Finite Group Databasebeta

← #13059861

Subgroups — #13059861

Order 1344.

692 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C3×D28C211yesmaximal, characteristic, complemented
2C2×D7×#67C211yesmaximal, characteristic, complemented
3#10621412C211yesmaximal, characteristic, complemented
4C7×#929C211yesmaximal, characteristic, complemented
5#10621126C211yesmaximal, characteristic, complemented
6#10621125C211yesmaximal, characteristic, complemented
7D7×#217C211yesmaximal, characteristic, complemented
8#10621036C212yesmaximal, complemented
9#10620971C212yesmaximal, complemented
10#10620970C212yesmaximal, complemented
11#10620969C212yesmaximal, complemented
12#90344—31nomaximal, Hall
13C23×C3×D7C2211yescharacteristic
14C23×C3×D7C2211yescharacteristic, complemented
15C22×D3×D7C2211yescharacteristic, complemented
16C22×C3×C4×C7C2211yescharacteristic, complemented
17C2×#4268C2211yescharacteristic
18C2×C7×#67C2211yescharacteristic
19C2×C3×D28—24no—
20C2×C3×D28C2212yes—
21C2×C3×D28C2212yescomplemented
22D7×#67—24no—
23C2×#4246C2211yescharacteristic, complemented
24C2×#4246C2212yescomplemented
25C2×#4245C2211yescharacteristic, complemented
26C2×#4245C2212yescomplemented
27C2×Dic3×D7C2211yescharacteristic, complemented
28C2×Dic3×D7C2212yescomplemented
29#66636C2211yescharacteristic, complemented
30#66631C2212yescomplemented
31#66629C2212yescomplemented
32C7×#217C2211yescharacteristic, complemented
33C7×(C2×D3)⋊C4C2212yescomplemented
34C7×Dic3⋊C4C2212yescomplemented
35(C2×D7)⋊Dic3C2212yescomplemented
36(C2×D7)⋊Dic3C2212yescomplemented
37(C2×D7)⋊Dic3C2212yescomplemented
38C2×D4×D7—31no—
39C22×D28D311yescharacteristic, complemented
40C7×#123—31no—
41#6500—31no—
42#6492—31no—
43#6458—32no—
44#6457—32no—
45#6447—32no—
46#6445—31no—
47#6445—32no—
48D7×C22⋊C4—31no—
49#5804—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×D3×C7C2311yescharacteristic
53C22×C3×D7—22no—
54C22×C3×D7—42no—
55C22×C3×D7—42no—
56C22×C3×D7D414yescomplemented
57C22×C3×D7C2312yes—
58C22×C3×D7C2311yescharacteristic
59C22×C3×D7C2311yescharacteristic
60C2×D3×D7—22no—
61C2×D3×D7—42no—
62C2×C3×C4×C7—22no—
63C2×C3×C4×C7C2312yes—
64#4268—22no—
65C2×Dic21C2311yescharacteristic
66C2×Dic21C2312yes—
67C7×#67—22no—
68C2×C7×Dic3C2311yescharacteristic
69C2×C7×Dic3C2312yes—
70C3×D28—24no—
71C3×D28—24no—
72#4246—22no—
73#4246—42no—
74#4245—22no—
75#4245—42no—
76Dic3×D7—22no—
77Dic3×D7—42no—
78C23×D7—31no—
79C23×D7C2×D311yescharacteristic
80C23×D7C2×D311yescharacteristic, complemented
81C2×C7×D4—31no—
82C22×C4×C7C2×D311yescharacteristic, complemented
83C2×C22⋊D7—31no—
84C2×C22⋊D7—31no—
85C2×C22⋊D7—32no—
86D4×D7—64no—
87C2×D28—24no—
88C2×D28—31no—
89C2×D28—32no—
90C2×D28C2×D312yes—
91C2×D28C2×D312yescomplemented
92C2×C4×D7—31no—
93C2×C4×D7—32no—
94C7×C4⋊C4—32no—
95C7×C22⋊C4—31no—
96C7×C22⋊C4—32no—
97C22⋊Dic7—31no—
98(C2×D7)⋊C4—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101(C2×D7)⋊C4—32no—
102Dic7⋊C4—32no—
103C22×C3×D4—71no—
104C22×#67—71no—
105#953—71no—
106#953—71no—
107C2×#217—71no—
108#939—72no—
109#939—72no—
110#937—72no—
111#935—72no—
112#929—71no—
113#929D711yescharacteristic, complemented
114C22×C3×C7—22no—
115C22×C3×C7C2×D412yes—
116C22×C3×C7C2411yescharacteristic
117C2×D21—21no—
118C2×D21—22no—
119C2×D3×C7—21no—
120C2×D3×C7—22no—
121C2×C3×D7—22no—
122C2×C3×D7—22no—
123C2×C3×D7—22no—
124C2×C3×D7—22no—
125C2×C3×D7—22no—
126C2×C3×D7—44no—
127C2×C3×D7—44no—
128C2×C3×D7C2×D412yes—
129C2×C3×D7C2×D412yes—
130D3×D7—44no—
131C3×C4×C7—22no—
132Dic21—21no—
133Dic21—22no—
134C7×Dic3—21no—
135C7×Dic3—22no—
136#534—211noSylow
137C23×C7—31no—
138C23×C7C22×D311yescharacteristic
139C22×D7—22no—
140C22×D7—31no—
141C22×D7—42no—
142C22×D7—42no—
143C22×D7—62no—
144C22×D7—122no—
145C22×D7#6714yescomplemented
146C22×D7C22×D312yes—
147C22×D7C22×D311yescharacteristic
148C22×D7C22×D311yescharacteristic
149C7×D4—62no—
150C2×C4×C7—22no—
151C2×C4×C7—32no—
152C2×C4×C7—31no—
153C2×C4×C7C22×D312yes—
154C22⋊D7—62no—
155C22⋊D7—62no—
156C22⋊D7—122no—
157C2×Dic7—31no—
158C2×Dic7—32no—
159D28—24no—
160D28—24no—
161D28—62no—
162D28—122no—
163C4×D7—62no—
164C4×D7—122no—
165C24×C3—71no—
166C24×C3—71no—
167C23×D3—71no—
168C2×C3×D4—72no—
169C2×C3×D4—72no—
170C2×C3×D4—144no—
171C22×C3×C4C2×D711yescharacteristic, complemented
172C2×#67—71no—
173C2×#67—71no—
174C2×#67—72no—
175C2×#67—71no—
176C2×#67—72no—
177C2×#67—144no—
178C2×#67C2×D711yescharacteristic
179C22×Dic3—71no—
180C22×Dic3—72no—
181#217—71no—
182#217—72no—
183#217—72no—
184#217—72no—
185#217C2×D711yescharacteristic, complemented
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4C2×D712yescomplemented
188Dic3⋊C4—72no—
189Dic3⋊C4C2×D712yescomplemented
190C2×C3×C7—22no—
191C2×C3×C7C22×D411yescharacteristic
192C2×C3×C7H(5,𝔽2)12yes—
193D21—41no—
194C3×D7—22no—
195C3×D7—22no—
196C3×D7—42no—
197D3×C7—41no—
198C22×D4—71no—
199C22×D4—211no—
200#123—31no—
201#123—212no—
202#123—211no—
203#121—212no—
204#121—212no—
205#120—212no—
206#120—211no—
207#120—211no—
208C2×C22⋊C4—211no—
209C22×C7—22no—
210C22×C7—62no—
211C22×C7—61no—
212C22×C7C2×#6712yes—
213C22×C7C23×D311yescharacteristic
214C2×D7—22no—
215C2×D7—22no—
216C2×D7—22no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—44no—
220C2×D7—44no—
221C2×D7—62no—
222C2×D7—61no—
223C2×D7—124no—
224C2×D7C2×#6712yes—
225C2×D7C2×#6712yes—
226C4×C7—22no—
227C4×C7—62no—
228C4×C7—61no—
229Dic7—62no—
230Dic7—61no—
231C23×C3—71no—
232C23×C3—71no—
233C23×C3—74no—
234C23×C3—72no—
235C23×C3—142no—
236C23×C3—282no—
237C23×C3—282no—
238C23×C3C22×D711yescharacteristic
239C22×D3—71no—
240C22×D3—142no—
241C22×D3—282no—
242C22×D3C22×D711yescharacteristic
243C3×D4—144no—
244C3×D4—144no—
245C2×C3×C4—22no—
246C2×C3×C4C22×D712yes—
247#67—22no—
248#67—142no—
249#67—142no—
250#67—142no—
251#67—282no—
252#67—282no—
253C2×Dic3—71no—
254C2×Dic3—72no—
255C2×Dic3—142no—
256C2×Dic3—282no—
257C2×Dic3C22×D711yescharacteristic
258C2×Dic3C22×D712yes—
259C3×C7#53411yescharacteristic, complemented, Hall
260C24—71no—
261C24—71no—
262C24—211no—
263C2×D4—31no—
264C2×D4—72no—
265C2×D4—72no—
266C2×D4—144no—
267C2×D4—211no—
268C2×D4—212no—
269C2×D4—211no—
270C2×D4—211no—
271C2×D4—212no—
272C2×D4—424no—
273C22×C4—211no—
274C22×C4—212no—
275C22×C4D3×D711yescharacteristic, complemented
276C4⋊C4—32no—
277C4⋊C4—212no—
278C22⋊C4—31no—
279C22⋊C4—32no—
280C22⋊C4—212no—
281C22⋊C4—212no—
282C22⋊C4—212no—
283C22⋊C4—211no—
284C22⋊C4—212no—
285C2×C7—22no—
286C2×C7—121no—
287C2×C7#100412yes—
288C2×C7C22×#6711yescharacteristic
289D7—22no—
290D7—22no—
291D7—42no—
292D7—121no—
293C22×C3—22no—
294C22×C3—72no—
295C22×C3—72no—
296C22×C3—142no—
297C22×C3—142no—
298C22×C3—142no—
299C22×C3—142no—
300C22×C3—142no—
301C22×C3—284no—
302C22×C3—284no—
303C22×C3D4×D712yes—
304C22×C3C23×D711yescharacteristic
305C2×D3—21no—
306C2×D3—22no—
307C2×D3—141no—
308C2×D3—142no—
309C2×D3—284no—
310C3×C4—22no—
311Dic3—21no—
312Dic3—22no—
313Dic3—141no—
314Dic3—142no—
315C23—31no—
316C23—71no—
317C23—71no—
318C23—74no—
319C23—72no—
320C23—142no—
321C23—211no—
322C23—282no—
323C23—282no—
324C23—422no—
325C23—842no—
326C23C2×D3×D711yescharacteristic
327D4—62no—
328D4—144no—
329D4—144no—
330D4—422no—
331D4—422no—
332D4—422no—
333D4—842no—
334D4—842no—
335C2×C4—22no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—212no—
339C2×C4—211no—
340C2×C4—422no—
341C2×C4—842no—
342C2×C4C2×D3×D712yes—
343C7#580411yescharacteristic, complemented, Sylow
344C2×C3—22no—
345C2×C3—142no—
346C2×C3—142no—
347C2×C3—282no—
348C2×C3C2×D4×D711yescharacteristic
349C2×C3#655312yes—
350D3—41no—
351D3—281no—
352C22—22no—
353C22—62no—
354C22—61no—
355C22—72no—
356C22—72no—
357C22—142no—
358C22—142no—
359C22—142no—
360C22—142no—
361C22—142no—
362C22—284no—
363C22—284no—
364C22—422no—
365C22—421no—
366C22—844no—
367C22D7×#6712yes—
368C22C22×D3×D711yescharacteristic
369C4—22no—
370C4—62no—
371C4—61no—
372C4—422no—
373C4—421no—
374C3#9034411yescharacteristic, complemented, Sylow
375C2—22no—
376C2—121no—
377C2—142no—
378C2—142no—
379C2—282no—
380C2—841no—
381C2#1062156412yes—
382C2C2×D7×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#534
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×D28C2semidirect
2. C2×D7×#67C2semidirect
3. #10621412C2semidirect
4. C7×#929C2semidirect
5. #10621126C2semidirect
6. #10621125C2semidirect
7. D7×#217C2semidirect
8. #10621036C2semidirect
9. #10620971C2semidirect
10. #10620970C2semidirect
11. #10620969C2semidirect
14. C23×C3×D7C22semidirect
15. C22×D3×D7C22semidirect
16. C22×C3×C4×C7C22semidirect
21. C2×C3×D28C22semidirect
23. C2×#4246C22semidirect
24. C2×#4246C22semidirect
25. C2×#4245C22semidirect
26. C2×#4245C22semidirect
27. C2×Dic3×D7C22semidirect
28. C2×Dic3×D7C22semidirect
29. #66636C22semidirect
30. #66631C22semidirect
31. #66629C22semidirect
32. C7×#217C22semidirect
33. C7×(C2×D3)⋊C4C22semidirect
34. C7×Dic3⋊C4C22semidirect
35. (C2×D7)⋊Dic3C22semidirect
36. (C2×D7)⋊Dic3C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
39. C22×D28D3semidirect
56. C22×C3×D7D4semidirect
80. C23×D7C2×D3semidirect
82. C22×C4×C7C2×D3semidirect
91. C2×D28C2×D3semidirect
113. #929D7semidirect
145. C22×D7#67semidirect
171. C22×C3×C4C2×D7semidirect
185. #217C2×D7semidirect
187. (C2×D3)⋊C4C2×D7semidirect
189. Dic3⋊C4C2×D7semidirect
259. C3×C7#534semidirect
275. C22×C4D3×D7semidirect
343. C7#5804semidirect
374. C3#90344semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D282
C2×D7×#672
#106214122
C7×#9292
#106211262
#106211252
D7×#2172
#106210362
#106209712
#106209702
#106209692
#903443
#58047