Finite Group Databasebeta

← #22586702

Subgroups — #22586702

Order 9088.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14556106C211yesmaximal, characteristic
2#17985961C211yesmaximal, characteristic
3C22⋊C8×D71C211yesmaximal, characteristic
4#17985725C211yesmaximal, characteristic, complemented
5#17985724C211yesmaximal, characteristic, complemented
6#17985724C211yesmaximal, characteristic, complemented
7#324×C71C211yesmaximal, characteristic, complemented
8C22×Dic142C411yescharacteristic
9C22×C4×D71C2211yescharacteristic
10C2×#11765848—21no—
11C2×C8×D71—21no—
12#14556106—21no—
13#14556106—21no—
14#14556106—21no—
15#14556106—21no—
16C2×Dic71⋊C4C411yescharacteristic
17C2×Dic71⋊C4C2211yescharacteristic
18C2×(C2×D71)⋊C4C411yescharacteristic
19C2×(C2×D71)⋊C4C411yescharacteristic
20C22⋊(C71⋊C8)C2211yescharacteristic
21C22⋊(C71⋊C8)C2211yescharacteristic
22(C2×D71)⋊C8—21no—
23(C2×D71)⋊C8—21no—
24C2×C4⋊C4×C71C2211yescharacteristic
25C22⋊C8×C71C2211yescharacteristic
26C22⋊C8×C71C2211yescharacteristic
27C23×D71C2×C411yescharacteristic
28C2×Dic142—21no—
29C2×Dic142—21no—
30C2×Dic142—42no—
31C22×Dic71C2×C411yescharacteristic
32C22×Dic71C2×C411yescharacteristic
33C22×Dic71C2×C411yescharacteristic
34C2×C4×D71—42no—
35C2×C4×D71D411yescharacteristic
36C2×C4×D71D411yescharacteristic
37C2×C4×D71D411yescharacteristic
38C2×C4×D71D411yescharacteristic
39#11765848—42no—
40C2×C71⋊C8—21no—
41C2×C71⋊C8—21no—
42C8×D71—42no—
43Dic71⋊C4—21no—
44Dic71⋊C4—21no—
45Dic71⋊C4—21no—
46Dic71⋊C4—21no—
47(C2×D71)⋊C4—21no—
48(C2×D71)⋊C4—21no—
49(C2×D71)⋊C4—21no—
50(C2×D71)⋊C4—21no—
51C22×C4×C71C2×C411yescharacteristic
52C22×C4×C71C2×C411yescharacteristic
53C22×C4×C71C2311yescharacteristic
54C2×C8×C71—21no—
55C2×C8×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71—21no—
58C23×C71C22×C411yescharacteristic
59C22×D71—21no—
60C22×D71—21no—
61C22×D71—22no—
62C22×D71—41no—
63C22×D71C22⋊C411yescharacteristic
64C22×D71C22⋊C411yescharacteristic
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×Dic71—22no—
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—21no—
77C2×Dic71—21no—
78C2×Dic71—41no—
79C2×Dic71C22⋊C411yescharacteristic
80C2×Dic71C22⋊C411yescharacteristic
81C4×D71—22no—
82C4×D71—22no—
83C4×D71—41no—
84C4×D71—41no—
85Dic142—42no—
86Dic142—42no—
87C8×C71—41no—
88C8×C71—41no—
89C71⋊C8—41no—
90C71⋊C8—41no—
91C22×C71—21no—
92C22×C71—21no—
93C22×C71C2×C22⋊C411yescharacteristic
94C22×C71#13111yescharacteristic
95C22×C71#13111yescharacteristic
96C2×D71—21no—
97C2×D71—21no—
98C2×D71—21no—
99C2×D71—21no—
100C2×D71—21no—
101C2×D71—42no—
102C2×D71—42no—
103C2×D71#10112yes—
104C4×C71—21no—
105C4×C71—21no—
106C4×C71—41no—
107C4×C71—41no—
108Dic71—21no—
109Dic71—41no—
110Dic71—41no—
111Dic71#9912yes—
112C2×C71—21no—
113C2×C71—21no—
114C2×C71#40811yescharacteristic
115C2×C71C2×#9911yescharacteristic
116C2×C71C2×#10111yescharacteristic
117D71—22no—
118D71—41no—
119#1472—711nomaximal, Sylow
120C71#147211yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—711no—
122#407—711no—
123C2×C22⋊C8—711no—
124#324—711no—
125#324—711no—
126#324—711no—
127#324D7111yescharacteristic, complemented
128C22×Q8—711no—
129C23×C4—711no—
130C2×M16—1421no—
131C22×C8—1421no—
132C22⋊Q8—1421no—
133C22⋊Q8—1421no—
134C22⋊Q8—1421no—
135C22⋊Q8—1421no—
136C2×C4⋊C4—711no—
137C2×C4⋊C4—711no—
138C2×C4⋊C4C2×D7111yescharacteristic
139C2×C22⋊C4—711no—
140C2×C22⋊C4—711no—
141C22⋊C8—711no—
142C22⋊C8—711no—
143C22⋊C8—1421no—
144C22⋊C8—1421no—
145C22⋊C8C2×D7111yescharacteristic
146C22⋊C8C2×D7111yescharacteristic
147C24—711no—
148C2×Q8—2842no—
149C2×Q8—1421no—
150C2×Q8—1421no—
151C22×C4—2842no—
152C22×C4—711no—
153C22×C4—711no—
154C22×C4—711no—
155C22×C4—711no—
156C22×C4—711no—
157C22×C4—711no—
158C22×C4—711no—
159C22×C4C4×D7111yescharacteristic
160C22×C4C4×D7111yescharacteristic
161C22×C4C22×D7111yescharacteristic
162M16—2842no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—2842no—
166C2×C8—1421no—
167C2×C8—1421no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—1421no—
171C4⋊C4—1421no—
172C4⋊C4—1421no—
173C4⋊C4—1421no—
174C22⋊C4—1421no—
175C22⋊C4—1421no—
176C22⋊C4—1421no—
177C22⋊C4—1421no—
178C23—2841no—
179C23—711no—
180C23—711no—
181C23—1422no—
182C23—1421no—
183C23—1421no—
184C23C2×C4×D7111yescharacteristic
185Q8—2842no—
186Q8—2842no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—2841no—
194C2×C4—2841no—
195C2×C4—2841no—
196C2×C4—711no—
197C2×C4—711no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1421no—
201C2×C4—1421no—
202C2×C4—1421no—
203C2×C4—1421no—
204C2×C4—1422no—
205C2×C4D4×D7111yescharacteristic
206C2×C4D4×D7111yescharacteristic
207C8—41no—
208C8—41no—
209C8—2841no—
210C8—2841no—
211C22—21no—
212C22—21no—
213C22—2842no—
214C22—2842no—
215C22—712no—
216C22—1421no—
217C22—1421no—
218C22—1421no—
219C22—1421no—
220C22—1421no—
221C22C22⋊C4×D7111yescharacteristic
222C22#1455613411yescharacteristic
223C22#1455613611yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—2841no—
229C4—2841no—
230C4—712no—
231C4—1421no—
232C2—21no—
233C2—21no—
234C2—2841no—
235C2—1422no—
236C2#1798596411yescharacteristic
237C2#99×D7111yescharacteristic
238C2#101×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC23×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#324×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#324D71
hypocenterC71#1472
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #17985725C2semidirect
5. #17985724C2semidirect
6. #17985724C2semidirect
7. #324×C71C2semidirect
120. C71#1472semidirect
127. #324D71semidirect