Finite Group Databasebeta

← #22586870

Subgroups — #22586870

Order 9088.

326 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556102C211yesmaximal, characteristic
2#17985862C211yesmaximal, characteristic, complemented
3#17985734C214yesmaximal, complemented
4#407×C71C211yesmaximal, characteristic, complemented
5C22×D284C411yescharacteristic, complemented
6C22×C22⋊D71—21no—
7#14556102—21no—
8#14556102—21no—
9#14556102C412yescomplemented
10#14556102C2212yes—
11C2×C4⋊Dic71C411yescharacteristic, complemented
12C2×C22⋊Dic71—21no—
13C2×(C2×D71)⋊C4—21no—
14C71⋊#95—21no—
15C71⋊#95C2212yescomplemented
16C23×C4×C71C2211yescharacteristic
17C2×M16×C71—21no—
18C22⋊C8×C71—21no—
19C22⋊C8×C71C2212yescomplemented
20C23×D71—21no—
21C2×D284—21no—
22C2×D284—42no—
23C2×D284C2×C412yes—
24C2×C22⋊D71—22no—
25C2×C22⋊D71—22no—
26C2×C22⋊D71—44no—
27C22×Dic71—21no—
28C22×Dic71—22no—
29C4⋊Dic71—21no—
30C4⋊Dic71C2×C412yes—
31C22⋊Dic71—42no—
32(C2×D71)⋊C4—22no—
33(C2×D71)⋊C4—22no—
34C24×C71D411yescharacteristic
35C22×C4×C71—42no—
36C22×C4×C71C2×C411yescharacteristic
37C22×C4×C71C2×C411yescharacteristic
38C22×C4×C71D411yescharacteristic
39C22×C4×C71D411yescharacteristic
40C22×C4×C71D411yescharacteristic
41C22×C4×C71C2311yescharacteristic
42M16×C71—42no—
43C2×C8×C71—22no—
44C23×C71—22no—
45C23×C71—22no—
46C23×C71—41no—
47C23×C71C22⋊C411yescharacteristic
48C23×C71C22⋊C411yescharacteristic
49C23×C71C2×D411yescharacteristic
50C22×D71—22no—
51C22×D71—42no—
52C22×D71—42no—
53C22×D71—42no—
54C2×C4×C71—22no—
55C2×C4×C71—22no—
56C2×C4×C71—41no—
57C2×C4×C71—41no—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71C22⋊C411yescharacteristic
61C2×C4×C71C22⋊C411yescharacteristic
62C2×C4×C71C22×C411yescharacteristic
63C2×C4×C71C2×D411yescharacteristic
64C22⋊D71—44no—
65C22⋊D71—44no—
66C2×Dic71—22no—
67C2×Dic71—42no—
68C2×Dic71—42no—
69C2×Dic71—42no—
70C2×Dic71—42no—
71D284—42no—
72D284—42no—
73C8×C71—42no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71—21no—
78C22×C71—21no—
79C22×C71—21no—
80C22×C71—21no—
81C22×C71—44no—
82C22×C71#9912yes—
83C22×C71C2×C22⋊C411yescharacteristic
84C22×C71AGL(1,ℤ/8)12yes—
85C2×D71—42no—
86C2×D71—42no—
87C2×D71—42no—
88C2×D71—84no—
89C4×C71—21no—
90C4×C71—21no—
91C4×C71—21no—
92C4×C71—21no—
93Dic71—42no—
94Dic71—82no—
95C2×C71—22no—
96C2×C71—22no—
97C2×C71—41no—
98C2×C71C2×#9911yescharacteristic
99C2×C71#41811yescharacteristic
100C2×C71#42111yescharacteristic
101D71—82no—
102#1510—711nomaximal, Sylow
103C71#151011yescharacteristic, complemented, Sylow
104C2×#121—711no—
105#407D7111yescharacteristic, complemented
106#381—711no—
107#327—714no—
108C22×D4—711no—
109C22×D4—1421no—
110C23×C4C2×D7111yescharacteristic
111C2×M16—21no—
112#121—712no—
113#121—712no—
114#121—1421no—
115#121—1421no—
116C2×C4⋊C4—711no—
117C2×C22⋊C4—1421no—
118C2×C22⋊C4—1421no—
119C22⋊C8—21no—
120C22⋊C8C2×D7112yescomplemented
121#95—712no—
122#95—1421no—
123C24—1421no—
124C24D28411yescharacteristic
125C2×D4—2842no—
126C2×D4—2844no—
127C2×D4—712no—
128C2×D4—1422no—
129C2×D4—1422no—
130C2×D4—1421no—
131C22×C4—42no—
132C22×C4—1422no—
133C22×C4—1421no—
134C22×C4C4×D7111yescharacteristic
135C22×C4C4×D7111yescharacteristic
136C22×C4D28411yescharacteristic
137C22×C4C22⋊D7111yescharacteristic
138C22×C4C22⋊D7111yescharacteristic
139C22×C4C22×D7111yescharacteristic
140M16—42no—
141C2×C8—22no—
142C4⋊C4—712no—
143C4⋊C4—1421no—
144C22⋊C4—2842no—
145C22⋊C4—1422no—
146C22⋊C4—1422no—
147C23—22no—
148C23—22no—
149C23—41no—
150C23—2842no—
151C23—2842no—
152C23—2842no—
153C23—1422no—
154C23(C2×D71)⋊C411yescharacteristic
155C23(C2×D71)⋊C411yescharacteristic
156C23C2×D28411yescharacteristic
157D4—2842no—
158D4—2842no—
159D4—2844no—
160D4—2844no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—2842no—
168C2×C4—2842no—
169C2×C4—2842no—
170C2×C4—2842no—
171C2×C4—1422no—
172C2×C4(C2×D71)⋊C411yescharacteristic
173C2×C4(C2×D71)⋊C411yescharacteristic
174C2×C4C2×C4×D7111yescharacteristic
175C2×C4C2×C22⋊D7111yescharacteristic
176C8—42no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—2842no—
186C22—2842no—
187C22—2842no—
188C22—5684no—
189C22#1455604212yes—
190C22C2×(C2×D71)⋊C411yescharacteristic
191C22#1455615612yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—2842no—
197C4—5682no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—5682no—
202C2C2×#1455604211yescharacteristic
203C2#1798600511yescharacteristic
204C2#1798601511yescharacteristic
subgroupG/H
centerC22C2×(C2×D71)⋊C4
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#407×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#407D71
hypocenterC71#1510
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #17985862C2semidirect
3. #17985734C2semidirect
4. #407×C71C2semidirect
5. C22×D284C4semidirect
9. #14556102C4semidirect
11. C2×C4⋊Dic71C4semidirect
15. C71⋊#95C22semidirect
19. C22⋊C8×C71C22semidirect
103. C71#1510semidirect
105. #407D71semidirect
120. C22⋊C8C2×D71semidirect