Finite Group Databasebeta

← #13054564

Subgroups — #13054564

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#10621415C211yesmaximal, characteristic, complemented
3C7×#936C211yesmaximal, characteristic, complemented
4C3×#6495C211yesmaximal, characteristic, complemented
5#10620515C211yesmaximal, characteristic, complemented
6#10620512C211yesmaximal, characteristic, complemented
7#10620497C211yesmaximal, characteristic, complemented
8#89834—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66683—22no—
14#66677—21no—
15D21⋊Q8—21no—
16#66670—21no—
17D7×Dic6—21no—
18#66633—21no—
19#66632—21no—
20#66627C2211yescharacteristic
21C7×#217—22no—
22C4×C7×Dic3C2211yescharacteristic
23C3×#1142—21no—
24C3×#1141—21no—
25C3×#1136C2211yescharacteristic
26#6551—31no—
27C7×#124—31no—
28#6495—31no—
29#6495D311yescharacteristic, complemented
30#6410—31no—
31#6379—31no—
32#6379—31no—
33#5201—71nomaximal, Hall
34C23×C3×C7—21no—
35C3×C7×D4—21no—
36C3×C7×D4—21no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—21no—
39D84D411yescharacteristic
40C4×D21—21no—
41Dic42D411yescharacteristic
42C2×C7×Dic3—21no—
43C2×C7×Dic3D412yes—
44C7×Dic6—21no—
45C7×Dic6—21no—
46C3×C22⋊D7—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—
55C3×C7⋊C8—21no—
56C2×C7×Q8—31no—
57C2×C7×D4C2×D311yescharacteristic
58#1161—61no—
59#1161—61no—
60Q8×D7—61no—
61Q8×D7—61no—
62#1157—31no—
63#1157—62no—
64#1157C2×D311yescharacteristic
65C7×C22⋊C4—62no—
66C42×C7—31no—
67#1142—21no—
68#1142—61no—
69#1141—21no—
70#1141—61no—
71#1136—31no—
72#1136C2×D311yescharacteristic
73#1010—71no—
74C3×AGL(1,ℤ/8)—71no—
75#936D711yescharacteristic, complemented
76#932—71no—
77#837—71no—
78#825—71no—
79#824—71no—
80C22×C3×C7—42no—
81C22×C3×C7—41no—
82C22×C3×C7C2×D411yescharacteristic
83C2×D21—21no—
84C2×C3×D7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88C3×Dic7—21no—
89C7×Dic3—22no—
90C7×Dic3—41no—
91#454—211noSylow
92C23×C7—21no—
93C7×Q8—61no—
94C7×Q8—61no—
95C7×D4—21no—
96C7×D4—21no—
97C2×C4×C7—32no—
98C2×C4×C7—61no—
99C2×C4×C7C22×D311yescharacteristic
100C22⋊D7—21no—
101C22⋊D7—61no—
102D28—31no—
103D28—62no—
104D28#6711yescharacteristic
105C4×D7—21no—
106C4×D7—62no—
107C4×D7—62no—
108C4×D7—61no—
109Dic14—31no—
110Dic14—62no—
111Dic14#6711yescharacteristic
112C7⋊C8—21no—
113C7⋊C8—61no—
114C3×#41—71no—
115C2×C3×D4C2×D711yescharacteristic
116D3×Q8—141no—
117#237—141no—
118#237—142no—
119#235—71no—
120#235—141no—
121C2×Dic6—141no—
122C2×Dic6C2×D711yescharacteristic
123C3×SD16—141no—
124C3×D8—141no—
125C3×M16—71no—
126#217—22no—
127#214—141no—
128#213—141no—
129C4×Dic3C2×D711yescharacteristic
130#208—71no—
131C2×C3×C7—21no—
132C2×C3×C7—81no—
133C2×C3×C7#12011yescharacteristic
134D21—41no—
135C3×D7—41no—
1362−1+4—211no—
137AGL(1,ℤ/8)—71no—
138AGL(1,ℤ/8)—211no—
139#124—31no—
140#104—211no—
141#104—211no—
142#101—211no—
143C22×C7—42no—
144C22×C7—41no—
145C22×C7D3×D411yescharacteristic
146C2×D7—21no—
147C2×D7—61no—
148C4×C7—62no—
149C4×C7—121no—
150C4×C7D3×D411yescharacteristic
151C4×C7C2×#6711yescharacteristic
152Dic7—21no—
153Dic7—61no—
154C23×C3—21no—
155C3×Q8—71no—
156C3×D4—21no—
157C3×D4—21no—
158C3×D4—71no—
159C3×D4—141no—
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—142no—
163#67—141no—
164C2×Dic3—21no—
165C2×Dic3—142no—
166C2×Dic3C22⋊D712yes—
167D12—71no—
168C4×D3—142no—
169C4×D3—141no—
170Dic6—21no—
171Dic6—21no—
172Dic6—71no—
173Dic6—142no—
174C3×C8—141no—
175C3⋊C8—141no—
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×D4D3×D711yescharacteristic
186SD16—141no—
187SD16—421no—
188D8—141no—
189D8—421no—
190M16—71no—
191M16—211no—
192C22⋊C4—62no—
193C42—31no—
194C2×C7—21no—
195C2×C7—81no—
196C2×C7#93711yescharacteristic
197D7—41no—
198D7—121no—
199C22×C3—42no—
200C22×C3—41no—
201C22×C3—141no—
202C22×C3C2×C22⋊D711yescharacteristic
203C2×D3—141no—
204C3×C4—141no—
205C3×C4D4×D711yescharacteristic
206C3×C4D4×D711yescharacteristic
207Dic3—22no—
208Dic3—41no—
209Dic3—141no—
210C23—21no—
211Q8—61no—
212Q8—61no—
213Q8—71no—
214Q8—211no—
215Q8—422no—
216D4—21no—
217D4—21no—
218D4—71no—
219D4—141no—
220D4—211no—
221D4—422no—
222D4—421no—
223C2×C4—32no—
224C2×C4—61no—
225C2×C4—141no—
226C2×C4—422no—
227C2×C4—422no—
228C2×C4—421no—
229C2×C4C2×D3×D711yescharacteristic
230C8—141no—
231C8—421no—
232C7#520111yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—81no—
235C2×C3—281no—
236C2×C3#650011yescharacteristic
237D3—281no—
238C22—42no—
239C22—41no—
240C22—141no—
241C22—421no—
242C22D3×C22⋊D711yescharacteristic
243C4—62no—
244C4—121no—
245C4—141no—
246C4—421no—
247C4D7×#6711yescharacteristic
248C4#6669411yescharacteristic
249C3#8983411yescharacteristic, complemented, Sylow
250C2—21no—
251C2—81no—
252C2—281no—
253C2—841no—
254C2#1062113911yescharacteristic
subgroupG/H
centerC2#10621139
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
hypercenterC2×D4D3×D7
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621547C2semidirect
2. #10621415C2semidirect
3. C7×#936C2semidirect
4. C3×#6495C2semidirect
5. #10620515C2semidirect
6. #10620512C2semidirect
7. #10620497C2semidirect
29. #6495D3semidirect
75. #936D7semidirect
176. C3×C7#454semidirect
232. C7#5201semidirect
249. C3#89834semidirect

Maximal subgroups

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

subgroupindex
#106215472
#106214152
C7×#9362
C3×#64952
#106205152
#106205122
#106204972
#898343
#52017