Finite Group Databasebeta

← #13061697

Subgroups — #13061697

Order 1344.

692 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 147 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×D21C212yesmaximal, complemented
2C3×C7×#121C211yesmaximal, characteristic, complemented
3#10621422C212yesmaximal, complemented
4#10621420C212yesmaximal, complemented
5#10621413C211yesmaximal, characteristic, complemented
6C2×#66631C211yesmaximal, characteristic, complemented
7#10621377C212yesmaximal, complemented
8#10621366C212yesmaximal, complemented
9#10621365C212yesmaximal, complemented
10#90296—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C23×D21C2212yescomplemented
13C2×C3×C7×D4C2212yescomplemented
14C2×C3×C7×D4C2211yescharacteristic, complemented
15C22×C3×C4×C7C2211yescharacteristic, complemented
16C2×#4268C2212yes—
17C2×#4268C2212yescomplemented
18C2×#4268C2212yescomplemented
19C22×Dic21C2211yescharacteristic, complemented
20D4×D21—28no—
21C2×D84C2212yes—
22C2×D84C2211yescharacteristic, complemented
23C2×C4×D21C2212yescomplemented
24C3×C7×C4⋊C4C2211yescharacteristic, complemented
25C3×C7×C22⋊C4C2212yescomplemented
26#66636C2212yescomplemented
27#66631C2212yescomplemented
28#66631C2212yescomplemented
29#66631C2212yescomplemented
30#66631C2212yescomplemented
31C4⋊Dic21C2211yescharacteristic, complemented
32#66629C2212yescomplemented
33C2×D4×D7—32no—
34C7×#121D311yescharacteristic, complemented
35#6502—32no—
36#6500—32no—
37#6493—31no—
38C2×(C2×D7)⋊C4—31no—
39#6457—32no—
40#6446—32no—
41#6445—32no—
42#5605—71nomaximal, Hall
43C23×C3×C7C2312yes—
44C23×C3×C7C2311yescharacteristic
45C22×D21—24no—
46C22×D21—41no—
47C22×D21—41no—
48C22×D21—42no—
49C22×D21—42no—
50C22×D21D414yescomplemented
51C22×D21C2312yes—
52C22×D21C2312yes—
53C3×C7×D4—24no—
54C3×C7×D4—41no—
55C2×C3×C4×C7—41no—
56C2×C3×C4×C7C2312yes—
57C2×C3×C4×C7C2311yescharacteristic
58C2×C3×C4×C7C2311yescharacteristic
59#4268—24no—
60#4268—24no—
61#4268—42no—
62C2×Dic21—41no—
63C2×Dic21C2312yes—
64C2×Dic21C2312yes—
65D84—24no—
66D84—41no—
67C4×D21—24no—
68C23×D7—31no—
69C23×D7—32no—
70C2×C7×D4C2×D312yescomplemented
71C2×C7×D4C2×D311yescharacteristic, complemented
72C22×C4×C7C2×D311yescharacteristic, complemented
73C2×C22⋊D7—32no—
74C2×C22⋊D7—32no—
75C2×C22⋊D7—32no—
76C22×Dic7—31no—
77D4×D7—68no—
78C2×D28—32no—
79C2×D28—31no—
80C2×C4×D7—32no—
81C7×C4⋊C4C2×D311yescharacteristic, complemented
82C7×C22⋊C4C2×D312yescomplemented
83C22⋊Dic7—32no—
84(C2×D7)⋊C4—32no—
85(C2×D7)⋊C4—32no—
86(C2×D7)⋊C4—32no—
87(C2×D7)⋊C4—32no—
88C4⋊Dic7—31no—
89Dic7⋊C4—32no—
90C2×D3×D4—72no—
91C3×#121D711yescharacteristic, complemented
92#939—72no—
93#937—72no—
94#930—71no—
95C2×(C2×D3)⋊C4—71no—
96#894—72no—
97#883—72no—
98#882—72no—
99C22×C3×C7—21no—
100C22×C3×C7—21no—
101C22×C3×C7—22no—
102C22×C3×C7—22no—
103C22×C3×C7—22no—
104C22×C3×C7C2×D412yes—
105C22×C3×C7C2411yescharacteristic
106C2×D21—22no—
107C2×D21—22no—
108C2×D21—22no—
109C2×D21—22no—
110C2×D21—22no—
111C2×D21—44no—
112C2×D21—48no—
113C2×D21C2×D414yes—
114C3×C4×C7—22no—
115C3×C4×C7—21no—
116C3×C4×C7—21no—
117Dic21—22no—
118Dic21—22no—
119#534—211noSylow
120C23×C7C22×D312yes—
121C23×C7C22×D311yescharacteristic
122C22×D7—32no—
123C22×D7—34no—
124C22×D7—32no—
125C22×D7—64no—
126C22×D7—121no—
127C22×D7—122no—
128C22×D7—121no—
129C22×D7—122no—
130C7×D4—24no—
131C7×D4—41no—
132C2×C4×C7—41no—
133C2×C4×C7C22×D312yes—
134C2×C4×C7C22×D311yescharacteristic
135C2×C4×C7C22×D311yescharacteristic
136C22⋊D7—64no—
137C22⋊D7—64no—
138C22⋊D7—122no—
139C2×Dic7—32no—
140C2×Dic7—32no—
141C2×Dic7—121no—
142D28—64no—
143D28—121no—
144C4×D7—64no—
145C23×D3—71no—
146C23×D3—72no—
147C2×C3×D4C2×D712yescomplemented
148C2×C3×D4C2×D711yescharacteristic, complemented
149C22×C3×C4C2×D711yescharacteristic, complemented
150C2×#67—72no—
151C2×#67—72no—
152C2×#67—72no—
153C22×Dic3—71no—
154D3×D4—148no—
155C2×D12—72no—
156C2×D12—71no—
157C2×C4×D3—72no—
158C3×C4⋊C4C2×D711yescharacteristic, complemented
159C3×C22⋊C4C2×D712yescomplemented
160#217—72no—
161(C2×D3)⋊C4—72no—
162(C2×D3)⋊C4—72no—
163(C2×D3)⋊C4—72no—
164(C2×D3)⋊C4—72no—
165C4⋊Dic3—71no—
166Dic3⋊C4—72no—
167C2×C3×C7—22no—
168C2×C3×C7—42no—
169C2×C3×C7C22×D411yescharacteristic
170C2×C3×C7H(5,𝔽2)11yescharacteristic
171C2×C3×C7H(5,𝔽2)11yescharacteristic
172D21—24no—
173D21—42no—
174C22×D4—212no—
175#123—212no—
176#123—212no—
177#121—212no—
178#121—211no—
179#121D2111yescharacteristic, complemented
180#120—212no—
181#120—212no—
182C2×C22⋊C4—211no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—22no—
186C22×C7—22no—
187C22×C7—22no—
188C22×C7D3×D412yes—
189C22×C7C23×D311yescharacteristic
190C2×D7—34no—
191C2×D7—62no—
192C2×D7—62no—
193C2×D7—62no—
194C2×D7—62no—
195C2×D7—62no—
196C2×D7—124no—
197C2×D7—128no—
198C4×C7—22no—
199C4×C7—21no—
200C4×C7—21no—
201Dic7—62no—
202Dic7—62no—
203C23×C3C22×D712yes—
204C23×C3C22×D711yescharacteristic
205C22×D3—72no—
206C22×D3—74no—
207C22×D3—72no—
208C22×D3—144no—
209C22×D3—281no—
210C22×D3—282no—
211C22×D3—281no—
212C22×D3—282no—
213C3×D4—24no—
214C3×D4—41no—
215C2×C3×C4—41no—
216C2×C3×C4C22×D712yes—
217C2×C3×C4C22×D711yescharacteristic
218C2×C3×C4C22×D711yescharacteristic
219#67—144no—
220#67—144no—
221#67—282no—
222C2×Dic3—72no—
223C2×Dic3—72no—
224C2×Dic3—281no—
225D12—144no—
226D12—281no—
227C4×D3—144no—
228C3×C7#53411yescharacteristic, complemented, Hall
229C24—211no—
230C24—212no—
231C2×D4—212no—
232C2×D4—212no—
233C2×D4—212no—
234C2×D4—212no—
235C2×D4—211no—
236C2×D4—428no—
237C2×D4C2×D2112yescomplemented
238C2×D4C2×D2111yescharacteristic, complemented
239C22×C4—211no—
240C22×C4—212no—
241C22×C4C2×D2111yescharacteristic, complemented
242C4⋊C4—212no—
243C4⋊C4—211no—
244C4⋊C4C2×D2111yescharacteristic, complemented
245C22⋊C4—212no—
246C22⋊C4—212no—
247C22⋊C4—212no—
248C22⋊C4—212no—
249C22⋊C4—212no—
250C22⋊C4C2×D2112yescomplemented
251C2×C7—22no—
252C2×C7—42no—
253C2×C7C2×D3×D411yescharacteristic
254C2×C7#100411yescharacteristic
255C2×C7#100911yescharacteristic
256D7—64no—
257D7—122no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3D4×D712yes—
264C22×C3C23×D711yescharacteristic
265C2×D3—74no—
266C2×D3—142no—
267C2×D3—142no—
268C2×D3—142no—
269C2×D3—142no—
270C2×D3—142no—
271C2×D3—284no—
272C2×D3—288no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—21no—
276Dic3—142no—
277Dic3—142no—
278C23—212no—
279C23—214no—
280C23—212no—
281C23—424no—
282C23—841no—
283C23—841no—
284C23—842no—
285C23—842no—
286C23C22×D2112yes—
287C23C22×D2111yescharacteristic
288D4—24no—
289D4—41no—
290D4—424no—
291D4—424no—
292D4—424no—
293D4—841no—
294D4—842no—
295C2×C4—41no—
296C2×C4—212no—
297C2×C4—212no—
298C2×C4—424no—
299C2×C4—841no—
300C2×C4C22×D2111yescharacteristic
301C2×C4C22×D2111yescharacteristic
302C2×C4C22×D2112yes—
303C7#560511yescharacteristic, complemented, Sylow
304C2×C3—22no—
305C2×C3—42no—
306C2×C3C2×D4×D711yescharacteristic
307C2×C3#654811yescharacteristic
308C2×C3#655311yescharacteristic
309D3—144no—
310D3—282no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—214no—
317C22—422no—
318C22—422no—
319C22—422no—
320C22—422no—
321C22—422no—
322C22—844no—
323C22—848no—
324C22D4×D2112yes—
325C22C23×D2111yescharacteristic
326C4—22no—
327C4—21no—
328C4—21no—
329C4—422no—
330C4—422no—
331C3#9029611yescharacteristic, complemented, Sylow
332C2—22no—
333C2—42no—
334C2—424no—
335C2—842no—
336C2C2×D4×D2111yescharacteristic
337C2#1062165711yescharacteristic
338C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#534
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D21C2semidirect
2. C3×C7×#121C2semidirect
3. #10621422C2semidirect
4. #10621420C2semidirect
5. #10621413C2semidirect
6. C2×#66631C2semidirect
7. #10621377C2semidirect
8. #10621366C2semidirect
9. #10621365C2semidirect
12. C23×D21C22semidirect
13. C2×C3×C7×D4C22semidirect
14. C2×C3×C7×D4C22semidirect
15. C22×C3×C4×C7C22semidirect
17. C2×#4268C22semidirect
18. C2×#4268C22semidirect
19. C22×Dic21C22semidirect
22. C2×D84C22semidirect
23. C2×C4×D21C22semidirect
24. C3×C7×C4⋊C4C22semidirect
25. C3×C7×C22⋊C4C22semidirect
26. #66636C22semidirect
27. #66631C22semidirect
28. #66631C22semidirect
29. #66631C22semidirect
30. #66631C22semidirect
31. C4⋊Dic21C22semidirect
32. #66629C22semidirect
34. C7×#121D3semidirect
50. C22×D21D4semidirect
70. C2×C7×D4C2×D3semidirect
71. C2×C7×D4C2×D3semidirect
72. C22×C4×C7C2×D3semidirect
81. C7×C4⋊C4C2×D3semidirect
82. C7×C22⋊C4C2×D3semidirect
91. C3×#121D7semidirect
147. C2×C3×D4C2×D7semidirect
148. C2×C3×D4C2×D7semidirect
149. C22×C3×C4C2×D7semidirect
158. C3×C4⋊C4C2×D7semidirect
159. C3×C22⋊C4C2×D7semidirect
179. #121D21semidirect
228. C3×C7#534semidirect
237. C2×D4C2×D21semidirect
238. C2×D4C2×D21semidirect
241. C22×C4C2×D21semidirect
244. C4⋊C4C2×D21semidirect
250. C22⋊C4C2×D21semidirect
303. C7#5605semidirect
331. C3#90296semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C3×C7×#1212
#106214222
#106214202
#106214132
C2×#666312
#106213772
#106213662
#106213652
#902963
#56057