Finite Group Databasebeta

← #22596396

Subgroups — #22596396

Order 9088.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556119C211yesmaximal, characteristic, complemented
2C2×#14556114C211yesmaximal, characteristic
3#17985936C211yesmaximal, characteristic, complemented
4#17985935C211yesmaximal, characteristic, complemented
5#17985925C211yesmaximal, characteristic, complemented
6#17985923C212yesmaximal, complemented
7#17985922C212yesmaximal, complemented
8#17985920C212yesmaximal, complemented
9#17985915C212yesmaximal
10#17985863C211yesmaximal, characteristic, complemented
11#397×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14#14556119—24no—
15#14556114—24no—
16C2×C4⋊Dic71C2212yes—
17C2×C4⋊Dic71C2211yescharacteristic
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2211yescharacteristic
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—42no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—42no—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2311yescharacteristic
47C4⋊Dic71—22no—
48C4⋊Dic71—24no—
49C4⋊Dic71—42no—
50Dic71⋊C4—24no—
51Dic71⋊C4—42no—
52Dic71⋊C4—42no—
53C22⋊Dic71—22no—
54C22⋊Dic71—42no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—42no—
58(C2×D71)⋊C4—42no—
59C4×Dic71D414yes—
60C24×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2312yes—
63C22×C4×C71C2312yes—
64C22×C4×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C22⋊C4×C71—22no—
67C22⋊C4×C71—42no—
68C23×C71—21no—
69C23×C71—21no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—21no—
73C23×C71C2411yescharacteristic
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71C2×D412yes—
89C22⋊D71—44no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71C2×D412yes—
102C2×Dic71C2×D412yes—
103Dic142—44no—
104C22×C71—41no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71—42no—
108C22×C71—42no—
109C22×C71C22×D411yescharacteristic
110C22×C71H(5,𝔽2)11yescharacteristic
111C22×C71H(5,𝔽2)12yes—
112C22×C712−1+412yes—
113C22×C712−1+411yescharacteristic
114C2×D71—41no—
115C2×D71—41no—
116C2×D71—41no—
117C2×D71—42no—
118C2×D71—42no—
119C4×C71—22no—
120C4×C71—41no—
121C4×C71—42no—
122C4×C71—42no—
123Dic71—22no—
124Dic71—22no—
125Dic71—41no—
126Dic71—42no—
127Dic71—42no—
128C2×C71—81no—
129C2×C71#53511yescharacteristic
130C2×C71#53611yescharacteristic
131C2×C71#53711yescharacteristic
132C2×C71#56212yes—
133C2×C71#56312yes—
134D71—81no—
135#2820—711nomaximal, Sylow
136C71#282011yescharacteristic, complemented, Sylow
137C2×#125—711no—
138C2×#124—711no—
139#399—711no—
140#399—711no—
141#397—711no—
142#397D7111yescharacteristic, complemented
143#396—712no—
144#396—712no—
145#395—712no—
146#394—712no—
147#382—711no—
148C22×Q8—711no—
149C22×D4—711no—
150#125—1424no—
151#124—1424no—
152C2×C4⋊C4—712no—
153C2×C4⋊C4—712no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4C2×D7111yescharacteristic
159C2×C22⋊C4—712no—
160C2×C22⋊C4—712no—
161C2×C22⋊C4—712no—
162C2×C22⋊C4—711no—
163C2×C22⋊C4—711no—
164C2×C22⋊C4—711no—
165C2×C22⋊C4C2×D7111yescharacteristic
166C2×C22⋊C4C2×D7112yes—
167C2×C42—711no—
168#95—712no—
169#95—711no—
170#95—712no—
171#95C2×D7111yescharacteristic
172#95C2×D7112yes—
173C24—711no—
174C24C22×D7111yescharacteristic
175C2×Q8—2842no—
176C2×Q8—1422no—
177C2×D4—2842no—
178C2×D4—1422no—
179C22×C4—711no—
180C22×C4—711no—
181C22×C4—712no—
182C22×C4—711no—
183C22×C4—712no—
184C22×C4C22×D7112yes—
185C22×C4C22×D7112yes—
186C22×C4C22×D7111yescharacteristic
187C22×C4C22×D7111yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—2842no—
190C4⋊C4—2842no—
191C4⋊C4—2842no—
192C4⋊C4—1424no—
193C4⋊C4—1424no—
194C4⋊C4—1422no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—2842no—
198C22⋊C4—2842no—
199C22⋊C4—2842no—
200C22⋊C4—1422no—
201C22⋊C4—1422no—
202C22⋊C4—1422no—
203C42—714no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—1421no—
210C23—1421no—
211C23—1422no—
212C23—1422no—
213C23—1421no—
214C23C23×D7111yescharacteristic
215Q8—2844no—
216D4—2844no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—712no—
227C2×C4—712no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4—1421no—
237C2×C4—1422no—
238C2×C4—1421no—
239C2×C4D4×D7112yes—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—2841no—
246C22—2841no—
247C22—2841no—
248C22—2842no—
249C22—2842no—
250C22C2×D4×D7111yescharacteristic
251C22#1455617311yescharacteristic
252C22#1455617412yes—
253C22#1455617512yes—
254C22#1455617511yescharacteristic
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—2841no—
260C4—2842no—
261C4—2842no—
262C4—1422no—
263C4—1422no—
264C2—81no—
265C2—5681no—
266C2#1798656711yescharacteristic
267C2#1798657111yescharacteristic
268C2#1798657811yescharacteristic
269C2#1798674612yes—
270C2#1798675112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2820
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14556119C2semidirect
3. #17985936C2semidirect
4. #17985935C2semidirect
5. #17985925C2semidirect
6. #17985923C2semidirect
7. #17985922C2semidirect
8. #17985920C2semidirect
10. #17985863C2semidirect
11. #397×C71C2semidirect
136. C71#2820semidirect
142. #397D71semidirect