Finite Group Databasebeta

← #22593246

Subgroups — #22593246

Order 9088.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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
2C2×#14556093C211yesmaximal, characteristic, complemented
3C2×#14556090C211yesmaximal, characteristic
4#17985930C211yesmaximal, characteristic, complemented
5#17985917C211yesmaximal, characteristic
6#17985916C211yesmaximal, characteristic
7#17985889C212yesmaximal, complemented
8#17985888C212yesmaximal, complemented
9#17985863C211yesmaximal, characteristic, complemented
10#17985858C212yesmaximal, complemented
11#398×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C4×D71C2212yes—
15#14556106—28no—
16#14556093—24no—
17#14556090—24no—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2211yescharacteristic
28C2×C4×Dic71C2211yescharacteristic
29C2×C4×Dic71C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C71⋊#95C2211yescharacteristic
33C2×C4⋊C4×C71C2212yes—
34C2×C4⋊C4×C71C2211yescharacteristic
35C2×C4⋊C4×C71C2211yescharacteristic
36#95×C71C2212yes—
37#95×C71C2211yescharacteristic
38C23×D71C2312yes—
39C2×Dic142—22no—
40C2×Dic142—22no—
41C2×Dic142D414yescomplemented
42C2×D284—24no—
43C2×D284—22no—
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—24no—
50C2×C4×D71—24no—
51C2×C4×D71—24no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—24no—
55Dic71⋊C4—24no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61C4×Dic71—22no—
62C4×Dic71—22no—
63C4×Dic71D414yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C4⋊C4×C71—24no—
72C23×C71C2411yescharacteristic
73C22×D71—22no—
74C22×D71—22no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—22no—
79C22×D71#4114yes—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71C2×D412yes—
92C2×C4×C71C2×D412yes—
93C2×Dic71—21no—
94C2×Dic71—21no—
95C2×Dic71—22no—
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71C2×D412yes—
102C2×Dic71C2×D412yes—
103C2×Dic71C2×D412yes—
104C2×Dic71C2×D412yes—
105C2×Dic71#4112yes—
106D284—28no—
107C4×D71—48no—
108Dic142—28no—
109C22×C71C22×D411yescharacteristic
110C22×C71C22×D411yescharacteristic
111C22×C71C2×#4111yescharacteristic
112C22×C71C2×#4112yes—
113C22×C712−1+412yes—
114C2×D71—24no—
115C2×D71—24no—
116C2×D71—24no—
117C2×D71—42no—
118C2×D71—42no—
119C2×D71—42no—
120C2×D71—42no—
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—42no—
124C4×C71—41no—
125C4×C71—42no—
126Dic71—22no—
127Dic71—22no—
128Dic71—41no—
129Dic71—42no—
130Dic71#12114yes—
131C2×C71C2×#12111yescharacteristic
132C2×C71#52511yescharacteristic
133C2×C71#53611yescharacteristic
134C2×C71#54812yes—
135C2×C71#55212yes—
136D71—44no—
137#2536—711nomaximal, Sylow
138C71#253611yescharacteristic, complemented, Sylow
139C2×C22⋊Q8—712no—
140C2×C4×Q8—711no—
141C2×C4×D4—711no—
142#398—711no—
143#398D7111yescharacteristic, complemented
144#394—711no—
145#394—711no—
146#388—712no—
147#388—712no—
148#382—711no—
149#380—712no—
150C22×Q8—711no—
151C22×D4—711no—
152C23×C4—712no—
153C22⋊Q8—1428no—
154C4×Q8—1424no—
155C4×D4—1424no—
156C2×C4⋊C4—711no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4—712no—
159C2×C4⋊C4—712no—
160C2×C4⋊C4C2×D7111yescharacteristic
161C2×C4⋊C4C2×D7111yescharacteristic
162C2×C4⋊C4C2×D7112yes—
163C2×C22⋊C4—712no—
164C2×C22⋊C4—712no—
165C2×C22⋊C4—712no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—712no—
168C2×C42—711no—
169C2×C42—711no—
170C2×C42—711no—
171#95—712no—
172#95—712no—
173#95—711no—
174#95C2×D7112yes—
175#95C2×D7111yescharacteristic
176C24—712no—
177C2×Q8—714no—
178C2×Q8—1422no—
179C2×Q8—1422no—
180C2×D4—1424no—
181C2×D4—1422no—
182C22×C4—711no—
183C22×C4—711no—
184C22×C4—711no—
185C22×C4—712no—
186C22×C4—711no—
187C22×C4—1424no—
188C22×C4—1424no—
189C22×C4—1424no—
190C22×C4C22×D7112yes—
191C22×C4C22×D7112yes—
192C22×C4C22×D7111yescharacteristic
193C22×C4C22×D7111yescharacteristic
194C22×C4C22×D7111yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—1422no—
199C4⋊C4—1424no—
200C4⋊C4—1424no—
201C4⋊C4—1422no—
202C22⋊C4—1424no—
203C22⋊C4—1424no—
204C22⋊C4—1424no—
205C22⋊C4—1424no—
206C22⋊C4—1424no—
207C42—714no—
208C42—1422no—
209C42—1422no—
210C23—714no—
211C23—1422no—
212C23—1422no—
213C23—1422no—
214C23—1422no—
215C23—1422no—
216C23—1422no—
217C23C23×D7111yescharacteristic
218Q8—1428no—
219D4—1428no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—2848no—
232C2×C4—712no—
233C2×C4—712no—
234C2×C4—712no—
235C2×C4—712no—
236C2×C4—712no—
237C2×C4—1421no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4—1422no—
242C2×C4—1421no—
243C2×C4—1421no—
244C2×C4—1422no—
245C2×C4D4×D7112yes—
246C2×C4D4×D7112yes—
247C22—2842no—
248C22—2842no—
249C22—2842no—
250C22—2842no—
251C22—1424no—
252C22—1424no—
253C22—1424no—
254C22C2×D4×D7111yescharacteristic
255C22C2×D4×D7111yescharacteristic
256C22#41×D7112yes—
257C22#41×D7111yescharacteristic
258C22#1455617612yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—2842no—
265C4—2841no—
266C4—714no—
267C4—1422no—
268C4—1422no—
269C2—2844no—
270C2#121×D7111yescharacteristic
271C2#1798650711yescharacteristic
272C2#1798657411yescharacteristic
273C2#1798665912yes—
274C2#1798668112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2536
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. C2×#14556093C2semidirect
4. #17985930C2semidirect
7. #17985889C2semidirect
8. #17985888C2semidirect
9. #17985863C2semidirect
10. #17985858C2semidirect
11. #398×C71C2semidirect
41. C2×Dic142D4semidirect
138. C71#2536semidirect
143. #398D71semidirect