Finite Group Databasebeta

← #22599776

Subgroups — #22599776

Order 9088.

370 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986602C212yesmaximal, complemented
2#17986301C212yesmaximal, complemented
3#17986298C212yesmaximal, complemented
4#17986252C211yesmaximal, characteristic, complemented
5#17986233C212yesmaximal, complemented
6#17986193C212yesmaximal, complemented
7#17986192C212yesmaximal, complemented
8#17986027C211yesmaximal, characteristic, complemented
9#469×C71C211yesmaximal, characteristic, complemented
10C2×D568C2211yescharacteristic
11C2×#11765850C2212yes—
12C22×C71⋊C8C2211yescharacteristic
13#14556124C2212yes—
14#14556119C2212yes—
15#14556112—22no—
16#14556110—22no—
17#14556102C2212yes—
18#14556100—22no—
19#14556099—22no—
20D4×Dic71C2212yes—
21#14556090C2212yes—
22#14556087C2211yescharacteristic
23#14556061C2212yes—
24#14556060C2211yescharacteristic
25Dic71⋊C8C2212yes—
26Dic71⋊C8C2212yes—
27#14556050C2212yes—
28#14556049C2212yes—
29#14556048C2212yes—
30C2×D8×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32#121×C71C2212yes—
33#106×C71C2211yescharacteristic
34#102×C71C2212yes—
35C2×D284C2312yes—
36C2×C22⋊D71—22no—
37C2×C22⋊D71—22no—
38C2×C22⋊D71—22no—
39C22×Dic71—22no—
40C2×C4×D71—22no—
41D568—41no—
42#11765850—42no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8C2312yes—
45C4⋊Dic71—22no—
46C4⋊Dic71D412yes—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—22no—
49C22⋊Dic71—22no—
50C22⋊Dic71D411yescharacteristic
51C22⋊Dic71D411yescharacteristic
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—22no—
54C4×Dic71C2312yes—
55C2×D4×C71—22no—
56C2×D4×C71C2312yes—
57C22×C4×C71C2311yescharacteristic
58D8×C71—41no—
59M16×C71—41no—
60C2×C8×C71C2311yescharacteristic
61C2×C8×C71C2311yescharacteristic
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C23×C71—22no—
65C23×C71C2×D411yescharacteristic
66C22×D71—22no—
67D4×C71—22no—
68D4×C71—22no—
69D4×C71—82no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2411yescharacteristic
75C22⋊D71—82no—
76C22⋊D71—82no—
77C22⋊D71—82no—
78C2×Dic71—22no—
79C2×Dic71—82no—
80C2×Dic71C2×D412yes—
81C2×Dic71C2×D412yes—
82D284—22no—
83D284—22no—
84C4×D71—82no—
85C8×C71—21no—
86C8×C71—21no—
87C71⋊C8—22no—
88C22×C71—21no—
89C22×C71—22no—
90C22×C71—42no—
91C22×C71—42no—
92C22×C71—42no—
93C22×C71C22×D411yescharacteristic
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C4×C71—21no—
98C4×C71—42no—
99C4×C71H(5,𝔽2)11yescharacteristic
100C4×C712−1+411yescharacteristic
101Dic71—22no—
102Dic71—22no—
103Dic71—42no—
104C2×C71—41no—
105C2×C71—82no—
106C2×C71#53711yescharacteristic
107C2×C71#57612yes—
108D71—82no—
109#3259—711nomaximal, Sylow
110C71#325911yescharacteristic, complemented, Sylow
111#540—712no—
112#478—712no—
113#478—712no—
114#469—711no—
115#469D7111yescharacteristic, complemented
116#466—712no—
117#459—712no—
118#459—712no—
119#424—711no—
120C2×D8—712no—
121C2×D8—711no—
122C2×D8C2×D7111yescharacteristic
123C2×M16C2×D7111yescharacteristic
124C22×C8—711no—
125#127—712no—
126#125—712no—
127#123—1422no—
128#123—1422no—
129#121—712no—
130#121—1422no—
131#121—1422no—
132#121C2×D7112yes—
133C4×D4—712no—
134C4×D4—712no—
135#117—711no—
136#107—712no—
137#106—711no—
138#106C2×D7111yescharacteristic
139C4⋊C8—712no—
140C4⋊C8—712no—
141#102—712no—
142#102—712no—
143#102—712no—
144#102C2×D7112yes—
145C2×D4—22no—
146C2×D4—712no—
147C2×D4—1422no—
148C2×D4—1422no—
149C2×D4—1422no—
150C2×D4C22×D7112yes—
151C22×C4—1422no—
152C22×C4—1422no—
153C22×C4C22×D7111yescharacteristic
154D8—41no—
155D8—2841no—
156D8—2842no—
157M16—41no—
158C2×C8—2841no—
159C2×C8—712no—
160C2×C8C22×D7111yescharacteristic
161C2×C8C22×D7111yescharacteristic
162C4⋊C4—712no—
163C4⋊C4—712no—
164C4⋊C4—1422no—
165C4⋊C4—1422no—
166C4⋊C4C22×D7112yes—
167C22⋊C4—22no—
168C22⋊C4—711no—
169C22⋊C4—711no—
170C22⋊C4—1422no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C42—712no—
174C23—22no—
175C23—1422no—
176C23D4×D7111yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—5682no—
181D4—5682no—
182D4—5682no—
183D4—1422no—
184D4—1422no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—5682no—
189C2×C4—5682no—
190C2×C4—712no—
191C2×C4—712no—
192C2×C4—1422no—
193C2×C4D4×D7111yescharacteristic
194C2×C4C23×D7111yescharacteristic
195C8—21no—
196C8—21no—
197C8—1422no—
198C22—21no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—2842no—
204C22—2842no—
205C22—2842no—
206C22C2×D4×D7111yescharacteristic
207C4—21no—
208C4—42no—
209C4—2842no—
210C4—1422no—
211C4—1422no—
212C4#1455617311yescharacteristic
213C4#1455617511yescharacteristic
214C2—41no—
215C2—82no—
216C2—5682no—
217C2#1798657611yescharacteristic
218C2#1798680312yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3259
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. #17986602C2semidirect
2. #17986301C2semidirect
3. #17986298C2semidirect
4. #17986252C2semidirect
5. #17986233C2semidirect
6. #17986193C2semidirect
7. #17986192C2semidirect
8. #17986027C2semidirect
9. #469×C71C2semidirect
110. C71#3259semidirect
115. #469D71semidirect