Finite Group Databasebeta

← #22594209

Subgroups — #22594209

Order 9088.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×C4×D284C211yesmaximal, characteristic
2#17985934C212yesmaximal, complemented
3#17985929C212yesmaximal, complemented
4#17985925C212yesmaximal, complemented
5#17985916C212yesmaximal
6#17985870C214yesmaximal, complemented
7#17985867C211yesmaximal, characteristic, complemented
8C2×#125×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C4×D71C2212yes—
11C4×D284—24no—
12C2×C4⋊Dic71C2211yescharacteristic
13C2×Dic71⋊C4C2214yes—
14C2×(C2×D71)⋊C4C2214yes—
15C2×(C2×D71)⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2214yes—
17C71⋊#95C2212yes—
18C71⋊#95C2212yes—
19C71⋊#95C2212yes—
20C71⋊#95C2214yes—
21#125×C71—24no—
22C2×C4⋊C4×C71C2212yes—
23C2×C4⋊C4×C71C2212yes—
24C2×C4⋊C4×C71C2211yescharacteristic
25C2×C4⋊C4×C71C2211yescharacteristic
26C2×C42×C71C2211yescharacteristic
27C23×D71C2312yes—
28C2×D284—24no—
29C2×D284—22no—
30C22×Dic71C2312yes—
31C22×Dic71C2314yes—
32C2×C4×D71—24no—
33C2×C4×D71—44no—
34C4⋊Dic71—22no—
35Dic71⋊C4—44no—
36(C2×D71)⋊C4—28no—
37(C2×D71)⋊C4—24no—
38(C2×D71)⋊C4—44no—
39C22×C4×C71C2312yes—
40C22×C4×C71C2312yes—
41C22×C4×C71C2311yescharacteristic
42C22×C4×C71C2311yescharacteristic
43C22×C4×C71C2311yescharacteristic
44C4⋊C4×C71—22no—
45C4⋊C4×C71—22no—
46C4⋊C4×C71—24no—
47C4⋊C4×C71—24no—
48C42×C71D414yes—
49C23×C71C2411yescharacteristic
50C22×D71—22no—
51C22×D71—22no—
52C22×D71—24no—
53C22×D71—24no—
54C22×D71#4114yes—
55C2×C4×C71—22no—
56C2×C4×C71—22no—
57C2×C4×C71—22no—
58C2×C4×C71—22no—
59C2×C4×C71—22no—
60C2×C4×C71—24no—
61C2×C4×C71—22no—
62C2×C4×C71—22no—
63C2×C4×C71C2×D412yes—
64C2×C4×C71C2×D412yes—
65C2×C4×C71C2×D412yes—
66C2×Dic71—22no—
67C2×Dic71—24no—
68C2×Dic71—24no—
69C2×Dic71—24no—
70C2×Dic71—24no—
71D284—28no—
72C4×D71—48no—
73C22×C71C22×D411yescharacteristic
74C22×C71C2×#4112yes—
75C22×C71H(5,𝔽2)12yes—
76C22×C712−1+412yes—
77C2×D71—24no—
78C2×D71—28no—
79C2×D71—42no—
80C2×D71—42no—
81C2×D71—44no—
82C4×C71—22no—
83C4×C71—22no—
84C4×C71—22no—
85C4×C71—42no—
86C4×C71—42no—
87Dic71—42no—
88Dic71—44no—
89C2×C71#52511yescharacteristic
90C2×C71#53511yescharacteristic
91C2×C71#53611yescharacteristic
92C2×C71#53914yes—
93D71—44no—
94#2633—711nomaximal, Sylow
95C71#263311yescharacteristic, complemented, Sylow
96C2×#125D7111yescharacteristic, complemented
97C2×C4×D4—711no—
98#399—712no—
99#398—712no—
100#397—712no—
101#394—712no—
102#385—714no—
103#383—711no—
104C22×D4—711no—
105C23×C4—712no—
106#125—24no—
107C4×D4—1424no—
108C2×C4⋊C4—714no—
109C2×C4⋊C4—711no—
110C2×C4⋊C4C2×D7112yes—
111C2×C4⋊C4C2×D7112yes—
112C2×C4⋊C4C2×D7111yescharacteristic
113C2×C4⋊C4C2×D7111yescharacteristic
114C2×C22⋊C4—714no—
115C2×C22⋊C4—712no—
116C2×C22⋊C4—714no—
117C2×C42C2×D7111yescharacteristic
118#95—712no—
119#95—712no—
120#95—712no—
121#95—714no—
122C24—712no—
123C2×D4—1424no—
124C2×D4—1422no—
125C22×C4—2844no—
126C22×C4—714no—
127C22×C4—712no—
128C22×C4—1424no—
129C22×C4C22×D7112yes—
130C22×C4C22×D7112yes—
131C22×C4C22×D7111yescharacteristic
132C22×C4C22×D7111yescharacteristic
133C22×C4C22×D7111yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—2844no—
139C4⋊C4—1422no—
140C22⋊C4—2844no—
141C22⋊C4—1428no—
142C22⋊C4—1424no—
143C42C22⋊D7114yes—
144C23—714no—
145C23—1422no—
146C23—1422no—
147C23—1424no—
148C23—1424no—
149C23C23×D7111yescharacteristic
150D4—1428no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—2848no—
160C2×C4—1422no—
161C2×C4—1424no—
162C2×C4—1424no—
163C2×C4—1424no—
164C2×C4—1424no—
165C2×C4C2×C22⋊D7112yes—
166C2×C4C2×C22⋊D7112yes—
167C2×C4C2×C22⋊D7112yes—
168C22—2842no—
169C22—2842no—
170C22—2844no—
171C22—1428no—
172C22—1424no—
173C22C22×C22⋊D7111yescharacteristic
174C22#41×D7112yes—
175C22#1455617412yes—
176C22#1455617612yes—
177C4—22no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—2842no—
183C4—2844no—
184C2—2844no—
185C2#1798650611yescharacteristic
186C2#1798656311yescharacteristic
187C2#1798657311yescharacteristic
188C2#1798659714yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#125×C71C2
hypercenterC2×#125D71
hypocenterC71#2633
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #17985934C2semidirect
3. #17985929C2semidirect
4. #17985925C2semidirect
6. #17985870C2semidirect
7. #17985867C2semidirect
8. C2×#125×C71C2semidirect
95. C71#2633semidirect
96. C2×#125D71semidirect