Finite Group Databasebeta

← #22595310

Subgroups — #22595310

Order 9088.

512 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14556106C212yesmaximal, complemented
2#17985940C212yesmaximal, complemented
3#17985928C212yesmaximal, complemented
4#17985916C211yesmaximal, characteristic
5#17985904C211yesmaximal, characteristic, complemented
6#17985889C212yesmaximal, complemented
7#17985877C212yesmaximal
8#17985870C212yesmaximal, complemented
9#398×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12C22×C4×D71C2212yes—
13#14556106—28no—
14C2×Dic71⋊C4C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2211yescharacteristic
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2211yescharacteristic
27C2×C4⋊C4×C71C2212yes—
28C2×C4⋊C4×C71C2211yescharacteristic
29C2×C4⋊C4×C71C2211yescharacteristic
30#95×C71C2212yes—
31#95×C71C2211yescharacteristic
32C23×D71C2312yes—
33C2×Dic142—41no—
34C2×Dic142—41no—
35C2×Dic142D414yescomplemented
36C2×D284—24no—
37C2×D284—41no—
38C22×Dic71C2312yes—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C2×C4×D71—24no—
42C2×C4×D71—42no—
43C2×C4×D71—42no—
44Dic71⋊C4—24no—
45Dic71⋊C4—24no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—24no—
48(C2×D71)⋊C4—24no—
49(C2×D71)⋊C4—24no—
50(C2×D71)⋊C4—42no—
51C4×Dic71—41no—
52C22×C4×C71C2312yes—
53C22×C4×C71C2311yescharacteristic
54C22×C4×C71C2312yes—
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C4⋊C4×C71—24no—
58C4⋊C4×C71—41no—
59C4⋊C4×C71—41no—
60C23×C71C2411yescharacteristic
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71—22no—
67C22×D71#4114yes—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D412yes—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71C2×D414yes—
90D284—44no—
91C4×D71—48no—
92Dic142—44no—
93C22×C71C22×D411yescharacteristic
94C22×C71C2×#4112yes—
95C22×C71H(5,𝔽2)11yescharacteristic
96C22×C71H(5,𝔽2)11yescharacteristic
97C22×C712−1+412yes—
98C2×D71—24no—
99C2×D71—24no—
100C2×D71—24no—
101C2×D71—42no—
102C2×D71—42no—
103C2×D71—42no—
104C2×D71—42no—
105C4×C71—22no—
106C4×C71—42no—
107C4×C71—42no—
108C4×C71—41no—
109C4×C71—41no—
110Dic71—24no—
111Dic71—42no—
112Dic71—42no—
113C2×C71#53411yescharacteristic
114C2×C71#53912yes—
115C2×C71#54812yes—
116C2×C71#55111yescharacteristic
117C2×C71#56311yescharacteristic
118D71—44no—
119#2728—711nomaximal, Sylow
120C71#272811yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—712no—
122#400—712no—
123#398D7111yescharacteristic, complemented
124#397—712no—
125#394—711no—
126#391—711no—
127#388—712no—
128#386—712no—
129#385—712no—
130C22×Q8—711no—
131C22×D4—711no—
132C23×C4—712no—
133C22⋊Q8—1428no—
134C2×C4⋊C4—712no—
135C2×C4⋊C4—712no—
136C2×C4⋊C4C2×D7112yes—
137C2×C4⋊C4C2×D7111yescharacteristic
138C2×C4⋊C4C2×D7111yescharacteristic
139C2×C22⋊C4—712no—
140C2×C22⋊C4—712no—
141C2×C22⋊C4—712no—
142C2×C22⋊C4—712no—
143C2×C22⋊C4—712no—
144C2×C42—711no—
145#95—712no—
146#95—712no—
147#95—712no—
148#95—712no—
149#95—711no—
150#95C2×D7112yes—
151#95C2×D7111yescharacteristic
152C24—712no—
153C2×Q8—2841no—
154C2×Q8—2841no—
155C2×Q8—714no—
156C2×D4—2841no—
157C2×D4—1424no—
158C22×C4—2842no—
159C22×C4—2842no—
160C22×C4—712no—
161C22×C4—712no—
162C22×C4—712no—
163C22×C4—1424no—
164C22×C4C22×D7111yescharacteristic
165C22×C4C22×D7111yescharacteristic
166C22×C4C22×D7112yes—
167C22×C4C22×D7112yes—
168C22×C4C22×D7111yescharacteristic
169C4⋊C4—24no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1424no—
173C4⋊C4—1424no—
174C22⋊C4—2842no—
175C22⋊C4—1424no—
176C22⋊C4—1424no—
177C22⋊C4—1424no—
178C22⋊C4—1424no—
179C42—2841no—
180C23—714no—
181C23—1422no—
182C23—1422no—
183C23—1422no—
184C23—1422no—
185C23—1422no—
186C23—1422no—
187C23C23×D7111yescharacteristic
188Q8—2844no—
189D4—2844no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—2848no—
203C2×C4—714no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4—1422no—
211C2×C4—1422no—
212C2×C4D4×D7112yes—
213C22—2842no—
214C22—2842no—
215C22—2842no—
216C22—2842no—
217C22—1424no—
218C22—1424no—
219C22—1424no—
220C22C2×D4×D7111yescharacteristic
221C22#41×D7112yes—
222C22#1455617311yescharacteristic
223C22#1455617411yescharacteristic
224C22#1455617612yes—
225C4—22no—
226C4—42no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—2842no—
231C4—2842no—
232C4—1424no—
233C2—2844no—
234C2#1798655711yescharacteristic
235C2#1798659712yes—
236C2#1798665912yes—
237C2#1798667711yescharacteristic
238C2#1798674811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2728
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14556106C2semidirect
2. #17985940C2semidirect
3. #17985928C2semidirect
5. #17985904C2semidirect
6. #17985889C2semidirect
8. #17985870C2semidirect
9. #398×C71C2semidirect
35. C2×Dic142D4semidirect
120. C71#2728semidirect
123. #398D71semidirect