Finite Group Databasebeta

← #22599742

Subgroups — #22599742

Order 9088.

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

normal subgroups only

This group has 354 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#17986535C212yesmaximal, complemented
2#17986304C212yesmaximal, complemented
3#17986298C212yesmaximal, complemented
4#17986252C211yesmaximal, characteristic
5#17986247C212yesmaximal, complemented
6#17986223C212yesmaximal, complemented
7#17986217C212yesmaximal, complemented
8#17986022C211yesmaximal, characteristic, complemented
9#470×C71C211yesmaximal, characteristic, complemented
10C2×#11765852C2212yes—
11C2×D568C2211yescharacteristic
12C2×#11765849C2211yescharacteristic
13#14556123—22no—
14#14556119C2212yes—
15#14556117C2212yes—
16#14556112—22no—
17#14556102C2212yes—
18Q8×Dic71C2212yes—
19#14556090C2212yes—
20(C22⋊D71)⋊C4—22no—
21#14556086—22no—
22C2×C4×Dic71C2211yescharacteristic
23#14556061C2212yes—
24#14556060C2211yescharacteristic
25Dic71⋊C8C2212yes—
26Dic71⋊C8C2212yes—
27#14556052C2212yes—
28#14556050C2212yes—
29#14556048C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#106×C71C2211yescharacteristic
34#103×C71C2212yes—
35C2×D284C2312yes—
36C2×C22⋊D71—22no—
37C22×Dic71D412yes—
38C2×C4×D71—22no—
39#11765852—42no—
40D568—41no—
41#11765849—41no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71—22no—
44C4⋊Dic71—22no—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—22no—
47(C2×D71)⋊C4—22no—
48(C2×D71)⋊C4—22no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51C4×Dic71—22no—
52C4×Dic71D411yescharacteristic
53C4×Dic71D411yescharacteristic
54C4×Dic71C2312yes—
55C2×Q8×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57Q16×C71—41no—
58M16×C71—41no—
59C2×C8×C71C2311yescharacteristic
60C2×C8×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66Q8×C71—22no—
67Q8×C71—22no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2411yescharacteristic
74C22⋊D71—44no—
75C2×Dic71—22no—
76C2×Dic71—42no—
77C2×Dic71C2×D412yes—
78C2×Dic71C2×D412yes—
79D284—22no—
80D284—22no—
81C4×D71—44no—
82C8×C71—21no—
83C8×C71—21no—
84C71⋊C8—22no—
85C22×C71—21no—
86C22×C71—22no—
87C22×C71C22×D411yescharacteristic
88C2×D71—42no—
89C2×D71—42no—
90C2×D71—42no—
91C4×C71—21no—
92C4×C71—42no—
93C4×C71—42no—
94C4×C71H(5,𝔽2)11yescharacteristic
95C4×C712−1+411yescharacteristic
96Dic71—22no—
97Dic71—22no—
98Dic71—42no—
99C2×C71—41no—
100C2×C71#53711yescharacteristic
101C2×C71#57512yes—
102D71—82no—
103#3256—711nomaximal, Sylow
104C71#325611yescharacteristic, complemented, Sylow
105#529—712no—
106#479—712no—
107#478—712no—
108#470D7111yescharacteristic, complemented
109#469—711no—
110#468—712no—
111#464—712no—
112#463—712no—
113#423—711no—
114C2×Q16C2×D7111yescharacteristic
115C2×SD16—712no—
116C2×D8—711no—
117C2×M16—711no—
118C2×M16C2×D7111yescharacteristic
119#126—1422no—
120#125—712no—
121#124—712no—
122#123—1422no—
123C22⋊Q8C2×D7112yes—
124#121—712no—
125C4×Q8—712no—
126C4×D4—712no—
127C4×D4—1422no—
128#117—1422no—
129C2×C42—711no—
130#107—712no—
131#106—711no—
132#106C2×D7111yescharacteristic
133C4⋊C8—712no—
134C4⋊C8—712no—
135#103—712no—
136#103C2×D7112yes—
137#102—712no—
138#102—712no—
139C2×Q8C22×D7112yes—
140C2×D4—712no—
141C2×D4—1422no—
142C22×C4—712no—
143C22×C4—1422no—
144C22×C4C22×D7111yescharacteristic
145Q16—41no—
146SD16—2842no—
147D8—2841no—
148M16—41no—
149M16—2841no—
150C2×C8—712no—
151C2×C8C22×D7111yescharacteristic
152C2×C8C22×D7111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—712no—
155C4⋊C4—1422no—
156C4⋊C4—1422no—
157C4⋊C4—1422no—
158C4⋊C4C22×D7112yes—
159C22⋊C4—22no—
160C22⋊C4—1422no—
161C22⋊C4—1422no—
162C22⋊C4—1422no—
163C22⋊C4—1422no—
164C42—711no—
165C42—711no—
166C42—712no—
167C42—1422no—
168C23—1422no—
169C23D4×D7111yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—2844no—
173D4—1422no—
174D4—1422no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—2844no—
180C2×C4—2842no—
181C2×C4—712no—
182C2×C4—712no—
183C2×C4—1422no—
184C2×C4D4×D7111yescharacteristic
185C2×C4C23×D7111yescharacteristic
186C8—21no—
187C8—21no—
188C8—1422no—
189C22—21no—
190C22—22no—
191C22—2842no—
192C22—2842no—
193C22—2842no—
194C22C2×D4×D7111yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—2842no—
199C4—1422no—
200C4—1422no—
201C4#1455617311yescharacteristic
202C4#1455617511yescharacteristic
203C2—41no—
204C2—5682no—
205C2#1798657611yescharacteristic
206C2#1798679812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3256
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #17986535C2semidirect
2. #17986304C2semidirect
3. #17986298C2semidirect
5. #17986247C2semidirect
6. #17986223C2semidirect
7. #17986217C2semidirect
8. #17986022C2semidirect
9. #470×C71C2semidirect
104. C71#3256semidirect
108. #470D71semidirect