Finite Group Databasebeta

← #22597818

Subgroups — #22597818

Order 9088.

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

normal subgroups only

This group has 584 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×#14556153C211yesmaximal, characteristic, complemented
2C2×#14556124C211yesmaximal, characteristic, complemented
3#41×Dic71C211yesmaximal, characteristic, complemented
4#17986197C214yesmaximal, complemented
5#17986192C214yesmaximal, complemented
6#17986022C211yesmaximal, characteristic, complemented
7#17986005C211yesmaximal, characteristic, complemented
8#17986004C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71—22no—
12#14556153—24no—
13C2×#11765852C2212yescomplemented
14C2×#11765850C2212yescomplemented
15C2×#11765849C2211yescharacteristic
16#14556124—21no—
17#14556124—21no—
18#14556124C2214yes—
19Q8×Dic71C2212yescomplemented
20D4×Dic71—22no—
21D4×Dic71C2212yescomplemented
22#14556087—21no—
23#14556087C2211yescharacteristic
24C2×C4×Dic71—21no—
25C2×C4×Dic71C2211yescharacteristic
26Dic71⋊C8C2214yescomplemented
27#14556050C2214yescomplemented
28#14556049C2214yescomplemented
29C2×#41×C71C2211yescharacteristic
30C22×D4×C71C2211yescharacteristic
31AGL(1,ℤ/8)×C71—24no—
32C2×SD16×C71C2212yescomplemented
33C2×D8×C71C2212yescomplemented
34C2×M16×C71C2211yescharacteristic
35C23×D71—21no—
36C2×D284—22no—
37C2×D284—22no—
38C2×D284—21no—
39C2×D284C2312yes—
40C2×C22⋊D71—24no—
41C2×C22⋊D71—24no—
42C2×C22⋊D71—48no—
43C22×Dic71—22no—
44C22×Dic71D412yes—
45#11765852—24no—
46#11765850—24no—
47#11765849—22no—
48C2×C71⋊C8C2312yes—
49C4⋊Dic71—22no—
50C4⋊Dic71C2312yes—
51C22⋊Dic71—21no—
52C22⋊Dic71—22no—
53C4×Dic71—21no—
54C4×Dic71—22no—
55C4×Dic71D411yescharacteristic
56C4×Dic71D411yescharacteristic
57C4×Dic71C2312yes—
58C24×C71—21no—
59#41×C71—22no—
60#41×C71D414yescomplemented
61C2×Q8×C71C2311yescharacteristic
62C2×D4×C71—21no—
63C2×D4×C71—22no—
64C2×D4×C71—21no—
65C2×D4×C71—22no—
66C2×D4×C71C2312yes—
67C2×D4×C71C2311yescharacteristic
68C22×C4×C71—21no—
69C22×C4×C71C2311yescharacteristic
70SD16×C71—24no—
71D8×C71—24no—
72M16×C71—22no—
73C2×C8×C71C2312yes—
74C23×C71—21no—
75C23×C71—22no—
76C23×C71—44no—
77C23×C71—42no—
78C23×C71C2×D411yescharacteristic
79C22×D71—22no—
80C22×D71—44no—
81C22×D71—42no—
82Q8×C71—21no—
83Q8×C71C2×D412yescomplemented
84D4×C71—24no—
85D4×C71—22no—
86D4×C71—21no—
87D4×C71—22no—
88D4×C71—42no—
89D4×C71C2×D412yescomplemented
90C2×C4×C71—21no—
91C2×C4×C71—24no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D412yes—
94C2×C4×C71#4112yes—
95C2×C4×C71C2411yescharacteristic
96C22⋊D71—48no—
97C22⋊D71—48no—
98C2×Dic71—24no—
99C2×Dic71—22no—
100C2×Dic71—44no—
101C2×Dic71C2×D412yes—
102C2×Dic71C2×D412yes—
103D284—22no—
104D284—22no—
105D284—42no—
106C8×C71—22no—
107C71⋊C8—22no—
108C22×C71—22no—
109C22×C71—22no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—44no—
113C22×C71—84no—
114C22×C71#12112yes—
115C22×C71C22×D411yescharacteristic
116C2×D71—42no—
117C2×D71—44no—
118C2×D71—84no—
119C4×C71—22no—
120C4×C71#12112yes—
121C4×C71C22×D411yescharacteristic
122C4×C71C2×#4111yescharacteristic
123Dic71—22no—
124Dic71—42no—
125Dic71AGL(1,ℤ/8)14yescomplemented
126C2×C71—22no—
127C2×C71—42no—
128C2×C71—82no—
129C2×C71C2×#12111yescharacteristic
130C2×C71C2×AGL(1,ℤ/8)12yes—
131D71—82no—
132#3038—711nomaximal, Sylow
133C71#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—711no—
135C2×AGL(1,ℤ/8)D7111yescharacteristic, complemented
136C2×#127—711no—
137C4×#41—711no—
138#460—714no—
139#459—714no—
140#423—711no—
141#418—711no—
142#418—711no—
143C2×#41C2×D7111yescharacteristic
144C22×D4—711no—
145C22×D4—1422no—
146C22×D4C2×D7111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—1424no—
149C2×SD16—712no—
150C2×SD16C2×D7112yescomplemented
151C2×D8—712no—
152C2×D8C2×D7112yescomplemented
153C2×M16—711no—
154C2×M16C2×D7111yescharacteristic
155#127—714no—
156#127—1421no—
157#127—1421no—
158C4×Q8—712no—
159C4×D4—712no—
160C4×D4—1422no—
161#117—711no—
162#117—1421no—
163C2×C42—711no—
164C2×C42—1421no—
165C4⋊C8—714no—
166#102—714no—
167#102—714no—
168C24—21no—
169C24—1421no—
170#41—22no—
171#41C22⋊D7114yescomplemented
172C2×Q8C22×D7111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2848no—
178C2×D4—712no—
179C2×D4—1424no—
180C2×D4—1424no—
181C2×D4—1421no—
182C2×D4—1422no—
183C2×D4—1422no—
184C2×D4C22×D7112yes—
185C2×D4C22×D7111yescharacteristic
186C22×C4—21no—
187C22×C4—712no—
188C22×C4—1422no—
189C22×C4C22×D7111yescharacteristic
190SD16—24no—
191SD16—1424no—
192D8—24no—
193D8—1424no—
194M16—22no—
195M16—1422no—
196C2×C8—712no—
197C2×C8C22×D7112yes—
198C4⋊C4—712no—
199C4⋊C4—1422no—
200C22⋊C4—1422no—
201C22⋊C4—1421no—
202C42—711no—
203C42—711no—
204C42—712no—
205C42—1422no—
206C42—1421no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—2844no—
212C23—2842no—
213C23—1422no—
214C23D4×D7111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D7112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—2848no—
223D4—2848no—
224D4—2842no—
225D4—1422no—
226D4—1422no—
227D4C2×C22⋊D7112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2844no—
231C2×C4—712no—
232C2×C4—712no—
233C2×C4—1422no—
234C2×C4—1424no—
235C2×C4D4×D7111yescharacteristic
236C2×C4C2×C22⋊D7112yes—
237C2×C4#1176586412yes—
238C2×C4C23×D7111yescharacteristic
239C8—22no—
240C8—1422no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—2842no—
248C22—2844no—
249C22—5684no—
250C22#1455610012yes—
251C22C2×D4×D7111yescharacteristic
252C4—22no—
253C4—2842no—
254C4—714no—
255C4—1422no—
256C4#1455610012yes—
257C4C22×C22⋊D7111yescharacteristic
258C4C2×#1176586411yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—5682no—
263C2C2×#1455610011yescharacteristic
264C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×AGL(1,ℤ/8)×C71C2
socleC22×C71C22×D4
hypercenterC2×AGL(1,ℤ/8)D71
hypocenterC71#3038
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14556153C2semidirect
2. C2×#14556124C2semidirect
3. #41×Dic71C2semidirect
4. #17986197C2semidirect
5. #17986192C2semidirect
6. #17986022C2semidirect
7. #17986005C2semidirect
8. #17986004C2semidirect
9. C2×AGL(1,ℤ/8)×C71C2semidirect
13. C2×#11765852C22semidirect
14. C2×#11765850C22semidirect
19. Q8×Dic71C22semidirect
21. D4×Dic71C22semidirect
26. Dic71⋊C8C22semidirect
27. #14556050C22semidirect
28. #14556049C22semidirect
32. C2×SD16×C71C22semidirect
33. C2×D8×C71C22semidirect
60. #41×C71D4semidirect
83. Q8×C71C2×D4semidirect
89. D4×C71C2×D4semidirect
125. Dic71AGL(1,ℤ/8)semidirect
133. C71#3038semidirect
135. C2×AGL(1,ℤ/8)D71semidirect
150. C2×SD16C2×D71semidirect
152. C2×D8C2×D71semidirect
171. #41C22⋊D71semidirect
216. Q8C2×C22⋊D71semidirect
227. D4C2×C22⋊D71semidirect