Finite Group Databasebeta

← #22592806

Subgroups — #22592806

Order 9088.

708 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14556113C212yesmaximal, complemented
2C2×#14556110C212yesmaximal, complemented
3C2×#14556101C211yesmaximal, characteristic
4C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
5#17985936C211yesmaximal, characteristic, complemented
6#17985935C211yesmaximal, characteristic, complemented
7#17985934C211yesmaximal, characteristic, complemented
8#17985899C212yesmaximal, complemented
9#17985868C212yesmaximal, complemented
10#17985846C211yesmaximal, characteristic, complemented
11#399×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C4×D71C2212yes—
16C22×C4×D71C2211yescharacteristic
17#14556113—28no—
18#14556110—28no—
19#14556101—24no—
20C2×C4⋊Dic71C2212yes—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×Dic71⋊C4C2211yescharacteristic
23C2×Dic71⋊C4C2212yes—
24C2×Dic71⋊C4C2212yes—
25C2×Dic71⋊C4C2212yes—
26C4⋊C4×D71—24no—
27C2×C22⋊Dic71C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2212yes—
32C2×C4×Dic71C2212yes—
33C71⋊#95C2211yescharacteristic
34C2×C4⋊C4×C71C2212yes—
35C2×C4⋊C4×C71C2211yescharacteristic
36C2×C22⋊C4×C71C2212yes—
37C2×C22⋊C4×C71C2211yescharacteristic
38#95×C71C2211yescharacteristic
39C23×D71C2311yescharacteristic
40C23×D71C2311yescharacteristic
41C2×D284—22no—
42C2×D284—24no—
43C2×C22⋊D71—22no—
44C2×C22⋊D71—24no—
45C2×C22⋊D71—22no—
46C2×C22⋊D71—24no—
47C22×Dic71C2312yes—
48C22×Dic71C2311yescharacteristic
49C22×Dic71C2312yes—
50C22×Dic71C2311yescharacteristic
51C2×C4×D71—24no—
52C2×C4×D71—24no—
53C2×C4×D71—24no—
54C2×C4×D71—24no—
55C2×C4×D71D414yes—
56C4⋊Dic71—22no—
57C4⋊Dic71—24no—
58Dic71⋊C4—24no—
59Dic71⋊C4—22no—
60Dic71⋊C4—24no—
61Dic71⋊C4—24no—
62C22⋊Dic71—24no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—24no—
65(C2×D71)⋊C4—24no—
66(C2×D71)⋊C4D418yescomplemented
67C4×Dic71—24no—
68C24×C71C2311yescharacteristic
69C22×C4×C71C2312yes—
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C22×C4×C71C2312yes—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—24no—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—24no—
77C23×C71—22no—
78C23×C71—21no—
79C23×C71—22no—
80C23×C71—21no—
81C23×C71—21no—
82C23×C71C2411yescharacteristic
83C22×D71—22no—
84C22×D71—21no—
85C22×D71—21no—
86C22×D71—22no—
87C22×D71—21no—
88C22×D71—22no—
89C22×D71—21no—
90C22×D71—21no—
91C22×D71C2×D414yes—
92C22×D71C2×D412yes—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—21no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71C2×D414yes—
102C2×C4×C71C2×D412yes—
103C22⋊D71—44no—
104C22⋊D71—44no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71C2×D414yes—
113C2×Dic71C2×D412yes—
114C2×Dic71#4114yes—
115D284—44no—
116C4×D71—28no—
117C4×D71—48no—
118C22×C71—41no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71—42no—
122C22×C71—42no—
123C22×C71C22×D411yescharacteristic
124C22×C71C22×D412yes—
125C22×C71C2×#4112yes—
126C22×C712−1+411yescharacteristic
127C22×C712−1+411yescharacteristic
128C2×D71—22no—
129C2×D71—22no—
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—41no—
133C2×D71—41no—
134C2×D71—41no—
135C2×D71—42no—
136C2×D71—42no—
137C2×D71#12714yes—
138C4×C71—22no—
139C4×C71—24no—
140C4×C71—42no—
141C4×C71—41no—
142Dic71—24no—
143Dic71—42no—
144Dic71—41no—
145Dic71#12314yescomplemented
146C2×C71—81no—
147C2×C71C2×#12311yescharacteristic
148C2×C71C2×#12711yescharacteristic
149C2×C71#53611yescharacteristic
150C2×C71#54712yes—
151C2×C71#54712yes—
152D71—24no—
153D71—81no—
154#2501—711nomaximal, Sylow
155C71#250111yescharacteristic, complemented, Sylow
156C2×#123—712no—
157C2×#123—712no—
158C2×#121—711no—
159C22×C4⋊C4—711no—
160#399—711no—
161#399—711no—
162#399—711no—
163#399D7111yescharacteristic, complemented
164#390—712no—
165#384—712no—
166C4×C22⋊C4—711no—
167C22×D4—711no—
168C22×D4—711no—
169C22×D4—711no—
170C23×C4—711no—
171C23×C4—712no—
172#123—1428no—
173#123—1428no—
174#121—1424no—
175C2×C4⋊C4—712no—
176C2×C4⋊C4—711no—
177C2×C4⋊C4—711no—
178C2×C4⋊C4—712no—
179C2×C4⋊C4—712no—
180C2×C4⋊C4—712no—
181C2×C4⋊C4—1424no—
182C2×C4⋊C4C2×D7111yescharacteristic
183C2×C4⋊C4C2×D7112yes—
184C2×C22⋊C4—712no—
185C2×C22⋊C4—712no—
186C2×C22⋊C4—711no—
187C2×C22⋊C4—712no—
188C2×C22⋊C4—712no—
189C2×C22⋊C4C2×D7111yescharacteristic
190C2×C22⋊C4C2×D7112yes—
191C2×C42—712no—
192#95—711no—
193#95C2×D7111yescharacteristic
194C24—711no—
195C24—711no—
196C24C22×D7111yescharacteristic
197C2×D4—1422no—
198C2×D4—1424no—
199C2×D4—1422no—
200C2×D4—1424no—
201C2×D4—1422no—
202C2×D4—1424no—
203C22×C4—711no—
204C22×C4—712no—
205C22×C4—714no—
206C22×C4—712no—
207C22×C4—711no—
208C22×C4—1424no—
209C22×C4—1424no—
210C22×C4—1424no—
211C22×C4—1424no—
212C22×C4C22×D7112yes—
213C22×C4C22×D7111yescharacteristic
214C22×C4C22×D7111yescharacteristic
215C22×C4C22×D7112yes—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—1424no—
219C4⋊C4—1422no—
220C4⋊C4—1424no—
221C4⋊C4—1424no—
222C4⋊C4—1422no—
223C4⋊C4—1424no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—718no—
227C22⋊C4—1424no—
228C22⋊C4—1424no—
229C22⋊C4—1422no—
230C22⋊C4—1424no—
231C42—1424no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—714no—
238C23—712no—
239C23—1422no—
240C23—1421no—
241C23—1422no—
242C23—1421no—
243C23—1421no—
244C23—1422no—
245C23—1421no—
246C23—1421no—
247C23C23×D7111yescharacteristic
248D4—2844no—
249D4—2844no—
250D4—2844no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2848no—
260C2×C4—712no—
261C2×C4—714no—
262C2×C4—714no—
263C2×C4—1428no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1421no—
267C2×C4—1422no—
268C2×C4—1422no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4D4×D7114yes—
272C2×C4D4×D7112yes—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—2841no—
279C22—2842no—
280C22—2842no—
281C22—2841no—
282C22—2841no—
283C22—714no—
284C22—1422no—
285C22—1424no—
286C22—1424no—
287C22—1422no—
288C22C2×D4×D7111yescharacteristic
289C22C2×D4×D7112yes—
290C22#41×D7112yes—
291C22#1455617511yescharacteristic
292C22#1455617611yescharacteristic
293C4—24no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—2842no—
298C4—2841no—
299C4—714no—
300C4—1424no—
301C2—81no—
302C2—5681no—
303C2—1424no—
304C2#123×D7111yescharacteristic
305C2#127×D7111yescharacteristic
306C2#1798657211yescharacteristic
307C2#1798664712yes—
308C2#1798665112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2501
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556113C2semidirect
2. C2×#14556110C2semidirect
4. C2×C4⋊C4×D71C2semidirect
5. #17985936C2semidirect
6. #17985935C2semidirect
7. #17985934C2semidirect
8. #17985899C2semidirect
9. #17985868C2semidirect
10. #17985846C2semidirect
11. #399×C71C2semidirect
66. (C2×D71)⋊C4D4semidirect
145. Dic71#123semidirect
155. C71#2501semidirect
163. #399D71semidirect