Finite Group Databasebeta

← #13054404

Subgroups — #13054404

Order 1344.

396 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 70 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×D4×D21C211yesmaximal, characteristic, complemented
2C2×C3×#1141C211yesmaximal, characteristic, complemented
3#10621027C211yesmaximal, characteristic, complemented
4C2×#66561C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620504C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89545—31nomaximal, Hall
9C23×D21—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×#4268—21no—
12D4×D21—22no—
13D4×D21—22no—
14C2×D84C2211yescharacteristic
15C2×C4×D21C2211yescharacteristic
16C2×C3×D28C2211yescharacteristic, complemented
17C2×#4246—21no—
18C7×C4⋊Dic3C2211yescharacteristic, complemented
19C2×C7×C3⋊C8C2211yescharacteristic, complemented
20C3×#1141—22no—
21C2×C3×C7⋊C8C2211yescharacteristic, complemented
22(C2×D7)⋊Dic3—21no—
23#66561—22no—
24C2×D4×D7—31no—
25C2×#1141D311yescharacteristic, complemented
26C2×D56—31no—
27#6458—31no—
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6383—31no—
31#5156—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—24no—
34C22×D21—21no—
35C22×D21—42no—
36C22×D21—42no—
37C22×D21D411yescharacteristic
38C22×C3×D7—21no—
39C3×C7×D4—21no—
40C3×C7×D4D412yescomplemented
41C2×C3×C4×C7C2311yescharacteristic
42#4268—24no—
43C2×Dic21D411yescharacteristic
44D84—21no—
45D84D412yescomplemented
46C4×D21—21no—
47C4×D21—21no—
48C2×C7×Dic3—21no—
49C3×D28—21no—
50C3×D28—21no—
51#4246—42no—
52C3×C7⋊C8—21no—
53C7×C3⋊C8—21no—
54C23×D7—61no—
55C2×C7×D4C2×D311yescharacteristic
56C2×C22⋊D7—61no—
57D4×D7—62no—
58D4×D7—62no—
59C2×D28—31no—
60C2×D28—61no—
61C2×D28C2×D311yescharacteristic, complemented
62C2×C4×D7—31no—
63C2×C7×C8—31no—
64C7×C4⋊C4—31no—
65#1141—22no—
66(C2×D7)⋊C4—61no—
67C2×C7⋊C8C2×D311yescharacteristic, complemented
68D56—62no—
69C2×D3×D4—71no—
70C2×C3×D8—71no—
71#938—71no—
72C2×#213—71no—
73#832D711yescharacteristic, complemented
74#821—71no—
75(C2×D3)⋊C8—71no—
76C22×C3×C7—22no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7C2×D411yescharacteristic
80C2×D21—21no—
81C2×D21—21no—
82C2×D21—22no—
83C2×D21—48no—
84C2×D21—41no—
85C2×D21—41no—
86C2×D21D812yescomplemented
87C2×C3×D7—41no—
88C2×C3×D7—41no—
89C2×C3×D7—41no—
90C3×C4×C7C2×D411yescharacteristic
91C3×C4×C7C2×D411yescharacteristic
92Dic21—21no—
93C7×Dic3—41no—
94#447—211noSylow
95C23×C7—21no—
96C22×D7—21no—
97C22×D7—31no—
98C22×D7—64no—
99C22×D7—61no—
100C22×D7—122no—
101C22×D7—122no—
102C7×D4—21no—
103C7×D4#6712yescomplemented
104C2×C4×C7—61no—
105C2×C4×C7C22×D311yescharacteristic
106C22⋊D7—64no—
107C2×Dic7—31no—
108D28—21no—
109D28—21no—
110D28—32no—
111D28—61no—
112D28—122no—
113C4×D7—61no—
114C4×D7—61no—
115C7×C8—61no—
116C7⋊C8—21no—
117C23×D3—141no—
118C2×C3×D4—71no—
119C2×C3×D4C2×D711yescharacteristic
120C2×#67—141no—
121C2×#67—141no—
122D3×D4—142no—
123D3×D4—142no—
124C2×D12—71no—
125C2×C4×D3—71no—
126C3×D8—142no—
127C2×C3×C8—71no—
128#217—141no—
129#213—142no—
130C4⋊Dic3C2×D711yescharacteristic, complemented
131C2×C3⋊C8C2×D711yescharacteristic, complemented
132C2×C3×C7—42no—
133C2×C3×C7#12011yescharacteristic
134C2×C3×C7C2×D811yescharacteristic
135C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
136D21—22no—
137D21—42no—
138C3×D7—81no—
139C22×D4—211no—
140C2×D8—71no—
141C2×D8—211no—
142#121—211no—
143#102—31no—
144#102—211no—
145C22⋊C8—211no—
146C22×C7—22no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7D3×D411yescharacteristic
150C2×D7—32no—
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—61no—
155C2×D7—61no—
156C2×D7—62no—
157C2×D7—128no—
158C2×D7—121no—
159C2×D7—121no—
160C4×C7—121no—
161C4×C7D3×D411yescharacteristic
162C4×C7C2×#6711yescharacteristic
163Dic7—61no—
164C23×C3—21no—
165C23×C3—141no—
166C22×D3—71no—
167C22×D3—144no—
168C22×D3—141no—
169C22×D3—282no—
170C22×D3—282no—
171C3×D4—21no—
172C3×D4—141no—
173C3×D4—141no—
174C3×D4D2812yescomplemented
175C2×C3×C4C22×D711yescharacteristic
176#67—144no—
177#67—282no—
178C2×Dic3—21no—
179C2×Dic3—71no—
180D12—72no—
181D12—141no—
182C4×D3—141no—
183C4×D3—141no—
184C3×C8—141no—
185C3⋊C8—21no—
186C3×C7#44711yescharacteristic, complemented, Hall
187C24—421no—
188C2×D4—71no—
189C2×D4—211no—
190C2×D4—421no—
191C2×D4—421no—
192C2×D4—422no—
193C2×D4—422no—
194C2×D4D3×D711yescharacteristic
195C22×C4—211no—
196D8—142no—
197D8—422no—
198C2×C8—31no—
199C2×C8—71no—
200C4⋊C4—31no—
201C22⋊C4—421no—
202C2×C7—42no—
203C2×C7D3×D811yescharacteristic
204C2×C7#91111yescharacteristic
205C2×C7#93711yescharacteristic
206D7—62no—
207D7—81no—
208D7—122no—
209C22×C3—22no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—281no—
213C22×C3—281no—
214C22×C3—281no—
215C22×C3D4×D711yescharacteristic
216C2×D3—72no—
217C2×D3—141no—
218C2×D3—141no—
219C2×D3—142no—
220C2×D3—288no—
221C2×D3—281no—
222C2×D3—281no—
223C3×C4C2×D2811yescharacteristic
224C3×C4D4×D711yescharacteristic
225Dic3—41no—
226Dic3—141no—
227C23—21no—
228C23—141no—
229C23—211no—
230C23—424no—
231C23—421no—
232C23—842no—
233C23—842no—
234D4—21no—
235D4—141no—
236D4—141no—
237D4—212no—
238D4—424no—
239D4—421no—
240D4—842no—
241D4#424612yescomplemented
242C2×C4—61no—
243C2×C4—211no—
244C2×C4—421no—
245C2×C4—421no—
246C2×C4C2×D3×D711yescharacteristic
247C8—61no—
248C8—141no—
249C7#515611yescharacteristic, complemented, Sylow
250C2×C3—42no—
251C2×C3—561no—
252C2×C3#644511yescharacteristic
253C2×C3D7×D811yescharacteristic
254C2×C3#647711yescharacteristic
255D3—142no—
256D3—282no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—212no—
261C22—281no—
262C22—281no—
263C22—281no—
264C22—421no—
265C22—421no—
266C22—422no—
267C22—848no—
268C22—841no—
269C22—841no—
270C22#6669411yescharacteristic
271C4—121no—
272C4—421no—
273C4C2×#424611yescharacteristic
274C4#6669411yescharacteristic
275C3#8954511yescharacteristic, complemented, Sylow
276C2—42no—
277C2—422no—
278C2—561no—
279C2—842no—
280C2#1062104311yescharacteristic
281C2#1062106011yescharacteristic
282C2#1062113411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#447
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×D21C2semidirect
2. C2×C3×#1141C2semidirect
3. #10621027C2semidirect
4. C2×#66561C2semidirect
5. C7×#832C2semidirect
6. #10620504C2semidirect
7. #10620492C2semidirect
16. C2×C3×D28C22semidirect
18. C7×C4⋊Dic3C22semidirect
19. C2×C7×C3⋊C8C22semidirect
21. C2×C3×C7⋊C8C22semidirect
25. C2×#1141D3semidirect
40. C3×C7×D4D4semidirect
45. D84D4semidirect
61. C2×D28C2×D3semidirect
67. C2×C7⋊C8C2×D3semidirect
73. #832D7semidirect
86. C2×D21D8semidirect
103. C7×D4#67semidirect
130. C4⋊Dic3C2×D7semidirect
131. C2×C3⋊C8C2×D7semidirect
174. C3×D4D28semidirect
186. C3×C7#447semidirect
241. D4#4246semidirect
249. C7#5156semidirect
275. C3#89545semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C2×C3×#11412
#106210272
C2×#665612
C7×#8322
#106205042
#106204922
#895453
#51567