Finite Group Databasebeta

← #13052898

Subgroups — #13052898

Order 1344.

324 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66670C211yesmaximal, characteristic, complemented
2#10621378C211yesmaximal, characteristic, complemented
3C2×#66568C211yesmaximal, characteristic, complemented
4C2×#66561C211yesmaximal, characteristic, complemented
5C7×#810C211yesmaximal, characteristic, complemented
6C3×#6383C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89583—31nomaximal, Hall
9C2×D84—21no—
10C2×D84C2211yescharacteristic
11C2×C4×D21C2211yescharacteristic
12C2×C7×Dic6C2211yescharacteristic
13C2×C3×D28C2211yescharacteristic
14C2×#4246—21no—
15C2×Dic3×D7—21no—
16#66670—22no—
17#66670—22no—
18C3×C7×C4⋊C4C2211yescharacteristic
19#66631—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66568—22no—
23#66561—22no—
24C2×#1161—31no—
25C2×#1143—31no—
26C2×D56—31no—
27#6458—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6383D311yescharacteristic, complemented
31#4781—71nomaximal, Hall
32C22×D21—21no—
33C22×D21D411yescharacteristic
34C22×C3×D7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21D411yescharacteristic
38D84—21no—
39D84—21no—
40D84—42no—
41C4×D21—21no—
42C4×D21—21no—
43C2×C7×Dic3—21no—
44C7×Dic6—21no—
45C7×Dic6D412yescomplemented
46C3×D28—21no—
47C3×D28D412yescomplemented
48#4246—24no—
49Dic3×D7—24no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8—31no—
53#1161—62no—
54#1161—62no—
55C2×D28—31no—
56C2×D28—61no—
57C2×D28—61no—
58C2×D28C2×D311yescharacteristic
59C2×C4×D7—31no—
60C2×C4×D7—61no—
61C2×C7×C8—31no—
62C7×C4⋊C4C2×D311yescharacteristic
63#1143—62no—
64(C2×D7)⋊C4—61no—
65C2×C7⋊C8C2×D311yescharacteristic
66D56—62no—
67C2×#237—71no—
68C2×#213—71no—
69C2×#204—71no—
70#895—71no—
71C3×#102—71no—
72(C2×D3)⋊C8—71no—
73#810D711yescharacteristic, complemented
74C22×C3×C7C2×D411yescharacteristic
75C2×D21—21no—
76C2×D21—21no—
77C2×D21—21no—
78C2×D21—41no—
79C2×D21—41no—
80C2×D21—41no—
81C2×C3×D7—22no—
82C2×C3×D7—41no—
83C2×C3×D7—41no—
84C3×C4×C7—41no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88C7×Dic3—22no—
89#449—211noSylow
90C22×D7—21no—
91C22×D7—31no—
92C22×D7—61no—
93C7×Q8—32no—
94C7×Q8—61no—
95C2×C4×C7—21no—
96C2×C4×C7—61no—
97C2×C4×C7C22×D311yescharacteristic
98C2×Dic7—31no—
99D28—21no—
100D28—64no—
101D28—61no—
102D28—61no—
103D28—122no—
104D28D1212yescomplemented
105C4×D7—64no—
106C4×D7—61no—
107C4×D7—61no—
108C7×C8—61no—
109C7⋊C8—21no—
110C2×C3×D4—71no—
111C2×#67—141no—
112C22×Dic3—141no—
113#237—142no—
114#237—142no—
115C2×D12—71no—
116C2×D12—141no—
117C2×C4×D3—71no—
118C2×Dic6C2×D711yescharacteristic
119C2×C3×C8—71no—
120C3×C4⋊C4C2×D711yescharacteristic
121#213—142no—
122(C2×D3)⋊C4—141no—
123C2×C3⋊C8C2×D711yescharacteristic
124#204—142no—
125C2×C3×C7#12011yescharacteristic
126C2×C3×C7#13511yescharacteristic
127C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
128D21—41no—
129D21—81no—
130C3×D7—42no—
131C2×#41—211no—
132C2×SD16—211no—
133C2×D8—211no—
134#121—211no—
135#103—31no—
136#102—71no—
137C22⋊C8—211no—
138C22×C7D3×D411yescharacteristic
139C2×D7—22no—
140C2×D7—41no—
141C2×D7—41no—
142C2×D7—61no—
143C2×D7—61no—
144C2×D7—61no—
145C2×D7—121no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—41no—
149C4×C7—62no—
150C4×C7C2×D1211yescharacteristic
151C4×C7D3×D411yescharacteristic
152Dic7—61no—
153C23×C3—141no—
154C22×D3—71no—
155C22×D3—141no—
156C3×D4—72no—
157C3×D4—141no—
158C2×C3×C4—21no—
159C2×C3×C4C22×D711yescharacteristic
160#67—144no—
161C2×Dic3—21no—
162C2×Dic3—71no—
163C2×Dic3—144no—
164D12—141no—
165D12—141no—
166D12—282no—
167C4×D3—141no—
168C4×D3—141no—
169Dic6—21no—
170Dic6D2812yescomplemented
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—31no—
177C2×D4—71no—
178C2×D4—211no—
179C2×D4—421no—
180C2×D4—421no—
181C22×C4—211no—
182C22×C4—421no—
183SD16—422no—
184D8—422no—
185C2×C8—31no—
186C2×C8—71no—
187C4⋊C4D3×D711yescharacteristic
188C22⋊C4—421no—
189C2×C7#88211yescharacteristic
190C2×C7#91111yescharacteristic
191C2×C7#91611yescharacteristic
192D7—42no—
193D7—121no—
194D7—241no—
195C22×C3—142no—
196C22×C3—281no—
197C22×C3—281no—
198C22×C3D4×D711yescharacteristic
199C2×D3—141no—
200C2×D3—141no—
201C2×D3—141no—
202C2×D3—281no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—41no—
206C3×C4C2×D2811yescharacteristic
207C3×C4D4×D711yescharacteristic
208Dic3—22no—
209Dic3—141no—
210C23—141no—
211C23—211no—
212C23—421no—
213Q8—32no—
214Q8—61no—
215D4—72no—
216D4—141no—
217D4—424no—
218D4—421no—
219D4—421no—
220D4—842no—
221C2×C4—21no—
222C2×C4—61no—
223C2×C4—211no—
224C2×C4—424no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—141no—
230C7#478111yescharacteristic, complemented, Sylow
231C2×C3—282no—
232C2×C3#644511yescharacteristic
233C2×C3#647711yescharacteristic
234C2×C3#648211yescharacteristic
235D3—281no—
236D3—561no—
237C22—142no—
238C22—281no—
239C22—281no—
240C22—421no—
241C22—421no—
242C22—421no—
243C22—841no—
244C22—841no—
245C22—841no—
246C22#6669411yescharacteristic
247C4—41no—
248C4—62no—
249C4—421no—
250C4D7×D1211yescharacteristic
251C4D3×D2811yescharacteristic
252C3#8958311yescharacteristic, complemented, Sylow
253C2—282no—
254C2—841no—
255C2—1681no—
256C2#1062103811yescharacteristic
257C2#1062106011yescharacteristic
258C2#1062108311yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#66670C2semidirect
2. #10621378C2semidirect
3. C2×#66568C2semidirect
4. C2×#66561C2semidirect
5. C7×#810C2semidirect
6. C3×#6383C2semidirect
7. #10620492C2semidirect
30. #6383D3semidirect
45. C7×Dic6D4semidirect
47. C3×D28D4semidirect
73. #810D7semidirect
104. D28D12semidirect
170. Dic6D28semidirect
173. C3×C7#449semidirect
230. C7#4781semidirect
252. C3#89583semidirect

Maximal subgroups

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

subgroupindex
C2×#666702
#106213782
C2×#665682
C2×#665612
C7×#8102
C3×#63832
#106204922
#895833
#47817