Finite Group Databasebeta

← #22599811

Subgroups — #22599811

Order 9088.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986620C212yesmaximal, complemented
2#17986290C212yesmaximal, complemented
3#17986285C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986252C211yesmaximal, characteristic
6#17986228C212yesmaximal, complemented
7#17986222C212yesmaximal, complemented
8#17986215C212yesmaximal, complemented
9#424×C71C211yesmaximal, characteristic, complemented
10C2×Dic284C2211yescharacteristic
11C2×#11765854C2212yes—
12C2×D568C2211yescharacteristic
13#14556128C2212yes—
14#14556123—22no—
15#14556116C2212yes—
16#14556114—22no—
17#14556110—22no—
18#14556107C2212yes—
19#14556106—22no—
20#14556102C2212yes—
21C4×Dic142C2212yes—
22C4×D284C2212yes—
23#14556060C2212yes—
24#14556060C2211yescharacteristic
25#14556060C2211yescharacteristic
26#14556053C2212yes—
27#14556053C2212yes—
28#14556050C2212yes—
29#14556050C2212yes—
30C2×M16×C71C2211yescharacteristic
31C22×C8×C71C2211yescharacteristic
32#117×C71C2211yescharacteristic
33C4⋊C8×C71C2212yes—
34C4⋊C8×C71C2212yes—
35C2×Dic142—22no—
36C2×Dic142C2312yes—
37C2×D284C2312yes—
38C2×C22⋊D71—22no—
39C2×C4×D71—22no—
40Dic284—41no—
41#11765854—42no—
42D568—41no—
43C4⋊Dic71—22no—
44C4⋊Dic71C2312yes—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48C22⋊Dic71—22no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C4×Dic71—22no—
54C22×C4×C71C2311yescharacteristic
55M16×C71—41no—
56C2×C8×C71—41no—
57C2×C8×C71C2312yes—
58C2×C8×C71C2311yescharacteristic
59C2×C8×C71C2311yescharacteristic
60C4⋊C4×C71D412yes—
61C22⋊C4×C71D411yescharacteristic
62C22⋊C4×C71D411yescharacteristic
63C42×C71C2312yes—
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71C2×D412yes—
69C2×C4×C71C2×D412yes—
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2411yescharacteristic
72C22⋊D71—82no—
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—22no—
76D284—22no—
77D284—22no—
78C4×D71—82no—
79Dic142—22no—
80Dic142—22no—
81Dic142—82no—
82C8×C71—22no—
83C8×C71—21no—
84C8×C71—21no—
85C22×C71—21no—
86C22×C71—22no—
87C22×C71C22×D411yescharacteristic
88C2×D71—42no—
89C2×D71—42no—
90C2×D71—42no—
91C4×C71—21no—
92C4×C71—22no—
93C4×C71—22no—
94C4×C71H(5,𝔽2)11yescharacteristic
95C4×C712−1+411yescharacteristic
96Dic71—42no—
97Dic71—42no—
98Dic71—42no—
99C2×C71—41no—
100C2×C71#53711yescharacteristic
101C2×C71#57712yes—
102D71—82no—
103#3263—711nomaximal, Sylow
104C71#326311yescharacteristic, complemented, Sylow
105#542—712no—
106#476—712no—
107#475—712no—
108#470—711no—
109#469—711no—
110#465—712no—
111#464—712no—
112#463—712no—
113#424D7111yescharacteristic, complemented
114C2×Q16—711no—
115C2×SD16—712no—
116C2×D8—711no—
117C2×M16C2×D7111yescharacteristic
118C22×C8C2×D7111yescharacteristic
119C4⋊Q8—712no—
120#126—1422no—
121#124—712no—
122#124—1422no—
123#123—1422no—
124C22⋊Q8—712no—
125C22⋊Q8—1422no—
126#121—712no—
127C4×Q8—712no—
128C4×D4—712no—
129#117C2×D7111yescharacteristic
130#106—712no—
131#106—711no—
132#106—711no—
133C4⋊C8C2×D7112yes—
134C4⋊C8C2×D7112yes—
135#103—712no—
136#103—712no—
137#102—712no—
138#102—712no—
139C2×Q8—712no—
140C2×Q8—1422no—
141C2×D4—712no—
142C2×D4—1422no—
143C22×C4—1422no—
144C22×C4C22×D7111yescharacteristic
145Q16—2841no—
146SD16—2842no—
147D8—2841no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D7111yescharacteristic
151C2×C8C22×D7112yes—
152C2×C8C22×D7111yescharacteristic
153C4⋊C4—712no—
154C4⋊C4—712no—
155C4⋊C4—1422no—
156C4⋊C4—1422no—
157C4⋊C4—1422no—
158C4⋊C4D28412yes—
159C22⋊C4—1422no—
160C22⋊C4—1422no—
161C22⋊C4—1422no—
162C22⋊C4—1422no—
163C22⋊C4—1422no—
164C22⋊C4D28411yescharacteristic
165C22⋊C4D28411yescharacteristic
166C42—1422no—
167C42C22×D7112yes—
168C23—1422no—
169C23C2×D28411yescharacteristic
170Q8—5682no—
171Q8—1422no—
172Q8—1422no—
173D4—5682no—
174D4—1422no—
175D4—1422no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—5682no—
179C2×C4—1422no—
180C2×C4—1422no—
181C2×C4—1422no—
182C2×C4C2×D28412yes—
183C2×C4C2×D28412yes—
184C2×C4C2×D28411yescharacteristic
185C2×C4C23×D7111yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—21no—
190C22—22no—
191C22—2842no—
192C22—2842no—
193C22—2842no—
194C22C22×D28411yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—2842no—
199C4—2842no—
200C4—2842no—
201C4#1455617311yescharacteristic
202C4#1455617611yescharacteristic
203C2—41no—
204C2—5682no—
205C2#1798658011yescharacteristic
206C2#1798681112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#424×C71C2
socleC22×C71C22×D4
hypercenter#424D71
hypocenterC71#3263
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #17986620C2semidirect
2. #17986290C2semidirect
3. #17986285C2semidirect
4. #17986257C2semidirect
6. #17986228C2semidirect
7. #17986222C2semidirect
8. #17986215C2semidirect
9. #424×C71C2semidirect
104. C71#3263semidirect
113. #424D71semidirect