Finite Group Databasebeta

← #11740839

Subgroups — #11740839

Order 896.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#6548C211yesmaximal, characteristic, complemented
2C7×#535C211yesmaximal, characteristic, complemented
3#90513C212yesmaximal, complemented
4C2×#6407C211yesmaximal, characteristic, complemented
5#89849C214yesmaximal, complemented
6#89833C214yesmaximal, complemented
7#89777C212yesmaximal, complemented
8C2×C7×#41C2211yescharacteristic, complemented
9C22×C7×D4C2211yescharacteristic
10C22×C22⋊D7—22no—
11#6548—22no—
12#6548—24no—
13#6548C2214yes—
14C2×#1159—22no—
15C2×D4×D7—22no—
16C2×#1157C2212yes—
17C7×#127C2212yescomplemented
18C7×#124C2212yescomplemented
19C7×#121—22no—
20C7×#120—22no—
21C7×#117C2211yescharacteristic, complemented
22#6514—22no—
23#6513—24no—
24#6512—22no—
25#6505C2214yescomplemented
26#6495C2214yescomplemented
27C2×#1136C2212yescomplemented
28#6407C2214yescomplemented
29#6379C2218yescomplemented
30#3019—71nomaximal, Sylow
31C24×C7D412yes—
32C23×D7—22no—
33C7×#41—22no—
34C7×#41—22no—
35C2×C7×Q8C2311yescharacteristic, complemented
36C2×C7×D4—21no—
37C2×C7×D4—24no—
38C2×C7×D4—22no—
39C2×C7×D4—22no—
40C2×C7×D4D412yes—
41C2×C7×D4C2312yes—
42C2×C7×D4C2311yescharacteristic, complemented
43C22×C4×C7—21no—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—24no—
46C2×C22⋊D7—28no—
47C2×C22⋊D7—22no—
48C2×C22⋊D7—28no—
49C22×Dic7—22no—
50#1159—28no—
51#1159—28no—
52D4×D7—28no—
53D4×D7—28no—
54#1157—22no—
55#1157—22no—
56#1157D414yescomplemented
57#1157C2314yes—
58C2×D28D412yescomplemented
59C2×C4×D7—22no—
60C2×Dic14D412yescomplemented
61C7×C4⋊C4—21no—
62C7×C22⋊C4—21no—
63C7×C22⋊C4—24no—
64C42×C7C2312yescomplemented
65#1144—24no—
66#1143—24no—
67#1142—24no—
68#1141—24no—
69#1136—22no—
70#1136C2314yescomplemented
71C2×C7⋊C8—22no—
72C2×H(5,𝔽2)—71no—
73#575—72no—
74#535D711yescharacteristic, complemented
75#455—74no—
76#453—74no—
77#421—72no—
78C2×#100—71no—
79C23×C7—22no—
80C23×C7—22no—
81C23×C7—21no—
82C23×C7—82no—
83C23×C7C2×D414yes—
84C23×C7C2×D411yescharacteristic
85C22×D7—22no—
86C22×D7—28no—
87C22×D7—84no—
88C7×Q8—22no—
89C7×D4—24no—
90C7×D4—24no—
91C7×D4—22no—
92C7×D4—41no—
93C7×D4—41no—
94C7×D4—84no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—42no—
98C2×C4×C7C2×D412yes—
99C2×C4×C7C2×D412yes—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—216no—
103C22⋊D7—24no—
104C22⋊D7—24no—
105C2×Dic7—28no—
106C2×Dic7—22no—
107D28—22no—
108D28C2×D414yes—
109C4×D7—24no—
110C4×D7—24no—
111Dic14—22no—
112Dic14C2×D414yes—
113C7⋊C8—24no—
114H(5,𝔽2)—74no—
115H(5,𝔽2)—144no—
116H(5,𝔽2)—142no—
117C2×#41—72no—
118C2×#41—142no—
119C2×#41C2×D711yescharacteristic, complemented
120C22×D4—142no—
121C22×D4—142no—
122C22×D4C2×D711yescharacteristic
123#137—74no—
124#137—142no—
125AGL(1,ℤ/8)—74no—
126AGL(1,ℤ/8)—142no—
127#135—144no—
128C2×M16—72no—
129#127C2×D712yescomplemented
130#124C2×D712yescomplemented
131#121—22no—
132#120—22no—
133#117C2×D711yescharacteristic, complemented
134#104—78no—
135#100—74no—
136C22×C7—24no—
137C22×C7—41no—
138C22×C7—44no—
139C22×C7—44no—
140C22×C7—42no—
141C22×C7—41no—
142C22×C7—82no—
143C22×C7#12011yescharacteristic
144C22×C7#12011yescharacteristic
145C22×C7C22×D411yescharacteristic
146C2×D7—24no—
147C2×D7—44no—
148C2×D7—88no—
149C4×C7—42no—
150C4×C7—41no—
151C4×C7#12012yes—
152C4×C7C22×D412yes—
153Dic7—24no—
154C24—142no—
155C24C22⋊D712yes—
156#41—22no—
157#41—22no—
158#41—74no—
159#41—74no—
160#41—148no—
161#41—148no—
162#41—142no—
163#41—142no—
164C2×Q8—72no—
165C2×Q8C22×D711yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—24no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—72no—
171C2×D4—148no—
172C2×D4—148no—
173C2×D4—144no—
174C2×D4—148no—
175C2×D4—142no—
176C2×D4—148no—
177C2×D4C22⋊D712yes—
178C2×D4C22×D712yes—
179C2×D4C22×D711yescharacteristic, complemented
180C22×C4—21no—
181C22×C4—142no—
182C22×C4—142no—
183C22×C4C22×D711yescharacteristic
184Q16—144no—
185SD16—144no—
186SD16—144no—
187D8—144no—
188M16—74no—
189M16—142no—
190C2×C8—142no—
191C4⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D712yescomplemented
195C2×C7—21no—
196C2×C7—21no—
197C2×C7—21no—
198C2×C7—44no—
199C2×C7—81no—
200C2×C7C2×#12011yescharacteristic
201D7—44no—
202C23—22no—
203C23—22no—
204C23—21no—
205C23—82no—
206C23—148no—
207C23—142no—
208C23—564no—
209C23C2×C22⋊D714yes—
210C23C2×C22⋊D711yescharacteristic
211Q8—22no—
212Q8—74no—
213Q8—142no—
214D4—24no—
215D4—24no—
216D4—22no—
217D4—41no—
218D4—41no—
219D4—74no—
220D4—84no—
221D4—1416no—
222D4—144no—
223D4—144no—
224D4—142no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—148no—
229C2×C4—142no—
230C2×C4—144no—
231C2×C4—144no—
232C2×C4D4×D712yes—
233C2×C4D4×D712yes—
234C2×C4C2×C22⋊D711yescharacteristic
235C2×C4C23×D711yescharacteristic
236C8—144no—
237C7#301911yescharacteristic, complemented, Sylow
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—144no—
246C22—284no—
247C22—568no—
248C22#650011yescharacteristic
249C22#650011yescharacteristic
250C22C22×C22⋊D711yescharacteristic
251C4—42no—
252C4—41no—
253C4—144no—
254C4#650012yes—
255C4C2×D4×D712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—284no—
262C2C2×#650011yescharacteristic
subgroupG/H
centerC2C2×#6500
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#535C2
socleC2×C7C2×#120
hypercenter#535D7
hypocenterC7#3019
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#6548C2semidirect
2. C7×#535C2semidirect
3. #90513C2semidirect
4. C2×#6407C2semidirect
5. #89849C2semidirect
6. #89833C2semidirect
7. #89777C2semidirect
8. C2×C7×#41C22semidirect
17. C7×#127C22semidirect
18. C7×#124C22semidirect
21. C7×#117C22semidirect
25. #6505C22semidirect
26. #6495C22semidirect
27. C2×#1136C22semidirect
28. #6407C22semidirect
29. #6379C22semidirect
35. C2×C7×Q8C23semidirect
42. C2×C7×D4C23semidirect
56. #1157D4semidirect
58. C2×D28D4semidirect
60. C2×Dic14D4semidirect
64. C42×C7C23semidirect
70. #1136C23semidirect
74. #535D7semidirect
119. C2×#41C2×D7semidirect
129. #127C2×D7semidirect
130. #124C2×D7semidirect
133. #117C2×D7semidirect
165. C2×Q8C22×D7semidirect
179. C2×D4C22×D7semidirect
194. C42C22×D7semidirect
237. C7#3019semidirect

Maximal subgroups

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

subgroupindex
C2×#65482
C7×#5352
#905132
C2×#64072
#898492
#898332
#897772
#30197