Finite Group Databasebeta

← #13053634

Subgroups — #13053634

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
2C3×#6471C211yesmaximal, characteristic, complemented
3#10620930C211yesmaximal, characteristic, complemented
4#10620874C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C7×#824C211yesmaximal, characteristic, complemented
7#10620511C211yesmaximal, characteristic, complemented
8#89525—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13#66683—22no—
14#66677—21no—
15D21⋊Q8—21no—
16#66670—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19C4×Dic21C2211yescharacteristic
20C7×#208C2211yescharacteristic
21C3×D56—21no—
22C3×#1132—21no—
23(C2×D7)⋊Dic3—22no—
24#66565—21no—
25#66561—21no—
26#6551—31no—
27#6510—31no—
28#6471—31no—
29#6471D311yescharacteristic, complemented
30C7×#101—31no—
31#6399—31no—
32#6399—31no—
33#5201—71nomaximal, Hall
34C22×C3×D7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37C2×Dic21—21no—
38D84D411yescharacteristic
39C4×D21—21no—
40Dic42D411yescharacteristic
41C2×C7×Dic3D412yes—
42C7×Dic6—21no—
43C7×Dic6—21no—
44C3×C22⋊D7—21no—
45C3×D28—21no—
46C3×D28—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×D28C2×D311yescharacteristic
65C7×M16—31no—
66C7×M16C2×D311yescharacteristic
67(C2×D7)⋊C4—62no—
68C4×Dic7—31no—
69D56—21no—
70D56—61no—
71#1132—21no—
72#1132—61no—
73#1010—71no—
74C3×AGL(1,ℤ/8)—71no—
75#936—71no—
76#932—71no—
77#837—71no—
78#825—71no—
79#824D711yescharacteristic, complemented
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×C3×D7—21no—
83C2×C3×D7—42no—
84C2×C3×D7—41no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88Dic21—41no—
89C3×Dic7—21no—
90C7×Dic3—22no—
91#454—211noSylow
92C22×D7—21no—
93C7×Q8—61no—
94C7×Q8—61no—
95C2×C4×C7—32no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—21no—
98C22⋊D7—61no—
99C2×Dic7—61no—
100D28—21no—
101D28—21no—
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×D4—71no—
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×M16C2×D711yescharacteristic
126#217—142no—
127#214—141no—
128#213—141no—
129C4×Dic3—71no—
130#208C2×D711yescharacteristic
131C2×C3×C7—21no—
132C2×C3×C7#12011yescharacteristic
133D21—41no—
134C3×D7—41no—
135C3×D7—81no—
1362−1+4—211no—
137AGL(1,ℤ/8)—71no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—211no—
141#104—211no—
142#101—31no—
143C22×C7D3×D411yescharacteristic
144C2×D7—21no—
145C2×D7—42no—
146C2×D7—41no—
147C2×D7—61no—
148C4×C7—62no—
149C4×C7D3×D411yescharacteristic
150C4×C7C2×#6711yescharacteristic
151Dic7—21no—
152Dic7—61no—
153Dic7—121no—
154C23×C3—141no—
155C3×Q8—71no—
156C3×D4—71no—
157C3×D4—141no—
158C3×D4—141no—
159C3×D4—141no—
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—142no—
163#67—141no—
164C2×Dic3—142no—
165C2×Dic3—141no—
166C2×Dic3D2812yes—
167D12—71no—
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—71no—
186SD16—141no—
187SD16—421no—
188D8—141no—
189D8—421no—
190M16—31no—
191M16D3×D711yescharacteristic
192C22⋊C4—422no—
193C42—211no—
194C2×C7—21no—
195C2×C7#93711yescharacteristic
196D7—41no—
197D7—81no—
198D7—121no—
199C22×C3—141no—
200C22×C3—282no—
201C22×C3—281no—
202C22×C3C2×D2811yescharacteristic
203C2×D3—141no—
204C3×C4—141no—
205C3×C4D4×D711yescharacteristic
206C3×C4D4×D711yescharacteristic
207Dic3—22no—
208Dic3—141no—
209Dic3—281no—
210C23—141no—
211Q8—61no—
212Q8—61no—
213Q8—71no—
214Q8—211no—
215Q8—422no—
216D4—71no—
217D4—141no—
218D4—141no—
219D4—141no—
220D4—211no—
221D4—422no—
222D4—421no—
223C2×C4—32no—
224C2×C4—141no—
225C2×C4—422no—
226C2×C4—422no—
227C2×C4—421no—
228C2×C4—421no—
229C2×C4C2×D3×D711yescharacteristic
230C8—21no—
231C8—61no—
232C7#520111yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—281no—
235C2×C3—561no—
236C2×C3#644511yescharacteristic
237D3—281no—
238C22—141no—
239C22—282no—
240C22—281no—
241C22—421no—
242C22D3×D2811yescharacteristic
243C4—62no—
244C4—141no—
245C4—421no—
246C4—841no—
247C4#6668111yescharacteristic
248C4D7×#6711yescharacteristic
249C3#8952511yescharacteristic, complemented, Sylow
250C2—21no—
251C2—281no—
252C2—561no—
253C2—841no—
254C2#1062103611yescharacteristic
subgroupG/H
centerC2#10621036
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. C3×#6471C2semidirect
3. #10620930C2semidirect
4. #10620874C2semidirect
5. #10620664C2semidirect
6. C7×#824C2semidirect
7. #10620511C2semidirect
29. #6471D3semidirect
79. #824D7semidirect
176. C3×C7#454semidirect
232. C7#5201semidirect
249. C3#89525semidirect

Maximal subgroups

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

subgroupindex
#106215472
C3×#64712
#106209302
#106208742
#106206642
C7×#8242
#106205112
#895253
#52017