Finite Group Databasebeta

← #13060129

Subgroups — #13060129

Order 1344.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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×#66730C211yesmaximal, characteristic, complemented
2C22×#4245C211yesmaximal, characteristic, complemented
3C7×#937C211yesmaximal, characteristic, complemented
4C3×#6500C211yesmaximal, characteristic, complemented
5#10621133C212yesmaximal, complemented
6#10621132C211yesmaximal, characteristic, complemented
7#10621129C212yesmaximal, complemented
8#10621126C211yesmaximal, characteristic, complemented
9#10621094C211yesmaximal, characteristic, complemented
10#10621024C212yesmaximal, complemented
11#10621023C212yesmaximal, complemented
12#90283—31nomaximal, Hall
13C23×D3×C7C2211yescharacteristic
14C23×C3×D7C2211yescharacteristic
15C2×C3×C7×D4C2211yescharacteristic
16C2×#4268C2211yescharacteristic
17C2×#4268C2211yescharacteristic, complemented
18C22×Dic21C2211yescharacteristic
19C22×Dic21C2211yescharacteristic, complemented
20#66730—24no—
21C2×C4×D21C2211yescharacteristic, complemented
22C2×Dic42C2211yescharacteristic, complemented
23C2×C7×#67C2212yescomplemented
24C2×C3×C22⋊D7C2212yescomplemented
25C2×#4247C2212yescomplemented
26C2×#4246C2212yescomplemented
27C2×#4245—24no—
28C2×#4245C2212yes—
29C2×#4245C2212yes—
30C7×#217C2211yescharacteristic
31C7×(C2×D3)⋊C4C2212yescomplemented
32C3×C22⋊Dic7C2211yescharacteristic
33C3×(C2×D7)⋊C4C2212yescomplemented
34#66577C2212yescomplemented
35(C2×D3)⋊Dic7C2212yescomplemented
36(C2×D7)⋊Dic3C2212yescomplemented
37Dic3×Dic7C2212yescomplemented
38C22×C22⋊D7—31no—
39C2×#1159—31no—
40C7×#120—31no—
41#6516—31no—
42#6503—32no—
43#6502—32no—
44#6500D311yescharacteristic, complemented
45#6448—32no—
46#6447—32no—
47#6445—31no—
48#6440—31no—
49#5592—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C23×C3×C7C2311yescharacteristic
52C22×D21C2311yescharacteristic
53C22×D3×C7—22no—
54C22×D3×C7—42no—
55C22×D3×C7C2312yes—
56C22×C3×D7—22no—
57C22×C3×D7—42no—
58C22×C3×D7C2312yes—
59C3×C7×D4—22no—
60C2×C3×C4×C7C2311yescharacteristic
61#4268—22no—
62#4268—22no—
63C2×Dic21—22no—
64C2×Dic21D414yescomplemented
65C2×Dic21C2311yescharacteristic
66C2×Dic21C2311yescharacteristic
67C2×Dic21C2311yescharacteristic
68C4×D21—22no—
69Dic42—22no—
70C7×#67—42no—
71C2×C7×Dic3C2312yes—
72C3×C22⋊D7—42no—
73C2×C3×Dic7C2312yes—
74#4247—42no—
75#4246—42no—
76#4245—24no—
77#4245—24no—
78C24×C7—31no—
79C23×D7C2×D311yescharacteristic
80C2×C7×D4—32no—
81C2×C7×D4C2×D311yescharacteristic
82C2×C22⋊D7—31no—
83C2×C22⋊D7—31no—
84C2×C22⋊D7—32no—
85C2×C22⋊D7—32no—
86C2×C22⋊D7—32no—
87C2×C22⋊D7—64no—
88C2×C22⋊D7C2×D312yescomplemented
89C22×Dic7—31no—
90C22×Dic7—31no—
91#1159—64no—
92C2×D28—32no—
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C22⋊C4—31no—
96C7×C22⋊C4—32no—
97C22⋊Dic7—32no—
98C22⋊Dic7C2×D311yescharacteristic
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4C2×D312yescomplemented
101Dic7⋊C4—32no—
102C4×Dic7—32no—
103C22×#67—71no—
104C2×#237—71no—
105C3×#120—71no—
106#953—71no—
107#940—72no—
108#939—72no—
109#937D711yescharacteristic, complemented
110#885—72no—
111#884—72no—
112#882—71no—
113#877—71no—
114C22×C3×C7—22no—
115C22×C3×C7—21no—
116C22×C3×C7—22no—
117C22×C3×C7C2×D412yes—
118C22×C3×C7C2411yescharacteristic
119C2×D21—21no—
120C2×D21—22no—
121C2×D3×C7—22no—
122C2×D3×C7—22no—
123C2×D3×C7—22no—
124C2×D3×C7—44no—
125C2×C3×D7—22no—
126C2×C3×D7—22no—
127C2×C3×D7—22no—
128C2×C3×D7—44no—
129C3×C4×C7—21no—
130Dic21—21no—
131Dic21C2×D412yes—
132Dic21C2×D412yes—
133C3×Dic7—22no—
134C7×Dic3—22no—
135#535—211noSylow
136C23×C7—32no—
137C23×C7—62no—
138C23×C7—122no—
139C23×C7C22×D311yescharacteristic
140C23×C7C22×D311yescharacteristic
141C22×D7—22no—
142C22×D7—31no—
143C22×D7—42no—
144C22×D7C22×D312yes—
145C7×D4—22no—
146C7×D4—122no—
147C2×C4×C7—32no—
148C2×C4×C7C22×D311yescharacteristic
149C22⋊D7—42no—
150C22⋊D7—64no—
151C22⋊D7—64no—
152C22⋊D7—62no—
153C22⋊D7—62no—
154C22⋊D7—122no—
155C2×Dic7—34no—
156C2×Dic7—31no—
157C2×Dic7—31no—
158C2×Dic7—31no—
159C2×Dic7—62no—
160C2×Dic7C22×D312yes—
161D28—122no—
162C4×D7—62no—
163Dic14—62no—
164C24×C3—71no—
165C23×D3C2×D711yescharacteristic
166C2×C3×D4—72no—
167C2×C3×D4C2×D711yescharacteristic
168C2×#67—71no—
169C2×#67—71no—
170C2×#67—72no—
171C2×#67—72no—
172C2×#67—72no—
173C2×#67—144no—
174C2×#67C2×D712yescomplemented
175C22×Dic3—71no—
176C22×Dic3—71no—
177#237—144no—
178C2×D12—72no—
179C2×C4×D3—71no—
180C2×Dic6—71no—
181C3×C22⋊C4—71no—
182C3×C22⋊C4—72no—
183#217—72no—
184#217C2×D711yescharacteristic
185(C2×D3)⋊C4—72no—
186(C2×D3)⋊C4C2×D712yescomplemented
187Dic3⋊C4—72no—
188C4×Dic3—72no—
189C2×C3×C7—22no—
190C2×C3×C7—41no—
191C2×C3×C7C22×D411yescharacteristic
192C2×C3×C7H(5,𝔽2)12yes—
193D21—41no—
194C3×D7—42no—
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—71no—
204#120—211no—
205#120—211no—
206#117—211no—
207C22×C7—22no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—62no—
211C22×C7—62no—
212C22×C7—62no—
213C22×C7—124no—
214C22×C7D3×D412yes—
215C22×C7C23×D311yescharacteristic
216C2×D7—22no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—44no—
220C2×D7—61no—
221C2×D7—62no—
222C4×C7—21no—
223C4×C7—62no—
224Dic7—22no—
225Dic7—32no—
226Dic7—32no—
227Dic7—61no—
228C23×C3—72no—
229C23×C3—142no—
230C23×C3—282no—
231C23×C3C22×D711yescharacteristic
232C23×C3C22×D711yescharacteristic
233C22×D3—22no—
234C22×D3—42no—
235C22×D3—71no—
236C22×D3C22×D712yes—
237C3×D4—22no—
238C3×D4—282no—
239C2×C3×C4—72no—
240C2×C3×C4C22×D711yescharacteristic
241#67—42no—
242#67—144no—
243#67—144no—
244#67—142no—
245#67—142no—
246#67—282no—
247C2×Dic3—74no—
248C2×Dic3—71no—
249C2×Dic3—71no—
250C2×Dic3—71no—
251C2×Dic3—142no—
252C2×Dic3C22×D712yes—
253D12—282no—
254C4×D3—142no—
255Dic6—142no—
256C3×C7#53511yescharacteristic, complemented, Hall
257C24—31no—
258C24—71no—
259#41—424no—
260C2×Q8—211no—
261C2×D4—32no—
262C2×D4—72no—
263C2×D4—212no—
264C2×D4—212no—
265C2×D4—212no—
266C2×D4—212no—
267C2×D4—211no—
268C2×D4—211no—
269C2×D4—424no—
270C2×D4D3×D711yescharacteristic
271C22×C4—211no—
272C22×C4—211no—
273C22×C4—211no—
274C4⋊C4—212no—
275C22⋊C4—31no—
276C22⋊C4—32no—
277C22⋊C4—71no—
278C22⋊C4—72no—
279C22⋊C4—212no—
280C22⋊C4—212no—
281C42—212no—
282C2×C7—22no—
283C2×C7—41no—
284C2×C7—122no—
285C2×C7C2×D3×D411yescharacteristic
286C2×C7#100412yes—
287D7—42no—
288D7—121no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—22no—
292C22×C3—142no—
293C22×C3—142no—
294C22×C3—142no—
295C22×C3—284no—
296C22×C3D4×D712yes—
297C22×C3C23×D711yescharacteristic
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—44no—
302C2×D3—141no—
303C2×D3—142no—
304C3×C4—21no—
305C3×C4—142no—
306Dic3—22no—
307Dic3—72no—
308Dic3—72no—
309Dic3—141no—
310C23—32no—
311C23—62no—
312C23—72no—
313C23—122no—
314C23—142no—
315C23—211no—
316C23—282no—
317C23C2×D3×D711yescharacteristic
318C23C2×D3×D711yescharacteristic
319Q8—422no—
320D4—22no—
321D4—122no—
322D4—282no—
323D4—422no—
324D4—424no—
325D4—424no—
326D4—422no—
327D4—842no—
328D4—842no—
329C2×C4—32no—
330C2×C4—72no—
331C2×C4—214no—
332C2×C4—211no—
333C2×C4—211no—
334C2×C4—211no—
335C2×C4—422no—
336C2×C4—422no—
337C2×C4C2×D3×D711yescharacteristic
338C7#559211yescharacteristic, complemented, Sylow
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—282no—
342C2×C3C2×D4×D711yescharacteristic
343C2×C3#654812yes—
344D3—42no—
345D3—281no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—124no—
353C22—142no—
354C22—142no—
355C22—142no—
356C22—284no—
357C22—422no—
358C22—421no—
359C22#6669412yes—
360C22C22×D3×D711yescharacteristic
361C4—21no—
362C4—62no—
363C4—142no—
364C4—212no—
365C4—212no—
366C4—421no—
367C3#9028311yescharacteristic, complemented, Sylow
368C2—22no—
369C2—41no—
370C2—122no—
371C2—282no—
372C2—841no—
373C2#1062157212yes—
374C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#535
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. C2×#66730C2semidirect
2. C22×#4245C2semidirect
3. C7×#937C2semidirect
4. C3×#6500C2semidirect
5. #10621133C2semidirect
6. #10621132C2semidirect
7. #10621129C2semidirect
8. #10621126C2semidirect
9. #10621094C2semidirect
10. #10621024C2semidirect
11. #10621023C2semidirect
17. C2×#4268C22semidirect
19. C22×Dic21C22semidirect
21. C2×C4×D21C22semidirect
22. C2×Dic42C22semidirect
23. C2×C7×#67C22semidirect
24. C2×C3×C22⋊D7C22semidirect
25. C2×#4247C22semidirect
26. C2×#4246C22semidirect
31. C7×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D7)⋊C4C22semidirect
34. #66577C22semidirect
35. (C2×D3)⋊Dic7C22semidirect
36. (C2×D7)⋊Dic3C22semidirect
37. Dic3×Dic7C22semidirect
44. #6500D3semidirect
64. C2×Dic21D4semidirect
88. C2×C22⋊D7C2×D3semidirect
100. (C2×D7)⋊C4C2×D3semidirect
109. #937D7semidirect
174. C2×#67C2×D7semidirect
186. (C2×D3)⋊C4C2×D7semidirect
256. C3×C7#535semidirect
338. C7#5592semidirect
367. C3#90283semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C22×#42452
C7×#9372
C3×#65002
#106211332
#106211322
#106211292
#106211262
#106210942
#106210242
#106210232
#902833
#55927