Finite Group Databasebeta

← #13058058

Subgroups — #13058058

Order 1344.

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

normal subgroups only

This group has 668 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×#4247C211yesmaximal, characteristic, complemented
2C2×#66686C211yesmaximal, characteristic, complemented
3#10621365C211yesmaximal, characteristic, complemented
4C7×#882C211yesmaximal, characteristic, complemented
5C3×#6440C211yesmaximal, characteristic, complemented
6#10621135C211yesmaximal, characteristic, complemented
7#10621132C211yesmaximal, characteristic, complemented
8#10621013C212yesmaximal, complemented
9#10620982C212yesmaximal, complemented
10#10620978C212yesmaximal, complemented
11#10620912C212yesmaximal, complemented
12#90283—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C23×D3×C7C2211yescharacteristic
15C2×#4268C2211yescharacteristic
16C2×D84C2212yescomplemented
17C2×C7×#67C2211yescharacteristic
18C2×C7×D12C2212yescomplemented
19C22×C3×Dic7C2211yescharacteristic
20C2×#4248C2211yescharacteristic, complemented
21C2×#4247—24no—
22C2×#4247C2212yes—
23C2×#4247C2212yes—
24C2×#4247C2211yescharacteristic, complemented
25C2×#4247C2212yescomplemented
26C2×#4247C2212yescomplemented
27C2×D21⋊C4C2211yescharacteristic, complemented
28#66686—24no—
29C2×D3×Dic7C2211yescharacteristic, complemented
30C3×C7×C22⋊C4C2211yescharacteristic
31#66631C2212yescomplemented
32C7×(C2×D3)⋊C4C2212yescomplemented
33C3×C22⋊Dic7C2211yescharacteristic
34C3×Dic7⋊C4C2212yescomplemented
35C3×C4×Dic7C2212yescomplemented
36#66575C2212yescomplemented
37(C2×D3)⋊Dic7C2212yescomplemented
38C22×C22⋊D7—31no—
39C2×#1159—31no—
40C7×#120—31no—
41#6516—31no—
42#6503—32no—
43#6502—32no—
44#6500—31no—
45#6448—32no—
46#6447—32no—
47#6445—31no—
48#6440D311yescharacteristic, complemented
49#5525—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—22no—
52C22×D21—42no—
53C22×D21C2312yes—
54C22×D21C2311yescharacteristic
55C22×D3×C7—22no—
56C22×D3×C7—42no—
57C22×D3×C7C2312yes—
58C22×D3×C7C2311yescharacteristic
59C2×C3×C4×C7C2312yes—
60#4268—22no—
61C2×Dic21C2311yescharacteristic
62D84—42no—
63C7×#67—22no—
64C2×C7×Dic3C2311yescharacteristic
65C7×D12—42no—
66C2×C3×Dic7D414yescomplemented
67C2×C3×Dic7C2312yes—
68C2×C3×Dic7C2311yescharacteristic
69C2×C3×Dic7C2311yescharacteristic
70#4248—22no—
71#4247—22no—
72#4247—24no—
73#4247—24no—
74#4247—42no—
75#4247—42no—
76D21⋊C4—22no—
77D3×Dic7—22no—
78C24×C7—31no—
79C23×D7—31no—
80C2×C7×D4—31no—
81C2×C7×D4—32no—
82C2×C22⋊D7—31no—
83C2×C22⋊D7—32no—
84C2×C22⋊D7—32no—
85C2×C22⋊D7—31no—
86C2×C22⋊D7—32no—
87C2×C22⋊D7—32no—
88C2×C22⋊D7—64no—
89C22×Dic7—31no—
90C22×Dic7C2×D311yescharacteristic
91#1159—64no—
92C2×D28—32no—
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C22⋊C4—32no—
96C7×C22⋊C4C2×D311yescharacteristic
97C22⋊Dic7—32no—
98C22⋊Dic7C2×D311yescharacteristic
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101Dic7⋊C4C2×D312yescomplemented
102C4×Dic7C2×D312yescomplemented
103C2×#235—71no—
104C22×D12—71no—
105C3×#117—71no—
106#895—72no—
107#895—72no—
108#882—71no—
109#882—71no—
110#882—71no—
111#882D711yescharacteristic, complemented
112#875—72no—
113#874—72no—
114C22×C3×C7—22no—
115C22×C3×C7C2×D412yes—
116C22×C3×C7C2411yescharacteristic
117C2×D21—21no—
118C2×D21—22no—
119C2×D21—22no—
120C2×D21—22no—
121C2×D21—22no—
122C2×D21—44no—
123C2×D3×C7—21no—
124C2×D3×C7—22no—
125C2×D3×C7—22no—
126C2×D3×C7—22no—
127C2×D3×C7—22no—
128C2×D3×C7—44no—
129C3×C4×C7—22no—
130Dic21—21no—
131C3×Dic7—22no—
132C3×Dic7C2×D412yes—
133C3×Dic7C2×D412yes—
134C7×Dic3—21no—
135#535—211noSylow
136C23×C7—32no—
137C23×C7—31no—
138C23×C7—62no—
139C23×C7—122no—
140C23×C7C22×D311yescharacteristic
141C22×D7—32no—
142C22×D7—31no—
143C22×D7—62no—
144C22×D7—122no—
145C7×D4—62no—
146C7×D4—122no—
147C2×C4×C7—31no—
148C2×C4×C7C22×D312yes—
149C22⋊D7—62no—
150C22⋊D7—62no—
151C22⋊D7—64no—
152C22⋊D7—64no—
153C22⋊D7—122no—
154C22⋊D7—122no—
155C2×Dic7—31no—
156C2×Dic7—62no—
157C2×Dic7D1214yescomplemented
158C2×Dic7C22×D312yes—
159C2×Dic7C22×D311yescharacteristic
160C2×Dic7C22×D311yescharacteristic
161D28—122no—
162C4×D7—62no—
163Dic14—62no—
164C23×D3—71no—
165C23×D3C2×D711yescharacteristic
166C22×C3×C4—71no—
167C2×#67—71no—
168C2×#67C2×D711yescharacteristic
169#235—144no—
170C2×D12—71no—
171C2×D12—72no—
172C2×D12—72no—
173C2×D12—72no—
174C2×D12—72no—
175C2×D12—72no—
176C2×D12—144no—
177C2×D12C2×D712yescomplemented
178C2×C4×D3—71no—
179C2×C4×D3—71no—
180C2×Dic6—71no—
181C3×C4⋊C4—72no—
182C3×C22⋊C4—71no—
183C3×C22⋊C4C2×D711yescharacteristic
184C3×C42—72no—
185(C2×D3)⋊C4—72no—
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4—72no—
188(C2×D3)⋊C4C2×D712yescomplemented
189C2×C3×C7—22no—
190C2×C3×C7C22×D411yescharacteristic
191C2×C3×C7H(5,𝔽2)12yes—
192D21—41no—
193D21—42no—
194D3×C7—41no—
195D3×C7—42no—
196C2×#41—211no—
197C22×D4—211no—
198#127—212no—
199#124—212no—
200#121—212no—
201#121—212no—
202#120—31no—
203#120—211no—
204#120—211no—
205#120—211no—
206#117—71no—
207C22×C7—22no—
208C22×C7—61no—
209C22×C7—62no—
210C22×C7—62no—
211C22×C7—62no—
212C22×C7—62no—
213C22×C7—124no—
214C22×C7C2×D1212yes—
215C22×C7C23×D311yescharacteristic
216C2×D7—61no—
217C2×D7—62no—
218C2×D7—62no—
219C2×D7—62no—
220C2×D7—62no—
221C2×D7—124no—
222C4×C7—22no—
223C4×C7—61no—
224Dic7—22no—
225Dic7—61no—
226Dic7C2×D1212yes—
227Dic7C2×D1212yes—
228C23×C3C22×D711yescharacteristic
229C22×D3—22no—
230C22×D3—42no—
231C22×D3—72no—
232C22×D3—71no—
233C22×D3—142no—
234C22×D3—282no—
235C22×D3C22×D712yes—
236C22×D3C22×D711yescharacteristic
237C2×C3×C4—74no—
238C2×C3×C4—71no—
239C2×C3×C4—71no—
240C2×C3×C4—72no—
241C2×C3×C4C22×D712yes—
242#67—22no—
243#67—142no—
244C2×Dic3—71no—
245C2×Dic3C22×D711yescharacteristic
246D12—42no—
247D12—144no—
248D12—144no—
249D12—142no—
250D12—282no—
251D12—282no—
252D12—282no—
253C4×D3—142no—
254C4×D3—142no—
255Dic6—142no—
256C3×C7#53511yescharacteristic, complemented, Hall
257C24—31no—
258C24—211no—
259#41—424no—
260C2×Q8—211no—
261C2×D4—31no—
262C2×D4—32no—
263C2×D4—212no—
264C2×D4—212no—
265C2×D4—211no—
266C2×D4—212no—
267C2×D4—212no—
268C2×D4—211no—
269C2×D4—212no—
270C2×D4—424no—
271C22×C4—71no—
272C22×C4—211no—
273C22×C4—211no—
274C4⋊C4—72no—
275C22⋊C4—32no—
276C22⋊C4—71no—
277C22⋊C4—212no—
278C22⋊C4—212no—
279C22⋊C4—212no—
280C22⋊C4D3×D711yescharacteristic
281C42—72no—
282C2×C7—22no—
283C2×C7—122no—
284C2×C7—121no—
285C2×C7C22×D1211yescharacteristic
286C2×C7#100912yes—
287D7—122no—
288D7—121no—
289C22×C3—22no—
290C22×C3D4×D712yes—
291C22×C3C23×D711yescharacteristic
292C2×D3—21no—
293C2×D3—22no—
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—44no—
298C2×D3—141no—
299C2×D3—142no—
300C2×D3—142no—
301C2×D3—142no—
302C2×D3—142no—
303C2×D3—284no—
304C3×C4—22no—
305C3×C4—72no—
306C3×C4—72no—
307C3×C4—142no—
308Dic3—21no—
309Dic3—141no—
310C23—32no—
311C23—31no—
312C23—62no—
313C23—122no—
314C23—212no—
315C23—211no—
316C23—422no—
317C23—842no—
318C23C2×D3×D711yescharacteristic
319Q8—422no—
320D4—62no—
321D4—122no—
322D4—424no—
323D4—424no—
324D4—422no—
325D4—422no—
326D4—842no—
327D4—842no—
328D4—842no—
329C2×C4—31no—
330C2×C4—74no—
331C2×C4—71no—
332C2×C4—71no—
333C2×C4—72no—
334C2×C4—211no—
335C2×C4—422no—
336C2×C4—422no—
337C2×C4C2×D3×D712yes—
338C7#552511yescharacteristic, complemented, Sylow
339C2×C3—22no—
340C2×C3C2×D4×D711yescharacteristic
341C2×C3#654812yes—
342D3—42no—
343D3—41no—
344D3—282no—
345D3—281no—
346C22—22no—
347C22—61no—
348C22—62no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—124no—
353C22—421no—
354C22—422no—
355C22—422no—
356C22—422no—
357C22—422no—
358C22—844no—
359C22D7×D1212yes—
360C22C22×D3×D711yescharacteristic
361C4—22no—
362C4—61no—
363C4—72no—
364C4—72no—
365C4—142no—
366C4—421no—
367C3#9028311yescharacteristic, complemented, Sylow
368C2—22no—
369C2—122no—
370C2—121no—
371C2—842no—
372C2—841no—
373C2C2×D7×D1211yescharacteristic
374C2#1062157412yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#535
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×#4247C2semidirect
2. C2×#66686C2semidirect
3. #10621365C2semidirect
4. C7×#882C2semidirect
5. C3×#6440C2semidirect
6. #10621135C2semidirect
7. #10621132C2semidirect
8. #10621013C2semidirect
9. #10620982C2semidirect
10. #10620978C2semidirect
11. #10620912C2semidirect
16. C2×D84C22semidirect
18. C2×C7×D12C22semidirect
20. C2×#4248C22semidirect
24. C2×#4247C22semidirect
25. C2×#4247C22semidirect
26. C2×#4247C22semidirect
27. C2×D21⋊C4C22semidirect
29. C2×D3×Dic7C22semidirect
31. #66631C22semidirect
32. C7×(C2×D3)⋊C4C22semidirect
34. C3×Dic7⋊C4C22semidirect
35. C3×C4×Dic7C22semidirect
36. #66575C22semidirect
37. (C2×D3)⋊Dic7C22semidirect
48. #6440D3semidirect
66. C2×C3×Dic7D4semidirect
101. Dic7⋊C4C2×D3semidirect
102. C4×Dic7C2×D3semidirect
111. #882D7semidirect
157. C2×Dic7D12semidirect
177. C2×D12C2×D7semidirect
188. (C2×D3)⋊C4C2×D7semidirect
256. C3×C7#535semidirect
338. C7#5525semidirect
367. C3#90283semidirect

Maximal subgroups

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

subgroupindex
C22×#42472
C2×#666862
#106213652
C7×#8822
C3×#64402
#106211352
#106211322
#106210132
#106209822
#106209782
#106209122
#902833
#55257