Finite Group Databasebeta

← #22600677

Subgroups — #22600677

Order 9088.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986710C211yesmaximal, characteristic, complemented
2#17986669C211yesmaximal, characteristic
3#17986311C212yesmaximal, complemented
4#17986291C211yesmaximal, characteristic, complemented
5#17986244C212yesmaximal, complemented
6#17986196C211yesmaximal, characteristic, complemented
7#17986162C212yesmaximal, complemented
8C4×Dic284C211yesmaximal, characteristic, complemented
9#17986078C211yesmaximal, characteristic, complemented
10#17985997C212yesmaximal, complemented
11#486×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2212yes—
13C2×Dic284C2211yescharacteristic
14C2×#11765853C2212yes—
15C2×#11765848C2212yes—
16#14556124—21no—
17#14556119—21no—
18#14556108C2212yes—
19#14556104—22no—
20#14556100—22no—
21#14556099C2212yes—
22C4×Dic142C2211yescharacteristic
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25D4×Dic71—21no—
26#14556090—21no—
27C4⋊C4×D71C2212yes—
28#14556062C2211yescharacteristic
29#14556061C2212yes—
30C4⋊(C71⋊C8)C2211yescharacteristic
31#14556053C2211yescharacteristic
32#14556053C2211yescharacteristic
33#14556051C2212yes—
34#14556049C2212yes—
35(C2×D71)⋊C8C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38#102×C71C2212yes—
39#102×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41#11765864—24no—
42#11765864—42no—
43C2×Dic142C2311yescharacteristic
44C2×Dic142C2311yescharacteristic
45C2×D284D411yescharacteristic
46C2×C22⋊D71—22no—
47C2×C22⋊D71—22no—
48C22×Dic71—22no—
49C2×C4×D71—22no—
50C2×C4×D71C2312yes—
51Dic284—22no—
52#11765853—42no—
53#11765848—42no—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71—22no—
56C4⋊Dic71D411yescharacteristic
57C4⋊Dic71C2311yescharacteristic
58C4⋊Dic71C2311yescharacteristic
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61Dic71⋊C4—22no—
62C22⋊Dic71—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4D412yes—
65C4×Dic71—21no—
66C4×Dic71—21no—
67C2×Q8×C71—21no—
68C2×D4×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71C2312yes—
71C2×C8×C71C2311yescharacteristic
72C2×C8×C71C2311yescharacteristic
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C42×C71C2311yescharacteristic
76C23×C71—22no—
77C22×D71C2×D412yes—
78Q8×C71—81no—
79D4×C71—22no—
80D4×C71—22no—
81D4×C71—42no—
82D4×C71—81no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C22⋊D71—44no—
88C22⋊D71—82no—
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—44no—
94C2×Dic71C2×D412yes—
95D284—22no—
96C4×D71—22no—
97C4×D71—22no—
98C4×D71—82no—
99Dic142—21no—
100Dic142—21no—
101Dic142—21no—
102Dic142#4112yes—
103C8×C71—21no—
104C8×C71—21no—
105C71⋊C8—22no—
106C22×C71—42no—
107C22×C71—42no—
108C22×C71—42no—
109C22×C71C22×D411yescharacteristic
110C2×D71—22no—
111C2×D71—22no—
112C2×D71—22no—
113C4×C71—21no—
114C4×C71—42no—
115C4×C71#13712yes—
116C4×C71C2×#4111yescharacteristic
117C4×C71H(5,𝔽2)11yescharacteristic
118Dic71—22no—
119Dic71—22no—
120Dic71—41no—
121Dic71—41no—
122Dic71—41no—
123C2×C71—82no—
124C2×C71#54611yescharacteristic
125C2×C71C2×#13711yescharacteristic
126C2×C71#57611yescharacteristic
127D71—42no—
128#3333—711nomaximal, Sylow
129C71#333311yescharacteristic, complemented, Sylow
130#554—711no—
131#550—711no—
132#486D7111yescharacteristic, complemented
133#480—712no—
134#477—711no—
135#468—712no—
136#460—711no—
137#452—712no—
138C4×Q16—711no—
139#436—711no—
140#417—712no—
141C2×#41—712no—
142C2×Q16—711no—
143C2×SD16—712no—
144C2×M16—712no—
145C4⋊Q8C2×D7111yescharacteristic
146#127—1421no—
147#127C2×D7111yescharacteristic
148#125—1421no—
149C22⋊Q8—712no—
150C22⋊Q8—1422no—
151#121—712no—
152#121—1422no—
153C4×Q8—711no—
154C4×Q8—711no—
155C4×D4—711no—
156C4×D4—1421no—
157C4×D4—1421no—
158C2×C4⋊C4—712no—
159#107—711no—
160#107—712no—
161C4⋊C8—711no—
162#103—711no—
163#103—711no—
164#103—712no—
165#102—712no—
166#102C2×D7112yes—
167#102C2×D7112yes—
168C22⋊C8—712no—
169C4×C8C2×D7111yescharacteristic
170#41—2842no—
171#41—1424no—
172C2×Q8—21no—
173C2×Q8—711no—
174C2×Q8—711no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—711no—
178C2×D4—1422no—
179C2×D4—1422no—
180C2×D4C22×D7112yes—
181C22×C4—712no—
182C22×C4—1422no—
183C22×C4—1422no—
184Q16—1422no—
185SD16—2842no—
186M16—2842no—
187C2×C8—712no—
188C2×C8C22×D7111yescharacteristic
189C2×C8C22×D7111yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—711no—
192C4⋊C4—711no—
193C4⋊C4—711no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4C22×D7112yes—
199C22⋊C4—712no—
200C22⋊C4—1422no—
201C22⋊C4—1422no—
202C42—1421no—
203C42—1421no—
204C42C22×D7111yescharacteristic
205C23—22no—
206C23—712no—
207Q8—81no—
208Q8—712no—
209Q8—1421no—
210Q8—1421no—
211Q8—1421no—
212D4—22no—
213D4—22no—
214D4—42no—
215D4—81no—
216D4—2844no—
217D4—5682no—
218D4—1422no—
219C2×C4—22no—
220C2×C4—2844no—
221C2×C4—5682no—
222C2×C4—712no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4D4×D7111yescharacteristic
230C2×C4D4×D7111yescharacteristic
231C2×C4C23×D7111yescharacteristic
232C8—21no—
233C8—21no—
234C8—1422no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—1422no—
239C22—1422no—
240C22—1422no—
241C22C2×D4×D7111yescharacteristic
242C4—21no—
243C4—42no—
244C4—2841no—
245C4—2841no—
246C4—2841no—
247C4—1422no—
248C4—1422no—
249C4#1455615712yes—
250C4#41×D7111yescharacteristic
251C4#1455617411yescharacteristic
252C2—82no—
253C2—2842no—
254C2#1798664211yescharacteristic
255C2C2×#1455615711yescharacteristic
256C2#1798680011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3333
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17986710C2semidirect
3. #17986311C2semidirect
4. #17986291C2semidirect
5. #17986244C2semidirect
6. #17986196C2semidirect
7. #17986162C2semidirect
8. C4×Dic284C2semidirect
9. #17986078C2semidirect
10. #17985997C2semidirect
11. #486×C71C2semidirect
129. C71#3333semidirect
132. #486D71semidirect