Finite Group Databasebeta

← #11738106

Subgroups — #11738106

Order 896.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6477C211yesmaximal, characteristic, complemented
2C2×#6464C211yesmaximal, characteristic, complemented
3#90320C211yesmaximal, characteristic, complemented
4#90298C211yesmaximal, characteristic, complemented
5C7×#465C211yesmaximal, characteristic, complemented
6#89946C211yesmaximal, characteristic, complemented
7#89887C211yesmaximal, characteristic, complemented
8#89821C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89637C211yesmaximal, characteristic, complemented
11#89633C211yesmaximal, characteristic, complemented
12#89579C211yesmaximal, characteristic, complemented
13#89576C211yesmaximal, characteristic, complemented
14#89547C211yesmaximal, characteristic, complemented
15D7×#102C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C22×C7×C8C2211yescharacteristic, complemented
21C7×C22⋊Q8C2211yescharacteristic, complemented
22C7×#121C2211yescharacteristic
23#6509C2211yescharacteristic, complemented
24C2×#1143C2211yescharacteristic
25#6503—21no—
26#6501C2211yescharacteristic
27#6500—21no—
28C2×#1141C2211yescharacteristic
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic, complemented
31C2×#1136C2211yescharacteristic, complemented
32#6477—24no—
33C2×D56C2211yescharacteristic
34#6464—24no—
35C2×#1131C2211yescharacteristic
36C2×#1131C2211yescharacteristic, complemented
37C2×C8×D7C2211yescharacteristic
38C2×C8×D7C2211yescharacteristic, complemented
39#6459—21no—
40#6458—21no—
41#6458C2211yescharacteristic
42#6458C2211yescharacteristic, complemented
43#6455C2211yescharacteristic
44D7×C4⋊C4C2211yescharacteristic
45#6448—21no—
46#6447—21no—
47#6447—21no—
48#6445—21no—
49C7×#106C2211yescharacteristic, complemented
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic
52#6409C2211yescharacteristic, complemented
53#6406C2211yescharacteristic
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic, complemented
56#6392C2211yescharacteristic, complemented
57#6383C2211yescharacteristic
58#6383C2211yescharacteristic, complemented
59#6382C2211yescharacteristic, complemented
60#6381C2211yescharacteristic, complemented
61#3062—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7D411yescharacteristic
72C2×C22⋊D7D411yescharacteristic
73#1161—22no—
74#1161—22no—
75D4×D7—22no—
76D4×D7—22no—
77#1157—22no—
78#1157—22no—
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28D411yescharacteristic
83C2×D28C2311yescharacteristic
84C2×D28C2311yescharacteristic
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7C2311yescharacteristic
88C2×C4×D7C2311yescharacteristic
89C2×Dic14D411yescharacteristic
90C7×SD16—22no—
91C2×C7×C8—22no—
92C2×C7×C8C2311yescharacteristic
93C2×C7×C8C2311yescharacteristic
94C7×C4⋊C4—21no—
95C7×C4⋊C4C2311yescharacteristic
96C7×C4⋊C4C2311yescharacteristic
97C7×C22⋊C4—21no—
98C7×C22⋊C4—21no—
99C22⋊Dic7—21no—
100#1143—22no—
101#1141—22no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108C4⋊Dic7C2311yescharacteristic
109C4⋊Dic7C2311yescharacteristic
110Dic7⋊C4—21no—
111Dic7⋊C4—21no—
112C4×Dic7—21no—
113#1136—22no—
114C2×C7⋊C8C2311yescharacteristic
115C2×C7⋊C8C2311yescharacteristic
116D56—22no—
117#1131—22no—
118#1131D414yescomplemented
119C8×D7—22no—
120C8×D7—22no—
121C2×AGL(1,ℤ/8)—71no—
122C2×#131—71no—
123#537—71no—
124#535—71no—
125#469—71no—
126#469—71no—
127#468—71no—
128#468—71no—
129#466—71no—
130#466—71no—
131#465—71no—
132#465D711yescharacteristic, complemented
133#428—71no—
134#416—71no—
135C2×#102—71no—
136C23×C7—21no—
137C23×C7C2×D411yescharacteristic
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—42no—
141C22×D7—81no—
142C22×D7—81no—
143C22×D7C2×D411yescharacteristic
144C22×D7C2×D411yescharacteristic
145C7×Q8—21no—
146C7×Q8—21no—
147C7×D4—21no—
148C7×D4—21no—
149C7×D4—81no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7C2×D411yescharacteristic
156C2×C4×C7C2411yescharacteristic
157C22⋊D7—22no—
158C22⋊D7—22no—
159C22⋊D7—42no—
160C22⋊D7—81no—
161C22⋊D7—81no—
162C22⋊D7—81no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7C2×D411yescharacteristic
166C2×Dic7C2×D411yescharacteristic
167D28—22no—
168D28—21no—
169D28—21no—
170D28—21no—
171D28—21no—
172D28—42no—
173D28—81no—
174D28—81no—
175C4×D7—21no—
176C4×D7—21no—
177C4×D7—42no—
178C4×D7—81no—
179C4×D7C2×D412yes—
180C4×D7#4112yes—
181Dic14—22no—
182C7×C8—21no—
183C7×C8C2×D412yes—
184C7⋊C8—21no—
185C7⋊C8C2×D412yes—
186C2×#41—71no—
187C2×#41—71no—
188C22×D4—71no—
189AGL(1,ℤ/8)—144no—
190C2×SD16—71no—
191C2×SD16C2×D711yescharacteristic
192C2×D8—71no—
193C2×D8—71no—
194#131—144no—
195C2×M16—71no—
196C2×M16—71no—
197C2×M16—71no—
198C22×C8—71no—
199C22×C8—71no—
200C22×C8C2×D711yescharacteristic, complemented
201#127—141no—
202#124—141no—
203C22⋊Q8—71no—
204C22⋊Q8—141no—
205C22⋊Q8C2×D711yescharacteristic, complemented
206#121—71no—
207#121—71no—
208#121—71no—
209#121—71no—
210#121—71no—
211#121—141no—
212#121—141no—
213#121—141no—
214#121C2×D711yescharacteristic
215#120—141no—
216#120—141no—
217#117—71no—
218C2×C4⋊C4—71no—
219#107—71no—
220#107—71no—
221#106—71no—
222#106C2×D711yescharacteristic, complemented
223#103—71no—
224#103C2×D711yescharacteristic, complemented
225#102—71no—
226#102—71no—
227#102—71no—
228#102—71no—
229#102—71no—
230#102C2×D711yescharacteristic
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7C22×D411yescharacteristic
238C2×D7—21no—
239C2×D7—21no—
240C2×D7—21no—
241C2×D7—21no—
242C2×D7—21no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—84no—
250C2×D7#12112yes—
251C4×C7—21no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—41no—
255C4×C7C22×D411yescharacteristic
256C4×C7C2×#4111yescharacteristic
257Dic7—21no—
258Dic7—41no—
259Dic7—41no—
260Dic7#12112yes—
261C24—141no—
262#41—142no—
263#41—142no—
264#41—142no—
265#41—142no—
266C2×Q8—71no—
267C2×Q8C22×D711yescharacteristic
268C2×D4—21no—
269C2×D4—71no—
270C2×D4—71no—
271C2×D4—71no—
272C2×D4—71no—
273C2×D4—71no—
274C2×D4—141no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—142no—
278C2×D4—142no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4—141no—
283C2×D4C22×D711yescharacteristic
284C22×C4—71no—
285C22×C4—71no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289SD16—22no—
290SD16—142no—
291D8—142no—
292D8—142no—
293M16—74no—
294M16—142no—
295M16—142no—
296C2×C8—22no—
297C2×C8—71no—
298C2×C8—71no—
299C2×C8—142no—
300C2×C8—142no—
301C2×C8C22×D711yescharacteristic
302C2×C8C22×D711yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—141no—
307C4⋊C4—141no—
308C4⋊C4C22×D711yescharacteristic
309C4⋊C4C22×D711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—141no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C42—141no—
320C2×C7—41no—
321C2×C7—81no—
322C2×C7C2×#12111yescharacteristic
323C2×C7#57611yescharacteristic
324C2×C7#57711yescharacteristic
325D7—22no—
326D7—41no—
327D7—81no—
328D7—81no—
329C23—21no—
330C23—71no—
331C23—71no—
332C23—141no—
333C23—141no—
334C23—282no—
335C23—561no—
336C23—561no—
337C23D4×D711yescharacteristic
338Q8—21no—
339Q8—21no—
340Q8—142no—
341D4—21no—
342D4—21no—
343D4—81no—
344D4—142no—
345D4—142no—
346D4—142no—
347D4—141no—
348D4—141no—
349D4—141no—
350D4—141no—
351D4—282no—
352D4—282no—
353D4—561no—
354D4—561no—
355D4—561no—
356D4—561no—
357D4—561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—71no—
364C2×C4—72no—
365C2×C4—72no—
366C2×C4—71no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—282no—
372C2×C4—561no—
373C2×C4D4×D711yescharacteristic
374C2×C4C23×D711yescharacteristic
375C8—21no—
376C8—72no—
377C8—141no—
378C8D4×D712yes—
379C7#306211yescharacteristic, complemented, Sylow
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—72no—
387C22—141no—
388C22—141no—
389C22—141no—
390C22—141no—
391C22—141no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—281no—
397C22—281no—
398C22—564no—
399C22C2×D4×D711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—72no—
405C4—141no—
406C4—281no—
407C4—281no—
408C4C2×D4×D711yescharacteristic
409C4D7×#4111yescharacteristic
410C2—41no—
411C2—81no—
412C2—142no—
413C2—281no—
414C2—561no—
415C2—561no—
416C2D7×#12111yescharacteristic
417C2#9045311yescharacteristic
418C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
hypercenter#465D7
hypocenterC7#3062
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#6477C2semidirect
2. C2×#6464C2semidirect
3. #90320C2semidirect
4. #90298C2semidirect
5. C7×#465C2semidirect
6. #89946C2semidirect
7. #89887C2semidirect
8. #89821C2semidirect
9. #89811C2semidirect
10. #89637C2semidirect
11. #89633C2semidirect
12. #89579C2semidirect
13. #89576C2semidirect
14. #89547C2semidirect
15. D7×#102C2semidirect
20. C22×C7×C8C22semidirect
21. C7×C22⋊Q8C22semidirect
23. #6509C22semidirect
30. #6493C22semidirect
31. C2×#1136C22semidirect
36. C2×#1131C22semidirect
38. C2×C8×D7C22semidirect
42. #6458C22semidirect
49. C7×#106C22semidirect
50. C7×#103C22semidirect
52. #6409C22semidirect
55. #6395C22semidirect
56. #6392C22semidirect
58. #6383C22semidirect
59. #6382C22semidirect
60. #6381C22semidirect
118. #1131D4semidirect
132. #465D7semidirect
200. C22×C8C2×D7semidirect
205. C22⋊Q8C2×D7semidirect
222. #106C2×D7semidirect
224. #103C2×D7semidirect
379. C7#3062semidirect

Maximal subgroups

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

subgroupindex
C2×#64772
C2×#64642
#903202
#902982
C7×#4652
#899462
#898872
#898212
#898112
#896372
#896332
#895792
#895762
#895472
D7×#1022
#30627