Finite Group Databasebeta

← #22599568

Subgroups — #22599568

Order 9088.

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

normal subgroups only

This group has 362 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
1C2×#14556127C211yesmaximal, characteristic, complemented
2#17986530C211yesmaximal, characteristic
3#17986406C212yesmaximal, complemented
4#17986357C212yesmaximal, complemented
5#17986335C211yesmaximal, characteristic, complemented
6#17986334C211yesmaximal, characteristic, complemented
7#17986315C211yesmaximal, characteristic
8#17986283C212yesmaximal, complemented
9#17986277C212yesmaximal, complemented
10#17986053C211yesmaximal, characteristic, complemented
11#481×C71C211yesmaximal, characteristic
12C22×Dic142—21no—
13#14556127—21no—
14#14556127—21no—
15#14556127C2212yes—
16#14556127C2212yes—
17#14556121—21no—
18#14556119C2211yescharacteristic
19#14556115C2211yescharacteristic
20#14556109—21no—
21#14556107C2211yescharacteristic
22#14556093C2211yescharacteristic
23D4×Dic71—21no—
24D4×Dic71C2211yescharacteristic
25#14556084—21no—
26C2×C4⋊Dic71C2211yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71C2211yescharacteristic
29#14556062C2212yes—
30#14556061C2212yes—
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33#14556053C2212yes—
34#14556051C2212yes—
35#14556049C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yes—
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#106×C71C2212yes—
41#102×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—22no—
44C2×Dic142—22no—
45C2×Dic142—44no—
46C2×Dic142C2311yescharacteristic
47C22×Dic71—21no—
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C22×Dic71D411yescharacteristic
51C2×C71⋊C8C2312yes—
52C4⋊Dic71—21no—
53C4⋊Dic71—21no—
54C4⋊Dic71C2312yes—
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—21no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61C22⋊Dic71—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64C22⋊Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71D411yescharacteristic
67C4×Dic71D411yescharacteristic
68C4×Dic71C2311yescharacteristic
69C4×Dic71C2311yescharacteristic
70C2×D4×C71—21no—
71C2×D4×C71C2311yescharacteristic
72C22×C4×C71—21no—
73C22×C4×C71C2311yescharacteristic
74C2×C8×C71C2312yes—
75C4⋊C4×C71—21no—
76C4⋊C4×C71C2312yes—
77C4⋊C4×C71C2311yescharacteristic
78C22⋊C4×C71—21no—
79C23×C71—21no—
80C23×C71C2×D411yescharacteristic
81D4×C71—22no—
82D4×C71—42no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2411yescharacteristic
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—42no—
95C2×Dic71—42no—
96C2×Dic71C2×D411yescharacteristic
97C2×Dic71C2×D411yescharacteristic
98C2×Dic71C2×D411yescharacteristic
99C2×Dic71C2×D411yescharacteristic
100Dic142—22no—
101Dic142—44no—
102Dic142—44no—
103C8×C71—22no—
104C71⋊C8—22no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—41no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71#13712yes—
111C22×C71C22×D411yescharacteristic
112C4×C71—21no—
113C4×C71—42no—
114C4×C71—41no—
115C4×C712−1+412yes—
116Dic71—21no—
117Dic71—21no—
118Dic71—21no—
119Dic71—41no—
120Dic71—42no—
121Dic71—41no—
122Dic71AGL(1,ℤ/8)12yes—
123C2×C71—22no—
124C2×C71—81no—
125C2×C71#53611yescharacteristic
126C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C71C2×#13711yescharacteristic
128#3239—711nomaximal, Sylow
129C71#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—711no—
131#529—711no—
132#501—712no—
133#489—712no—
134#484—711no—
135#484—711no—
136#481—711no—
137#481D7111yescharacteristic
138#475—712no—
139#474—712no—
140#432—711no—
141C22×Q8—1421no—
142C4⋊Q8—712no—
143C4⋊Q8—712no—
144C4⋊Q8—1421no—
145C4⋊Q8—1421no—
146#126—1421no—
147#125—711no—
148#124—711no—
149#123—1421no—
150C22⋊Q8—711no—
151#121C2×D7111yescharacteristic
152C4×Q8—711no—
153C4×D4—711no—
154C4×D4—1421no—
155#117—1421no—
156C2×C4⋊C4—711no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4C2×D7111yescharacteristic
159C2×C42—711no—
160#107—712no—
161#107—712no—
162#106—712no—
163#106C2×D7112yes—
164C4⋊C8—712no—
165#103—712no—
166#103—712no—
167#102—712no—
168#102C2×D7112yes—
169C22⋊C8—711no—
170C22⋊C8C2×D7111yescharacteristic
171#97—712no—
172C2×Q8—2844no—
173C2×Q8—711no—
174C2×Q8—1422no—
175C2×Q8—1422no—
176C2×D4—21no—
177C2×D4C22×D7111yescharacteristic
178C22×C4—21no—
179C22×C4—711no—
180C22×C4—711no—
181C22×C4—1421no—
182C22×C4—1421no—
183C22×C4C22×D7111yescharacteristic
184C2×C8—712no—
185C2×C8C22×D7112yes—
186C4⋊C4—21no—
187C4⋊C4—712no—
188C4⋊C4—711no—
189C4⋊C4—1422no—
190C4⋊C4—1422no—
191C4⋊C4—1421no—
192C4⋊C4—1421no—
193C4⋊C4—1421no—
194C4⋊C4—1421no—
195C4⋊C4—1421no—
196C4⋊C4C22×D7112yes—
197C4⋊C4C22×D7111yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C22⋊C4—1421no—
203C42—711no—
204C42—711no—
205C42—711no—
206C42—711no—
207C42—1421no—
208C23—21no—
209C23D4×D7111yescharacteristic
210Q8—2844no—
211Q8—2844no—
212Q8—1422no—
213D4—22no—
214D4—42no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—2842no—
220C2×C4—2842no—
221C2×C4—711no—
222C2×C4—711no—
223C2×C4—711no—
224C2×C4—711no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4D4×D7111yescharacteristic
231C2×C4C23×D7111yescharacteristic
232C8—22no—
233C8—1422no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22#1455615712yes—
240C22C2×D4×D7111yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—2841no—
245C4—2842no—
246C4—2841no—
247C4—712no—
248C4—1421no—
249C4—1421no—
250C4—1421no—
251C4#1455617512yes—
252C2—22no—
253C2—81no—
254C2#1798657111yescharacteristic
255C2AGL(1,ℤ/8)×D7111yescharacteristic
256C2C2×#1455615711yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#481×C71C2
socleC22×C71C22×D4
hypercenter#481D71
hypocenterC71#3239
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. C2×#14556127C2semidirect
3. #17986406C2semidirect
4. #17986357C2semidirect
5. #17986335C2semidirect
6. #17986334C2semidirect
8. #17986283C2semidirect
9. #17986277C2semidirect
10. #17986053C2semidirect
129. C71#3239semidirect