Finite Group Databasebeta

← #11741455

Subgroups — #11741455

Order 896.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C7×#576C211yesmaximal, characteristic, complemented
2#90526C212yesmaximal, complemented
3#90460C211yesmaximal, characteristic, complemented
4#90013C212yesmaximal, complemented
5#89971C212yesmaximal, complemented
6#89970C212yesmaximal, complemented
7#89968C211yesmaximal, characteristic, complemented
8#89932C211yesmaximal, characteristic, complemented
9#89923C211yesmaximal, characteristic, complemented
10#89916C211yesmaximal, characteristic, complemented
11#89914C211yesmaximal, characteristic, complemented
12C7×H(5,𝔽2)C2212yescomplemented
13#6553C2212yescomplemented
14C7×AGL(1,ℤ/8)—22no—
15C7×AGL(1,ℤ/8)C2212yescomplemented
16C7×#135—21no—
17C7×#135C2211yescharacteristic
18C2×C7×D8—21no—
19C2×C7×D8C2211yescharacteristic, complemented
20C7×#131C2211yescharacteristic, complemented
21#6513—22no—
22#6513C2211yescharacteristic
23#6512—22no—
24#6512C2212yescomplemented
25#6512C2212yescomplemented
26#6511C2212yescomplemented
27#6503C2212yescomplemented
28#6495—22no—
29#6495C2212yescomplemented
30C2×#1141—22no—
31C2×#1141C2212yescomplemented
32#6482—21no—
33#6477—22no—
34D7×D8—21no—
35C2×D56C2211yescharacteristic, complemented
36#6464C2211yescharacteristic, complemented
37#6411C2211yescharacteristic, complemented
38#6411C2212yescomplemented
39#6407C2212yescomplemented
40#6399C2211yescharacteristic, complemented
41#6397C2212yescomplemented
42#6396C2211yescharacteristic, complemented
43#6393C2211yescharacteristic, complemented
44C8×Dic7C2211yescharacteristic, complemented
45#3292—71nomaximal, Sylow
46C7×#41—22no—
47C7×#41—22no—
48C7×#41C2312yes—
49C7×#41C2311yescharacteristic
50C2×C7×D4—22no—
51C2×C7×D4—22no—
52C2×C7×D4—42no—
53C2×C7×D4C2312yes—
54C2×C22⋊D7—42no—
55#1161—22no—
56D4×D7—22no—
57D4×D7—42no—
58#1157—22no—
59#1157C2311yescharacteristic
60C2×D28—22no—
61C2×D28C2312yes—
62C7×Q16D411yescharacteristic, complemented
63C7×SD16—22no—
64C7×SD16D412yescomplemented
65C7×D8—22no—
66C7×D8—21no—
67C7×D8—21no—
68C7×D8D411yescharacteristic, complemented
69C7×M16—21no—
70C7×M16C2311yescharacteristic
71C2×C7×C8—21no—
72C2×C7×C8C2311yescharacteristic
73#1144D411yescharacteristic, complemented
74#1143—22no—
75#1143—22no—
76#1143D411yescharacteristic, complemented
77#1142—22no—
78#1142D411yescharacteristic, complemented
79#1141—22no—
80#1141—22no—
81#1141—22no—
82#1141—22no—
83#1141—22no—
84#1141D411yescharacteristic, complemented
85C4×Dic7C2311yescharacteristic
86#1136—22no—
87#1136C2311yescharacteristic
88#1136C2312yes—
89C2×C7⋊C8—22no—
90C2×C7⋊C8C2311yescharacteristic
91D56—21no—
92D56—21no—
93#1131—21no—
94C8×D7—21no—
95#576—72no—
96#576—71no—
97#576D711yescharacteristic, complemented
98#493—71no—
99#472—72no—
100#472—71no—
101#472—71no—
102#453—72no—
103#453—72no—
104#443—71no—
105#443—71no—
106C23×C7—22no—
107C23×C7—42no—
108C22×D7—22no—
109C22×D7—42no—
110C7×Q8C2×D411yescharacteristic
111C7×Q8C2×D412yes—
112C7×D4—22no—
113C7×D4—21no—
114C7×D4—22no—
115C7×D4—22no—
116C7×D4—42no—
117C7×D4—42no—
118C7×D4C2×D411yescharacteristic
119C7×D4C2×D412yes—
120C2×C4×C7—22no—
121C2×C4×C7—21no—
122C2×C4×C7—42no—
123C2×C4×C7C2411yescharacteristic
124C22⋊D7—21no—
125C22⋊D7—42no—
126C22⋊D7—84no—
127C2×Dic7—21no—
128D28—22no—
129D28—22no—
130D28—42no—
131D28C2×D411yescharacteristic
132C4×D7—21no—
133C4×D7—42no—
134Dic14C2×D411yescharacteristic
135C7×C8—21no—
136C7×C8C2×D411yescharacteristic
137C7×C8C2×D411yescharacteristic
138C7⋊C8—21no—
139C7⋊C8—22no—
140C7⋊C8C2×D411yescharacteristic
141C7⋊C8C2×D411yescharacteristic
142H(5,𝔽2)—72no—
143H(5,𝔽2)C2×D712yescomplemented
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—72no—
146AGL(1,ℤ/8)—72no—
147AGL(1,ℤ/8)—72no—
148AGL(1,ℤ/8)—142no—
149AGL(1,ℤ/8)—142no—
150AGL(1,ℤ/8)—142no—
151AGL(1,ℤ/8)C2×D712yescomplemented
152#135—21no—
153#135—71no—
154#135—142no—
155#135—141no—
156#135C2×D711yescharacteristic
157C2×D8—21no—
158C2×D8—72no—
159C2×D8—71no—
160C2×D8—142no—
161C2×D8—141no—
162C2×D8C2×D711yescharacteristic, complemented
163#131—72no—
164#131—71no—
165#131C2×D711yescharacteristic, complemented
166#127—72no—
167#108—71no—
168#108—71no—
169#104—71no—
170#104—72no—
171#104—71no—
172#100—72no—
173#100—72no—
174C4×C8—71no—
175C22×C7—22no—
176C22×C7—21no—
177C22×C7—42no—
178C22×C7—82no—
179C22×C7—162no—
180C22×C7H(5,𝔽2)11yescharacteristic
181C2×D7—21no—
182C2×D7—42no—
183C2×D7—82no—
184C2×D7—162no—
185C4×C7—22no—
186C4×C7—21no—
187C4×C7C22×D411yescharacteristic
188C4×C7C22×D411yescharacteristic
189Dic7—22no—
190Dic7—21no—
191#41—22no—
192#41—22no—
193#41—71no—
194#41—142no—
195#41—142no—
196#41C22×D711yescharacteristic
197#41C22×D712yes—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—42no—
201C2×D4—72no—
202C2×D4—142no—
203C2×D4—142no—
204C2×D4—282no—
205C2×D4—282no—
206C2×D4C22×D712yes—
207Q16—71no—
208Q16C22⋊D711yescharacteristic, complemented
209SD16—22no—
210SD16—71no—
211SD16—71no—
212SD16—142no—
213SD16—142no—
214SD16—142no—
215SD16C22⋊D712yescomplemented
216D8—22no—
217D8—21no—
218D8—21no—
219D8—71no—
220D8—142no—
221D8—142no—
222D8—142no—
223D8—142no—
224D8—142no—
225D8—141no—
226D8—141no—
227D8C22⋊D711yescharacteristic, complemented
228M16—21no—
229M16—72no—
230M16—71no—
231M16—142no—
232M16—141no—
233M16C22×D711yescharacteristic
234C2×C8—21no—
235C2×C8—71no—
236C2×C8—142no—
237C2×C8—141no—
238C2×C8C22×D711yescharacteristic
239C42—71no—
240C2×C7—21no—
241C2×C7—42no—
242C2×C7—41no—
243C2×C7—82no—
244C2×C7D4211yescharacteristic
245D7—41no—
246D7—82no—
247C23—22no—
248C23—42no—
249C23—142no—
250C23—282no—
251Q8—71no—
252Q8D4×D711yescharacteristic
253Q8C2×C22⋊D712yes—
254D4—21no—
255D4—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—42no—
260D4—71no—
261D4—141no—
262D4—142no—
263D4—142no—
264D4—282no—
265D4—282no—
266D4—564no—
267D4D4×D711yescharacteristic
268D4C2×C22⋊D712yes—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—282no—
275C2×C4C23×D711yescharacteristic
276C8—21no—
277C8—71no—
278C8—71no—
279C8—142no—
280C8—141no—
281C8C2×C22⋊D711yescharacteristic
282C8C2×C22⋊D711yescharacteristic
283C7#329211yescharacteristic, complemented, Sylow
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—141no—
289C22—162no—
290C22—282no—
291C22—562no—
292C22—1122no—
293C22#654811yescharacteristic
294C4—21no—
295C4—22no—
296C4—142no—
297C4—141no—
298C4C2×D4×D711yescharacteristic
299C4C22×C22⋊D711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—281no—
305C2—562no—
306C2D4×C22⋊D711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#576C2
socleC2×C7D42
hypercenter#576D7
hypocenterC7#3292
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. C7×#576C2semidirect
2. #90526C2semidirect
3. #90460C2semidirect
4. #90013C2semidirect
5. #89971C2semidirect
6. #89970C2semidirect
7. #89968C2semidirect
8. #89932C2semidirect
9. #89923C2semidirect
10. #89916C2semidirect
11. #89914C2semidirect
12. C7×H(5,𝔽2)C22semidirect
13. #6553C22semidirect
15. C7×AGL(1,ℤ/8)C22semidirect
19. C2×C7×D8C22semidirect
20. C7×#131C22semidirect
24. #6512C22semidirect
25. #6512C22semidirect
26. #6511C22semidirect
27. #6503C22semidirect
29. #6495C22semidirect
31. C2×#1141C22semidirect
35. C2×D56C22semidirect
36. #6464C22semidirect
37. #6411C22semidirect
38. #6411C22semidirect
39. #6407C22semidirect
40. #6399C22semidirect
41. #6397C22semidirect
42. #6396C22semidirect
43. #6393C22semidirect
44. C8×Dic7C22semidirect
62. C7×Q16D4semidirect
64. C7×SD16D4semidirect
68. C7×D8D4semidirect
73. #1144D4semidirect
76. #1143D4semidirect
78. #1142D4semidirect
84. #1141D4semidirect
97. #576D7semidirect
143. H(5,𝔽2)C2×D7semidirect
151. AGL(1,ℤ/8)C2×D7semidirect
162. C2×D8C2×D7semidirect
165. #131C2×D7semidirect
208. Q16C22⋊D7semidirect
215. SD16C22⋊D7semidirect
227. D8C22⋊D7semidirect
283. C7#3292semidirect

Maximal subgroups

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

subgroupindex
C7×#5762
#905262
#904602
#900132
#899712
#899702
#899682
#899322
#899232
#899162
#899142
#32927