Finite Group Databasebeta

← #22586686

Subgroups — #22586686

Order 9088.

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

normal subgroups only

This group has 280 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×#14556111C211yesmaximal, characteristic
2#17985963C211yesmaximal, characteristic, complemented
3#17985724C212yesmaximal, complemented
4#17985722C212yesmaximal, complemented
5C2×C22⋊C8×C71C211yesmaximal, characteristic, complemented
6C22×C22⋊D71C411yescharacteristic
7C2×#11765849—21no—
8#14556111—21no—
9#14556111—21no—
10#14556111—21no—
11#14556111—21no—
12C2×Dic71⋊C4C411yescharacteristic
13C2×Dic71⋊C4C2211yescharacteristic
14C2×C22⋊Dic71C411yescharacteristic
15C2×(C2×D71)⋊C4C411yescharacteristic
16C2×(C2×D71)⋊C4C2211yescharacteristic
17C22⋊(C71⋊C8)—21no—
18C22⋊(C71⋊C8)C2212yes—
19C23×C4×C71C2211yescharacteristic
20C22×C8×C71—21no—
21C22⋊C8×C71—21no—
22C22⋊C8×C71C2212yes—
23C23×D71C2×C411yescharacteristic
24C2×C22⋊D71—21no—
25C2×C22⋊D71—21no—
26C2×C22⋊D71—42no—
27C22×Dic71C2×C411yescharacteristic
28C22×Dic71C2×C411yescharacteristic
29C22×Dic71C2×C411yescharacteristic
30#11765849—42no—
31C2×C71⋊C8—22no—
32Dic71⋊C4—21no—
33Dic71⋊C4—21no—
34Dic71⋊C4—21no—
35Dic71⋊C4—21no—
36C22⋊Dic71—21no—
37C22⋊Dic71—21no—
38(C2×D71)⋊C4—21no—
39(C2×D71)⋊C4—21no—
40(C2×D71)⋊C4—21no—
41(C2×D71)⋊C4—21no—
42C24×C71C2×C411yescharacteristic
43C22×C4×C71—42no—
44C22×C4×C71C2×C411yescharacteristic
45C22×C4×C71D411yescharacteristic
46C22×C4×C71D411yescharacteristic
47C22×C4×C71D411yescharacteristic
48C22×C4×C71D411yescharacteristic
49C22×C4×C71C2311yescharacteristic
50C2×C8×C71—22no—
51C2×C8×C71—42no—
52C23×C71—22no—
53C23×C71—22no—
54C23×C71—41no—
55C23×C71C22⋊C411yescharacteristic
56C23×C71C22⋊C411yescharacteristic
57C23×C71C22×C411yescharacteristic
58C22×D71—21no—
59C22×D71—22no—
60C22×D71—41no—
61C22×D71—41no—
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71C22⋊C411yescharacteristic
69C2×C4×C71C22⋊C411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C22⋊D71—42no—
73C22⋊D71—42no—
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—21no—
77C2×Dic71—22no—
78C2×Dic71—41no—
79C2×Dic71—41no—
80C8×C71—42no—
81C71⋊C8—42no—
82C22×C71—21no—
83C22×C71—21no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71—21no—
89C22×C71—44no—
90C22×C71#9912yes—
91C22×C71C2×C22⋊C411yescharacteristic
92C22×C71#13112yes—
93C2×D71—41no—
94C2×D71—41no—
95C2×D71—41no—
96C2×D71—82no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101Dic71—41no—
102Dic71—41no—
103Dic71—41no—
104C2×C71—22no—
105C2×C71—22no—
106C2×C71—41no—
107C2×C71#40811yescharacteristic
108C2×C71C2×#9911yescharacteristic
109C2×C71#41311yescharacteristic
110D71—81no—
111#1468—711nomaximal, Sylow
112C71#146811yescharacteristic, complemented, Sylow
113C2×#123—711no—
114#407—711no—
115C2×C22⋊C8D7111yescharacteristic, complemented
116#324—712no—
117#323—712no—
118C22×D4—711no—
119C23×C4C2×D7111yescharacteristic
120C2×M16—1421no—
121C22×C8—21no—
122#123—1421no—
123#123—1421no—
124#123—1421no—
125#123—1421no—
126C2×C4⋊C4—711no—
127C2×C4⋊C4—711no—
128C2×C22⋊C4—711no—
129C2×C22⋊C4—711no—
130C2×C22⋊C4—711no—
131C22⋊C8—21no—
132C22⋊C8—712no—
133C22⋊C8—1421no—
134C22⋊C8C2×D7112yes—
135C24—711no—
136C24C4×D7111yescharacteristic
137C2×D4—2842no—
138C2×D4—1421no—
139C2×D4—1421no—
140C22×C4—42no—
141C22×C4—711no—
142C22×C4—711no—
143C22×C4—711no—
144C22×C4C4×D7111yescharacteristic
145C22×C4D28411yescharacteristic
146C22×C4D28411yescharacteristic
147C22×C4C22⋊D7111yescharacteristic
148C22×C4C22⋊D7111yescharacteristic
149C22×C4C22×D7111yescharacteristic
150M16—2842no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—1422no—
154C4⋊C4—1421no—
155C4⋊C4—1421no—
156C4⋊C4—1421no—
157C4⋊C4—1421no—
158C22⋊C4—1421no—
159C22⋊C4—1421no—
160C22⋊C4—1421no—
161C22⋊C4—1421no—
162C22⋊C4—1421no—
163C22⋊C4—1421no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—2841no—
168C23—2841no—
169C23—1421no—
170C23—1422no—
171C23(C2×D71)⋊C411yescharacteristic
172C23(C2×D71)⋊C411yescharacteristic
173C23C2×C4×D7111yescharacteristic
174D4—2842no—
175D4—2842no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—2841no—
183C2×C4—2841no—
184C2×C4—1421no—
185C2×C4—1421no—
186C2×C4—1421no—
187C2×C4—1422no—
188C2×C4(C2×D71)⋊C411yescharacteristic
189C2×C4(C2×D71)⋊C411yescharacteristic
190C2×C4C2×C22⋊D7111yescharacteristic
191C2×C4C2×D28411yescharacteristic
192C8—42no—
193C8—2842no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—2841no—
203C22—2841no—
204C22—2841no—
205C22—5682no—
206C22#1455604212yes—
207C22C2×(C2×D71)⋊C411yescharacteristic
208C22#1455613412yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2841no—
214C4—2841no—
215C4—2841no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—5681no—
220C2#1798596511yescharacteristic
221C2C2×#1455604211yescharacteristic
222C2#1798598111yescharacteristic
subgroupG/H
centerC22C2×(C2×D71)⋊C4
commutatorC23×C71C22×C4
FrattiniC22×C4C22×D71
FittingC2×C22⋊C8×C71C2
socleC22×C71C2×C22⋊C4
hypercenterC2×C22⋊C8D71
hypocenterC71#1468
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #17985963C2semidirect
3. #17985724C2semidirect
4. #17985722C2semidirect
5. C2×C22⋊C8×C71C2semidirect
112. C71#1468semidirect
115. C2×C22⋊C8D71semidirect