Finite Group Databasebeta

← #11737726

Subgroups — #11737726

Order 896.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#90527C212yesmaximal, complemented
2#90466C212yesmaximal, complemented
3C7×#453C211yesmaximal, characteristic, complemented
4#89972C212yesmaximal, complemented
5#89849C211yesmaximal, characteristic, complemented
6#89848C211yesmaximal, characteristic, complemented
7#89598C212yesmaximal, complemented
8#89596C212yesmaximal, complemented
9#89520C212yesmaximal, complemented
10C7×H(5,𝔽2)C2211yescharacteristic, complemented
11#6554C2212yes—
12#6548C2211yescharacteristic, complemented
13C7×AGL(1,ℤ/8)C2212yes—
14C7×#127C2211yescharacteristic
15#6514—22no—
16#6514C2212yescomplemented
17#6513—22no—
18#6513C2212yes—
19#6511C2212yescomplemented
20C2×#1142—22no—
21C2×#1142C2212yescomplemented
22C2×#1142C2212yescomplemented
23#6495—22no—
24#6495C2212yes—
25#6478—22no—
26D7×SD16—22no—
27#6475—22no—
28#6474—22no—
29#6472C2212yescomplemented
30#6470C2212yescomplemented
31#6432C2211yescharacteristic, complemented
32C7×#104C2212yescomplemented
33C7×#100C2211yescharacteristic, complemented
34#6407C2211yescharacteristic, complemented
35#6398C2212yescomplemented
36#6396C2212yescomplemented
37#6379C2212yescomplemented
38C4×C7⋊C8C2211yescharacteristic, complemented
39#3291—71nomaximal, Sylow
40C7×#41—22no—
41C7×#41C2312yes—
42C2×C7×D4—22no—
43C2×C7×D4—41no—
44C2×C7×D4—41no—
45C2×C7×D4C2311yescharacteristic
46C2×C7×D4C2311yescharacteristic
47C2×C22⋊D7—41no—
48Q8×D7—22no—
49#1159—22no—
50#1159—22no—
51#1159—42no—
52D4×D7—22no—
53#1157—22no—
54#1157C2312yes—
55C2×Dic14—22no—
56C2×Dic14C2312yes—
57C7×SD16—22no—
58C7×D8—22no—
59C7×M16C2312yes—
60C42×C7C2311yescharacteristic
61#1144—22no—
62#1144D412yescomplemented
63#1143D412yescomplemented
64#1142—22no—
65#1142—22no—
66#1142—22no—
67#1142—22no—
68#1142—22no—
69#1142—22no—
70#1142D412yescomplemented
71#1141—22no—
72#1141D412yescomplemented
73C4⋊Dic7—41no—
74#1136—22no—
75#1136C2312yes—
76C2×C7⋊C8—22no—
77C2×C7⋊C8C2312yes—
78Dic28—22no—
79#1132—22no—
80#1131—22no—
81C8×D7—22no—
82#577—72no—
83#577—72no—
84#492—71no—
85#471—72no—
86#471—72no—
87#456—72no—
88#453—71no—
89#453D711yescharacteristic, complemented
90#443—72no—
91C23×C7—21no—
92C23×C7—21no—
93C23×C7—41no—
94C22×D7—41no—
95C7×Q8C2×D412yes—
96C7×D4—22no—
97C7×D4—22no—
98C7×D4—22no—
99C7×D4—42no—
100C7×D4—82no—
101C7×D4C2×D412yes—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—41no—
105C2×C4×C7C2411yescharacteristic
106C22⋊D7—22no—
107C22⋊D7—42no—
108C22⋊D7—42no—
109C2×Dic7—22no—
110C2×Dic7—41no—
111C2×Dic7—42no—
112D28C2×D412yes—
113C4×D7—22no—
114C4×D7—42no—
115Dic14—22no—
116Dic14—22no—
117Dic14—42no—
118Dic14C2×D412yes—
119C7×C8—22no—
120C7⋊C8—22no—
121C7⋊C8C2×D412yes—
122C7⋊C8C2×D412yes—
1232−1+4—72no—
124H(5,𝔽2)—71no—
125H(5,𝔽2)C2×D711yescharacteristic, complemented
126#137—72no—
127#137—72no—
128#137—142no—
129#137—142no—
130AGL(1,ℤ/8)—72no—
131AGL(1,ℤ/8)—142no—
132AGL(1,ℤ/8)—142no—
133AGL(1,ℤ/8)C2×D712yes—
134#135—72no—
135#135—142no—
136#135—142no—
137C2×SD16—72no—
138C2×SD16—72no—
139C2×SD16—142no—
140C2×SD16—142no—
141#131—72no—
142#131—72no—
143C4⋊Q8—71no—
144#127C2×D711yescharacteristic
145#108—72no—
146#104—72no—
147#104C2×D712yescomplemented
148#101—72no—
149#100—71no—
150#100C2×D711yescharacteristic, complemented
151C4×C8—71no—
152C22×C7—22no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7—81no—
156C22×C7—81no—
157C22×C7—161no—
158C22×C7H(5,𝔽2)11yescharacteristic
159C2×D7—22no—
160C2×D7—161no—
161C4×C7—22no—
162C4×C7—22no—
163C4×C7C22×D412yes—
164Dic7—22no—
165Dic7—42no—
166#41—22no—
167#41—72no—
168#41—142no—
169#41—142no—
170#41—142no—
171#41—282no—
172#41C22×D712yes—
173C2×Q8—72no—
174C2×Q8—142no—
175C2×Q8—142no—
176C2×D4—22no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—142no—
180C2×D4—281no—
181C2×D4C22×D711yescharacteristic
182C2×D4C22×D711yescharacteristic
183Q16—72no—
184Q16—142no—
185Q16—142no—
186SD16—22no—
187SD16—72no—
188SD16—72no—
189SD16—142no—
190SD16—142no—
191SD16—142no—
192SD16—142no—
193SD16—142no—
194SD16—142no—
195SD16—142no—
196D8—22no—
197D8—72no—
198D8—142no—
199M16—72no—
200M16—142no—
201M16—142no—
202M16C22×D712yes—
203C2×C8—72no—
204C2×C8—142no—
205C2×C8—142no—
206C4⋊C4—281no—
207C42C22×D711yescharacteristic
208C2×C7—21no—
209C2×C7—42no—
210C2×C7—81no—
211C2×C7—81no—
212C2×C7D4211yescharacteristic
213D7—42no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—281no—
218Q8—72no—
219Q8—142no—
220Q8—142no—
221Q8—282no—
222Q8D4×D712yes—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—42no—
227D4—72no—
228D4—82no—
229D4—142no—
230D4—282no—
231D4—282no—
232D4D4×D712yes—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—282no—
239C2×C4—281no—
240C2×C4—282no—
241C2×C4C23×D711yescharacteristic
242C8—22no—
243C8—72no—
244C8—72no—
245C8—142no—
246C7#329111yescharacteristic, complemented, Sylow
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—142no—
253C22—161no—
254C22—1121no—
255C22#654811yescharacteristic
256C4—22no—
257C4—22no—
258C4—142no—
259C4—282no—
260C4C2×D4×D712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—282no—
266C2#9028111yescharacteristic
subgroupG/H
centerC2#90281
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#453C2
socleC2×C7D42
hypercenter#453D7
hypocenterC7#3291
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #90527C2semidirect
2. #90466C2semidirect
3. C7×#453C2semidirect
4. #89972C2semidirect
5. #89849C2semidirect
6. #89848C2semidirect
7. #89598C2semidirect
8. #89596C2semidirect
9. #89520C2semidirect
10. C7×H(5,𝔽2)C22semidirect
12. #6548C22semidirect
16. #6514C22semidirect
19. #6511C22semidirect
21. C2×#1142C22semidirect
22. C2×#1142C22semidirect
29. #6472C22semidirect
30. #6470C22semidirect
31. #6432C22semidirect
32. C7×#104C22semidirect
33. C7×#100C22semidirect
34. #6407C22semidirect
35. #6398C22semidirect
36. #6396C22semidirect
37. #6379C22semidirect
38. C4×C7⋊C8C22semidirect
62. #1144D4semidirect
63. #1143D4semidirect
70. #1142D4semidirect
72. #1141D4semidirect
89. #453D7semidirect
125. H(5,𝔽2)C2×D7semidirect
147. #104C2×D7semidirect
150. #100C2×D7semidirect
246. C7#3291semidirect

Maximal subgroups

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

subgroupindex
#905272
#904662
C7×#4532
#899722
#898492
#898482
#895982
#895962
#895202
#32917