Finite Group Databasebeta

← #22599360

Subgroups — #22599360

Order 9088.

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

normal subgroups only

This group has 490 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
1C2×#14556125C211yesmaximal, characteristic, complemented
2#17986533C211yesmaximal, characteristic, complemented
3#17986388C212yesmaximal, complemented
4#17986364C212yesmaximal, complemented
5#17986200C212yesmaximal, complemented
6#17986195C212yesmaximal, complemented
7#17986152C212yesmaximal, complemented
8#17986135C212yesmaximal, complemented
9#432×C71C211yesmaximal, characteristic, complemented
10C22×D284—21no—
11C22×D284—21no—
12C22×D284C2212yes—
13C2×#11765854C2214yescomplemented
14C2×D568C2214yescomplemented
15#14556125—21no—
16#14556125—21no—
17#14556125C2212yes—
18#14556125C2212yes—
19#14556122—21no—
20#14556120C2211yescharacteristic, complemented
21#14556116C2211yescharacteristic, complemented
22#14556111—21no—
23#14556107C2211yescharacteristic
24#14556102C2211yescharacteristic
25C4×Dic142C2211yescharacteristic, complemented
26C4×D284C2211yescharacteristic, complemented
27C4×C22⋊D71—21no—
28#14556085—21no—
29#14556050C2214yescomplemented
30#14556050C2214yescomplemented
31C2×C42×C71C2211yescharacteristic
32C4⋊C8×C71C2212yescomplemented
33C22⋊C8×C71C2212yes—
34#97×C71C2212yescomplemented
35C23×D71—22no—
36C2×Dic142C2311yescharacteristic
37C2×D284—22no—
38C2×D284—22no—
39C2×D284—22no—
40C2×D284—22no—
41C2×D284—22no—
42C2×D284—44no—
43C2×D284—44no—
44C2×D284—44no—
45C2×D284C2314yes—
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C4×D71—21no—
49#11765854—44no—
50D568—44no—
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53Dic71⋊C4—21no—
54Dic71⋊C4—21no—
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57C22⋊Dic71—21no—
58(C2×D71)⋊C4—21no—
59(C2×D71)⋊C4—21no—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—21no—
62C4×Dic71—21no—
63C22×C4×C71D411yescharacteristic
64C22×C4×C71D411yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C2×C8×C71C2314yes—
67C42×C71D411yescharacteristic
68C42×C71D411yescharacteristic
69C42×C71C2311yescharacteristic
70C42×C71C2311yescharacteristic
71C23×C71C2×D411yescharacteristic
72C22×D71—21no—
73C22×D71—24no—
74C22×D71—44no—
75C22×D71—44no—
76C22×D71—44no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C22⋊D71—42no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90D284—22no—
91D284—24no—
92D284—24no—
93D284—44no—
94D284—44no—
95D284—44no—
96D284—44no—
97D284—44no—
98C4×D71—42no—
99Dic142—22no—
100C8×C71—24no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71AGL(1,ℤ/8)12yes—
104C22×C71C22×D411yescharacteristic
105C2×D71—44no—
106C2×D71—41no—
107C2×D71—41no—
108C2×D71—44no—
109C2×D71—44no—
110C2×D71—41no—
111C2×D71—88no—
112C4×C71—21no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—21no—
116C4×C71AGL(1,ℤ/8)12yes—
117C4×C71H(5,𝔽2)12yes—
118Dic71—41no—
119Dic71—41no—
120Dic71—41no—
121C2×C71—22no—
122C2×C71#53511yescharacteristic
123C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
125D71—81no—
126D71—84no—
127#3217—711nomaximal, Sylow
128C71#321711yescharacteristic, complemented, Sylow
129C2×#127—711no—
130#529—711no—
131#496—712no—
132#490—712no—
133#460—712no—
134#459—712no—
135#450—712no—
136#447—712no—
137#432D7111yescharacteristic, complemented
138C22×D4—712no—
139C22×D4—1421no—
140C22×D4—1421no—
141C2×SD16—714no—
142C2×D8—714no—
143#127—712no—
144#127—712no—
145#127—1421no—
146#127—1421no—
147#126—1421no—
148#125—711no—
149#124—711no—
150#123—1421no—
151C22⋊Q8—711no—
152#121—711no—
153C4×Q8—711no—
154C4×D4—711no—
155C4×D4—1421no—
156#117—1421no—
157C2×C42C2×D7111yescharacteristic
158C4⋊C8C2×D7112yescomplemented
159#102—714no—
160#102—714no—
161C22⋊C8C2×D7112yes—
162#97C2×D7112yescomplemented
163C24—1422no—
164C2×Q8—711no—
165C2×D4—2844no—
166C2×D4—2844no—
167C2×D4—2844no—
168C2×D4—714no—
169C2×D4—711no—
170C2×D4—1421no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4—1422no—
174C2×D4—1422no—
175C2×D4—1422no—
176C22×C4—1421no—
177C22×C4D28411yescharacteristic
178C22×C4D28411yescharacteristic
179C22×C4C22×D7111yescharacteristic
180SD16—2844no—
181D8—2844no—
182C2×C8C22×D7114yes—
183C4⋊C4—711no—
184C4⋊C4—711no—
185C4⋊C4—1421no—
186C4⋊C4—1421no—
187C4⋊C4—1421no—
188C4⋊C4—1421no—
189C22⋊C4—1421no—
190C22⋊C4—1421no—
191C22⋊C4—1421no—
192C22⋊C4—1421no—
193C22⋊C4—1421no—
194C42—1421no—
195C42D28411yescharacteristic
196C42D28411yescharacteristic
197C42C22×D7111yescharacteristic
198C42C22×D7111yescharacteristic
199C23—2844no—
200C23—2844no—
201C23—2844no—
202C23—1421no—
203C23—1424no—
204C23C2×D28411yescharacteristic
205Q8—1422no—
206D4—2844no—
207D4—2844no—
208D4—2844no—
209D4—2844no—
210D4—2842no—
211D4—2844no—
212D4—1422no—
213D4—1424no—
214D4—1424no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—2842no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4C2×D28411yescharacteristic
223C2×C4C2×D28411yescharacteristic
224C2×C4C2×D28411yescharacteristic
225C2×C4C2×D28411yescharacteristic
226C2×C4C2×D28411yescharacteristic
227C2×C4C23×D7111yescharacteristic
228C8—24no—
229C22—21no—
230C22—21no—
231C22—2844no—
232C22—2841no—
233C22—2841no—
234C22—2844no—
235C22—2844no—
236C22—2841no—
237C22—5688no—
238C22#1455615612yes—
239C22C22×D28411yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—2841no—
245C4—2841no—
246C4—2841no—
247C4#1455615612yes—
248C4#1455617412yes—
249C2—22no—
250C2—5684no—
251C2—5681no—
252C2#1798656811yescharacteristic
253C2C2×#1455615611yescharacteristic
254C2C2×#1455615611yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3217
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#14556125C2semidirect
2. #17986533C2semidirect
3. #17986388C2semidirect
4. #17986364C2semidirect
5. #17986200C2semidirect
6. #17986195C2semidirect
7. #17986152C2semidirect
8. #17986135C2semidirect
9. #432×C71C2semidirect
13. C2×#11765854C22semidirect
14. C2×D568C22semidirect
20. #14556120C22semidirect
21. #14556116C22semidirect
25. C4×Dic142C22semidirect
26. C4×D284C22semidirect
29. #14556050C22semidirect
30. #14556050C22semidirect
32. C4⋊C8×C71C22semidirect
34. #97×C71C22semidirect
128. C71#3217semidirect
137. #432D71semidirect
158. C4⋊C8C2×D71semidirect
162. #97C2×D71semidirect