Finite Group Databasebeta

← #13060126

Subgroups — #13060126

Order 1344.

524 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 128 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
2#10621563C211yesmaximal, characteristic, complemented
3C7×#932C211yesmaximal, characteristic, complemented
4C3×#6495C211yesmaximal, characteristic, complemented
5#10621061C212yesmaximal, complemented
6#10621059C212yesmaximal, complemented
7#10621050C212yesmaximal, complemented
8#10621044C212yesmaximal, complemented
9#10620884C211yesmaximal, characteristic, complemented
10#10620875C211yesmaximal, characteristic, complemented
11#10620854C211yesmaximal, characteristic, complemented
12#90464—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic, complemented
14C2×#4268—21no—
15C22×Dic21—21no—
16#66730—21no—
17#66730—21no—
18#66730C2212yes—
19#66730C2212yescomplemented
20C2×C4×D21C2211yescharacteristic
21C2×Dic42C2211yescharacteristic, complemented
22C7×#235C2211yescharacteristic
23C3×#1157C2211yescharacteristic
24#66694—21no—
25#66684—21no—
26#66681C2211yescharacteristic, complemented
27C4×D3×D7—21no—
28#66675—21no—
29D21⋊Q8C2211yescharacteristic, complemented
30#66672C2211yescharacteristic, complemented
31#66670C2211yescharacteristic, complemented
32C7×#214C2212yescomplemented
33C7×#213C2212yescomplemented
34C7×#208C2211yescharacteristic, complemented
35C3×#1142C2212yescomplemented
36C3×#1141C2212yescomplemented
37C3×#1136C2211yescharacteristic, complemented
38#66571C2212yescomplemented
39#66570C2212yescomplemented
40#66567C2212yescomplemented
41#66565C2212yescomplemented
42#66559C2212yescomplemented
43D21⋊C8C2212yescomplemented
44D7×#41—31no—
45C2×#1159—31no—
46C7×AGL(1,ℤ/8)—31no—
47#6514—31no—
48#6495D311yescharacteristic, complemented
49#6479—32no—
50#6478—32no—
51#6475—32no—
52#6474—32no—
53#6472—31no—
54D7×M16—31no—
55#5773—71nomaximal, Hall
56C23×C3×C7—21no—
57C22×D21D411yescharacteristic
58C2×D3×D7—21no—
59C3×C7×D4—21no—
60C3×C7×D4C2312yes—
61C2×C3×C4×C7C2311yescharacteristic
62#4268—22no—
63#4268—22no—
64C2×Dic21—21no—
65C2×Dic21—22no—
66C2×Dic21—22no—
67C2×Dic21D411yescharacteristic
68C4×D21D411yescharacteristic
69C4×D21D411yescharacteristic
70C4×D21C2311yescharacteristic
71C4×D21C2311yescharacteristic
72Dic42—21no—
73Dic42C2312yes—
74C7×#67—21no—
75C7×D12C2311yescharacteristic
76C4×D3×C7—21no—
77C7×Dic6C2311yescharacteristic
78C3×C22⋊D7—21no—
79C3×D28C2311yescharacteristic
80C3×C4×D7—21no—
81C3×Dic14C2311yescharacteristic
82#4248—21no—
83#4247—21no—
84#4246—21no—
85#4245—21no—
86D21⋊C4—21no—
87D3×Dic7—21no—
88Dic3×D7—21no—
89C3×C7⋊C8C2312yes—
90C7×C3⋊C8C2312yes—
91C7×#41—31no—
92C2×C7×D4C2×D311yescharacteristic, complemented
93C2×C22⋊D7—61no—
94C22×Dic7—61no—
95#1161—31no—
96Q8×D7—31no—
97#1159—32no—
98#1159—32no—
99#1159—31no—
100#1159—61no—
101#1159—61no—
102#1159—61no—
103D4×D7—31no—
104D4×D7—61no—
105#1157—61no—
106#1157C2×D311yescharacteristic
107C2×C4×D7—31no—
108C2×C4×D7—61no—
109C2×Dic14—31no—
110C7×SD16—32no—
111C7×D8—32no—
112C7×M16—31no—
113#1144—32no—
114#1142—32no—
115#1142C2×D312yescomplemented
116#1141C2×D312yescomplemented
117#1136C2×D311yescharacteristic, complemented
118Dic28—32no—
119#1132—32no—
120#1131—32no—
121C8×D7—32no—
122D3×#41—71no—
123C2×#237—71no—
124C3×AGL(1,ℤ/8)—71no—
125#951—71no—
126#932D711yescharacteristic, complemented
127#916—72no—
128#915—72no—
129#912—72no—
130#911—72no—
131#909—71no—
132D3×M16—71no—
133C22×C3×C7—22no—
134C22×C3×C7—42no—
135C22×C3×C7C2×D411yescharacteristic
136C2×D21—41no—
137C2×D21C2×D411yescharacteristic
138C2×D21C2×D411yescharacteristic
139C2×D3×C7—21no—
140C2×C3×D7—21no—
141D3×D7—42no—
142C3×C4×C7C2×D411yescharacteristic
143C3×C4×C7C2411yescharacteristic
144Dic21—22no—
145Dic21C2×D411yescharacteristic
146Dic21C2×D411yescharacteristic
147C3×Dic7—21no—
148C7×Dic3—21no—
149#575—211noSylow
150C23×C7—21no—
151C22×D7—31no—
152C22×D7—61no—
153C7×Q8—31no—
154C7×D4—21no—
155C7×D4—31no—
156C7×D4—61no—
157C7×D4C22×D312yes—
158C2×C4×C7—61no—
159C2×C4×C7C22×D311yescharacteristic
160C22⋊D7—21no—
161C22⋊D7—61no—
162C22⋊D7—61no—
163C22⋊D7—62no—
164C22⋊D7—62no—
165C2×Dic7—31no—
166C2×Dic7—61no—
167C2×Dic7—62no—
168C2×Dic7—62no—
169C2×Dic7—61no—
170D28—61no—
171D28C22×D311yescharacteristic
172C4×D7—21no—
173C4×D7—31no—
174C4×D7—31no—
175C4×D7—31no—
176C4×D7—31no—
177C4×D7—61no—
178C4×D7—61no—
179Dic14—32no—
180Dic14—61no—
181Dic14—61no—
182Dic14C22×D311yescharacteristic
183C7×C8—32no—
184C7⋊C8C22×D312yes—
185C3×#41—71no—
186C2×C3×D4C2×D711yescharacteristic, complemented
187C2×#67—141no—
188C22×Dic3—141no—
189#239—71no—
190D3×Q8—71no—
191#237—72no—
192#237—72no—
193#237—71no—
194#237—141no—
195#237—141no—
196#237—141no—
197D3×D4—71no—
198D3×D4—141no—
199#235—141no—
200#235C2×D711yescharacteristic
201C2×C4×D3—71no—
202C2×C4×D3—141no—
203C2×Dic6—71no—
204C3×SD16—72no—
205C3×D8—72no—
206C3×M16—71no—
207#216—72no—
208#214—72no—
209#214C2×D712yescomplemented
210#213C2×D712yescomplemented
211#208C2×D711yescharacteristic, complemented
212Dic12—72no—
213#204—72no—
214ΓL(1,𝔽25)—72no—
215D3×C8—72no—
216C2×C3×C7—21no—
217C2×C3×C7—42no—
218C2×C3×C7C22×D411yescharacteristic
219D21—21no—
220D21—21no—
221C3×D7—41no—
222D3×C7—41no—
223C2×#41—211no—
224C2×#41—211no—
225#137—211no—
226#137—211no—
227#137—212no—
228AGL(1,ℤ/8)—31no—
229AGL(1,ℤ/8)—71no—
230AGL(1,ℤ/8)—212no—
231#135—212no—
232#135—212no—
233C2×M16—211no—
234C22×C7—22no—
235C22×C7—42no—
236C22×C7—61no—
237C22×C7D3×D411yescharacteristic
238C2×D7—21no—
239C2×D7—31no—
240C2×D7—31no—
241C2×D7—122no—
242C2×D7—121no—
243C4×C7—61no—
244C4×C7D3×D411yescharacteristic
245C4×C7C23×D311yescharacteristic
246Dic7—21no—
247Dic7—31no—
248Dic7—31no—
249Dic7—62no—
250C23×C3—21no—
251C22×D3—71no—
252C22×D3—141no—
253C3×Q8—71no—
254C3×D4—21no—
255C3×D4—71no—
256C3×D4—141no—
257C3×D4C22×D712yes—
258C2×C3×C4—141no—
259C2×C3×C4C22×D711yescharacteristic
260#67—21no—
261#67—141no—
262#67—141no—
263#67—142no—
264#67—142no—
265C2×Dic3—71no—
266C2×Dic3—141no—
267C2×Dic3—142no—
268C2×Dic3—142no—
269C2×Dic3—141no—
270D12—141no—
271D12C22×D711yescharacteristic
272C4×D3—21no—
273C4×D3—71no—
274C4×D3—71no—
275C4×D3—71no—
276C4×D3—71no—
277C4×D3—141no—
278C4×D3—141no—
279Dic6—72no—
280Dic6—141no—
281Dic6—141no—
282Dic6C22×D711yescharacteristic
283C3×C8—72no—
284C3⋊C8C22×D712yes—
285C3×C7#57511yescharacteristic, complemented, Hall
286#41—31no—
287#41—71no—
288#41—211no—
289#41—211no—
290#41—212no—
291#41—212no—
292#41—421no—
293#41—421no—
294#41—421no—
295#41—421no—
296C2×Q8—211no—
297C2×Q8—211no—
298C2×D4—211no—
299C2×D4—421no—
300C2×D4—421no—
301C2×D4D3×D711yescharacteristic, complemented
302C22×C4—211no—
303C22×C4—421no—
304C22×C4—421no—
305Q16—212no—
306Q16—212no—
307SD16—32no—
308SD16—72no—
309SD16—212no—
310SD16—212no—
311D8—32no—
312D8—72no—
313M16—31no—
314M16—71no—
315M16—212no—
316C2×C8—212no—
317C2×C7—21no—
318C2×C7—42no—
319C2×C7—121no—
320C2×C7C2×D3×D411yescharacteristic
321D7—41no—
322D7—61no—
323D7—61no—
324C22×C3—22no—
325C22×C3—42no—
326C22×C3—141no—
327C22×C3D4×D711yescharacteristic
328C2×D3—21no—
329C2×D3—71no—
330C2×D3—71no—
331C2×D3—282no—
332C2×D3—281no—
333C3×C4—141no—
334C3×C4D4×D711yescharacteristic
335C3×C4C23×D711yescharacteristic
336Dic3—21no—
337Dic3—71no—
338Dic3—71no—
339Dic3—142no—
340C23—21no—
341C23—211no—
342C23—421no—
343Q8—31no—
344Q8—71no—
345Q8—212no—
346Q8—421no—
347Q8—421no—
348D4—21no—
349D4—31no—
350D4—61no—
351D4—71no—
352D4—141no—
353D4—421no—
354D4—421no—
355D4—421no—
356D4—422no—
357D4—422no—
358D4C2×D3×D712yes—
359C2×C4—61no—
360C2×C4—141no—
361C2×C4—211no—
362C2×C4—211no—
363C2×C4—211no—
364C2×C4—211no—
365C2×C4—211no—
366C2×C4—421no—
367C2×C4—421no—
368C2×C4—421no—
369C2×C4—422no—
370C2×C4—422no—
371C2×C4—421no—
372C2×C4C2×D3×D711yescharacteristic
373C8—32no—
374C8—72no—
375C7#577311yescharacteristic, complemented, Sylow
376C2×C3—21no—
377C2×C3—42no—
378C2×C3—281no—
379C2×C3C2×D4×D711yescharacteristic
380D3—41no—
381D3—141no—
382D3—141no—
383C22—22no—
384C22—42no—
385C22—61no—
386C22—141no—
387C22—211no—
388C22—211no—
389C22—841no—
390C22—842no—
391C22#6669411yescharacteristic
392C4—61no—
393C4—141no—
394C4—211no—
395C4—211no—
396C4—422no—
397C4#6669411yescharacteristic
398C4C22×D3×D711yescharacteristic
399C3#9046411yescharacteristic, complemented, Sylow
400C2—21no—
401C2—42no—
402C2—121no—
403C2—281no—
404C2—421no—
405C2—421no—
406C2C2×#6669411yescharacteristic
subgroupG/H
centerC2C2×#66694
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C22×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#575
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66730C2semidirect
2. #10621563C2semidirect
3. C7×#932C2semidirect
4. C3×#6495C2semidirect
5. #10621061C2semidirect
6. #10621059C2semidirect
7. #10621050C2semidirect
8. #10621044C2semidirect
9. #10620884C2semidirect
10. #10620875C2semidirect
11. #10620854C2semidirect
13. C2×C3×C7×D4C22semidirect
19. #66730C22semidirect
21. C2×Dic42C22semidirect
26. #66681C22semidirect
29. D21⋊Q8C22semidirect
30. #66672C22semidirect
31. #66670C22semidirect
32. C7×#214C22semidirect
33. C7×#213C22semidirect
34. C7×#208C22semidirect
35. C3×#1142C22semidirect
36. C3×#1141C22semidirect
37. C3×#1136C22semidirect
38. #66571C22semidirect
39. #66570C22semidirect
40. #66567C22semidirect
41. #66565C22semidirect
42. #66559C22semidirect
43. D21⋊C8C22semidirect
48. #6495D3semidirect
92. C2×C7×D4C2×D3semidirect
115. #1142C2×D3semidirect
116. #1141C2×D3semidirect
117. #1136C2×D3semidirect
126. #932D7semidirect
186. C2×C3×D4C2×D7semidirect
209. #214C2×D7semidirect
210. #213C2×D7semidirect
211. #208C2×D7semidirect
285. C3×C7#575semidirect
301. C2×D4D3×D7semidirect
375. C7#5773semidirect
399. C3#90464semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
#106215632
C7×#9322
C3×#64952
#106210612
#106210592
#106210502
#106210442
#106208842
#106208752
#106208542
#904643
#57737