximal, Hall
34C22×C3×D7D412yes—
35C2×D3×D7—22no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—21no—
38C2×Dic21—21no—
39D84D411yescharacteristic
40C4×D21—21no—
41Dic42D411yescharacteristic
42C7×#67—21no—
43C2×C7×Dic3—21no—
44C7×D12D411yescharacteristic
45C4×D3×C7—21no—
46C7×Dic6—21no—
47C7×Dic6—21no—
48C7×Dic6D411yescharacteristic
49C3×D28—21no—
50C3×D28—21no—
51#4246—22no—
52#4245—22no—
53Dic3×D7—22no—
54C3×C7×C8—21no—
55C3×C7⋊C8—21no—
56C7×#41—31no—
57C2×C7×Q8—31no—
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×M16C2×D311yescharacteristic, complemented
68#1144—61no—
69#1143—61no—
70(C2×D7)⋊C4—62no—
71C4×Dic7—31no—
72#1136C2×D311yescharacteristic, complemented
73#1004—71no—
74#936—71no—
75#909—71no—
76#909D711yescharacteristic, complemented
77C3×#100—71no—
78#825—71no—
79#825—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×C7C2×D411yescharacteristic
87C3×C4×C7C2×D411yescharacteristic
88Dic21—21no—
89Dic21—41no—
90C7×Dic3—21no—
91C7×Dic3—41no—
92#455—211noSylow
93C22×D7—62no—
94C22×D7D1212yes—
95C7×Q8—31no—
96C7×Q8—61no—
97C7×Q8—61no—
98C7×D4—31no—
99C7×D4—61no—
100C2×C4×C7—61no—
101C2×C4×C7—61no—
102C2×C4×C7C22×D311yescharacteristic
103C22⋊D7—62no—
104C22⋊D7—61no—
105C2×Dic7—61no—
106D28—21no—
107D28—21no—
108D28—31no—
109D28—62no—
110C4×D7—62no—
111C4×D7—61no—
112Dic14—31no—
113C7×C8—21no—
114C7⋊C8—21no—
115C2×C3×D4—71no—
116C2×#67—142no—
117#237—141no—
118#237—141no—
119D3×D4—141no—
120D3×D4—141no—
121#235—71no—
122#235C2×D711yescharacteristic
123C2×Dic6C2×D711yescharacteristic, complemented
124C3×M16—71no—
125C3×M16C2×D711yescharacteristic, complemented
126#217—142no—
127C4×Dic3—71no—
128Dic12—21no—
129Dic12—141no—
130#204—21no—
131#204—141no—
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—211no—
141#104—211no—
142#104—211no—
143#100—71no—
144C22×C7—61no—
145C22×C7C2×D1211yescharacteristic
146C2×D7—22no—
147C2×D7—42no—
148C2×D7—61no—
149C2×D7—242no—
150C4×C7—61no—
151C4×C7—121no—
152C4×C7D3×D411yescharacteristic
153C4×C7D3×D411yescharacteristic
154Dic7—61no—
155Dic7—121no—
156C23×C3—72no—
157C22×D3—142no—
158C3×D4—141no—
159C3×D4—141no—
160C2×C3×C4C22×D711yescharacteristic
161#67—21no—
162#67—142no—
163#67—142no—
164#67—141no—
165C2×Dic3—21no—
166C2×Dic3—142no—
167C2×Dic3—141no—
168D12—71no—
169D12C22⋊D711yescharacteristic
170C4×D3—21no—
171C4×D3—141no—
172Dic6—21no—
173Dic6—21no—
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×Q8—31no—
184C2×D4—71no—
185C2×D4—421no—
186C2×D4—421no—
187C2×D4—422no—
188Q16—61no—
189Q16—421no—
190SD16—61no—
191SD16—421no—
192M16—71no—
193M16D3×D711yescharacteristic, complemented
194C22⋊C4—422no—
195C42—211no—
196C2×C7—21no—
197C2×C7—121no—
198C2×C7#88211yescharacteristic
199D7—42no—
200D7—121no—
201C22×C3—142no—
202C22×C3—282no—
203C22×C3D4×D711yescharacteristic
204C2×D3—21no—
205C2×D3—141no—
206C2×D3—562no—
207C3×C4D4×D711yescharacteristic
208C3×C4C2×C22⋊D711yescharacteristic
209Dic3—21no—
210Dic3—41no—
211Dic3—141no—
212Dic3—281no—
213C23—72no—
214C23—422no—
215Q8—31no—
216Q8—61no—
217Q8—61no—
218Q8—211no—
219D4—31no—
220D4—61no—
221D4—141no—
222D4—141no—
223D4—211no—
224D4—422no—
225D4—422no—
226D4—421no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—422no—
230C2×C4—421no—
231C2×C4—421no—
232C2×C4C2×D3×D711yescharacteristic
233C8—21no—
234C8—141no—
235C7#474611yescharacteristic, 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×D1211yescharacteristic
247C4—61no—
248C4—121no—
249C4—421no—
250C4—841no—
251C4#6668111yescharacteristic
252C4D3×C22⋊D711yescharacteristic
253C3#8997411yescharacteristic, complemented, Sylow
254C2—21no—
255C2—121no—
256C2—282no—
257C2—841no—
258C2#1062103711yescharacteristic
subgroupG/H
centerC2#10621037
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
prime radical#10621564C2
hypercenterM16D3×D7
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C7×#909C2semidirect
3. #10620930C2semidirect
4. #10620883C2semidirect
5. #10620664C2semidirect
6. C3×#6397C2semidirect
7. #10620513C2semidirect
11. C2×C7×Dic6C22semidirect
18. C3×C7×M16C22semidirect
19. C4×Dic21C22semidirect
22. C3×#1136C22semidirect
32. #6397D3semidirect
67. C7×M16C2×D3semidirect
72. #1136C2×D3semidirect
76. #909D7semidirect
123. C2×Dic6C2×D7semidirect
125. C3×M16C2×D7semidirect
178. C3×C7#455semidirect
193. M16D3×D7semidirect
235. C7#4746semidirect
253. C3#89974semidirect

Maximal subgroups

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

subgroupindex
#106215642
C7×#9092
#106209302
#106208832
#106206642
C3×#63972
#106205132
#899743
#47467