Finite Group Databasebeta

← #13060178

Subgroups — #13060178

Order 1344.

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

normal subgroups only

This group has 520 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#10621419C211yesmaximal, characteristic, complemented
2C7×#937C211yesmaximal, characteristic, complemented
3C3×#6500C211yesmaximal, characteristic, complemented
4#10621130C212yesmaximal, complemented
5#10621125C212yesmaximal, complemented
6#10621096C212yesmaximal, complemented
7#10621087C212yesmaximal, complemented
8#10621026C212yesmaximal, complemented
9#10621024C212yesmaximal, complemented
10#90285—31nomaximal, Hall
11C23×D3×C7C2211yescharacteristic, complemented
12C23×C3×D7C2211yescharacteristic, complemented
13C2×C3×C7×D4C2211yescharacteristic
14C2×Dic42C2211yescharacteristic, complemented
15C2×C7×#67C2212yes—
16C2×C3×C22⋊D7C2212yes—
17C2×#4248C2212yescomplemented
18C2×#4245C2212yes—
19C2×#4245C2212yes—
20#66636C2212yescomplemented
21#66636C2212yescomplemented
22C4×Dic21C2211yescharacteristic, complemented
23C7×#217C2211yescharacteristic, complemented
24C7×(C2×D3)⋊C4C2212yescomplemented
25C3×C22⋊Dic7C2211yescharacteristic, complemented
26C3×(C2×D7)⋊C4C2212yescomplemented
27(C2×D3)⋊Dic7C2212yescomplemented
28(C2×D3)⋊Dic7C2212yescomplemented
29(C2×D7)⋊Dic3C2212yescomplemented
30(C2×D7)⋊Dic3C2212yescomplemented
31Dic3×Dic7C2212yescomplemented
32C7×#120—31no—
33#6516—32no—
34#6500—32no—
35#6500D311yescharacteristic, complemented
36#6499—31no—
37#6499—32no—
38#6448—32no—
39#6448—32no—
40#6448—32no—
41#5594—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D3×C7—41no—
45C22×D3×C7—42no—
46C22×D3×C7C2312yes—
47C22×C3×D7—41no—
48C22×C3×D7—42no—
49C22×C3×D7C2312yes—
50C3×C7×D4—41no—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2312yes—
53C2×Dic21C2312yes—
54Dic42—41no—
55C7×#67—42no—
56C2×C7×Dic3C2312yes—
57C3×C22⋊D7—42no—
58C2×C3×Dic7C2312yes—
59#4248—42no—
60#4245—42no—
61#4245—42no—
62C24×C7—31no—
63C23×D7C2×D311yescharacteristic, complemented
64C2×C7×D4—32no—
65C2×C7×D4C2×D311yescharacteristic
66C2×C22⋊D7—32no—
67C2×C22⋊D7—32no—
68C2×C22⋊D7C2×D312yes—
69C2×Dic14—31no—
70C2×Dic14—32no—
71C7×C22⋊C4—31no—
72C7×C22⋊C4—32no—
73C22⋊Dic7—32no—
74C22⋊Dic7—32no—
75C22⋊Dic7—32no—
76C22⋊Dic7—32no—
77C22⋊Dic7C2×D311yescharacteristic, complemented
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4C2×D312yescomplemented
81C4×Dic7—31no—
82C4×Dic7—32no—
83C3×#120—71no—
84#953—72no—
85#937—72no—
86#937D711yescharacteristic, complemented
87#936—71no—
88#936—72no—
89#885—72no—
90#885—72no—
91#885—72no—
92C22×C3×C7—22no—
93C22×C3×C7—22no—
94C22×C3×C7—21no—
95C22×C3×C7—21no—
96C22×C3×C7C2411yescharacteristic
97C2×D3×C7—22no—
98C2×D3×C7—22no—
99C2×D3×C7—22no—
100C2×D3×C7—44no—
101C2×C3×D7—22no—
102C2×C3×D7—22no—
103C2×C3×D7—22no—
104C2×C3×D7—44no—
105C3×C4×C7—21no—
106Dic21—22no—
107Dic21—22no—
108C3×Dic7—22no—
109C7×Dic3—22no—
110H(3,𝔽4)—211noSylow
111C23×C7—32no—
112C23×C7—122no—
113C23×C7—121no—
114C23×C7C22×D311yescharacteristic
115C23×C7C22×D311yescharacteristic
116C22×D7—42no—
117C22×D7—41no—
118C22×D7C22×D312yes—
119C7×D4—41no—
120C7×D4—122no—
121C2×C4×C7—32no—
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—42no—
124C22⋊D7—122no—
125C22⋊D7—122no—
126C2×Dic7—32no—
127C2×Dic7—32no—
128C2×Dic7C22×D312yes—
129Dic14—122no—
130Dic14—121no—
131C24×C3—71no—
132C23×D3C2×D711yescharacteristic, complemented
133C2×C3×D4—72no—
134C2×C3×D4C2×D711yescharacteristic
135C2×#67—72no—
136C2×#67—72no—
137C2×#67C2×D712yes—
138C2×Dic6—71no—
139C2×Dic6—72no—
140C3×C22⋊C4—71no—
141C3×C22⋊C4—72no—
142#217—72no—
143#217—72no—
144#217—72no—
145#217—72no—
146#217C2×D711yescharacteristic, complemented
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4C2×D712yescomplemented
150C4×Dic3—71no—
151C4×Dic3—72no—
152C2×C3×C7—41no—
153C2×C3×C7—41no—
154C2×C3×C7H(5,𝔽2)12yes—
155C2×C3×C7H(5,𝔽2)11yescharacteristic
156C3×D7—42no—
157D3×C7—42no—
158#124—212no—
159#124—212no—
160#124—212no—
161#124—212no—
162#124—211no—
163#120—31no—
164#120—71no—
165#120—212no—
166#120—212no—
167C22×C7—22no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—22no—
171C22×C7—62no—
172C22×C7—62no—
173C22×C7—62no—
174C22×C7—124no—
175C22×C7C23×D311yescharacteristic
176C2×D7—22no—
177C2×D7—22no—
178C2×D7—22no—
179C2×D7—44no—
180C4×C7—21no—
181C4×C7—62no—
182Dic7—22no—
183Dic7—62no—
184Dic7—62no—
185C23×C3—72no—
186C23×C3—282no—
187C23×C3—281no—
188C23×C3C22×D711yescharacteristic
189C23×C3C22×D711yescharacteristic
190C22×D3—42no—
191C22×D3—41no—
192C22×D3C22×D712yes—
193C3×D4—41no—
194C3×D4—282no—
195C2×C3×C4—72no—
196C2×C3×C4C22×D711yescharacteristic
197#67—42no—
198#67—282no—
199#67—282no—
200C2×Dic3—72no—
201C2×Dic3—72no—
202C2×Dic3C22×D712yes—
203Dic6—282no—
204Dic6—281no—
205C3×C7H(3,𝔽4)11yescharacteristic, complemented, Hall
206C24—31no—
207C24—71no—
208C2×Q8—212no—
209C2×Q8—211no—
210C2×D4—32no—
211C2×D4—72no—
212C2×D4—212no—
213C2×D4—212no—
214C2×D4D3×D711yescharacteristic
215C22⋊C4—31no—
216C22⋊C4—32no—
217C22⋊C4—71no—
218C22⋊C4—72no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C42—212no—
226C42—211no—
227C2×C7—41no—
228C2×C7—41no—
229C2×C7—122no—
230C2×C7#100412yes—
231C2×C7#100411yescharacteristic
232D7—42no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—22no—
237C22×C3—142no—
238C22×C3—142no—
239C22×C3—142no—
240C22×C3—284no—
241C22×C3C23×D711yescharacteristic
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—44no—
246C3×C4—21no—
247C3×C4—142no—
248Dic3—22no—
249Dic3—142no—
250Dic3—142no—
251C23—32no—
252C23—72no—
253C23—122no—
254C23—121no—
255C23—282no—
256C23—281no—
257C23C2×D3×D711yescharacteristic
258C23C2×D3×D711yescharacteristic
259Q8—841no—
260Q8—842no—
261D4—41no—
262D4—122no—
263D4—282no—
264D4—842no—
265D4—842no—
266C2×C4—32no—
267C2×C4—72no—
268C2×C4—212no—
269C2×C4—212no—
270C2×C4C2×D3×D711yescharacteristic
271C7#559411yescharacteristic, complemented, Sylow
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—282no—
275C2×C3#654812yes—
276C2×C3#654811yescharacteristic
277D3—42no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—124no—
286C22—142no—
287C22—142no—
288C22—142no—
289C22—284no—
290C22C22×D3×D711yescharacteristic
291C4—21no—
292C4—62no—
293C4—142no—
294C4—422no—
295C4—422no—
296C3#9028511yescharacteristic, complemented, Sylow
297C2—41no—
298C2—41no—
299C2—122no—
300C2—282no—
301C2#1062157212yes—
302C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7H(3,𝔽4)
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. #10621419C2semidirect
2. C7×#937C2semidirect
3. C3×#6500C2semidirect
4. #10621130C2semidirect
5. #10621125C2semidirect
6. #10621096C2semidirect
7. #10621087C2semidirect
8. #10621026C2semidirect
9. #10621024C2semidirect
11. C23×D3×C7C22semidirect
12. C23×C3×D7C22semidirect
14. C2×Dic42C22semidirect
17. C2×#4248C22semidirect
20. #66636C22semidirect
21. #66636C22semidirect
22. C4×Dic21C22semidirect
23. C7×#217C22semidirect
24. C7×(C2×D3)⋊C4C22semidirect
25. C3×C22⋊Dic7C22semidirect
26. C3×(C2×D7)⋊C4C22semidirect
27. (C2×D3)⋊Dic7C22semidirect
28. (C2×D3)⋊Dic7C22semidirect
29. (C2×D7)⋊Dic3C22semidirect
30. (C2×D7)⋊Dic3C22semidirect
31. Dic3×Dic7C22semidirect
35. #6500D3semidirect
63. C23×D7C2×D3semidirect
77. C22⋊Dic7C2×D3semidirect
80. (C2×D7)⋊C4C2×D3semidirect
86. #937D7semidirect
132. C23×D3C2×D7semidirect
146. #217C2×D7semidirect
149. (C2×D3)⋊C4C2×D7semidirect
205. C3×C7H(3,𝔽4)semidirect
271. C7#5594semidirect
296. C3#90285semidirect

Maximal subgroups

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

subgroupindex
#106214192
C7×#9372
C3×#65002
#106211302
#106211252
#106210962
#106210872
#106210262
#106210242
#902853
#55947