Finite Group Databasebeta

← #22594159

Subgroups — #22594159

Order 9088.

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

normal subgroups only

This group has 508 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×#14556105C211yesmaximal, characteristic
2C22×Dic71⋊C4C211yesmaximal, characteristic
3#17985940C212yesmaximal, complemented
4#17985933C212yesmaximal, complemented
5#17985930C212yesmaximal, complemented
6#17985919C212yesmaximal, complemented
7#17985872C211yesmaximal, characteristic
8#17985860C211yesmaximal, characteristic
9#17985860C211yesmaximal, characteristic
10#17985857C211yesmaximal, characteristic
11#385×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13C23×Dic71C2211yescharacteristic
14C23×Dic71C2211yescharacteristic
15#14556105—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×Dic71⋊C4—24no—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×Dic71⋊C4C2212yes—
24C2×C22⋊Dic71C2211yescharacteristic
25C2×C22⋊Dic71C2211yescharacteristic
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C23×C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2212yes—
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2211yescharacteristic
36#95×C71C2212yes—
37C2×Dic142—22no—
38C2×Dic142—42no—
39C22×Dic71—22no—
40C22×Dic71—22no—
41C22×Dic71—42no—
42C22×Dic71D414yes—
43C22×Dic71Q814yescomplemented
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C22×Dic71C2312yes—
50C4⋊Dic71—22no—
51Dic71⋊C4—24no—
52Dic71⋊C4—24no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4—42no—
56Dic71⋊C4—42no—
57C22⋊Dic71—22no—
58C22⋊Dic71—22no—
59C24×C71C2311yescharacteristic
60C22×C4×C71—22no—
61C22×C4×C71—42no—
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C4⋊C4×C71—42no—
67C22⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71C2×D412yes—
72C23×C71C2×Q812yes—
73C23×C71#4112yes—
74C23×C71C2411yescharacteristic
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—44no—
86C2×Dic71—28no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—44no—
99C2×Dic71C2×D412yes—
100C2×Dic71C2×D412yes—
101C2×Dic71C2×Q814yes—
102Dic142—44no—
103C22×C71—22no—
104C22×C71—22no—
105C22×C71—24no—
106C22×C71—24no—
107C22×C71C22⋊Q814yes—
108C22×C71C22×D411yescharacteristic
109C22×C71C22×Q811yescharacteristic
110C22×C71C2×#4111yescharacteristic
111C22×C71H(5,𝔽2)12yes—
112C22×C712−1+412yes—
113C4×C71—41no—
114C4×C71—42no—
115C4×C71—41no—
116C4×C71—42no—
117Dic71—24no—
118Dic71—22no—
119Dic71—22no—
120Dic71—41no—
121Dic71—41no—
122Dic71—42no—
123C2×C71—24no—
124C2×C71C2×C22⋊Q811yescharacteristic
125C2×C71#53411yescharacteristic
126C2×C71#53611yescharacteristic
127C2×C71#53912yes—
128C2×C71#54412yes—
129#2627—711nomaximal, Sylow
130C71#262711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—711no—
132C22×C4⋊C4—711no—
133#400—712no—
134#398—712no—
135#398—712no—
136#395—712no—
137#385—711no—
138#385D7111yescharacteristic
139#381—711no—
140#381—711no—
141#380—711no—
142C22×Q8—711no—
143C23×C4—711no—
144C23×C4—711no—
145C23×C4C2×D7111yescharacteristic
146C22⋊Q8—1424no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—712no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—712no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—1424no—
155C2×C4⋊C4C2×D7112yes—
156C2×C22⋊C4—711no—
157C2×C22⋊C4—711no—
158C2×C22⋊C4C2×D7111yescharacteristic
159C2×C22⋊C4C2×D7111yescharacteristic
160#95—712no—
161#95—712no—
162#95—712no—
163#95—712no—
164#95—712no—
165#95—712no—
166#95C2×D7112yes—
167C24C22×D7111yescharacteristic
168C2×Q8—2842no—
169C2×Q8—1422no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—2842no—
173C22×C4—714no—
174C22×C4—714no—
175C22×C4—712no—
176C22×C4—711no—
177C22×C4—711no—
178C22×C4—711no—
179C22×C4—712no—
180C22×C4—711no—
181C22×C4—1422no—
182C22×C4—1422no—
183C22×C4C22×D7112yes—
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186C22×C4C22×D7112yes—
187C4⋊C4—42no—
188C4⋊C4—2842no—
189C4⋊C4—2842no—
190C4⋊C4—1424no—
191C4⋊C4—1424no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—1422no—
198C22⋊C4—1422no—
199C23—22no—
200C23—22no—
201C23D4×D7112yes—
202C23Q8×D7112yes—
203C23#1176586512yes—
204C23C23×D7111yescharacteristic
205Q8—2844no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—2844no—
218C2×C4—712no—
219C2×C4—712no—
220C2×C4—714no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1428no—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—22no—
237C22#1455610614yes—
238C22C2×D4×D7111yescharacteristic
239C22C2×Q8×D7111yescharacteristic
240C22C2×#1176586511yescharacteristic
241C22#1455617312yes—
242C22#1455617512yes—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—41no—
247C4—2841no—
248C4—2842no—
249C4—2841no—
250C4—1424no—
251C4—1422no—
252C4—1422no—
253C2—24no—
254C2C2×#1455610611yescharacteristic
255C2#1798655911yescharacteristic
256C2#1798657111yescharacteristic
257C2#1798659112yes—
258C2#1798662812yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#385×C71C2
hypercenter#385D71
hypocenterC71#2627
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #17985940C2semidirect
4. #17985933C2semidirect
5. #17985930C2semidirect
6. #17985919C2semidirect
43. C22×Dic71Q8semidirect
130. C71#2627semidirect