Finite Group Databasebeta

← #13053639

Subgroups — #13053639

Order 1344.

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

normal subgroups only

This group has 292 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#10621547C211yesmaximal, characteristic, complemented
2#10621391C211yesmaximal, characteristic, complemented
3#10620937C211yesmaximal, characteristic, complemented
4#10620863C211yesmaximal, characteristic, complemented
5C7×#824C211yesmaximal, characteristic, complemented
6C3×#6399C211yesmaximal, characteristic, complemented
7#10620516C211yesmaximal, characteristic, complemented
8#89525—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66683—22no—
14#66677—21no—
15D21⋊Q8—21no—
16#66670—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19D168—21no—
20#66623—21no—
21C7×#208C2211yescharacteristic
22C3×C4×Dic7C2211yescharacteristic
23#66575—22no—
24#66566—21no—
25#66561—21no—
26#6551—31no—
27#6510—31no—
28#6471—31no—
29#6471—31no—
30C7×#101—31no—
31#6399—31no—
32#6399D311yescharacteristic, complemented
33#4827—71nomaximal, Hall
34C22×D21—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37D84—21no—
38D84—21no—
39D84D411yescharacteristic
40C4×D21—21no—
41Dic42D411yescharacteristic
42C2×C7×Dic3D412yes—
43C7×Dic6—21no—
44C7×Dic6—21no—
45C3×C22⋊D7—21no—
46C2×C3×Dic7—21no—
47C3×D28D411yescharacteristic
48C3×C4×D7—21no—
49C3×Dic14D411yescharacteristic
50#4248—22no—
51#4246—22no—
52D21⋊C4—22no—
53Dic3×D7—22no—
54C3×C7×C8—21no—
55C7×C3⋊C8—21no—
56C2×C7×Q8—31no—
57#1161—61no—
58#1161—61no—
59Q8×D7—61no—
60Q8×D7—61no—
61#1157—31no—
62#1157—62no—
63#1157C2×D311yescharacteristic
64C2×D28—31no—
65C7×M16—31no—
66C7×M16C2×D311yescharacteristic
67(C2×D7)⋊C4—62no—
68C4×Dic7C2×D311yescharacteristic
69D56—61no—
70D56—61no—
71#1132—61no—
72#1132—61no—
73#1010—71no—
74#949—71no—
75#908—71no—
76#875—71no—
77C3×#104—71no—
78#824D711yescharacteristic, complemented
79#805—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D21—42no—
83C2×D21—41no—
84C2×C3×D7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88C3×Dic7—21no—
89C3×Dic7—41no—
90C7×Dic3—22no—
91#454—211noSylow
92C22×D7—61no—
93C7×Q8—61no—
94C7×Q8—61no—
95C2×C4×C7—32no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—21no—
98C22⋊D7—61no—
99C2×Dic7—21no—
100D28—31no—
101D28—62no—
102D28—61no—
103D28—61no—
104D28D1211yescharacteristic
105C4×D7—21no—
106C4×D7—62no—
107C4×D7—62no—
108C4×D7—61no—
109Dic14—31no—
110Dic14—62no—
111Dic14D1211yescharacteristic
112C7×C8—21no—
113C7×C8—61no—
114C3×#41—71no—
115D3×Q8—141no—
116#237—141no—
117#237—142no—
118#235—71no—
119#235—141no—
120C2×D12—71no—
121C2×Dic6—141no—
122C2×Dic6C2×D711yescharacteristic
123C3×M16C2×D711yescharacteristic
124C3×C42—71no—
125#215—141no—
126#213—141no—
127(C2×D3)⋊C4—142no—
128#208C2×D711yescharacteristic
129D24—141no—
130#204—141no—
131C2×C3×C7—21no—
132C2×C3×C7#12011yescharacteristic
133D21—41no—
134D21—81no—
135C3×D7—41no—
1362−1+4—211no—
137AGL(1,ℤ/8)—211no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—71no—
141#104—211no—
142#101—31no—
143C22×C7D3×D411yescharacteristic
144C2×D7—21no—
145C2×D7—61no—
146C2×D7—122no—
147C2×D7—121no—
148C4×C7—62no—
149C4×C7C2×D1211yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—21no—
152Dic7—41no—
153Dic7—61no—
154C22×D3—141no—
155C3×Q8—71no—
156C3×D4—71no—
157C3×D4—141no—
158C2×C3×C4—141no—
159C2×C3×C4—141no—
160C2×C3×C4C22×D711yescharacteristic
161#67—142no—
162#67—141no—
163C2×Dic3—142no—
164C2×Dic3D2812yes—
165D12—71no—
166D12—141no—
167D12—141no—
168C4×D3—142no—
169C4×D3—141no—
170Dic6—21no—
171Dic6—21no—
172Dic6—71no—
173Dic6—142no—
174C3×C8—21no—
175C3⋊C8—21no—
176C3×C7#45411yescharacteristic, complemented, Hall
177#41—71no—
178#41—211no—
179#41—422no—
180#41—421no—
181#41—421no—
182C2×Q8—31no—
183C2×Q8—421no—
184C2×Q8—421no—
185C2×D4—211no—
186SD16—421no—
187SD16—421no—
188D8—421no—
189D8—421no—
190M16—31no—
191M16D3×D711yescharacteristic
192C22⋊C4—422no—
193C42—71no—
194C2×C7—21no—
195C2×C7#88211yescharacteristic
196D7—41no—
197D7—121no—
198D7—241no—
199C22×C3—141no—
200C22×C3C2×D2811yescharacteristic
201C2×D3—141no—
202C2×D3—282no—
203C2×D3—281no—
204C3×C4—141no—
205C3×C4—281no—
206C3×C4D4×D711yescharacteristic
207C3×C4D4×D711yescharacteristic
208Dic3—22no—
209Dic3—141no—
210C23—421no—
211Q8—61no—
212Q8—61no—
213Q8—71no—
214Q8—211no—
215Q8—422no—
216D4—71no—
217D4—141no—
218D4—211no—
219D4—422no—
220D4—421no—
221D4—421no—
222D4—421no—
223C2×C4—32no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—422no—
227C2×C4—422no—
228C2×C4—421no—
229C2×C4C2×D3×D711yescharacteristic
230C8—21no—
231C8—61no—
232C7#482711yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—281no—
235C2×C3#644511yescharacteristic
236D3—281no—
237D3—561no—
238C22—141no—
239C22—421no—
240C22—842no—
241C22—841no—
242C22D3×D2811yescharacteristic
243C4—62no—
244C4—141no—
245C4—281no—
246C4—421no—
247C4D7×D1211yescharacteristic
248C4#6669411yescharacteristic
249C3#8952511yescharacteristic, complemented, Sylow
250C2—21no—
251C2—281no—
252C2—841no—
253C2—1681no—
254C2#1062103811yescharacteristic
subgroupG/H
centerC2#10621038
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
hypercenterM16D3×D7
hypocenterC3×C7#454
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
1. #10621547C2semidirect
2. #10621391C2semidirect
3. #10620937C2semidirect
4. #10620863C2semidirect
5. C7×#824C2semidirect
6. C3×#6399C2semidirect
7. #10620516C2semidirect
32. #6399D3semidirect
78. #824D7semidirect
176. C3×C7#454semidirect
232. C7#4827semidirect
249. C3#89525semidirect

Maximal subgroups

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

subgroupindex
#106215472
#106213912
#106209372
#106208632
C7×#8242
C3×#63992
#106205162
#895253
#48277