Finite Group Databasebeta

← #22592799

Subgroups — #22592799

Order 9088.

568 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556103C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic, complemented
3#17985916C214yesmaximal, complemented
4#17985901C211yesmaximal, characteristic, complemented
5#17985900C211yesmaximal, characteristic, complemented
6#17985889C214yesmaximal, complemented
7#17985864C211yesmaximal, characteristic, complemented
8C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
9#384×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×D284C2212yes—
12C22×C4×D71C2211yescharacteristic
13#14556103—24no—
14#14556086—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2214yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2214yes—
21C2×C4×Dic71C2211yescharacteristic
22C2×C4×Dic71C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2214yes—
25C2×C4⋊C4×C71C2211yescharacteristic
26C2×C4⋊C4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2214yes—
28C2×C42×C71C2211yescharacteristic
29C23×D71C2312yes—
30C23×D71C2311yescharacteristic
31C2×D284—24no—
32C2×D284—22no—
33C2×D284—22no—
34C2×D284—22no—
35C2×D284—44no—
36C22×Dic71C2312yes—
37C22×Dic71C2312yes—
38C22×Dic71C2311yescharacteristic
39C2×C4×D71—22no—
40C2×C4×D71—22no—
41C2×C4×D71D414yes—
42C4⋊Dic71—22no—
43(C2×D71)⋊C4—24no—
44(C2×D71)⋊C4—24no—
45(C2×D71)⋊C4—28no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—44no—
48C4×Dic71—22no—
49C4×Dic71—24no—
50C22×C4×C71C2312yes—
51C22×C4×C71C2312yes—
52C22×C4×C71C2312yes—
53C22×C4×C71C2311yescharacteristic
54C4⋊C4×C71—22no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—44no—
57C42×C71—22no—
58C23×C71C2411yescharacteristic
59C22×D71—22no—
60C22×D71—21no—
61C22×D71—21no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—24no—
65C22×D71—22no—
66C22×D71—22no—
67C22×D71—24no—
68C22×D71C2×D412yes—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—24no—
74C2×C4×C71—24no—
75C2×C4×C71C2×D412yes—
76C2×C4×C71#4112yes—
77C2×C4×C71#4114yes—
78C2×C4×C71#4112yes—
79C2×Dic71—21no—
80C2×Dic71—21no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—24no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71C2×D412yes—
87D284—48no—
88D284—44no—
89C4×D71—28no—
90C22×C71C22×D411yescharacteristic
91C22×C71C2×#4111yescharacteristic
92C22×C71C2×#4111yescharacteristic
93C22×C71C2×#4112yes—
94C22×C71H(5,𝔽2)12yes—
95C2×D71—22no—
96C2×D71—22no—
97C2×D71—22no—
98C2×D71—42no—
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—42no—
103C2×D71—44no—
104C2×D71—44no—
105C4×C71—24no—
106C4×C71—42no—
107C4×C71—42no—
108C4×C71#12314yes—
109Dic71—22no—
110Dic71—42no—
111Dic71—42no—
112C2×C71C2×#12311yescharacteristic
113C2×C71#53211yescharacteristic
114C2×C71#53511yescharacteristic
115C2×C71#55312yes—
116C2×C71#55512yes—
117D71—42no—
118D71—82no—
119#2500—711nomaximal, Sylow
120C71#250011yescharacteristic, complemented, Sylow
121C2×#121—711no—
122C2×#117—711no—
123#394—714no—
124#390—711no—
125#390—711no—
126#388—714no—
127#384D7111yescharacteristic, complemented
128#382—711no—
129C4×C22⋊C4—711no—
130C22×D4—712no—
131C22×D4—711no—
132C23×C4—711no—
133#121—1424no—
134#117—1424no—
135C2×C4⋊C4—711no—
136C2×C4⋊C4C2×D7111yescharacteristic
137C2×C4⋊C4C2×D7111yescharacteristic
138C2×C4⋊C4C2×D7114yes—
139C2×C22⋊C4—714no—
140C2×C22⋊C4—712no—
141C2×C22⋊C4—712no—
142C2×C22⋊C4—712no—
143C2×C22⋊C4—714no—
144C2×C42—712no—
145C2×C42—711no—
146C2×C42C2×D7111yescharacteristic
147#95—712no—
148#95—714no—
149C24—711no—
150C24—712no—
151C2×D4—2844no—
152C2×D4—1424no—
153C2×D4—1422no—
154C2×D4—1422no—
155C2×D4—1422no—
156C22×C4—711no—
157C22×C4—714no—
158C22×C4—712no—
159C22×C4—712no—
160C22×C4—1422no—
161C22×C4—1422no—
162C22×C4C22×D7112yes—
163C22×C4C22×D7112yes—
164C22×C4C22×D7112yes—
165C22×C4C22×D7111yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—44no—
169C4⋊C4—1422no—
170C22⋊C4—2844no—
171C22⋊C4—1424no—
172C22⋊C4—1424no—
173C22⋊C4—1428no—
174C22⋊C4—1424no—
175C42—22no—
176C42—1424no—
177C42—1422no—
178C23—712no—
179C23—1422no—
180C23—1422no—
181C23—1424no—
182C23—1422no—
183C23—1424no—
184C23—1421no—
185C23—1422no—
186C23—1421no—
187C23—1422no—
188C23C23×D7111yescharacteristic
189D4—2848no—
190D4—2844no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—712no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1424no—
202C2×C4—1422no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4—1428no—
206C2×C4D4×D7112yes—
207C2×C4#1176586512yes—
208C2×C4#1176586614yes—
209C2×C4#1176586612yes—
210C22—2842no—
211C22—2842no—
212C22—2842no—
213C22—2844no—
214C22—2844no—
215C22—2842no—
216C22—2842no—
217C22—1422no—
218C22—1422no—
219C22—1422no—
220C22C2×D4×D7111yescharacteristic
221C22C2×#1176586511yescharacteristic
222C22C2×#1176586612yes—
223C22C2×#1176586611yescharacteristic
224C22#1455617312yes—
225C4—24no—
226C4—42no—
227C4—42no—
228C4—2842no—
229C4—2842no—
230C4—1422no—
231C4#1455611314yes—
232C2—2842no—
233C2—5682no—
234C2C2×#1455611311yescharacteristic
235C2#1798655011yescharacteristic
236C2#1798656211yescharacteristic
237C2#1798669712yes—
238C2#1798671512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#384×C71C2
hypercenter#384D71
hypocenterC71#2500
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#14556086C2semidirect
3. #17985916C2semidirect
4. #17985901C2semidirect
5. #17985900C2semidirect
6. #17985889C2semidirect
7. #17985864C2semidirect
8. C4×(C2×D71)⋊C4C2semidirect
9. #384×C71C2semidirect
120. C71#2500semidirect
127. #384D71semidirect