Finite Group Databasebeta

← #22593270

Subgroups — #22593270

Order 9088.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14556100C211yesmaximal, characteristic, complemented
3C2×#14556098C211yesmaximal, characteristic, complemented
4C2×#14556096C211yesmaximal, characteristic, complemented
5C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
6#17985934C212yesmaximal, complemented
7#17985908C212yesmaximal, complemented
8#17985900C212yesmaximal, complemented
9#17985899C212yesmaximal, complemented
10C22⋊C4×Dic71C211yesmaximal, characteristic, complemented
11C2×#123×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×D284C2212yes—
14C22×C22⋊D71C2211yescharacteristic
15C22×C22⋊D71C2212yes—
16C22×C22⋊D71C2211yescharacteristic, complemented
17C23×Dic71C2211yescharacteristic
18#14556111—24no—
19#14556100—24no—
20#14556098—24no—
21#14556096—24no—
22C2×Dic71⋊C4C2212yescomplemented
23C2×Dic71⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic71C2211yescharacteristic
25C2×C22⋊Dic71C2211yescharacteristic, complemented
26C2×(C2×D71)⋊C4—24no—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C2×(C2×D71)⋊C4C2212yes—
31C2×(C2×D71)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D71)⋊C4C2212yescomplemented
33C2×C4×Dic71C2212yescomplemented
34C71⋊#95C2212yescomplemented
35C22×D4×C71C2211yescharacteristic
36C23×C4×C71C2211yescharacteristic
37#123×C71—24no—
38C2×C4⋊C4×C71C2212yescomplemented
39C2×C22⋊C4×C71C2211yescharacteristic
40C2×C22⋊C4×C71C2212yescomplemented
41C23×D71—22no—
42C23×D71—24no—
43C23×D71—22no—
44C23×D71—22no—
45C23×D71—42no—
46C23×D71C2311yescharacteristic
47C23×D71C2312yes—
48C2×D284—24no—
49C2×D284—24no—
50C2×D284—42no—
51C2×C22⋊D71—22no—
52C2×C22⋊D71—22no—
53C2×C22⋊D71—22no—
54C2×C22⋊D71—24no—
55C2×C22⋊D71—24no—
56C2×C22⋊D71—22no—
57C2×C22⋊D71—42no—
58C2×C22⋊D71—42no—
59C22×Dic71—42no—
60C22×Dic71D414yescomplemented
61C22×Dic71C2311yescharacteristic
62C22×Dic71C2311yescharacteristic
63C22×Dic71C2311yescharacteristic
64C22×Dic71C2312yes—
65Dic71⋊C4—22no—
66Dic71⋊C4—24no—
67C22⋊Dic71—22no—
68C22⋊Dic71D414yescomplemented
69(C2×D71)⋊C4—24no—
70(C2×D71)⋊C4—24no—
71(C2×D71)⋊C4—24no—
72(C2×D71)⋊C4—24no—
73(C2×D71)⋊C4—22no—
74(C2×D71)⋊C4—24no—
75C4×Dic71—42no—
76C24×C71C2311yescharacteristic
77C24×C71C2311yescharacteristic
78C2×D4×C71—22no—
79C2×D4×C71—22no—
80C2×D4×C71—22no—
81C22×C4×C71—22no—
82C22×C4×C71—22no—
83C22×C4×C71—22no—
84C22×C4×C71C2312yes—
85C22×C4×C71C2311yescharacteristic
86C22×C4×C71C2312yes—
87C4⋊C4×C71—24no—
88C22⋊C4×C71—24no—
89C22⋊C4×C71D414yescomplemented
90C23×C71—22no—
91C23×C71—22no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—22no—
96C23×C71—22no—
97C23×C71C2×D412yes—
98C23×C71C2×D412yes—
99C23×C71C2×D412yes—
100C23×C71C2411yescharacteristic
101C22×D71—28no—
102C22×D71—22no—
103C22×D71—22no—
104C22×D71—22no—
105C22×D71—22no—
106C22×D71—22no—
107C22×D71—22no—
108C22×D71—21no—
109C22×D71—21no—
110C22×D71—21no—
111C22×D71—22no—
112C22×D71—22no—
113C22×D71—48no—
114C22×D71—48no—
115C22×D71—44no—
116C22×D71—44no—
117C22×D71#4114yes—
118D4×C71—44no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—44no—
128C2×C4×C71C2×D414yes—
129C22⋊D71—48no—
130C22⋊D71—44no—
131C22⋊D71—44no—
132C2×Dic71—22no—
133C2×Dic71—22no—
134C2×Dic71—22no—
135C2×Dic71—22no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—44no—
140C2×Dic71C2×D412yes—
141C2×Dic71C2×D414yes—
142C2×Dic71C2×D412yes—
143D284—48no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—24no—
147C22×C71—24no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—42no—
152C22×C71—42no—
153C22×C71#12714yes—
154C22×C71C22×D411yescharacteristic
155C22×C71C22×D411yescharacteristic
156C22×C71C22×D411yescharacteristic
157C22×C71C2×#4112yes—
158C22×C71H(5,𝔽2)12yes—
159C2×D71—24no—
160C2×D71—24no—
161C2×D71—24no—
162C2×D71—416no—
163C2×D71—42no—
164C2×D71—42no—
165C2×D71—42no—
166C2×D71—42no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—42no—
171C2×D71—42no—
172C4×C71—24no—
173C4×C71—42no—
174C4×C71—41no—
175Dic71—22no—
176Dic71—22no—
177Dic71—24no—
178Dic71—41no—
179C2×C71—24no—
180C2×C71—81no—
181C2×C71#52511yescharacteristic
182C2×C71C2×#12711yescharacteristic
183C2×C71#53411yescharacteristic
184C2×C71#54612yes—
185C2×C71#54612yes—
186D71—44no—
187D71—81no—
188#2539—711nomaximal, Sylow
189C71#253911yescharacteristic, complemented, Sylow
190C2×#123—711no—
191C2×#123D7111yescharacteristic, complemented
192C2×#121—711no—
193C2×#120—711no—
194C2×#120—711no—
195C22×C22⋊C4—711no—
196#399—712no—
197#392—712no—
198#390—712no—
199#390—712no—
200C4×C22⋊C4—711no—
201C25—711no—
202C22×D4—711no—
203C22×D4—712no—
204C22×D4—711no—
205C22×D4—712no—
206C22×D4C2×D7111yescharacteristic
207C23×C4—711no—
208C23×C4C2×D7111yescharacteristic
209#123—24no—
210#123—1424no—
211#121—1424no—
212#120—1424no—
213#120—1424no—
214C2×C4⋊C4—711no—
215C2×C4⋊C4—712no—
216C2×C4⋊C4C2×D7112yescomplemented
217C2×C22⋊C4—711no—
218C2×C22⋊C4—711no—
219C2×C22⋊C4—712no—
220C2×C22⋊C4—712no—
221C2×C22⋊C4—712no—
222C2×C22⋊C4—712no—
223C2×C22⋊C4—711no—
224C2×C22⋊C4—712no—
225C2×C22⋊C4—1424no—
226C2×C22⋊C4C2×D7111yescharacteristic
227C2×C22⋊C4C2×D7112yescomplemented
228C2×C42—712no—
229#95—712no—
230C24—2842no—
231C24—712no—
232C24—711no—
233C24—1424no—
234C24—1422no—
235C24—1422no—
236C24—1422no—
237C24C22×D7111yescharacteristic
238C24C22×D7111yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—2842no—
243C2×D4—2842no—
244C2×D4—2842no—
245C2×D4—1424no—
246C2×D4—1424no—
247C2×D4—1422no—
248C2×D4—1424no—
249C2×D4—1422no—
250C2×D4—1422no—
251C2×D4—1422no—
252C2×D4—1424no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—2842no—
257C22×C4—714no—
258C22×C4—711no—
259C22×C4—712no—
260C22×C4—711no—
261C22×C4—711no—
262C22×C4C22×D7112yes—
263C22×C4C22×D7111yescharacteristic
264C22×C4C22×D7112yes—
265C4⋊C4—24no—
266C4⋊C4—1422no—
267C4⋊C4—1424no—
268C22⋊C4—24no—
269C22⋊C4—714no—
270C22⋊C4—1424no—
271C22⋊C4—1424no—
272C22⋊C4—1424no—
273C22⋊C4—1422no—
274C22⋊C4—1424no—
275C22⋊C4—1422no—
276C22⋊C4—1424no—
277C22⋊C4C22⋊D7114yescomplemented
278C42—2842no—
279C23—22no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—2844no—
287C23—2848no—
288C23—2848no—
289C23—2844no—
290C23—714no—
291C23—1428no—
292C23—1421no—
293C23—1421no—
294C23—1421no—
295C23—1422no—
296C23—1422no—
297C23—1422no—
298C23—1422no—
299C23—1422no—
300C23—1422no—
301C23—1422no—
302C23—1422no—
303C23D4×D7112yes—
304C23D4×D7112yes—
305C23C2×C22⋊D7112yes—
306C23C23×D7111yescharacteristic
307D4—44no—
308D4—2848no—
309D4—2844no—
310D4—2844no—
311D4—2848no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—2844no—
322C2×C4—714no—
323C2×C4—712no—
324C2×C4—712no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1422no—
329C2×C4—1422no—
330C2×C4—1422no—
331C2×C4—1422no—
332C2×C4C2×C22⋊D7114yes—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—22no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—28416no—
343C22—2842no—
344C22—2842no—
345C22—2842no—
346C22—2842no—
347C22—2841no—
348C22—2842no—
349C22—2842no—
350C22—2841no—
351C22—2841no—
352C22—1424no—
353C22—1424no—
354C22—1424no—
355C22#1455612414yes—
356C22C2×D4×D7111yescharacteristic
357C22C2×D4×D7111yescharacteristic
358C22C22×C22⋊D7111yescharacteristic
359C22#41×D7112yes—
360C22#1455617412yes—
361C4—24no—
362C4—42no—
363C4—41no—
364C4—2841no—
365C4—1422no—
366C4—1422no—
367C4—1424no—
368C2—24no—
369C2—81no—
370C2—2844no—
371C2—5681no—
372C2#1798650711yescharacteristic
373C2C2×#1455612411yescharacteristic
374C2#1798656011yescharacteristic
375C2#1798663712yes—
376C2#1798664112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#123×C71C2
hypercenterC2×#123D71
hypocenterC71#2539
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14556111C2semidirect
2. C2×#14556100C2semidirect
3. C2×#14556098C2semidirect
4. C2×#14556096C2semidirect
5. C22×(C2×D71)⋊C4C2semidirect
6. #17985934C2semidirect
7. #17985908C2semidirect
8. #17985900C2semidirect
9. #17985899C2semidirect
10. C22⋊C4×Dic71C2semidirect
11. C2×#123×C71C2semidirect
16. C22×C22⋊D71C22semidirect
22. C2×Dic71⋊C4C22semidirect
23. C2×Dic71⋊C4C22semidirect
25. C2×C22⋊Dic71C22semidirect
31. C2×(C2×D71)⋊C4C22semidirect
32. C2×(C2×D71)⋊C4C22semidirect
33. C2×C4×Dic71C22semidirect
34. C71⋊#95C22semidirect
38. C2×C4⋊C4×C71C22semidirect
40. C2×C22⋊C4×C71C22semidirect
60. C22×Dic71D4semidirect
68. C22⋊Dic71D4semidirect
89. C22⋊C4×C71D4semidirect
189. C71#2539semidirect
191. C2×#123D71semidirect
216. C2×C4⋊C4C2×D71semidirect
227. C2×C22⋊C4C2×D71semidirect
277. C22⋊C4C22⋊D71semidirect