Finite Group Databasebeta

← #22598465

Subgroups — #22598465

Order 9088.

380 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986551C211yesmaximal, characteristic
2C2×Q8×Dic71C211yesmaximal, characteristic
3#17986330C211yesmaximal, characteristic
4#17986327C211yesmaximal, characteristic
5#17986282C212yesmaximal, complemented
6#17986207C212yesmaximal, complemented
7#17986155C211yesmaximal, characteristic
8#17986106C212yesmaximal, complemented
9#17986103C212yesmaximal, complemented
10#17986053C211yesmaximal, characteristic
11#451×C71C211yesmaximal, characteristic
12C2×#11765855C2212yes—
13#14556127C2211yescharacteristic
14#14556126C2211yescharacteristic
15#14556118—21no—
16#14556107C2211yescharacteristic
17#14556105—21no—
18Q8×Dic71—21no—
19Q8×Dic71—21no—
20Q8×Dic71C2212yes—
21Q8×Dic71C2211yescharacteristic
22Q8×Dic71C2212yes—
23#14556093—21no—
24#14556093C2211yescharacteristic
25#14556084—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2211yescharacteristic
28C2×C4×Dic71—21no—
29C2×C4×Dic71C2211yescharacteristic
30#14556060C2212yes—
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33#14556053C2212yes—
34#14556052C2212yes—
35#14556052C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yes—
38C22×Q8×C71C2211yescharacteristic
39C2×Q16×C71C2212yes—
40C22⋊Q8×C71C2211yescharacteristic
41#103×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142C2311yescharacteristic
45C22×Dic71—21no—
46C22×Dic71D411yescharacteristic
47C22×Dic71D411yescharacteristic
48#11765855—24no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71—22no—
53C4⋊Dic71—21no—
54C4⋊Dic71C2311yescharacteristic
55C4⋊Dic71C2312yes—
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61C22⋊Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71—22no—
64C4×Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71D411yescharacteristic
67C4×Dic71D411yescharacteristic
68C4×Dic71C2311yescharacteristic
69C4×Dic71C2311yescharacteristic
70C2×Q8×C71—24no—
71C2×Q8×C71—21no—
72C2×Q8×C71C2311yescharacteristic
73C2×Q8×C71C2312yes—
74C22×C4×C71—21no—
75C22×C4×C71C2311yescharacteristic
76Q16×C71—24no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2311yescharacteristic
80C22⋊C4×C71—21no—
81C23×C71C2×D411yescharacteristic
82Q8×C71—24no—
83Q8×C71—22no—
84Q8×C71—22no—
85Q8×C71#4114yes—
86C2×C4×C71—24no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2411yescharacteristic
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—42no—
99C2×Dic71C2×D411yescharacteristic
100C2×Dic71C2×D411yescharacteristic
101C2×Dic71C2×D411yescharacteristic
102C2×Dic71C2×D411yescharacteristic
103Dic142—22no—
104Dic142—42no—
105C8×C71—22no—
106C71⋊C8—22no—
107C22×C71—21no—
108C22×C71—21no—
109C22×C71#13712yes—
110C22×C71C22×D411yescharacteristic
111C4×C71—24no—
112C4×C71—21no—
113C4×C71—41no—
114C4×C71—41no—
115C4×C71C2×#4112yes—
116Dic71—21no—
117Dic71—21no—
118Dic71—21no—
119Dic71—41no—
120Dic71—41no—
121Dic71—42no—
122Dic71#13712yes—
123C2×C71—22no—
124C2×C71#52511yescharacteristic
125C2×C71C2×#13711yescharacteristic
126C2×C71C2×#13711yescharacteristic
127#3113—711nomaximal, Sylow
128C71#311311yescharacteristic, complemented, Sylow
129#533—711no—
130C2×C4×Q8—711no—
131#483—711no—
132#483—711no—
133#475—712no—
134#462—712no—
135#451—711no—
136#451D7111yescharacteristic
137#441—712no—
138#441—712no—
139#432—711no—
140C22×Q8C2×D7111yescharacteristic
141C2×Q16—712no—
142C2×Q16C2×D7112yes—
143C4⋊Q8—711no—
144C4⋊Q8—711no—
145#125—1421no—
146C22⋊Q8—711no—
147C22⋊Q8—1421no—
148C22⋊Q8C2×D7111yescharacteristic
149C4×Q8—712no—
150C4×Q8—712no—
151C4×Q8—711no—
152C4×Q8—711no—
153C4×Q8—1421no—
154C4×Q8—1421no—
155C4×Q8—1421no—
156#117—1421no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4—1421no—
159C2×C42—711no—
160C2×C42—1421no—
161#106—712no—
162#106—712no—
163C4⋊C8—712no—
164#103—712no—
165#103—712no—
166#103—712no—
167#103C2×D7112yes—
168C22⋊C8—711no—
169C22⋊C8C2×D7111yescharacteristic
170#97—712no—
171C2×Q8—24no—
172C2×Q8—21no—
173C2×Q8—711no—
174C2×Q8—1421no—
175C2×Q8C22×D7111yescharacteristic
176C2×Q8C22×D7112yes—
177C22×C4—21no—
178C22×C4—711no—
179C22×C4—711no—
180C22×C4—1421no—
181C22×C4C22×D7111yescharacteristic
182Q16—24no—
183Q16—1424no—
184C2×C8—712no—
185C2×C8C22×D7112yes—
186C4⋊C4—21no—
187C4⋊C4—712no—
188C4⋊C4—711no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4—1421no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4C22×D7111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—1421no—
201C42—711no—
202C42—711no—
203C42—711no—
204C42—711no—
205C42—1422no—
206C42—1422no—
207C42—1421no—
208C42—1421no—
209C23D4×D7111yescharacteristic
210Q8—24no—
211Q8—22no—
212Q8—22no—
213Q8—2842no—
214Q8—1422no—
215Q8#1176586414yes—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—2842no—
222C2×C4—711no—
223C2×C4—711no—
224C2×C4—711no—
225C2×C4—711no—
226C2×C4—1422no—
227C2×C4—1421no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4D4×D7111yescharacteristic
232C2×C4C23×D7111yescharacteristic
233C8—22no—
234C8—1422no—
235C22—21no—
236C22—21no—
237C22#1455616012yes—
238C22C2×D4×D7111yescharacteristic
239C4—24no—
240C4—21no—
241C4—41no—
242C4—41no—
243C4—2841no—
244C4—2841no—
245C4—2842no—
246C4—712no—
247C4—1421no—
248C4—1421no—
249C4—1421no—
250C4C2×#1176586412yes—
251C2—22no—
252C2#1798650511yescharacteristic
253C2#137×D7111yescharacteristic
254C2C2×#1455616011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#451×C71C2
socleC22×C71C22×D4
hypercenter#451D71
hypocenterC71#3113
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #17986282C2semidirect
6. #17986207C2semidirect
8. #17986106C2semidirect
9. #17986103C2semidirect
128. C71#3113semidirect