Finite Group Databasebeta

← #22599358

Subgroups — #22599358

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
1#17986533C211yesmaximal, characteristic, complemented
2#17986386C212yesmaximal, complemented
3#17986363C212yesmaximal, complemented
4#17986196C212yesmaximal, complemented
5#17986191C212yesmaximal, complemented
6#17986150C212yesmaximal, complemented
7#17986133C212yesmaximal, complemented
8#17986056C211yesmaximal, characteristic, complemented
9C2×#127×C71C211yesmaximal, characteristic, complemented
10C2×#11765853C2214yescomplemented
11C2×#11765850C2214yescomplemented
12#14556122—21no—
13#14556120C2211yescharacteristic, complemented
14#14556116C2211yescharacteristic, complemented
15#14556111—21no—
16#14556107C2211yescharacteristic
17#14556102C2211yescharacteristic
18C4×Dic142C2211yescharacteristic, complemented
19C4×D284C2211yescharacteristic, complemented
20C4×C22⋊D71—21no—
21#14556085—21no—
22C4⋊(C71⋊C8)C2212yescomplemented
23#14556049C2214yescomplemented
24#14556049C2214yescomplemented
25C22⋊(C71⋊C8)C2212yes—
26#14556039C2212yescomplemented
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71C2212yes—
30#127×C71—21no—
31#127×C71—21no—
32#127×C71C2212yes—
33#127×C71C2212yes—
34C2×C42×C71C2211yescharacteristic
35C2×Dic142C2311yescharacteristic
36C2×D284C2311yescharacteristic
37C2×C22⋊D71—21no—
38C2×C4×D71—21no—
39#11765853—44no—
40#11765850—44no—
41C2×C71⋊C8C2314yes—
42C4⋊Dic71C2311yescharacteristic
43C4⋊Dic71C2311yescharacteristic
44Dic71⋊C4—21no—
45Dic71⋊C4—21no—
46Dic71⋊C4—21no—
47Dic71⋊C4—21no—
48C22⋊Dic71—21no—
49(C2×D71)⋊C4—21no—
50(C2×D71)⋊C4—21no—
51(C2×D71)⋊C4—21no—
52(C2×D71)⋊C4—21no—
53C4×Dic71—21no—
54C24×C71—22no—
55C2×D4×C71—22no—
56C2×D4×C71—22no—
57C2×D4×C71—22no—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—44no—
61C2×D4×C71—44no—
62C2×D4×C71—44no—
63C2×D4×C71C2314yes—
64C22×C4×C71D411yescharacteristic
65C22×C4×C71D411yescharacteristic
66C22×C4×C71C2311yescharacteristic
67C42×C71D411yescharacteristic
68C42×C71D411yescharacteristic
69C42×C71C2311yescharacteristic
70C42×C71C2311yescharacteristic
71C23×C71—24no—
72C23×C71—44no—
73C23×C71—44no—
74C23×C71—44no—
75C23×C71C2×D411yescharacteristic
76C22×D71—21no—
77D4×C71—24no—
78D4×C71—24no—
79D4×C71—44no—
80D4×C71—44no—
81D4×C71—44no—
82D4×C71—44no—
83D4×C71—44no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2411yescharacteristic
93C22⋊D71—42no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97D284—22no—
98C4×D71—42no—
99Dic142—22no—
100C71⋊C8—24no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71—44no—
104C22×C71—44no—
105C22×C71—44no—
106C22×C71—88no—
107C22×C71AGL(1,ℤ/8)12yes—
108C22×C71C22×D411yescharacteristic
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
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—84no—
123C2×C71#53511yescharacteristic
124C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
126D71—81no—
127#3217—711nomaximal, Sylow
128C71#321711yescharacteristic, complemented, Sylow
129C2×#127D7111yescharacteristic, complemented
130#529—711no—
131#496—712no—
132#490—712no—
133#460—712no—
134#459—712no—
135#450—712no—
136#447—712no—
137#432—711no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C2×D7112yes—
141C2×SD16—714no—
142C2×D8—714no—
143#127—21no—
144#127—21no—
145#127C2×D7112yes—
146#127C2×D7112yes—
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⋊C8—712no—
159#102—714no—
160#102—714no—
161C22⋊C8—712no—
162#97—712no—
163C24—22no—
164C2×Q8—711no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—711no—
174C2×D4—1421no—
175C2×D4C22×D7114yes—
176C22×C4—1421no—
177C22×C4C22⋊D7111yescharacteristic
178C22×C4C22⋊D7111yescharacteristic
179C22×C4C22×D7111yescharacteristic
180SD16—2844no—
181D8—2844no—
182C2×C8—714no—
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—
195C42C22⋊D7111yescharacteristic
196C42C22⋊D7111yescharacteristic
197C42C22×D7111yescharacteristic
198C42C22×D7111yescharacteristic
199C23—24no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—1421no—
204C23C2×C22⋊D7111yescharacteristic
205Q8—1422no—
206D4—24no—
207D4—24no—
208D4—44no—
209D4—44no—
210D4—44no—
211D4—44no—
212D4—44no—
213D4—2842no—
214D4—1422no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—2842no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4C2×C22⋊D7111yescharacteristic
223C2×C4C2×C22⋊D7111yescharacteristic
224C2×C4C2×C22⋊D7111yescharacteristic
225C2×C4C2×C22⋊D7111yescharacteristic
226C2×C4C2×C22⋊D7111yescharacteristic
227C2×C4C23×D7111yescharacteristic
228C8—1424no—
229C22—21no—
230C22—21no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—88no—
235C22—2841no—
236C22—2841no—
237C22—2841no—
238C22#1455615412yes—
239C22C22×C22⋊D7111yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—2841no—
245C4—2841no—
246C4—2841no—
247C4#1455615412yes—
248C4#1455617312yes—
249C2—22no—
250C2—84no—
251C2—5681no—
252C2#1798656611yescharacteristic
253C2C2×#1455615411yescharacteristic
254C2C2×#1455615411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#127×C71C2
socleC22×C71C22×D4
hypercenterC2×#127D71
hypocenterC71#3217
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #17986533C2semidirect
2. #17986386C2semidirect
3. #17986363C2semidirect
4. #17986196C2semidirect
5. #17986191C2semidirect
6. #17986150C2semidirect
7. #17986133C2semidirect
8. #17986056C2semidirect
9. C2×#127×C71C2semidirect
10. C2×#11765853C22semidirect
11. C2×#11765850C22semidirect
13. #14556120C22semidirect
14. #14556116C22semidirect
18. C4×Dic142C22semidirect
19. C4×D284C22semidirect
22. C4⋊(C71⋊C8)C22semidirect
23. #14556049C22semidirect
24. #14556049C22semidirect
26. #14556039C22semidirect
128. C71#3217semidirect
129. C2×#127D71semidirect