Finite Group Databasebeta

← #22594466

Subgroups — #22594466

Order 9088.

598 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556111C211yesmaximal, characteristic, complemented
2C2×#14556102C211yesmaximal, characteristic, complemented
3#17985931C211yesmaximal, characteristic, complemented
4#17985926C211yesmaximal, characteristic, complemented
5#17985917C211yesmaximal, characteristic, complemented
6#17985915C212yesmaximal, complemented
7#17985909C211yesmaximal, characteristic, complemented
8#17985908C212yesmaximal, complemented
9#17985901C212yesmaximal, complemented
10#17985890C212yesmaximal, complemented
11C4×C22⋊C4×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C22×D284C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C22⋊D71C2212yes—
16#14556111—24no—
17#14556102—24no—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×C4⋊Dic71C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2211yescharacteristic
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C23×C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2212yes—
34C2×C42×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C23×D71C2312yes—
39C2×D284—22no—
40C2×D284—24no—
41C2×D284—42no—
42C2×D284—42no—
43C2×D284—42no—
44C2×C22⋊D71—24no—
45C2×C22⋊D71—22no—
46C2×C22⋊D71—42no—
47C2×C22⋊D71—42no—
48C2×C22⋊D71—42no—
49C22×Dic71C2311yescharacteristic
50C22×Dic71C2312yes—
51C22×Dic71C2311yescharacteristic
52C22×Dic71C2311yescharacteristic
53C4⋊Dic71—22no—
54C4⋊Dic71—42no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57C22⋊Dic71—22no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—42no—
63(C2×D71)⋊C4—42no—
64(C2×D71)⋊C4—42no—
65C24×C71C2311yescharacteristic
66C22×C4×C71—42no—
67C22×C4×C71D414yescomplemented
68C22×C4×C71C2312yes—
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2312yes—
72C22⋊C4×C71—24no—
73C42×C71—42no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—21no—
77C23×C71—21no—
78C23×C71C2×D412yes—
79C23×C71C2411yescharacteristic
80C22×D71—21no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—21no—
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—22no—
90C22×D71—22no—
91C22×D71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—44no—
100C2×C4×C71C2×D412yes—
101C2×C4×C71C2×D412yes—
102C2×C4×C71#4114yes—
103C22⋊D71—44no—
104C22⋊D71—48no—
105C2×Dic71—21no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—21no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—21no—
114C2×Dic71—22no—
115D284—44no—
116D284—48no—
117C22×C71—22no—
118C22×C71—22no—
119C22×C71—22no—
120C22×C71—42no—
121C22×C71—42no—
122C22×C71C22×D411yescharacteristic
123C22×C71C2×#4112yes—
124C22×C71H(5,𝔽2)12yes—
125C22×C71H(5,𝔽2)11yescharacteristic
126C22×C71H(5,𝔽2)11yescharacteristic
127C2×D71—41no—
128C2×D71—42no—
129C2×D71—42no—
130C2×D71—41no—
131C2×D71—42no—
132C2×D71—41no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—42no—
137C2×D71—42no—
138C2×D71—42no—
139C4×C71—22no—
140C4×C71—22no—
141C4×C71—24no—
142C4×C71—42no—
143Dic71—41no—
144Dic71—41no—
145Dic71—41no—
146Dic71—42no—
147C2×C71—42no—
148C2×C71#53211yescharacteristic
149C2×C71#53411yescharacteristic
150C2×C71#53511yescharacteristic
151C2×C71#53812yes—
152C2×C71#54012yes—
153D71—82no—
154D71—81no—
155#2661—711nomaximal, Sylow
156C71#266111yescharacteristic, complemented, Sylow
157C2×#123—711no—
158C2×#121—711no—
159#398—711no—
160#397—711no—
161#394—712no—
162#394—711no—
163#392—712no—
164#392—711no—
165#390—712no—
166#388—712no—
167C4×C22⋊C4D7111yescharacteristic, complemented
168C22×D4—712no—
169C22×D4—711no—
170C22×D4—711no—
171C22×D4—712no—
172C23×C4C2×D7111yescharacteristic
173#123—1424no—
174#121—1424no—
175C2×C4⋊C4—712no—
176C2×C4⋊C4—711no—
177C2×C4⋊C4—711no—
178C2×C4⋊C4—711no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—711no—
183C2×C22⋊C4—712no—
184C2×C22⋊C4—711no—
185C2×C22⋊C4—711no—
186C2×C22⋊C4—712no—
187C2×C22⋊C4C2×D7112yes—
188C2×C42C2×D7112yes—
189#95—712no—
190#95—712no—
191#95C2×D7111yescharacteristic
192#95C2×D7111yescharacteristic
193C24—711no—
194C24—712no—
195C24C22×D7111yescharacteristic
196C2×D4—2842no—
197C2×D4—2842no—
198C2×D4—2842no—
199C2×D4—2842no—
200C2×D4—2842no—
201C2×D4—2842no—
202C2×D4—1424no—
203C2×D4—1422no—
204C2×D4—1422no—
205C2×D4—1424no—
206C22×C4—42no—
207C22×C4—711no—
208C22×C4—712no—
209C22×C4—711no—
210C22×C4—711no—
211C22×C4D28414yescomplemented
212C22×C4C22×D7112yes—
213C22×C4C22×D7111yescharacteristic
214C22×C4C22×D7111yescharacteristic
215C22×C4C22×D7112yes—
216C4⋊C4—2842no—
217C4⋊C4—1422no—
218C4⋊C4—1422no—
219C4⋊C4—1422no—
220C22⋊C4—24no—
221C22⋊C4—2842no—
222C22⋊C4—2842no—
223C22⋊C4—2842no—
224C22⋊C4—1422no—
225C22⋊C4—1424no—
226C22⋊C4—1422no—
227C22⋊C4—1422no—
228C22⋊C4—1424no—
229C42—42no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—1421no—
235C23—1422no—
236C23—1422no—
237C23—1422no—
238C23—1422no—
239C23—1422no—
240C23—1422no—
241C23—1422no—
242C23—1421no—
243C23—1422no—
244C23—1422no—
245C23—1421no—
246C23C2×D28412yes—
247C23C23×D7111yescharacteristic
248D4—2844no—
249D4—2848no—
250D4—2844no—
251D4—2848no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—44no—
260C2×C4—1421no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4—1421no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1421no—
267C2×C4—1422no—
268C2×C4—1421no—
269C2×C4—1422no—
270C2×C4C2×D28412yes—
271C2×C4C2×D28412yes—
272C2×C4#1176586514yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—2842no—
279C22—2841no—
280C22—2842no—
281C22—2841no—
282C22—2842no—
283C22—2842no—
284C22—2842no—
285C22—2842no—
286C22—2842no—
287C22—2842no—
288C22—2841no—
289C22—2842no—
290C22C22×D28411yescharacteristic
291C22C2×#1176586512yes—
292C22#1455617312yes—
293C22#1455617411yescharacteristic
294C22#1455617411yescharacteristic
295C4—22no—
296C4—22no—
297C4—24no—
298C4—42no—
299C4—2841no—
300C4—2842no—
301C4—2841no—
302C4—2841no—
303C2—42no—
304C2—5682no—
305C2—5681no—
306C2#1798654911yescharacteristic
307C2#1798656111yescharacteristic
308C2#1798656811yescharacteristic
309C2#1798658212yes—
310C2#1798660012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC4×C22⋊C4×C71C2
hypercenterC4×C22⋊C4D71
hypocenterC71#2661
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14556111C2semidirect
2. C2×#14556102C2semidirect
3. #17985931C2semidirect
4. #17985926C2semidirect
5. #17985917C2semidirect
6. #17985915C2semidirect
7. #17985909C2semidirect
8. #17985908C2semidirect
9. #17985901C2semidirect
10. #17985890C2semidirect
11. C4×C22⋊C4×C71C2semidirect
67. C22×C4×C71D4semidirect
156. C71#2661semidirect
167. C4×C22⋊C4D71semidirect
211. C22×C4D284semidirect