Finite Group Databasebeta

← #13060098

Subgroups — #13060098

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1159C211yesmaximal, characteristic, complemented
2D7×#235C211yesmaximal, characteristic, complemented
3#10621415C211yesmaximal, characteristic, complemented
4C7×#932C211yesmaximal, characteristic, complemented
5#10621054C212yesmaximal, complemented
6#10621052C212yesmaximal, complemented
7#10621046C212yesmaximal, complemented
8#10621040C212yesmaximal, complemented
9#10620877C211yesmaximal, characteristic, complemented
10#10620871C211yesmaximal, characteristic, complemented
11D7×#208C211yesmaximal, characteristic, complemented
12#90464—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic, complemented
14#66728C2211yescharacteristic
15C7×#235C2211yescharacteristic
16C2×C3×C22⋊D7—21no—
17C22×C3×Dic7—21no—
18C3×#1159—21no—
19C3×#1159—21no—
20C3×#1159C2212yes—
21C3×#1159C2212yescomplemented
22C2×C3×C4×D7C2211yescharacteristic
23C2×C3×Dic14C2211yescharacteristic, complemented
24D7×#67—21no—
25#66686—21no—
26D7×D12C2211yescharacteristic, complemented
27C4×D3×D7—21no—
28#66677C2211yescharacteristic, complemented
29#66676C2211yescharacteristic, complemented
30#66675—21no—
31D7×Dic6C2211yescharacteristic, complemented
32#66633C2212yescomplemented
33#66632C2212yescomplemented
34#66627C2211yescharacteristic, complemented
35C7×#214C2212yescomplemented
36C7×#213C2212yescomplemented
37C7×#208C2211yescharacteristic, complemented
38#66570C2212yescomplemented
39#66569C2212yescomplemented
40#66566C2212yescomplemented
41#66564C2212yescomplemented
42#66557C2212yescomplemented
43D7×C3⋊C8C2212yescomplemented
44D7×#41—31no—
45C2×#1159D311yescharacteristic, complemented
46C7×AGL(1,ℤ/8)—31no—
47#6514—31no—
48#6495—31no—
49#6479—32no—
50#6478—32no—
51#6475—32no—
52#6474—32no—
53#6472—31no—
54D7×M16—31no—
55#5822—71nomaximal, Hall
56C23×C3×C7—21no—
57C22×C3×D7D411yescharacteristic
58C2×D3×D7—21no—
59C3×C7×D4—21no—
60C3×C7×D4C2312yes—
61C2×C3×C4×C7C2311yescharacteristic
62#4268—21no—
63D84C2311yescharacteristic
64C4×D21—21no—
65Dic42C2311yescharacteristic
66C7×#67—21no—
67C7×D12C2311yescharacteristic
68C4×D3×C7—21no—
69C7×Dic6C2311yescharacteristic
70C3×C22⋊D7—22no—
71C3×C22⋊D7—22no—
72C2×C3×Dic7—21no—
73C2×C3×Dic7—22no—
74C2×C3×Dic7—22no—
75C2×C3×Dic7D411yescharacteristic
76C3×C4×D7D411yescharacteristic
77C3×C4×D7D411yescharacteristic
78C3×C4×D7C2311yescharacteristic
79C3×C4×D7C2311yescharacteristic
80C3×Dic14—21no—
81C3×Dic14C2312yes—
82#4248—21no—
83#4247—21no—
84#4246—21no—
85#4245—21no—
86D21⋊C4—21no—
87D3×Dic7—21no—
88Dic3×D7—21no—
89C3⋊(C7⋊C8)C2312yes—
90C7×C3⋊C8C2312yes—
91C7×#41—31no—
92C2×C7×D4C2×D311yescharacteristic, complemented
93C2×C22⋊D7—21no—
94C22×Dic7—21no—
95#1161—31no—
96Q8×D7—31no—
97#1159—21no—
98#1159—21no—
99#1159—31no—
100#1159—61no—
101#1159C2×D312yes—
102#1159C2×D312yescomplemented
103D4×D7—31no—
104D4×D7—61no—
105#1157—31no—
106#1157—61no—
107C2×C4×D7—61no—
108C2×C4×D7C2×D311yescharacteristic
109C2×Dic14C2×D311yescharacteristic, complemented
110C7×SD16—32no—
111C7×D8—32no—
112C7×M16—31no—
113#1144—32no—
114#1142—32no—
115#1142—32no—
116#1141—32no—
117#1136—31no—
118Dic28—32no—
119#1132—32no—
120#1131—32no—
121C8×D7—32no—
122C2×C3×#41—71no—
123C2×#235—71no—
124#951—72no—
125#950—72no—
126#950—72no—
127#949—72no—
128#942—71no—
129#942—71no—
130#932—71no—
131#932D711yescharacteristic, complemented
132C2×#208—71no—
133C22×C3×C7—22no—
134C22×C3×C7—42no—
135C22×C3×C7C2×D411yescharacteristic
136C2×D21—21no—
137C2×D3×C7—21no—
138C2×C3×D7—41no—
139C2×C3×D7C2×D411yescharacteristic
140C2×C3×D7C2×D411yescharacteristic
141D3×D7—42no—
142C3×C4×C7C2×D411yescharacteristic
143C3×C4×C7C2411yescharacteristic
144Dic21—21no—
145C3×Dic7—22no—
146C3×Dic7C2×D411yescharacteristic
147C3×Dic7C2×D411yescharacteristic
148C7×Dic3—21no—
149#575—211noSylow
150C23×C7—21no—
151C22×D7—61no—
152C22×D7#6711yescharacteristic
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—22no—
161C22⋊D7—22no—
162C22⋊D7—61no—
163C22⋊D7—61no—
164C22⋊D7—61no—
165C2×Dic7—21no—
166C2×Dic7—22no—
167C2×Dic7—22no—
168C2×Dic7—61no—
169C2×Dic7#6711yescharacteristic
170D28—31no—
171D28—61no—
172C4×D7—61no—
173C4×D7—61no—
174C4×D7—61no—
175C4×D7#6711yescharacteristic
176C4×D7#6711yescharacteristic
177C4×D7C22×D311yescharacteristic
178C4×D7C22×D311yescharacteristic
179Dic14—21no—
180Dic14—31no—
181Dic14—61no—
182Dic14C22×D312yes—
183C7×C8—32no—
184C7⋊C8—32no—
185C3×#41—72no—
186C3×#41—72no—
187C3×#41—141no—
188C3×#41—141no—
189C2×C3×Q8—71no—
190C2×C3×D4—141no—
191C2×C3×D4C2×D711yescharacteristic, complemented
192C22×C3×C4—71no—
193C22×C3×C4—141no—
194C2×#67—141no—
195#235—71no—
196#235—71no—
197#235—71no—
198#235—141no—
199#235—141no—
200#235C2×D711yescharacteristic
201C2×D12—71no—
202C2×C4×D3—141no—
203C2×Dic6—71no—
204#216—72no—
205#216—72no—
206#215—72no—
207#215—72no—
208#214—72no—
209#214C2×D712yescomplemented
210#213—72no—
211#213C2×D712yescomplemented
212#208—71no—
213#208—72no—
214#208C2×D711yescharacteristic, complemented
215C2×C3⋊C8—72no—
216C2×C3×C7—21no—
217C2×C3×C7—42no—
218C2×C3×C7C22×D411yescharacteristic
219D21—41no—
220C3×D7—21no—
221C3×D7—21no—
222D3×C7—41no—
223C2×#41—71no—
224C2×#41—211no—
225#137—211no—
226#137—212no—
227#137—211no—
228AGL(1,ℤ/8)—31no—
229AGL(1,ℤ/8)—212no—
230AGL(1,ℤ/8)—211no—
231#135—212no—
232#135—212no—
233C2×M16—211no—
234C22×C7—22no—
235C22×C7—42no—
236C22×C7—61no—
237C22×C7C2×#6711yescharacteristic
238C2×D7—41no—
239C2×D7—61no—
240C2×D7—122no—
241C2×D7C2×#6711yescharacteristic
242C2×D7C2×#6711yescharacteristic
243C4×C7—61no—
244C4×C7C2×#6711yescharacteristic
245C4×C7C23×D311yescharacteristic
246Dic7—22no—
247Dic7—61no—
248Dic7C2×#6711yescharacteristic
249Dic7C2×#6711yescharacteristic
250C23×C3—21no—
251C23×C3—71no—
252C22×D3—141no—
253C3×Q8—72no—
254C3×Q8—141no—
255C3×D4—21no—
256C3×D4—142no—
257C3×D4—142no—
258C3×D4C22×D712yes—
259C2×C3×C4—71no—
260C2×C3×C4—71no—
261C2×C3×C4—71no—
262C2×C3×C4—71no—
263C2×C3×C4—71no—
264C2×C3×C4—142no—
265C2×C3×C4—142no—
266C2×C3×C4—141no—
267C2×C3×C4C22×D711yescharacteristic
268#67—21no—
269#67—141no—
270#67—141no—
271#67—141no—
272C2×Dic3—141no—
273D12—71no—
274D12—141no—
275D12C22×D711yescharacteristic
276C4×D3—21no—
277C4×D3—141no—
278C4×D3—141no—
279C4×D3—141no—
280Dic6—71no—
281Dic6—141no—
282Dic6C22×D711yescharacteristic
283C3⋊C8—72no—
284C3⋊C8C22×D712yes—
285C3×C7#57511yescharacteristic, complemented, Hall
286#41—31no—
287#41—72no—
288#41—72no—
289#41—141no—
290#41—141no—
291#41—211no—
292#41—211no—
293#41—211no—
294#41—421no—
295#41—421no—
296C2×Q8—71no—
297C2×Q8—211no—
298C2×D4—141no—
299C2×D4—211no—
300C2×D4—421no—
301C2×D4D3×D711yescharacteristic, complemented
302C22×C4—71no—
303C22×C4—141no—
304C22×C4—421no—
305Q16—212no—
306Q16—212no—
307SD16—32no—
308SD16—212no—
309SD16—212no—
310SD16—212no—
311D8—32no—
312D8—212no—
313M16—31no—
314M16—212no—
315M16—211no—
316C2×C8—212no—
317C2×C7—21no—
318C2×C7—42no—
319C2×C7—121no—
320C2×C7C22×#6711yescharacteristic
321D7—21no—
322D7—21no—
323D7—121no—
324C22×C3—22no—
325C22×C3—42no—
326C22×C3—71no—
327C22×C3—71no—
328C22×C3—281no—
329C22×C3D4×D711yescharacteristic
330C2×D3—21no—
331C2×D3—141no—
332C2×D3—282no—
333C3×C4—71no—
334C3×C4—71no—
335C3×C4—142no—
336C3×C4D4×D711yescharacteristic
337C3×C4C23×D711yescharacteristic
338Dic3—21no—
339Dic3—141no—
340C23—21no—
341C23—71no—
342C23—421no—
343Q8—31no—
344Q8—72no—
345Q8—141no—
346Q8—211no—
347Q8—421no—
348D4—21no—
349D4—31no—
350D4—61no—
351D4—142no—
352D4—142no—
353D4—211no—
354D4—421no—
355D4—421no—
356D4—421no—
357D4—421no—
358D4C2×D3×D712yes—
359C2×C4—61no—
360C2×C4—71no—
361C2×C4—71no—
362C2×C4—71no—
363C2×C4—71no—
364C2×C4—71no—
365C2×C4—142no—
366C2×C4—142no—
367C2×C4—141no—
368C2×C4—421no—
369C2×C4—421no—
370C2×C4—421no—
371C2×C4—421no—
372C2×C4C2×D3×D711yescharacteristic
373C8—32no—
374C8—212no—
375C7#582211yescharacteristic, complemented, Sylow
376C2×C3—21no—
377C2×C3—42no—
378C2×C3—141no—
379C2×C3—141no—
380C2×C3C2×D4×D711yescharacteristic
381D3—41no—
382D3—281no—
383C22—22no—
384C22—42no—
385C22—61no—
386C22—71no—
387C22—71no—
388C22—281no—
389C22—421no—
390C22—842no—
391C22D7×#6711yescharacteristic
392C4—61no—
393C4—71no—
394C4—71no—
395C4—142no—
396C4—421no—
397C4D7×#6711yescharacteristic
398C4C22×D3×D711yescharacteristic
399C3#9046411yescharacteristic, complemented, Sylow
400C2—21no—
401C2—42no—
402C2—121no—
403C2—141no—
404C2—141no—
405C2—841no—
406C2C2×D7×#6711yescharacteristic
subgroupG/H
centerC2C2×D7×#67
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×C3×#1159C2semidirect
2. D7×#235C2semidirect
3. #10621415C2semidirect
4. C7×#932C2semidirect
5. #10621054C2semidirect
6. #10621052C2semidirect
7. #10621046C2semidirect
8. #10621040C2semidirect
9. #10620877C2semidirect
10. #10620871C2semidirect
11. D7×#208C2semidirect
13. C2×C3×C7×D4C22semidirect
21. C3×#1159C22semidirect
23. C2×C3×Dic14C22semidirect
26. D7×D12C22semidirect
28. #66677C22semidirect
29. #66676C22semidirect
31. D7×Dic6C22semidirect
32. #66633C22semidirect
33. #66632C22semidirect
34. #66627C22semidirect
35. C7×#214C22semidirect
36. C7×#213C22semidirect
37. C7×#208C22semidirect
38. #66570C22semidirect
39. #66569C22semidirect
40. #66566C22semidirect
41. #66564C22semidirect
42. #66557C22semidirect
43. D7×C3⋊C8C22semidirect
45. C2×#1159D3semidirect
92. C2×C7×D4C2×D3semidirect
102. #1159C2×D3semidirect
109. C2×Dic14C2×D3semidirect
131. #932D7semidirect
191. C2×C3×D4C2×D7semidirect
209. #214C2×D7semidirect
211. #213C2×D7semidirect
214. #208C2×D7semidirect
285. C3×C7#575semidirect
301. C2×D4D3×D7semidirect
375. C7#5822semidirect
399. C3#90464semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
D7×#2352
#106214152
C7×#9322
#106210542
#106210522
#106210462
#106210402
#106208772
#106208712
D7×#2082
#904643
#58227