Finite Group Databasebeta

← #13060568

Subgroups — #13060568

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#1009C211yesmaximal, characteristic, complemented
2#10621551C211yesmaximal, characteristic, complemented
3#10621433C211yesmaximal, characteristic, complemented
4C3×#6514C211yesmaximal, characteristic, complemented
5#10621081C212yesmaximal, complemented
6#10621072C212yesmaximal, complemented
7#10621058C212yesmaximal, complemented
8D3×#1142C212yesmaximal, complemented
9C2×#66564C211yesmaximal, characteristic, complemented
10#10620866C211yesmaximal, characteristic, complemented
11#10620850C211yesmaximal, characteristic, complemented
12#90527—31nomaximal, Hall
13C3×C7×#41C2211yescharacteristic
14#66728C2211yescharacteristic, complemented
15C7×#239C2212yes—
16D3×C7×D4—22no—
17D3×C7×D4C2212yes—
18C7×#235—21no—
19C7×#235C2211yescharacteristic
20C2×C7×D12—21no—
21C2×C7×D12C2211yescharacteristic
22C2×C3×Dic14C2211yescharacteristic, complemented
23#66686—22no—
24D21⋊Q8—21no—
25#66673C2212yescomplemented
26#66672—21no—
27D3×Dic14C2212yescomplemented
28#66635C2211yescharacteristic, complemented
29#66634C2211yescharacteristic, complemented
30#66633C2211yescharacteristic, complemented
31#66632C2211yescharacteristic, complemented
32C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
33C3×#1144C2212yescomplemented
34C3×#1142C2212yescomplemented
35C3×#1136C2211yescharacteristic, complemented
36#66567C2212yescomplemented
37#66564C2212yescomplemented
38#66564C2212yescomplemented
39#66562C2212yescomplemented
40#66558C2212yescomplemented
41D3×C7⋊C8C2212yescomplemented
42C7×H(5,𝔽2)—31no—
43#6554—31no—
44#6514—32no—
45#6514D311yescharacteristic, complemented
46#6513—31no—
47#6513—32no—
48#6511—31no—
49C2×#1142—31no—
50C2×#1142—32no—
51#6495—32no—
52#6495—31no—
53#5778—71nomaximal, Hall
54C22×D3×C7—21no—
55C22×D3×C7—22no—
56C3×C7×Q8C2311yescharacteristic
57C3×C7×D4—21no—
58C3×C7×D4C2311yescharacteristic
59C2×C3×C4×C7—21no—
60C2×C3×C4×C7C2311yescharacteristic
61#4268—21no—
62D84C2311yescharacteristic
63C4×D21—21no—
64Dic42C2311yescharacteristic
65C7×#67—22no—
66C7×#67D412yes—
67C7×D12—22no—
68C7×D12—21no—
69C7×D12D411yescharacteristic
70C7×D12C2312yes—
71C4×D3×C7—22no—
72C4×D3×C7C2312yes—
73C7×Dic6D411yescharacteristic
74C2×C3×Dic7—21no—
75C3×Dic14—21no—
76C3×Dic14C2312yes—
77#4248—22no—
78#4247—22no—
79D21⋊C4—22no—
80D3×Dic7—22no—
81C3⋊(C7⋊C8)C2311yescharacteristic
82C3⋊(C7⋊C8)C2311yescharacteristic
83C3×C7⋊C8C2312yes—
84C7×#41—31no—
85C7×#41—32no—
86C7×#41—61no—
87C7×#41C2×D311yescharacteristic
88C2×C7×D4—32no—
89C2×C7×D4—31no—
90C2×C7×D4—61no—
91C2×C7×D4—62no—
92Q8×D7—61no—
93#1159—61no—
94#1159—62no—
95#1157—31no—
96#1157—32no—
97C2×Dic14—32no—
98C2×Dic14C2×D311yescharacteristic, complemented
99#1144—31no—
100#1144C2×D312yescomplemented
101#1143—31no—
102#1142—31no—
103#1142—32no—
104#1142—32no—
105#1142—32no—
106#1142C2×D312yescomplemented
107#1141—31no—
108#1141—32no—
109#1136—32no—
110#1136C2×D311yescharacteristic, complemented
111C2×C7⋊C8—31no—
112C2×C7⋊C8—32no—
113#1009D711yescharacteristic, complemented
114#1007—71no—
115C3×#137—71no—
116#950—71no—
117C2×#215—71no—
118#919—72no—
119#918—72no—
120#914—72no—
121D3×SD16—72no—
122#908—71no—
123#907—71no—
124C22×C3×C7—21no—
125C22×C3×C7C2×D411yescharacteristic
126C2×D21—21no—
127C2×D3×C7—22no—
128C2×D3×C7—82no—
129C2×D3×C7—81no—
130C2×D3×C7C2×D412yes—
131C3×C4×C7—21no—
132C3×C4×C7C2×D411yescharacteristic
133C3×C4×C7C2411yescharacteristic
134Dic21—21no—
135C3×Dic7—22no—
136C7×Dic3C2×D412yes—
137#577—211noSylow
138C23×C7—62no—
139C23×C7—61no—
140C7×Q8—31no—
141C7×Q8C22×D311yescharacteristic
142C7×D4—21no—
143C7×D4—32no—
144C7×D4—31no—
145C7×D4—32no—
146C7×D4—62no—
147C7×D4—62no—
148C7×D4—61no—
149C7×D4C22×D311yescharacteristic
150C2×C4×C7—21no—
151C2×C4×C7—32no—
152C2×C4×C7—62no—
153C2×C4×C7C22×D311yescharacteristic
154C22⋊D7—62no—
155C22⋊D7—61no—
156C2×Dic7—21no—
157C2×Dic7—62no—
158D28—31no—
159C4×D7—62no—
160C4×D7—61no—
161Dic14—21no—
162Dic14—31no—
163Dic14—62no—
164Dic14C22×D312yes—
165C7⋊C8—31no—
166C7⋊C8—31no—
167C7⋊C8C22×D312yes—
168C3×#41C2×D711yescharacteristic
169C2×C3×Q8—71no—
170#239—72no—
171#239—141no—
172#239C2×D712yes—
173D3×Q8—72no—
174D3×Q8—141no—
175D3×D4—22no—
176D3×D4C2×D712yes—
177#235—21no—
178#235—71no—
179#235—142no—
180#235C2×D711yescharacteristic
181C2×D12—21no—
182C2×D12C2×D711yescharacteristic
183C3×Q16—72no—
184C3×SD16—72no—
185C3×M16—71no—
186#216—71no—
187#215—71no—
188#215—72no—
189#215—72no—
190#214—71no—
191#213—71no—
192C2×C3⋊C8—71no—
193D24—72no—
194#204—72no—
195ΓL(1,𝔽25)—72no—
196D3×C8—72no—
197C2×C3×C7—21no—
198C2×C3×C7—41no—
199C2×C3×C7C22×D411yescharacteristic
200D21—41no—
201D3×C7—22no—
202D3×C7—42no—
2032−1+4—211no—
204H(5,𝔽2)—31no—
205#137—71no—
206#137—212no—
207AGL(1,ℤ/8)—211no—
208AGL(1,ℤ/8)—212no—
209#135—212no—
210#135—211no—
211C2×SD16—212no—
212C2×SD16—211no—
213#131—211no—
214C22×C7—21no—
215C22×C7—32no—
216C22×C7—62no—
217C22×C7—241no—
218C22×C7—242no—
219C22×C7D3×D411yescharacteristic
220C2×D7—61no—
221C4×C7—21no—
222C4×C7—32no—
223C4×C7D3×D411yescharacteristic
224C4×C7C23×D311yescharacteristic
225Dic7—22no—
226Dic7—61no—
227C22×D3—21no—
228C22×D3—22no—
229C3×Q8—72no—
230C3×Q8—141no—
231C3×Q8C22×D711yescharacteristic
232C3×D4—21no—
233C3×D4C22×D711yescharacteristic
234C2×C3×C4—21no—
235C2×C3×C4—141no—
236C2×C3×C4C22×D711yescharacteristic
237#67—22no—
238#67—141no—
239#67C22⋊D712yes—
240D12—22no—
241D12—21no—
242D12—71no—
243D12—142no—
244D12C22⋊D711yescharacteristic
245D12C22×D712yes—
246C4×D3—22no—
247C4×D3—142no—
248C4×D3—141no—
249C4×D3—142no—
250C4×D3C22×D712yes—
251Dic6—71no—
252Dic6—142no—
253Dic6C22⋊D711yescharacteristic
254C3×C8—72no—
255C3⋊C8—71no—
256C3⋊C8—71no—
257C3×C7#57711yescharacteristic, complemented, Hall
258#41—31no—
259#41—32no—
260#41—61no—
261#41—212no—
262#41—211no—
263#41—422no—
264#41—421no—
265#41D3×D711yescharacteristic
266C2×Q8—71no—
267C2×Q8—212no—
268C2×Q8—421no—
269C2×D4—32no—
270C2×D4—31no—
271C2×D4—61no—
272C2×D4—62no—
273Q16—72no—
274Q16—211no—
275SD16—72no—
276SD16—212no—
277SD16—212no—
278SD16—212no—
279SD16—211no—
280SD16—211no—
281D8—212no—
282D8—211no—
283M16—71no—
284M16—212no—
285C2×C8—212no—
286C2×C8—211no—
287C2×C7—21no—
288C2×C7—41no—
289C2×C7—62no—
290C2×C7—122no—
291C2×C7C2×D3×D411yescharacteristic
292D7—121no—
293C22×C3—21no—
294C22×C3C2×C22⋊D711yescharacteristic
295C2×D3—22no—
296C2×D3—82no—
297C2×D3—81no—
298C2×D3—141no—
299C2×D3C2×C22⋊D712yes—
300C3×C4—21no—
301C3×C4—142no—
302C3×C4C2×C22⋊D711yescharacteristic
303C3×C4C23×D711yescharacteristic
304Dic3—141no—
305Dic3C2×C22⋊D712yes—
306C23—62no—
307C23—61no—
308Q8—31no—
309Q8—72no—
310Q8—141no—
311Q8—211no—
312Q8—422no—
313Q8C2×D3×D711yescharacteristic
314D4—21no—
315D4—32no—
316D4—31no—
317D4—32no—
318D4—62no—
319D4—62no—
320D4—61no—
321D4—211no—
322D4—422no—
323D4—421no—
324D4C2×D3×D711yescharacteristic
325C2×C4—21no—
326C2×C4—32no—
327C2×C4—62no—
328C2×C4—141no—
329C2×C4—422no—
330C2×C4—422no—
331C2×C4—421no—
332C2×C4C2×D3×D711yescharacteristic
333C8—72no—
334C8—211no—
335C8—211no—
336C7#577811yescharacteristic, complemented, Sylow
337C2×C3—21no—
338C2×C3—41no—
339C2×C3C22×C22⋊D711yescharacteristic
340D3—22no—
341D3—42no—
342D3—281no—
343C22—21no—
344C22—32no—
345C22—62no—
346C22—241no—
347C22—242no—
348C22—421no—
349C22D3×C22⋊D711yescharacteristic
350C4—21no—
351C4—32no—
352C4—142no—
353C4—421no—
354C4D3×C22⋊D711yescharacteristic
355C4C22×D3×D711yescharacteristic
356C3#9052711yescharacteristic, complemented, Sylow
357C2—21no—
358C2—41no—
359C2—62no—
360C2—122no—
361C2—841no—
362C2C2×D3×C22⋊D711yescharacteristic
subgroupG/H
centerC2C2×D3×C22⋊D7
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7C22×D4
hypercenter#41D3×D7
hypocenterC3×C7#577
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C7×#1009C2semidirect
2. #10621551C2semidirect
3. #10621433C2semidirect
4. C3×#6514C2semidirect
5. #10621081C2semidirect
6. #10621072C2semidirect
7. #10621058C2semidirect
8. D3×#1142C2semidirect
9. C2×#66564C2semidirect
10. #10620866C2semidirect
11. #10620850C2semidirect
14. #66728C22semidirect
22. C2×C3×Dic14C22semidirect
25. #66673C22semidirect
27. D3×Dic14C22semidirect
28. #66635C22semidirect
29. #66634C22semidirect
30. #66633C22semidirect
31. #66632C22semidirect
32. C2×C3⋊(C7⋊C8)C22semidirect
33. C3×#1144C22semidirect
34. C3×#1142C22semidirect
35. C3×#1136C22semidirect
36. #66567C22semidirect
37. #66564C22semidirect
38. #66564C22semidirect
39. #66562C22semidirect
40. #66558C22semidirect
41. D3×C7⋊C8C22semidirect
45. #6514D3semidirect
98. C2×Dic14C2×D3semidirect
100. #1144C2×D3semidirect
106. #1142C2×D3semidirect
110. #1136C2×D3semidirect
113. #1009D7semidirect
257. C3×C7#577semidirect
336. C7#5778semidirect
356. C3#90527semidirect

Maximal subgroups

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

subgroupindex
C7×#10092
#106215512
#106214332
C3×#65142
#106210812
#106210722
#106210582
D3×#11422
C2×#665642
#106208662
#106208502
#905273
#57787