Finite Group Databasebeta

← #13054641

Subgroups — #13054641

Order 1344.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#66732C211yesmaximal, characteristic, complemented
2C2×C7×#215C211yesmaximal, characteristic, complemented
3#10621028C211yesmaximal, characteristic, complemented
4C2×#66562C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620503C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89944—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66732—22no—
11#66732—22no—
12C2×D84—21no—
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×C4×D21C2211yescharacteristic
16C2×C7×D12C2211yescharacteristic
17C2×#4247—21no—
18C7×#215—22no—
19C2×C7×C3⋊C8C2211yescharacteristic
20C3×C4⋊Dic7C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22(C2×D3)⋊Dic7—21no—
23#66562—22no—
24C2×#1161—31no—
25C2×C7×SD16—31no—
26#6501—31no—
27C2×#1141—31no—
28#6409D311yescharacteristic, complemented
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#4810—71nomaximal, Hall
32C22×D21—21no—
33C22×D21D411yescharacteristic
34C22×D3×C7—21no—
35C3×C7×Q8—21no—
36C3×C7×Q8D412yescomplemented
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21D411yescharacteristic
40D84—24no—
41D84—21no—
42D84D412yescomplemented
43C4×D21—24no—
44C4×D21—21no—
45C4×D21—21no—
46C7×D12—21no—
47C7×D12—21no—
48C2×C3×Dic7—21no—
49#4247—42no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8C2×D311yescharacteristic
53C2×C7×D4—31no—
54C2×C22⋊D7—61no—
55#1161—62no—
56#1161—62no—
57C2×D28—31no—
58C2×D28—61no—
59C2×C4×D7—31no—
60C2×C4×D7—61no—
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1141—62no—
65C4⋊Dic7C2×D311yescharacteristic
66C2×C7⋊C8C2×D311yescharacteristic
67C2×#239—71no—
68C2×#215D711yescharacteristic, complemented
69C2×D24—71no—
70#895—71no—
71C3×#103—71no—
72(C2×D3)⋊C8—71no—
73#809—71no—
74C22×C3×C7C2×D411yescharacteristic
75C2×D21—21no—
76C2×D21—22no—
77C2×D21—21no—
78C2×D21—21no—
79C2×D21—41no—
80C2×D21—41no—
81C2×D3×C7—41no—
82C2×D3×C7—41no—
83C2×D3×C7—41no—
84C3×C4×C7—22no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88C3×Dic7—41no—
89#449—211noSylow
90C23×C7—61no—
91C22×D7—31no—
92C22×D7—61no—
93C7×Q8—21no—
94C7×Q8D1212yescomplemented
95C7×D4—61no—
96C7×D4—61no—
97C2×C4×C7—21no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—122no—
100C2×Dic7—21no—
101C2×Dic7—31no—
102D28—32no—
103D28—64no—
104D28—61no—
105C4×D7—64no—
106C4×D7—61no—
107C4×D7—61no—
108C7×C8—61no—
109C7⋊C8—21no—
110C2×C3×Q8C2×D711yescharacteristic
111#239—142no—
112#239—142no—
113C2×D12—71no—
114C2×D12—141no—
115C2×D12—141no—
116C2×D12C2×D711yescharacteristic
117C2×C4×D3—71no—
118C2×C4×D3—141no—
119C2×C3×C8—71no—
120C3×C4⋊C4—71no—
121#215—22no—
122(C2×D3)⋊C4—141no—
123C2×C3⋊C8C2×D711yescharacteristic
124D24—142no—
125C2×C3×C7#12011yescharacteristic
126C2×C3×C7#13511yescharacteristic
127C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
128D21—41no—
129D21—42no—
130D3×C7—81no—
131C2×#41—211no—
132C2×SD16—31no—
133C2×D8—211no—
134#121—211no—
135#103—71no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—121no—
139C22×C7—121no—
140C22×C7—121no—
141C22×C7D3×D411yescharacteristic
142C2×D7—62no—
143C2×D7—61no—
144C2×D7—61no—
145C2×D7—61no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—22no—
149C4×C7C2×D1211yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—41no—
152Dic7—61no—
153C22×D3—21no—
154C22×D3—71no—
155C22×D3—141no—
156C3×Q8—21no—
157C3×Q8C22⋊D712yescomplemented
158C2×C3×C4—21no—
159C2×C3×C4—141no—
160C2×C3×C4C22×D711yescharacteristic
161C2×Dic3—71no—
162D12—21no—
163D12—21no—
164D12—72no—
165D12—144no—
166D12—141no—
167D12—282no—
168C4×D3—144no—
169C4×D3—141no—
170C4×D3—141no—
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8D3×D711yescharacteristic
177C2×D4—31no—
178C2×D4—211no—
179C2×D4—421no—
180C2×D4—421no—
181C22×C4—211no—
182C22×C4—421no—
183SD16—62no—
184D8—422no—
185C2×C8—31no—
186C2×C8—71no—
187C4⋊C4—71no—
188C22⋊C4—421no—
189C2×C7—241no—
190C2×C7#88211yescharacteristic
191C2×C7#91411yescharacteristic
192C2×C7#91911yescharacteristic
193D7—121no—
194D7—122no—
195C22×C3D4×D711yescharacteristic
196C2×D3—41no—
197C2×D3—41no—
198C2×D3—41no—
199C2×D3—141no—
200C2×D3—141no—
201C2×D3—142no—
202C2×D3—141no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—22no—
206C3×C4—281no—
207C3×C4D4×D711yescharacteristic
208C3×C4C2×C22⋊D711yescharacteristic
209Dic3—141no—
210C23—61no—
211C23—211no—
212C23—421no—
213Q8—21no—
214Q8#424712yescomplemented
215D4—61no—
216D4—61no—
217D4—212no—
218D4—424no—
219D4—421no—
220D4—842no—
221C2×C4—21no—
222C2×C4—141no—
223C2×C4—211no—
224C2×C4—424no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—141no—
230C7#481011yescharacteristic, complemented, Sylow
231C2×C3#647711yescharacteristic
232C2×C3#647911yescharacteristic
233C2×C3#650011yescharacteristic
234D3—81no—
235D3—281no—
236D3—282no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22—421no—
241C22—421no—
242C22—422no—
243C22—421no—
244C22—841no—
245C22—841no—
246C22#6669411yescharacteristic
247C4—22no—
248C4—281no—
249C4—421no—
250C4C2×#424711yescharacteristic
251C4#6669411yescharacteristic
252C3#8994411yescharacteristic, complemented, Sylow
253C2—241no—
254C2—841no—
255C2—842no—
256C2#1062106811yescharacteristic
257C2#1062108511yescharacteristic
258C2#1062113511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#66732C2semidirect
2. C2×C7×#215C2semidirect
3. #10621028C2semidirect
4. C2×#66562C2semidirect
5. C3×#6409C2semidirect
6. #10620503C2semidirect
7. #10620492C2semidirect
28. #6409D3semidirect
36. C3×C7×Q8D4semidirect
42. D84D4semidirect
68. C2×#215D7semidirect
94. C7×Q8D12semidirect
157. C3×Q8C22⋊D7semidirect
173. C3×C7#449semidirect
214. Q8#4247semidirect
230. C7#4810semidirect
252. C3#89944semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×C7×#2152
#106210282
C2×#665622
C3×#64092
#106205032
#106204922
#899443
#48107