Finite Group Databasebeta

← #13060134

Subgroups — #13060134

Order 1344.

668 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 138 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×D21C211yesmaximal, characteristic, complemented
2C2×#66677C211yesmaximal, characteristic, complemented
3C7×#936C211yesmaximal, characteristic, complemented
4C3×#6501C211yesmaximal, characteristic, complemented
5#10621134C212yesmaximal, complemented
6#10621124C212yesmaximal, complemented
7#10621123C212yesmaximal, complemented
8#10621095C212yesmaximal, complemented
9C4×#4246C211yesmaximal, characteristic, complemented
10#10620990C211yesmaximal, characteristic, complemented
11#10620945C211yesmaximal, characteristic, complemented
12#90367—31nomaximal, Hall
13C23×D21C2212yes—
14C2×C3×C7×D4C2211yescharacteristic
15C2×#4268C2212yes—
16D4×D21—24no—
17C2×D84C2211yescharacteristic
18C2×C4×D21C2211yescharacteristic
19C2×C7×Dic6C2211yescharacteristic, complemented
20C2×C3×C22⋊D7C2212yescomplemented
21C2×C3×C4×D7C2211yescharacteristic, complemented
22C2×#4246C2211yescharacteristic
23C2×#4246C2212yescomplemented
24C2×D21⋊C4C2212yescomplemented
25C2×D21⋊C4C2212yescomplemented
26#66677—24no—
27C7×#217C2212yescomplemented
28C7×#217C2212yescomplemented
29C4×C7×Dic3C2211yescharacteristic, complemented
30C3×C22⋊Dic7C2212yescomplemented
31C3×C4⋊Dic7C2211yescharacteristic, complemented
32Dic3⋊Dic7C2211yescharacteristic, complemented
33Dic7⋊Dic3C2212yescomplemented
34#66575C2212yes—
35#66575C2211yescharacteristic
36#66575C2212yescomplemented
37(C2×D7)⋊Dic3C2211yescharacteristic, complemented
38C2×#1161—31no—
39C2×D4×D7—31no—
40C7×#124—31no—
41#6509—31no—
42#6501D311yescharacteristic, complemented
43#6447—32no—
44#6446—32no—
45#6445—32no—
46D7×C22⋊C4—32no—
47C4×D28—31no—
48C4×D28—31no—
49#5612—71nomaximal, Hall
50C23×C3×C7C2312yes—
51C22×D21—24no—
52C22×D21—24no—
53C22×D21—42no—
54C22×D21C2312yes—
55C22×D21C2311yescharacteristic
56C22×C3×D7C2311yescharacteristic
57C3×C7×D4—22no—
58C2×C3×C4×C7C2311yescharacteristic
59#4268—24no—
60C2×Dic21C2311yescharacteristic
61D84D414yescomplemented
62C4×D21—22no—
63C2×C7×Dic3C2311yescharacteristic
64C2×C7×Dic3C2312yes—
65C2×C7×Dic3C2311yescharacteristic
66C7×Dic6—22no—
67C3×C22⋊D7—42no—
68C2×C3×Dic7C2312yes—
69C2×C3×Dic7C2311yescharacteristic
70C3×C4×D7—22no—
71#4246—24no—
72#4246—22no—
73D21⋊C4—24no—
74D21⋊C4—42no—
75C23×D7—32no—
76C2×C7×Q8—31no—
77C2×C7×D4C2×D311yescharacteristic
78C2×C22⋊D7—32no—
79C2×C22⋊D7C2×D312yescomplemented
80#1161—64no—
81D4×D7—64no—
82C2×D28—31no—
83C2×D28—31no—
84C2×D28—32no—
85C2×C4×D7—31no—
86C2×C4×D7—32no—
87C2×C4×D7—32no—
88C2×C4×D7C2×D311yescharacteristic, complemented
89C7×C22⋊C4—32no—
90C7×C22⋊C4—32no—
91C42×C7—31no—
92C22⋊Dic7C2×D312yescomplemented
93(C2×D7)⋊C4—31no—
94(C2×D7)⋊C4—31no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4—32no—
97C4⋊Dic7—31no—
98C4⋊Dic7C2×D311yescharacteristic, complemented
99Dic7⋊C4—32no—
100C2×D3×D4—71no—
101C2×#235—71no—
102C3×#121—71no—
103#938—72no—
104#937—72no—
105#936D711yescharacteristic, complemented
106C4×#67—71no—
107#897—71no—
108#893—71no—
109#883—72no—
110D3×C22⋊C4—72no—
111C22×C3×C7—22no—
112C22×C3×C7—22no—
113C22×C3×C7—22no—
114C22×C3×C7C2411yescharacteristic
115C2×D21—21no—
116C2×D21—22no—
117C2×D21—22no—
118C2×D21—21no—
119C2×D21—48no—
120C2×D21C2×D414yes—
121C2×D21#4112yescomplemented
122C2×C3×D7—21no—
123C2×C3×D7—21no—
124C2×C3×D7—21no—
125C3×C4×C7C2×D412yes—
126Dic21—21no—
127C3×Dic7—21no—
128C3×Dic7—22no—
129C7×Dic3—22no—
130C7×Dic3—21no—
131C7×Dic3—21no—
132#546—211noSylow
133C23×C7C22×D312yes—
134C22×D7—31no—
135C22×D7—32no—
136C22×D7—64no—
137C22×D7—64no—
138C22×D7—122no—
139C22×D7C22×D311yescharacteristic
140C7×Q8—62no—
141C7×D4—22no—
142C2×C4×C7—32no—
143C2×C4×C7—31no—
144C2×C4×C7—31no—
145C2×C4×C7C22×D311yescharacteristic
146C22⋊D7—42no—
147C22⋊D7—64no—
148C2×Dic7—31no—
149C2×Dic7C22×D312yes—
150C2×Dic7C22×D311yescharacteristic
151D28—34no—
152D28—64no—
153D28—62no—
154C4×D7—22no—
155C4×D7—64no—
156C4×D7—62no—
157C4×D7—122no—
158C23×D3—72no—
159C2×C3×D4—72no—
160C2×C3×D4C2×D711yescharacteristic
161C22×C3×C4—71no—
162C2×#67—72no—
163C2×#67—71no—
164C2×#67—72no—
165D3×D4—144no—
166#235—144no—
167C2×D12—71no—
168C2×C4×D3—71no—
169C2×C4×D3—72no—
170C2×C4×D3—72no—
171C2×Dic6C2×D711yescharacteristic, complemented
172C3×C4⋊C4—71no—
173C3×C22⋊C4—72no—
174#217—71no—
175#217C2×D712yescomplemented
176#217C2×D712yescomplemented
177(C2×D3)⋊C4—71no—
178(C2×D3)⋊C4—72no—
179(C2×D3)⋊C4—72no—
180C4⋊Dic3—72no—
181Dic3⋊C4—71no—
182C4×Dic3C2×D711yescharacteristic, complemented
183C2×C3×C7—42no—
184C2×C3×C7C22×D411yescharacteristic
185C2×C3×C7C2×#4111yescharacteristic
186C2×C3×C7H(5,𝔽2)11yescharacteristic
187D21—22no—
188D21—24no—
189C3×D7—41no—
190C2×#41—211no—
191C22×D4—211no—
192#124—31no—
193#123—212no—
194C22⋊Q8—211no—
195#121—71no—
196#121—212no—
197#120—212no—
198C4×D4—211no—
199C4×D4—211no—
200C2×C22⋊C4—212no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7C23×D311yescharacteristic
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—34no—
209C2×D7—32no—
210C2×D7—61no—
211C2×D7—62no—
212C2×D7—61no—
213C2×D7—62no—
214C2×D7—128no—
215C4×C7—62no—
216C4×C7—61no—
217C4×C7—61no—
218C4×C7D3×D412yes—
219Dic7—21no—
220Dic7—22no—
221Dic7—61no—
222C23×C3—71no—
223C23×C3C22×D712yes—
224C22×D3—71no—
225C22×D3—72no—
226C22×D3—144no—
227C22×D3—144no—
228C22×D3—282no—
229C3×D4—22no—
230C3×D4—282no—
231C2×C3×C4—71no—
232C2×C3×C4—72no—
233C2×C3×C4—142no—
234C2×C3×C4C22×D711yescharacteristic
235#67—144no—
236#67—142no—
237#67—144no—
238C2×Dic3—71no—
239C2×Dic3C22×D711yescharacteristic
240C2×Dic3C22×D711yescharacteristic
241C2×Dic3C22×D712yes—
242D12—74no—
243C4×D3—144no—
244C4×D3—142no—
245C4×D3—282no—
246Dic6—22no—
247C3×C7#54611yescharacteristic, complemented, Hall
248C24—212no—
249#41—424no—
250C2×Q8—31no—
251C2×D4—72no—
252C2×D4—211no—
253C2×D4—212no—
254C2×D4—212no—
255C2×D4—211no—
256C2×D4—424no—
257C2×D4D3×D711yescharacteristic
258C22×C4—71no—
259C22×C4—212no—
260C22×C4—212no—
261C22×C4—211no—
262C4⋊C4—71no—
263C4⋊C4—212no—
264C4⋊C4—211no—
265C22⋊C4—32no—
266C22⋊C4—32no—
267C22⋊C4—72no—
268C22⋊C4—211no—
269C22⋊C4—212no—
270C22⋊C4—212no—
271C22⋊C4—211no—
272C42—31no—
273C2×C7—42no—
274C2×C7C2×D3×D411yescharacteristic
275C2×C7#100411yescharacteristic
276C2×C7D3×#4111yescharacteristic
277D7—41no—
278D7—62no—
279D7—64no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—141no—
284C22×C3—141no—
285C22×C3—141no—
286C22×C3C23×D711yescharacteristic
287C2×D3—74no—
288C2×D3—72no—
289C2×D3—141no—
290C2×D3—142no—
291C2×D3—142no—
292C2×D3—141no—
293C2×D3—288no—
294C3×C4—141no—
295C3×C4—142no—
296C3×C4D4×D712yes—
297Dic3—22no—
298Dic3—21no—
299Dic3—21no—
300Dic3—141no—
301C23—71no—
302C23—211no—
303C23—212no—
304C23—424no—
305C23—424no—
306C23—842no—
307C23C2×D3×D712yes—
308Q8—62no—
309D4—22no—
310D4—214no—
311D4—282no—
312D4—424no—
313D4—422no—
314D4—424no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—71no—
319C2×C4—72no—
320C2×C4—142no—
321C2×C4—211no—
322C2×C4—424no—
323C2×C4—422no—
324C2×C4—842no—
325C2×C4C2×D3×D711yescharacteristic
326C7#561211yescharacteristic, complemented, Sylow
327C2×C3—42no—
328C2×C3—281no—
329C2×C3C2×D4×D711yescharacteristic
330C2×C3D7×#4111yescharacteristic
331C2×C3#655311yescharacteristic
332D3—142no—
333D3—144no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—212no—
341C22—214no—
342C22—422no—
343C22—422no—
344C22—421no—
345C22—421no—
346C22—848no—
347C22C22×D3×D711yescharacteristic
348C4—61no—
349C4—61no—
350C4—62no—
351C4—142no—
352C4—141no—
353C4—421no—
354C4#6669412yes—
355C3#9036711yescharacteristic, complemented, Sylow
356C2—42no—
357C2—281no—
358C2—422no—
359C2—424no—
360C2#1062157111yescharacteristic
361C2#1062157311yescharacteristic
362C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#546
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×D4×D21C2semidirect
2. C2×#66677C2semidirect
3. C7×#936C2semidirect
4. C3×#6501C2semidirect
5. #10621134C2semidirect
6. #10621124C2semidirect
7. #10621123C2semidirect
8. #10621095C2semidirect
9. C4×#4246C2semidirect
10. #10620990C2semidirect
11. #10620945C2semidirect
19. C2×C7×Dic6C22semidirect
20. C2×C3×C22⋊D7C22semidirect
21. C2×C3×C4×D7C22semidirect
23. C2×#4246C22semidirect
24. C2×D21⋊C4C22semidirect
25. C2×D21⋊C4C22semidirect
27. C7×#217C22semidirect
28. C7×#217C22semidirect
29. C4×C7×Dic3C22semidirect
30. C3×C22⋊Dic7C22semidirect
31. C3×C4⋊Dic7C22semidirect
32. Dic3⋊Dic7C22semidirect
33. Dic7⋊Dic3C22semidirect
36. #66575C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
42. #6501D3semidirect
61. D84D4semidirect
79. C2×C22⋊D7C2×D3semidirect
88. C2×C4×D7C2×D3semidirect
92. C22⋊Dic7C2×D3semidirect
98. C4⋊Dic7C2×D3semidirect
105. #936D7semidirect
121. C2×D21#41semidirect
171. C2×Dic6C2×D7semidirect
175. #217C2×D7semidirect
176. #217C2×D7semidirect
182. C4×Dic3C2×D7semidirect
247. C3×C7#546semidirect
326. C7#5612semidirect
355. C3#90367semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C2×#666772
C7×#9362
C3×#65012
#106211342
#106211242
#106211232
#106210952
C4×#42462
#106209902
#106209452
#903673
#56127