Finite Group Databasebeta

← #22586659

Subgroups — #22586659

Order 9088.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×(C22⋊D71)⋊C4C211yesmaximal, characteristic
2#17986049C212yesmaximal, complemented
3#17985723C211yesmaximal, characteristic, complemented
4#17985722C212yesmaximal, complemented
5#323×C71C211yesmaximal, characteristic, complemented
6C22×C22⋊D71C411yescharacteristic
7C23×Dic71C411yescharacteristic
8C22×C4×D71C411yescharacteristic
9(C22⋊D71)⋊C4—22no—
10(C22⋊D71)⋊C4—22no—
11C2×Dic71⋊C4C411yescharacteristic
12C2×(C2×D71)⋊C4C2211yescharacteristic
13C2×C4×Dic71C2211yescharacteristic
14Dic71⋊C8—22no—
15C22⋊(C71⋊C8)C2212yes—
16C8×Dic71—22no—
17C2×C22⋊C4×C71C2211yescharacteristic
18C22⋊C8×C71C2212yes—
19C23×D71C2×C411yescharacteristic
20C2×C22⋊D71—22no—
21C2×C22⋊D71—22no—
22C2×C22⋊D71C814yescomplemented
23C22×Dic71—21no—
24C22×Dic71—21no—
25C22×Dic71—42no—
26C22×Dic71C2×C411yescharacteristic
27C22×Dic71C2×C411yescharacteristic
28C22×Dic71C2×C411yescharacteristic
29C2×C4×D71—21no—
30C2×C4×D71—21no—
31C2×C4×D71—42no—
32C2×C71⋊C8—22no—
33Dic71⋊C4—22no—
34(C2×D71)⋊C4—22no—
35C4×Dic71D412yes—
36C4×Dic71D412yes—
37C24×C71C2×C411yescharacteristic
38C22×C4×C71C2×C411yescharacteristic
39C22×C4×C71C2311yescharacteristic
40C2×C8×C71—22no—
41C22⋊C4×C71—22no—
42C23×C71—21no—
43C23×C71—21no—
44C23×C71—41no—
45C23×C71—41no—
46C23×C71C2×C812yes—
47C23×C71C22×C411yescharacteristic
48C22×D71—21no—
49C22×D71—21no—
50C22×D71—41no—
51C22×D71—41no—
52C22×D71C2×C812yes—
53C2×C4×C71—21no—
54C2×C4×C71—41no—
55C2×C4×C71—41no—
56C2×C4×C71C2×D412yes—
57C22⋊D71—28no—
58C2×Dic71—22no—
59C2×Dic71—21no—
60C2×Dic71—42no—
61C2×Dic71—42no—
62C2×Dic71—41no—
63C2×Dic71C22⋊C411yescharacteristic
64C2×Dic71C22⋊C412yes—
65C2×Dic71C22⋊C411yescharacteristic
66C2×Dic71C2×C812yes—
67C2×Dic71M1612yes—
68C4×D71—42no—
69C4×D71—42no—
70C8×C71—42no—
71C71⋊C8—42no—
72C22×C71—21no—
73C22×C71—21no—
74C22×C71—22no—
75C22×C71—42no—
76C22×C71—42no—
77C22×C71—41no—
78C22×C71—41no—
79C22×C71C2×C22⋊C411yescharacteristic
80C22×C71C22×C811yescharacteristic
81C22×C71C2×M1611yescharacteristic
82C2×D71—22no—
83C2×D71—42no—
84C2×D71—41no—
85C2×D71—41no—
86C2×D71—42no—
87C4×C71—22no—
88C4×C71—41no—
89Dic71—21no—
90Dic71—41no—
91Dic71C22⋊C814yes—
92Dic71#10012yes—
93C2×C71—21no—
94C2×C71—21no—
95C2×C71—42no—
96C2×C71C2×C22⋊C811yescharacteristic
97C2×C71#41011yescharacteristic
98C2×C71C2×#10011yescharacteristic
99D71—42no—
100#1460—711nomaximal, Sylow
101C71#146011yescharacteristic, complemented, Sylow
102C2×C4×D4—711no—
103#431—712no—
104#323—711no—
105#323—712no—
106#323D7111yescharacteristic, complemented
107C22×D4—711no—
108C23×C4—711no—
109C23×C4—711no—
110C4×D4—1422no—
111C4×D4—1422no—
112C2×C4⋊C4—711no—
113C2×C22⋊C4—711no—
114C2×C22⋊C4C2×D7111yescharacteristic
115C2×C42—711no—
116C4⋊C8—1422no—
117C22⋊C8—712no—
118C22⋊C8C2×D7112yes—
119C4×C8—1422no—
120C24—711no—
121C24C4×D7111yescharacteristic
122C2×D4—714no—
123C2×D4—1422no—
124C2×D4—1422no—
125C22×C4—2842no—
126C22×C4—2842no—
127C22×C4—711no—
128C22×C4—711no—
129C22×C4—711no—
130C22×C4—1421no—
131C22×C4—1421no—
132C22×C4—1421no—
133C22×C4—1421no—
134C22×C4C4×D7111yescharacteristic
135C22×C4C22×D7111yescharacteristic
136C2×C8—22no—
137C2×C8—1422no—
138C4⋊C4—1422no—
139C22⋊C4—22no—
140C22⋊C4—1422no—
141C42—712no—
142C42—712no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—2841no—
148C23—2841no—
149C23—712no—
150C23—1421no—
151C23—1421no—
152C23C8×D7112yes—
153C23C2×C4×D7111yescharacteristic
154D4—1428no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—2842no—
159C2×C4—2842no—
160C2×C4—2842no—
161C2×C4—2842no—
162C2×C4—2841no—
163C2×C4—712no—
164C2×C4—711no—
165C2×C4—712no—
166C2×C4—712no—
167C2×C4—711no—
168C2×C4—1421no—
169C2×C4—1422no—
170C2×C4D4×D7112yes—
171C8—42no—
172C8—2842no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—2842no—
181C22—2842no—
182C22—2841no—
183C22—2841no—
184C22—1422no—
185C22C22⋊C4×D7111yescharacteristic
186C22C2×C8×D7111yescharacteristic
187C22M16×D7111yescharacteristic
188C4—22no—
189C4—41no—
190C4—2841no—
191C4—714no—
192C4—712no—
193C4—1421no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—2842no—
198C2C22⋊C8×D7111yescharacteristic
199C2#1798597111yescharacteristic
200C2#100×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC22×C71C22×C8
FrattiniC22×C4C22×D71
Fitting#323×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#323D71
hypocenterC71#1460
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #17986049C2semidirect
3. #17985723C2semidirect
4. #17985722C2semidirect
5. #323×C71C2semidirect
22. C2×C22⋊D71C8semidirect
101. C71#1460semidirect
106. #323D71semidirect