Finite Group Databasebeta

← #22598952

Subgroups — #22598952

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
1#17986616C212yesmaximal, complemented
2#17986396C211yesmaximal, characteristic, complemented
3#17986382C211yesmaximal, characteristic, complemented
4#17986357C212yesmaximal, complemented
5#17986257C211yesmaximal, characteristic, complemented
6#17986226C212yesmaximal, complemented
7#17986111C211yesmaximal, characteristic, complemented
8#17986105C211yesmaximal, characteristic, complemented
9#17986090C212yesmaximal, complemented
10#17985957C211yesmaximal, characteristic, complemented
11#469×C71C211yesmaximal, characteristic
12C2×Dic284C2211yescharacteristic
13C2×#11765853C2212yes—
14C22×C71⋊C8C2211yescharacteristic
15#14556127C2212yes—
16#14556121—22no—
17#14556115—22no—
18#14556115C2212yes—
19#14556109—22no—
20#14556107C2212yes—
21#14556105—22no—
22#14556093C2212yes—
23D4×Dic71C2212yes—
24#14556087C2211yescharacteristic
25#14556060C2211yescharacteristic
26#14556059C2212yes—
27#14556053C2212yes—
28#14556051C2212yes—
29#14556049C2212yes—
30#14556038C2211yescharacteristic
31#14556038C2211yescharacteristic
32C8×Dic71C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×D8×C71C2211yescharacteristic
35C2×M16×C71C2211yescharacteristic
36#121×C71C2212yes—
37#106×C71C2211yescharacteristic
38#102×C71C2212yes—
39C2×Dic142—22no—
40C2×Dic142C2312yes—
41C22×Dic71—22no—
42Dic284—22no—
43#11765853—24no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2311yescharacteristic
47C4⋊Dic71D412yes—
48C4⋊Dic71C2312yes—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52C22⋊Dic71—22no—
53C22⋊Dic71—22no—
54C22⋊Dic71—22no—
55C22⋊Dic71—22no—
56C22⋊Dic71D412yes—
57C4×Dic71—22no—
58C4×Dic71C2311yescharacteristic
59C4×Dic71C2311yescharacteristic
60C2×D4×C71—22no—
61C2×D4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63D8×C71—22no—
64M16×C71—22no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C22⋊C4×C71—22no—
69C23×C71—22no—
70C23×C71C2×D411yescharacteristic
71D4×C71—22no—
72D4×C71—22no—
73D4×C71—82no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—82no—
82C2×Dic71C2×D412yes—
83C2×Dic71C2×D412yes—
84Dic142—22no—
85Dic142—22no—
86Dic142—82no—
87C8×C71—21no—
88C8×C71#4112yes—
89C71⋊C8—21no—
90C71⋊C8#4112yes—
91C22×C71—21no—
92C22×C71—22no—
93C22×C71—42no—
94C22×C71—42no—
95C22×C71—42no—
96C22×C71C22×D411yescharacteristic
97C4×C71—21no—
98C4×C71—42no—
99C4×C71C2×#4111yescharacteristic
100C4×C71C2×#4111yescharacteristic
101Dic71—22no—
102Dic71—22no—
103Dic71—42no—
104Dic71—42no—
105C2×C71—41no—
106C2×C71—82no—
107C2×C71#53211yescharacteristic
108C2×C71#57712yes—
109#3173—711nomaximal, Sylow
110C71#317311yescharacteristic, complemented, Sylow
111#542—712no—
112#498—711no—
113#495—711no—
114#489—712no—
115#470—711no—
116#469D7111yescharacteristic
117#465—712no—
118#442—711no—
119#441—711no—
120C4×SD16—712no—
121#405—711no—
122C2×Q16—711no—
123C2×SD16—712no—
124C2×D8C2×D7111yescharacteristic
125C2×M16C2×D7111yescharacteristic
126C22×C8—711no—
127C4⋊Q8—712no—
128#126—1422no—
129#124—712no—
130#124—1422no—
131#123—1422no—
132C22⋊Q8—712no—
133C22⋊Q8—1422no—
134#121C2×D7112yes—
135C4×Q8—712no—
136C4×D4—712no—
137#117—711no—
138#106—711no—
139#106—712no—
140#106C2×D7111yescharacteristic
141#103—712no—
142#103—712no—
143#102—712no—
144#102C2×D7112yes—
145#97—711no—
146#97—711no—
147C4×C8—711no—
148C4×C8—711no—
149C2×Q8—712no—
150C2×Q8—1422no—
151C2×D4—22no—
152C2×D4C22×D7112yes—
153C22×C4—1422no—
154C22×C4C22×D7111yescharacteristic
155Q16—1422no—
156SD16—1424no—
157D8—22no—
158M16—22no—
159C2×C8—711no—
160C2×C8—711no—
161C2×C8—1422no—
162C2×C8C22×D7111yescharacteristic
163C2×C8C22×D7111yescharacteristic
164C4⋊C4—712no—
165C4⋊C4—712no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4C22×D7112yes—
170C22⋊C4—22no—
171C22⋊C4—712no—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C22⋊C4—1422no—
175C22⋊C4—1422no—
176C42—711no—
177C42—711no—
178C42—1422no—
179C23—22no—
180C23D4×D7111yescharacteristic
181Q8—5682no—
182Q8—1422no—
183Q8—1422no—
184D4—22no—
185D4—22no—
186D4—82no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—5682no—
191C2×C4—712no—
192C2×C4—712no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4D4×D7111yescharacteristic
196C2×C4C23×D7111yescharacteristic
197C8—21no—
198C8—712no—
199C8—1421no—
200C8#1176586412yes—
201C22—21no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C2×D4×D7111yescharacteristic
207C4—21no—
208C4—42no—
209C4—2842no—
210C4—2842no—
211C4—1422no—
212C4—1422no—
213C4#41×D7111yescharacteristic
214C4C2×#1176586411yescharacteristic
215C2—41no—
216C2—82no—
217C2#1798654411yescharacteristic
218C2#1798680912yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3173
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #17986616C2semidirect
2. #17986396C2semidirect
3. #17986382C2semidirect
4. #17986357C2semidirect
5. #17986257C2semidirect
6. #17986226C2semidirect
7. #17986111C2semidirect
8. #17986105C2semidirect
9. #17986090C2semidirect
10. #17985957C2semidirect
110. C71#3173semidirect