Finite Group Databasebeta

← #11739477

Subgroups — #11739477

Order 896.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#90462C211yesmaximal, characteristic, complemented
2#90460C211yesmaximal, characteristic, complemented
3C7×#508C211yesmaximal, characteristic, complemented
4#89967C211yesmaximal, characteristic, complemented
5#89965C211yesmaximal, characteristic, complemented
6#89690C212yesmaximal, complemented
7#89688C212yesmaximal, complemented
8D7×SD32C212yesmaximal, complemented
9#89683C212yesmaximal, complemented
10C2×#6374C211yesmaximal, characteristic, complemented
11#89673C211yesmaximal, characteristic, complemented
12#6554—21no—
13#6553—21no—
14C7×#135C2211yescharacteristic
15C7×#135C2211yescharacteristic, complemented
16#6514—21no—
17#6512—21no—
18#6482C2212yes—
19D7×Q16C2212yescomplemented
20#6478—22no—
21#6477—22no—
22#6475C2212yescomplemented
23D7×D8C2212yes—
24C2×Dic28C2211yescharacteristic, complemented
25C2×D56C2211yescharacteristic
26#6464C2211yescharacteristic
27C7×Q32C2211yescharacteristic, complemented
28C7×SD32C2211yescharacteristic, complemented
29C7×SD32C2211yescharacteristic, complemented
30C7×D16C2211yescharacteristic, complemented
31C2×C7×C16C2211yescharacteristic, complemented
32#6403C2212yescomplemented
33#6402C2212yescomplemented
34#6401C2212yescomplemented
35#6400C2212yescomplemented
36#6386C2211yescharacteristic, complemented
37#6374C2212yescomplemented
38#6374C2212yescomplemented
39#6372C2212yescomplemented
40D7×C16C2212yescomplemented
41#3416—71nomaximal, Sylow
42C7×#41—21no—
43C7×#41—21no—
44#1161—22no—
45Q8×D7—22no—
46#1159—22no—
47#1159—42no—
48D4×D7—22no—
49D4×D7—42no—
50#1157—41no—
51#1157—41no—
52#1157D411yescharacteristic
53C2×D28—21no—
54C2×D28—41no—
55C2×Dic14—21no—
56C2×Dic14—41no—
57C7×Q16C2311yescharacteristic
58C7×Q16C2311yescharacteristic
59C7×SD16—21no—
60C7×SD16—21no—
61C7×D8C2311yescharacteristic
62C7×D8C2311yescharacteristic
63C2×C7×C8C2311yescharacteristic
64#1144—22no—
65#1143—22no—
66#1142—22no—
67#1141—22no—
68#1136D411yescharacteristic
69Dic28—21no—
70Dic28C2312yes—
71D56—21no—
72D56C2312yes—
73#1131D412yes—
74C8×D7C2312yes—
75C7×C16C2311yescharacteristic
76C7×C16C2311yescharacteristic
77C7⋊C16C2312yes—
78#578—71no—
79#576—71no—
80#510—71no—
81#510—72no—
82#509—71no—
83#509—72no—
84#508—72no—
85#508D711yescharacteristic, complemented
86C2×SD32—72no—
87C2×SD32—71no—
88#504—71no—
89C22×D7—41no—
90C22×D7—42no—
91C7×Q8—21no—
92C7×Q8—21no—
93C7×D4—21no—
94C7×D4—21no—
95C7×D4—41no—
96C7×D4—41no—
97C2×C4×C7—41no—
98C2×C4×C7—41no—
99C2×C4×C7C2×D411yescharacteristic
100C22⋊D7—42no—
101C22⋊D7—42no—
102C22⋊D7D812yes—
103C2×Dic7—41no—
104C2×Dic7—42no—
105D28—22no—
106D28—42no—
107D28—41no—
108D28D811yescharacteristic
109C4×D7—42no—
110C4×D7—42no—
111C4×D7C2×D412yes—
112Dic14—22no—
113Dic14—41no—
114Dic14—42no—
115Dic14D811yescharacteristic
116C7×C8C2×D411yescharacteristic
117C7×C8C2411yescharacteristic
118C7⋊C8C2×D412yes—
1192−1+4—141no—
120H(5,𝔽2)—141no—
121#137—142no—
122#137—141no—
123AGL(1,ℤ/8)—142no—
124AGL(1,ℤ/8)—141no—
125#135—72no—
126#135—72no—
127#135C2×D711yescharacteristic
128#135C2×D711yescharacteristic, complemented
129C2×Q16—72no—
130C2×Q16—71no—
131C2×D8—72no—
132C2×D8—71no—
133#131—71no—
134Q32—72no—
135Q32C2×D711yescharacteristic, complemented
136SD32—72no—
137SD32—72no—
138SD32—72no—
139SD32—72no—
140SD32C2×D711yescharacteristic, complemented
141SD32C2×D711yescharacteristic, complemented
142D16—72no—
143D16C2×D711yescharacteristic, complemented
144M32—71no—
145M32—72no—
146C2×C16—72no—
147C2×C16C2×D711yescharacteristic, complemented
148C22×C7—41no—
149C22×C7—41no—
150C22×C7C2×D811yescharacteristic
151C2×D7—42no—
152C2×D7—162no—
153C2×D7—161no—
154C2×D7C2×D812yes—
155C4×C7—41no—
156C4×C7—41no—
157C4×C7C2×D811yescharacteristic
158C4×C7C22×D411yescharacteristic
159Dic7—42no—
160Dic7C2×D812yes—
161#41—21no—
162#41—21no—
163#41—71no—
164#41—142no—
165#41—142no—
166#41—281no—
167#41—281no—
168#41—282no—
169C2×Q8—142no—
170C2×Q8—141no—
171C2×Q8—281no—
172C2×D4—142no—
173C2×D4—141no—
174C2×D4—281no—
175C2×D4—282no—
176Q16—72no—
177Q16—142no—
178Q16—141no—
179Q16C22×D711yescharacteristic
180Q16C22×D711yescharacteristic
181SD16—21no—
182SD16—21no—
183SD16—142no—
184SD16—142no—
185D8—72no—
186D8—142no—
187D8—141no—
188D8C22×D711yescharacteristic
189D8C22×D711yescharacteristic
190M16—71no—
191M16—72no—
192C2×C8—72no—
193C2×C8C22×D711yescharacteristic
194C16—72no—
195C16C22×D711yescharacteristic
196C16C22×D711yescharacteristic
197C2×C7—21no—
198C2×C7—81no—
199C2×C7—81no—
200C2×C7C22×D811yescharacteristic
201D7—22no—
202D7—82no—
203C23—282no—
204C23—281no—
205Q8—21no—
206Q8—21no—
207Q8—71no—
208Q8—142no—
209Q8—282no—
210Q8—281no—
211D4—21no—
212D4—21no—
213D4—41no—
214D4—41no—
215D4—72no—
216D4—71no—
217D4—142no—
218D4—282no—
219D4—282no—
220D4—282no—
221D4—281no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—72no—
225C2×C4—282no—
226C2×C4—282no—
227C2×C4—282no—
228C2×C4—281no—
229C2×C4D4×D711yescharacteristic
230C8—72no—
231C8D4×D711yescharacteristic
232C8C23×D711yescharacteristic
233C7#341611yescharacteristic, complemented, Sylow
234C22—41no—
235C22—41no—
236C22—72no—
237C22—282no—
238C22—1121no—
239C22—1122no—
240C22D7×D811yescharacteristic
241C4—41no—
242C4—41no—
243C4—72no—
244C4—282no—
245C4D7×D811yescharacteristic
246C4C2×D4×D711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—142no—
251C2—562no—
252C2C2×D7×D811yescharacteristic
subgroupG/H
centerC2C2×D7×D8
commutator, cosocleC7×C8C24
FrattiniC8C23×D7
FittingC7×#508C2
socleC2×C7C22×D8
hypercenter#508D7
hypocenterC7#3416
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #90462C2semidirect
2. #90460C2semidirect
3. C7×#508C2semidirect
4. #89967C2semidirect
5. #89965C2semidirect
6. #89690C2semidirect
7. #89688C2semidirect
8. D7×SD32C2semidirect
9. #89683C2semidirect
10. C2×#6374C2semidirect
11. #89673C2semidirect
15. C7×#135C22semidirect
19. D7×Q16C22semidirect
22. #6475C22semidirect
24. C2×Dic28C22semidirect
27. C7×Q32C22semidirect
28. C7×SD32C22semidirect
29. C7×SD32C22semidirect
30. C7×D16C22semidirect
31. C2×C7×C16C22semidirect
32. #6403C22semidirect
33. #6402C22semidirect
34. #6401C22semidirect
35. #6400C22semidirect
36. #6386C22semidirect
37. #6374C22semidirect
38. #6374C22semidirect
39. #6372C22semidirect
40. D7×C16C22semidirect
85. #508D7semidirect
128. #135C2×D7semidirect
135. Q32C2×D7semidirect
140. SD32C2×D7semidirect
141. SD32C2×D7semidirect
143. D16C2×D7semidirect
147. C2×C16C2×D7semidirect
233. C7#3416semidirect

Maximal subgroups

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

subgroupindex
#904622
#904602
C7×#5082
#899672
#899652
#896902
#896882
D7×SD322
#896832
C2×#63742
#896732
#34167