Finite Group Databasebeta

← #22598712

Subgroups — #22598712

Order 9088.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#17986552C211yesmaximal, characteristic, complemented
2C2×#14556124C211yesmaximal, characteristic
3#17986362C214yesmaximal, complemented
4#17986344C214yesmaximal, complemented
5#17986005C211yesmaximal, characteristic, complemented
6#17986004C211yesmaximal, characteristic, complemented
7#17986003C211yesmaximal, characteristic, complemented
8M16×Dic71C211yesmaximal, characteristic, complemented
9#418×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71—22no—
12C2×#11765849C2211yescharacteristic
13#14556127C2212yescomplemented
14#14556124—22no—
15#14556124C2212yes—
16#14556124C2212yes—
17#14556119C2212yescomplemented
18#14556105—22no—
19C4×Dic142—22no—
20#14556087C2211yescharacteristic
21C2×C4×Dic71C2211yescharacteristic
22#14556050C2214yescomplemented
23#14556049C2214yescomplemented
24#14556048C2214yescomplemented
25#14556038C2212yescomplemented
26C8×Dic71C2212yescomplemented
27C22×D4×C71C2211yescharacteristic
28C2×M16×C71C2211yescharacteristic
29#117×C71C2211yescharacteristic
30#102×C71C2214yescomplemented
31C23×D71—21no—
32C2×Dic142—22no—
33C2×D284—22no—
34C2×D284—22no—
35C2×D284—21no—
36C2×D284C2312yes—
37C2×C22⋊D71—24no—
38C2×C22⋊D71—24no—
39C2×C22⋊D71—48no—
40C22×Dic71D412yes—
41#11765849—22no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71—22no—
44C4⋊Dic71C2312yes—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47C22⋊Dic71—21no—
48C4×Dic71—21no—
49C4×Dic71D412yes—
50C4×Dic71C2311yescharacteristic
51C4×Dic71C2311yescharacteristic
52C24×C71—21no—
53C2×D4×C71—21no—
54C2×D4×C71—22no—
55C2×D4×C71—22no—
56C2×D4×C71C2312yes—
57C22×C4×C71C2311yescharacteristic
58M16×C71—22no—
59C2×C8×C71C2312yes—
60C4⋊C4×C71C2312yes—
61C22⋊C4×C71—21no—
62C42×C71—21no—
63C23×C71—22no—
64C23×C71—44no—
65C23×C71—42no—
66C23×C71C2×D411yescharacteristic
67C22×D71—22no—
68C22×D71—44no—
69C22×D71—42no—
70D4×C71—22no—
71D4×C71—22no—
72D4×C71—42no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71#4112yes—
76C2×C4×C71#4112yes—
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—48no—
79C22⋊D71—48no—
80C2×Dic71—24no—
81C2×Dic71—22no—
82C2×Dic71C2×D412yes—
83C2×Dic71C2×D412yes—
84D284—22no—
85D284—22no—
86D284—42no—
87Dic142—44no—
88C8×C71—22no—
89C71⋊C8—22no—
90C22×C71—22no—
91C22×C71—42no—
92C22×C71—44no—
93C22×C71—84no—
94C22×C71#12412yes—
95C22×C71C22×D411yescharacteristic
96C2×D71—42no—
97C2×D71—44no—
98C2×D71—84no—
99C4×C71—42no—
100C4×C71#12412yes—
101C4×C71C2×#4111yescharacteristic
102C4×C71C2×#4111yescharacteristic
103Dic71—22no—
104Dic71—42no—
105Dic71AGL(1,ℤ/8)14yes—
106C2×C71—22no—
107C2×C71—82no—
108C2×C71C2×#12411yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)12yes—
110D71—82no—
111#3138—711nomaximal, Sylow
112C71#313811yescharacteristic, complemented, Sylow
113#533—711no—
114C2×#127—711no—
115#490—714no—
116#486—714no—
117#418—711no—
118#418—711no—
119#418—711no—
120#418D7111yescharacteristic, complemented
121C4×M16—711no—
122C22×D4—711no—
123C22×D4—1422no—
124C22×D4C2×D7111yescharacteristic
125C2×M16—711no—
126C2×M16C2×D7111yescharacteristic
127C4⋊Q8—712no—
128#127—712no—
129#127—712no—
130#127—1422no—
131#125—712no—
132C22⋊Q8—1422no—
133C4×Q8—1422no—
134#117—711no—
135#117C2×D7111yescharacteristic
136C2×C42—711no—
137#102—714no—
138#102—714no—
139#102—714no—
140#102C2×D7114yescomplemented
141#97—712no—
142C4×C8—712no—
143C24—21no—
144C24—1421no—
145C2×Q8—1422no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—2848no—
150C2×D4—712no—
151C2×D4—1424no—
152C2×D4—1424no—
153C2×D4—1421no—
154C2×D4—1422no—
155C2×D4—1422no—
156C2×D4C22×D7112yes—
157C22×C4—712no—
158C22×C4C22×D7111yescharacteristic
159M16—22no—
160M16—1422no—
161C2×C8—712no—
162C2×C8C22×D7112yes—
163C4⋊C4—712no—
164C4⋊C4—1422no—
165C4⋊C4—1422no—
166C4⋊C4—1422no—
167C4⋊C4C22×D7112yes—
168C22⋊C4—21no—
169C22⋊C4—1421no—
170C42—21no—
171C42—712no—
172C42—711no—
173C42—711no—
174C42—1421no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—2844no—
179C23—2842no—
180C23—1422no—
181C23D4×D7111yescharacteristic
182Q8—2844no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—2848no—
187D4—2848no—
188D4—2842no—
189D4—1422no—
190D4—1422no—
191C2×C4—22no—
192C2×C4—712no—
193C2×C4—712no—
194C2×C4—1422no—
195C2×C4—1424no—
196C2×C4D4×D7111yescharacteristic
197C2×C4#1176586412yes—
198C2×C4#1176586512yes—
199C2×C4C23×D7111yescharacteristic
200C8—22no—
201C8—1422no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—2842no—
207C22—2844no—
208C22—5684no—
209C22#1455611412yes—
210C22C2×D4×D7111yescharacteristic
211C4—42no—
212C4—2842no—
213C4—714no—
214C4—1422no—
215C4#1455611412yes—
216C4C2×#1176586411yescharacteristic
217C4C2×#1176586511yescharacteristic
218C2—22no—
219C2—82no—
220C2—5682no—
221C2C2×#1455611411yescharacteristic
222C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#418×C71C2
socleC22×C71C22×D4
hypercenter#418D71
hypocenterC71#3138
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #17986552C2semidirect
3. #17986362C2semidirect
4. #17986344C2semidirect
5. #17986005C2semidirect
6. #17986004C2semidirect
7. #17986003C2semidirect
8. M16×Dic71C2semidirect
9. #418×C71C2semidirect
13. #14556127C22semidirect
17. #14556119C22semidirect
22. #14556050C22semidirect
23. #14556049C22semidirect
24. #14556048C22semidirect
25. #14556038C22semidirect
26. C8×Dic71C22semidirect
30. #102×C71C22semidirect
112. C71#3138semidirect
120. #418D71semidirect
140. #102C2×D71semidirect