Finite Group Databasebeta

← #22593907

Subgroups — #22593907

Order 9088.

604 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 135 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×#14556090C211yesmaximal, characteristic, complemented
2C2×#14556084C211yesmaximal, characteristic, complemented
3#17985937C211yesmaximal, characteristic, complemented
4#17985917C211yesmaximal, characteristic, complemented
5#17985908C211yesmaximal, characteristic
6#17985889C212yesmaximal, complemented
7#17985876C212yesmaximal, complemented
8#17985875C212yesmaximal, complemented
9#17985870C212yesmaximal, complemented
10#17985866C211yesmaximal, characteristic, complemented
11#399×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C22⋊D71C2212yes—
14C23×Dic71C2211yescharacteristic
15C22×C4×D71C2212yes—
16#14556090—24no—
17#14556084—24no—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×C22⋊Dic71C2211yescharacteristic
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×C4×Dic71C2211yescharacteristic
26C2×C4×Dic71C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2212yes—
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yes—
36#95×C71C2211yescharacteristic
37C23×D71C2312yes—
38C2×D284—24no—
39C2×D284—22no—
40C2×C22⋊D71—24no—
41C2×C22⋊D71—24no—
42C2×C22⋊D71—42no—
43C22×Dic71—22no—
44C22×Dic71—22no—
45C22×Dic71—22no—
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C22×Dic71C2311yescharacteristic
50C22×Dic71C2311yescharacteristic
51C2×C4×D71—24no—
52C2×C4×D71—42no—
53C2×C4×D71—42no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56C22⋊Dic71—22no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—42no—
61C4×Dic71—22no—
62C4×Dic71D414yescomplemented
63C24×C71C2311yescharacteristic
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C4⋊C4×C71—22no—
69C4⋊C4×C71—42no—
70C22⋊C4×C71—22no—
71C22⋊C4×C71—42no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71#4112yes—
77C23×C71C2411yescharacteristic
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71#4114yes—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71C2×D412yes—
95C22⋊D71—48no—
96C2×Dic71—28no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71C2×D412yes—
108C2×Dic71C2×D412yes—
109C2×Dic71#4112yes—
110D284—28no—
111C4×D71—48no—
112C22×C71—22no—
113C22×C71—24no—
114C22×C71—41no—
115C22×C71—41no—
116C22×C71—42no—
117C22×C71C22×D411yescharacteristic
118C22×C71C2×#4112yes—
119C22×C71C2×#4111yescharacteristic
120C22×C71C2×#4111yescharacteristic
121C22×C71H(5,𝔽2)12yes—
122C2×D71—24no—
123C2×D71—24no—
124C2×D71—24no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—42no—
128C2×D71—42no—
129C4×C71—22no—
130C4×C71—42no—
131C4×C71—41no—
132C4×C71—42no—
133Dic71—22no—
134Dic71—22no—
135Dic71—22no—
136Dic71—41no—
137Dic71—42no—
138C2×C71—42no—
139C2×C71#52511yescharacteristic
140C2×C71#52511yescharacteristic
141C2×C71#53511yescharacteristic
142C2×C71#55112yes—
143C2×C71#55312yes—
144D71—44no—
145#2598—711nomaximal, Sylow
146C71#259811yescharacteristic, complemented, Sylow
147C2×C4×D4—711no—
148C2×#117—711no—
149#400—711no—
150#399D7111yescharacteristic, complemented
151#394—711no—
152#392—711no—
153#388—712no—
154#386—712no—
155#386—712no—
156#385—712no—
157#383—711no—
158C22×D4—712no—
159C22×D4—711no—
160C23×C4—712no—
161C23×C4—711no—
162C4×D4—1424no—
163#117—1424no—
164C2×C4⋊C4—711no—
165C2×C4⋊C4—711no—
166C2×C4⋊C4C2×D7111yescharacteristic
167C2×C4⋊C4C2×D7112yes—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—711no—
170C2×C22⋊C4—712no—
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4C2×D7111yescharacteristic
174C2×C22⋊C4C2×D7112yes—
175C2×C42—711no—
176C2×C42—711no—
177#95—712no—
178#95—712no—
179#95—711no—
180#95—712no—
181#95—712no—
182#95C2×D7111yescharacteristic
183C24—712no—
184C24C22×D7111yescharacteristic
185C2×D4—2842no—
186C2×D4—1424no—
187C2×D4—1424no—
188C2×D4—1424no—
189C2×D4—1422no—
190C22×C4—2842no—
191C22×C4—2842no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—711no—
195C22×C4—712no—
196C22×C4—711no—
197C22×C4—1424no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4—1422no—
201C22×C4C22×D7112yes—
202C22×C4C22×D7112yes—
203C22×C4C22×D7111yescharacteristic
204C22×C4C22×D7111yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—1422no—
208C4⋊C4—1422no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—2842no—
212C22⋊C4—1424no—
213C22⋊C4—1424no—
214C22⋊C4—1424no—
215C22⋊C4—1422no—
216C42—714no—
217C42—1422no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—714no—
223C23—1422no—
224C23—1422no—
225C23—1422no—
226C23—1422no—
227C23—1422no—
228C23—1422no—
229C23#1176586612yes—
230C23C23×D7111yescharacteristic
231D4—2848no—
232D4—1428no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—2848no—
243C2×C4—712no—
244C2×C4—712no—
245C2×C4—712no—
246C2×C4—1421no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4—1428no—
252C2×C4—1421no—
253C2×C4—1421no—
254C2×C4—1421no—
255C2×C4—1421no—
256C2×C4—1422no—
257C2×C4D4×D7112yes—
258C22—24no—
259C22—22no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—2842no—
264C22—2842no—
265C22—2842no—
266C22—2842no—
267C22—1424no—
268C22—1424no—
269C22—1424no—
270C22C2×D4×D7111yescharacteristic
271C22#41×D7112yes—
272C22#41×D7111yescharacteristic
273C22C2×#1176586611yescharacteristic
274C22#1455617312yes—
275C4—22no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—2842no—
280C4—2841no—
281C4—1422no—
282C4—1422no—
283C4—1422no—
284C2—42no—
285C2—2844no—
286C2#1798650711yescharacteristic
287C2#1798651211yescharacteristic
288C2#1798656211yescharacteristic
289C2#1798667612yes—
290C2#1798669412yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2598
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556090C2semidirect
2. C2×#14556084C2semidirect
3. #17985937C2semidirect
4. #17985917C2semidirect
6. #17985889C2semidirect
7. #17985876C2semidirect
8. #17985875C2semidirect
9. #17985870C2semidirect
10. #17985866C2semidirect
11. #399×C71C2semidirect
62. C4×Dic71D4semidirect
146. C71#2598semidirect
150. #399D71semidirect