Finite Group Databasebeta

← #13053632

Subgroups — #13053632

Order 1344.

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

normal subgroups only

This group has 284 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
2C3×#6472C211yesmaximal, characteristic, complemented
3#10620947C211yesmaximal, characteristic, complemented
4#10620877C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C7×#824C211yesmaximal, characteristic, complemented
7#10620511C211yesmaximal, characteristic, complemented
8#89524—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66683—22no—
14#66677—21no—
15D21⋊Q8—21no—
16#66670—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19C4×Dic21C2211yescharacteristic
20C7×#208C2211yescharacteristic
21C3×Dic28—21no—
22C3×#1132—21no—
23#66577—22no—
24#66570—21no—
25#66566—21no—
26#6551—31no—
27#6507—31no—
28#6472—31no—
29#6472D311yescharacteristic, complemented
30C7×#101—31no—
31#6399—31no—
32#6399—31no—
33#5205—71nomaximal, Hall
34C2×C3×C4×C7C2311yescharacteristic
35#4268—21no—
36C2×Dic21—21no—
37D84D411yescharacteristic
38C4×D21—21no—
39Dic42D411yescharacteristic
40C2×C7×Dic3D412yes—
41C7×Dic6—21no—
42C7×Dic6—21no—
43C3×C22⋊D7—21no—
44C2×C3×Dic7—21no—
45C3×D28D411yescharacteristic
46C3×C4×D7—21no—
47C3×Dic14—21no—
48C3×Dic14—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×Dic14C2×D311yescharacteristic
65C7×M16—31no—
66C7×M16C2×D311yescharacteristic
67Dic7⋊C4—62no—
68C4×Dic7—31no—
69Dic28—21no—
70Dic28—61no—
71#1132—21no—
72#1132—61no—
73#1010—71no—
74C3×#137—71no—
75#944—71no—
76#942—71no—
77#837—71no—
78#825—71no—
79#824D711yescharacteristic, complemented
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×C3×D7—21no—
83C3×C4×C7C2×D411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85Dic21—22no—
86Dic21—21no—
87C3×Dic7—21no—
88C3×Dic7—41no—
89C7×Dic3—22no—
90#456—211noSylow
91C7×Q8—61no—
92C7×Q8—61no—
93C2×C4×C7—32no—
94C2×C4×C7C22×D311yescharacteristic
95C22⋊D7—21no—
96C22⋊D7—61no—
97C2×Dic7—21no—
98C2×Dic7—61no—
99D28—31no—
100D28—62no—
101D28#6711yescharacteristic
102C4×D7—21no—
103C4×D7—62no—
104C4×D7—62no—
105C4×D7—61no—
106Dic14—21no—
107Dic14—21no—
108Dic14—31no—
109Dic14—62no—
110Dic14#6711yescharacteristic
111C7×C8—21no—
112C7×C8—61no—
113C3×#41—71no—
114C2×C3×Q8—71no—
115D3×Q8—141no—
116#237—141no—
117#237—142no—
118#235—71no—
119#235—141no—
120C2×Dic6—141no—
121C2×Dic6C2×D711yescharacteristic
122C3×Q16—141no—
123C3×SD16—141no—
124C3×M16C2×D711yescharacteristic
125#216—141no—
126#215—141no—
127Dic3⋊C4—142no—
128C4×Dic3—71no—
129#208C2×D711yescharacteristic
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132D21—41no—
133C3×D7—41no—
1342−1+4—211no—
135#137—71no—
136#137—211no—
137C4⋊Q8—211no—
138#104—211no—
139#104—211no—
140#101—31no—
141C22×C7D3×D411yescharacteristic
142C2×D7—21no—
143C2×D7—61no—
144C4×C7—62no—
145C4×C7D3×D411yescharacteristic
146C4×C7C2×#6711yescharacteristic
147Dic7—21no—
148Dic7—41no—
149Dic7—62no—
150Dic7—61no—
151C3×Q8—71no—
152C3×Q8—141no—
153C3×Q8—141no—
154C3×D4—71no—
155C3×D4—141no—
156C2×C3×C4—141no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159#67—142no—
160#67—141no—
161C2×Dic3—142no—
162C2×Dic3—141no—
163C2×Dic3D2812yes—
164D12—71no—
165C4×D3—142no—
166C4×D3—141no—
167Dic6—21no—
168Dic6—21no—
169Dic6—71no—
170Dic6—142no—
171C3×C8—21no—
172C3⋊C8—21no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—71no—
175#41—211no—
176#41—422no—
177#41—421no—
178#41—421no—
179C2×Q8—31no—
180C2×Q8—71no—
181C2×Q8—421no—
182C2×Q8—421no—
183Q16—141no—
184Q16—421no—
185SD16—141no—
186SD16—421no—
187M16—31no—
188M16D3×D711yescharacteristic
189C4⋊C4—422no—
190C42—211no—
191C2×C7—21no—
192C2×C7#93711yescharacteristic
193D7—41no—
194D7—121no—
195C22×C3—141no—
196C22×C3C2×D2811yescharacteristic
197C2×D3—141no—
198C3×C4—141no—
199C3×C4—281no—
200C3×C4D4×D711yescharacteristic
201C3×C4D4×D711yescharacteristic
202Dic3—22no—
203Dic3—142no—
204Dic3—141no—
205Q8—61no—
206Q8—61no—
207Q8—71no—
208Q8—141no—
209Q8—141no—
210Q8—211no—
211Q8—422no—
212D4—71no—
213D4—141no—
214D4—211no—
215D4—422no—
216D4—421no—
217C2×C4—32no—
218C2×C4—141no—
219C2×C4—141no—
220C2×C4—422no—
221C2×C4—422no—
222C2×C4—421no—
223C2×C4—421no—
224C2×C4C2×D3×D711yescharacteristic
225C8—21no—
226C8—61no—
227C7#520511yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—281no—
230C2×C3#644511yescharacteristic
231D3—281no—
232C22—141no—
233C22—421no—
234C22D3×D2811yescharacteristic
235C4—62no—
236C4—141no—
237C4—281no—
238C4—422no—
239C4—421no—
240C4#6668111yescharacteristic
241C4D7×#6711yescharacteristic
242C3#8952411yescharacteristic, complemented, Sylow
243C2—21no—
244C2—281no—
245C2—841no—
246C2#1062103611yescharacteristic
subgroupG/H
centerC2#10621036
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
prime radical#10621547C2
hypercenterM16D3×D7
hypocenterC3×C7#456
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. C3×#6472C2semidirect
3. #10620947C2semidirect
4. #10620877C2semidirect
5. #10620664C2semidirect
6. C7×#824C2semidirect
7. #10620511C2semidirect
29. #6472D3semidirect
79. #824D7semidirect
173. C3×C7#456semidirect
227. C7#5205semidirect
242. C3#89524semidirect

Maximal subgroups

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

subgroupindex
#106215472
C3×#64722
#106209472
#106208772
#106206642
C7×#8242
#106205112
#895243
#52057