Finite Group Databasebeta

← #13060197

Subgroups — #13060197

Order 1344.

504 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 120 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#10621420C211yesmaximal, characteristic, complemented
2C7×#940C211yesmaximal, characteristic, complemented
3C3×#6498C211yesmaximal, characteristic, complemented
4#10621135C212yesmaximal, complemented
5#10621128C212yesmaximal, complemented
6#10621120C212yesmaximal, complemented
7#10621117C212yesmaximal, complemented
8#10620978C212yesmaximal, complemented
9#10620942C212yesmaximal, complemented
10#90411—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C2×C7×#67C2212yes—
15C2×C7×#67C2212yescomplemented
16C2×C7×D12C2211yescharacteristic, complemented
17C22×C3×Dic7C2211yescharacteristic, complemented
18C2×#4247C2212yes—
19C2×#4247C2212yes—
20C2×D3×Dic7C2212yescomplemented
21#66636C2211yescharacteristic
22#66631C2212yes—
23C4×C7×Dic3C2211yescharacteristic, complemented
24C3×C22⋊Dic7C2212yes—
25C3×C22⋊Dic7C2211yescharacteristic, complemented
26C3×Dic7⋊C4C2212yescomplemented
27#66577C2212yescomplemented
28Dic7⋊Dic3C2212yescomplemented
29#66575C2212yes—
30#66575C2212yes—
31(C2×D3)⋊Dic7C2212yescomplemented
32C7×#127—31no—
33#6502—32no—
34#6502—32no—
35#6502—32no—
36#6500—31no—
37#6500—32no—
38#6498—32no—
39#6498D311yescharacteristic, complemented
40#6439—32no—
41#5666—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D21—41no—
45C22×D21—42no—
46C22×D21C2312yes—
47C22×D3×C7C2312yes—
48C3×C7×D4—41no—
49C2×C3×C4×C7C2311yescharacteristic
50#4268—42no—
51C2×Dic21C2312yes—
52C7×#67—42no—
53C7×#67—42no—
54C2×C7×Dic3C2312yes—
55C7×D12—41no—
56C2×C3×Dic7—41no—
57C2×C3×Dic7C2312yes—
58C2×C3×Dic7C2312yes—
59#4247—42no—
60#4247—42no—
61D3×Dic7—42no—
62C23×D7—31no—
63C2×C7×D4—32no—
64C2×C7×D4—32no—
65C2×C7×D4—31no—
66C2×C7×D4C2×D311yescharacteristic
67C2×C22⋊D7—32no—
68C2×C22⋊D7—32no—
69C2×C22⋊D7—32no—
70C22×Dic7—32no—
71C22×Dic7C2×D311yescharacteristic, complemented
72C42×C7—31no—
73C22⋊Dic7—31no—
74C22⋊Dic7—32no—
75C22⋊Dic7C2×D312yes—
76C22⋊Dic7C2×D311yescharacteristic, complemented
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80Dic7⋊C4—32no—
81Dic7⋊C4—32no—
82Dic7⋊C4C2×D312yescomplemented
83C3×#123—71no—
84#940D711yescharacteristic, complemented
85#937—71no—
86#930—72no—
87#898—72no—
88#895—72no—
89#884—72no—
90#883—72no—
91#882—72no—
92C22×C3×C7—21no—
93C22×C3×C7—22no—
94C22×C3×C7—22no—
95C22×C3×C7—21no—
96C22×C3×C7C2411yescharacteristic
97C2×D21—22no—
98C2×D21—22no—
99C2×D21—22no—
100C2×D21—44no—
101C2×D3×C7—22no—
102C2×D3×C7—22no—
103C2×D3×C7—22no—
104C3×C4×C7—21no—
105Dic21—22no—
106C3×Dic7—22no—
107C3×Dic7—22no—
108C7×Dic3—22no—
109#560—211noSylow
110C23×C7—32no—
111C23×C7C22×D311yescharacteristic
112C23×C7C22×D311yescharacteristic
113C22×D7—32no—
114C22×D7—121no—
115C22×D7—122no—
116C7×D4—41no—
117C7×D4—121no—
118C7×D4—122no—
119C7×D4—122no—
120C2×C4×C7—32no—
121C2×C4×C7C22×D311yescharacteristic
122C22⋊D7—122no—
123C22⋊D7—122no—
124C22⋊D7—122no—
125C2×Dic7—32no—
126C2×Dic7—41no—
127C2×Dic7—122no—
128C2×Dic7C22×D312yes—
129C2×Dic7C22×D312yes—
130C23×D3—71no—
131C2×C3×D4C2×D711yescharacteristic
132C22×C3×C4—71no—
133C2×#67—72no—
134C2×#67C2×D712yes—
135C2×#67C2×D712yescomplemented
136C2×D12—72no—
137C2×D12—72no—
138C2×D12C2×D711yescharacteristic, complemented
139C2×C4×D3—72no—
140C3×C4⋊C4—72no—
141C3×C22⋊C4—72no—
142C3×C22⋊C4—71no—
143#217—71no—
144(C2×D3)⋊C4—72no—
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148C4⋊Dic3—72no—
149Dic3⋊C4—72no—
150C4×Dic3C2×D711yescharacteristic, complemented
151C2×C3×C7—41no—
152C2×C3×C7—41no—
153C2×C3×C7H(5,𝔽2)11yescharacteristic
154C2×C3×C7H(5,𝔽2)12yes—
155D21—42no—
156D3×C7—42no—
157#127—31no—
158#126—212no—
159#123—71no—
160#123—212no—
161#121—212no—
162#121—212no—
163#121—212no—
164#120—212no—
165#120—211no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—62no—
171C22×C7—62no—
172C22×C7—62no—
173C22×C7C23×D311yescharacteristic
174C2×D7—62no—
175C2×D7—62no—
176C2×D7—62no—
177C2×D7—124no—
178C4×C7—21no—
179C4×C7—62no—
180Dic7—22no—
181Dic7—22no—
182Dic7—62no—
183C23×C3C22×D711yescharacteristic
184C23×C3C22×D711yescharacteristic
185C22×D3—72no—
186C22×D3—281no—
187C22×D3—282no—
188C22×D3C22×D712yes—
189C3×D4—41no—
190C2×C3×C4—72no—
191C2×C3×C4—72no—
192C2×C3×C4—281no—
193C2×C3×C4C22×D711yescharacteristic
194#67—42no—
195#67—42no—
196#67—282no—
197C2×Dic3—72no—
198C2×Dic3C22×D712yes—
199D12—41no—
200D12—282no—
201D12—282no—
202C4×D3—282no—
203C3×C7#56011yescharacteristic, complemented, Hall
204C24—211no—
205C2×D4—32no—
206C2×D4—32no—
207C2×D4—31no—
208C2×D4—212no—
209C2×D4—212no—
210C2×D4—212no—
211C2×D4D3×D711yescharacteristic
212C22×C4—71no—
213C22×C4—212no—
214C4⋊C4—72no—
215C4⋊C4—212no—
216C4⋊C4—212no—
217C22⋊C4—72no—
218C22⋊C4—71no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—211no—
223C22⋊C4—212no—
224C42—31no—
225C2×C7—41no—
226C2×C7—41no—
227C2×C7—122no—
228C2×C7#100411yescharacteristic
229C2×C7#100912yes—
230D7—122no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3C23×D711yescharacteristic
236C2×D3—22no—
237C2×D3—22no—
238C2×D3—22no—
239C2×D3—142no—
240C2×D3—142no—
241C2×D3—142no—
242C2×D3—284no—
243C3×C4—21no—
244C3×C4—142no—
245C3×C4—142no—
246Dic3—22no—
247Dic3—142no—
248C23—32no—
249C23—212no—
250C23—842no—
251C23—841no—
252C23C2×D3×D711yescharacteristic
253C23C2×D3×D711yescharacteristic
254D4—41no—
255D4—121no—
256D4—122no—
257D4—122no—
258D4—842no—
259D4—842no—
260D4—842no—
261C2×C4—32no—
262C2×C4—72no—
263C2×C4—72no—
264C2×C4—212no—
265C2×C4—281no—
266C2×C4—842no—
267C2×C4C2×D3×D711yescharacteristic
268C7#566611yescharacteristic, complemented, Sylow
269C2×C3—41no—
270C2×C3—41no—
271C2×C3#654812yes—
272C2×C3#654811yescharacteristic
273D3—42no—
274D3—282no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—21no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—422no—
283C22—422no—
284C22—422no—
285C22—844no—
286C22C22×D3×D711yescharacteristic
287C4—21no—
288C4—62no—
289C4—142no—
290C4—142no—
291C4—422no—
292C3#9041111yescharacteristic, complemented, Sylow
293C2—41no—
294C2—41no—
295C2—122no—
296C2—842no—
297C2#1062157412yes—
298C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#560
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621420C2semidirect
2. C7×#940C2semidirect
3. C3×#6498C2semidirect
4. #10621135C2semidirect
5. #10621128C2semidirect
6. #10621120C2semidirect
7. #10621117C2semidirect
8. #10620978C2semidirect
9. #10620942C2semidirect
15. C2×C7×#67C22semidirect
16. C2×C7×D12C22semidirect
17. C22×C3×Dic7C22semidirect
20. C2×D3×Dic7C22semidirect
23. C4×C7×Dic3C22semidirect
25. C3×C22⋊Dic7C22semidirect
26. C3×Dic7⋊C4C22semidirect
27. #66577C22semidirect
28. Dic7⋊Dic3C22semidirect
31. (C2×D3)⋊Dic7C22semidirect
39. #6498D3semidirect
71. C22×Dic7C2×D3semidirect
76. C22⋊Dic7C2×D3semidirect
82. Dic7⋊C4C2×D3semidirect
84. #940D7semidirect
135. C2×#67C2×D7semidirect
138. C2×D12C2×D7semidirect
150. C4×Dic3C2×D7semidirect
203. C3×C7#560semidirect
268. C7#5666semidirect
292. C3#90411semidirect

Maximal subgroups

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

subgroupindex
#106214202
C7×#9402
C3×#64982
#106211352
#106211282
#106211202
#106211172
#106209782
#106209422
#904113
#56667