Finite Group Databasebeta

← C22×#89591 #14220587

Subgroups — C22×#89591 #14220587

Order 1792.

3350 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6510C211yesmaximal, characteristic, complemented
2C23×#1143C211yesmaximal, characteristic, complemented
3C23×D56C211yesmaximal, characteristic
4C22×#6456C211yesmaximal, characteristic, complemented
5C22×C7×#103C211yesmaximal, characteristic, complemented
6C22×Dic7⋊C8C211yesmaximal, characteristic, complemented
7C22×#6383C211yesmaximal, characteristic, complemented
8C2×#89591C2124yesmaximal, direct product factor
9C23×C7×Q8C2211yescharacteristic
10C23×D28C2211yescharacteristic
11C23×D28C2211yescharacteristic
12C23×C4×D7—21no—
13C23×C7×C8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6510—26no—
16C2×#6510C22112yescomplemented
17C22×#1143—28no—
18C22×#1143C22112yescomplemented
19C22×(C2×D7)⋊C4—21no—
20C22×(C2×D7)⋊C4—21no—
21C22×(C2×D7)⋊C4—21no—
22C22×C4×Dic7C2211yescharacteristic
23C23×C7⋊C8C2211yescharacteristic
24C22×D56—28no—
25C22×D56C22112yes—
26C2×#6456—26no—
27C2×#6456C22112yescomplemented
28C2×C7×#103C22112yescomplemented
29C2×Dic7⋊C8C22112yescomplemented
30C2×#6383C22112yescomplemented
31#89591C22164yesdirect product factor
32C22×#463—71nomaximal, Sylow
33C24×D7—21no—
34C24×D7—21no—
35C22×C7×Q8—23no—
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8C2316yes—
39C23×C4×C7—21no—
40C23×C4×C7—21no—
41C23×C4×C7C2311yescharacteristic
42C23×Dic7D411yescharacteristic
43C23×Dic7D411yescharacteristic
44C22×D28—23no—
45C22×D28—24no—
46C22×D28—24no—
47C22×D28—24no—
48C22×D28—23no—
49C22×D28D418yes—
50C22×D28C2316yes—
51C22×D28C2316yes—
52C22×C4×D7—212no—
53C22×C4×D7—48no—
54C22×C7×C8—24no—
55C22×C7×C8C2316yes—
56C2×C7×C4⋊C4—23no—
57C2×C7×C4⋊C4C2316yes—
58#6510—224no—
59#6510C23116yescomplemented
60C2×#1143—248no—
61C2×#1143C23116yescomplemented
62C2×(C2×D7)⋊C4—212no—
63C2×(C2×D7)⋊C4—212no—
64C2×(C2×D7)⋊C4—212no—
65C2×C4×Dic7D416yes—
66C2×C4×Dic7C2316yes—
67C22×C7⋊C8—24no—
68C22×C7⋊C8C2316yes—
69C2×D56—248no—
70C2×D56C23116yes—
71#6456—224no—
72#6456C23116yescomplemented
73C7×#103C23116yescomplemented
74Dic7⋊C8C23116yescomplemented
75#6383C23116yescomplemented
76C23×SD16—71no—
77C23×D8—71no—
78C22×#124—71no—
79C22×C4×D4—71no—
80C2×#463—724no—
81C22×C4⋊C8—71no—
82C22×#103D711yescharacteristic, complemented
83C22×#102—71no—
84C24×C7C2×D411yescharacteristic
85C23×D7—28no—
86C23×D7—26no—
87C23×D7—26no—
88C23×D7—44no—
89C23×D7—44no—
90C23×D7—44no—
91C23×D7—44no—
92C23×D7—44no—
93C2×C7×Q8—26no—
94C2×C7×Q8—212no—
95C2×C7×Q8—212no—
96C2×C7×Q8—412no—
97C2×C7×Q8C2414yes—
98C22×C4×C7—26no—
99C22×C4×C7—26no—
100C22×C4×C7—44no—
101C22×C4×C7—44no—
102C22×C4×C7C2×D413yes—
103C22×C4×C7C2×D414yes—
104C22×C4×C7#4114yes—
105C22×C4×C7C2413yes—
106C22×Dic7—24no—
107C22×Dic7—24no—
108C22×Dic7C2×D416yes—
109C22×Dic7C2×D416yes—
110C2×D28—224no—
111C2×D28—26no—
112C2×D28—212no—
113C2×D28—212no—
114C2×D28—212no—
115C2×D28—26no—
116C2×D28—412no—
117C2×D28C2×D4124yes—
118C2×D28C2414yes—
119C2×D28C2414yes—
120C2×C4×D7—216no—
121C2×C4×D7—448no—
122C2×C7×C8—212no—
123C2×C7×C8C2414yes—
124C7×C4⋊C4—26no—
125C7×C4⋊C4C2414yes—
126#1143—232no—
127(C2×D7)⋊C4—216no—
128(C2×D7)⋊C4—216no—
129(C2×D7)⋊C4—216no—
130C4×Dic7C2×D4112yes—
131C4×Dic7C2414yes—
132C2×C7⋊C8—212no—
133C2×C7⋊C8C2414yes—
134D56—232no—
135C23×Q8C2×D711yescharacteristic
136C23×D4—71no—
137C23×D4—71no—
138C24×C4—141no—
139C22×SD16—712no—
140C22×SD16—148no—
141C22×D8—712no—
142C22×D8—148no—
143C23×C8—71no—
144C23×C8C2×D711yescharacteristic
145C2×#124—712no—
146C2×#124—146no—
147C2×C4×D4—712no—
148C2×C4×D4—146no—
149C22×C4⋊C4C2×D711yescharacteristic
150C22×C22⋊C4—141no—
151C22×C22⋊C4—141no—
152C22×C22⋊C4—141no—
153C22×C42—71no—
154#463—764no—
155C2×C4⋊C8—712no—
156C2×#103C2×D7112yescomplemented
157C2×#102—712no—
158C23×C7#12114yes—
159C23×C7#13514yes—
160C23×C7AGL(1,ℤ/8)14yes—
161C23×C7C22×D413yes—
162C22×D7—224no—
163C22×D7—24no—
164C22×D7—24no—
165C22×D7—412no—
166C22×D7—412no—
167C22×D7—432no—
168C22×D7—412no—
169C22×D7—412no—
170C22×D7—412no—
171C22×D7—816no—
172C7×Q8—24no—
173C7×Q8—24no—
174C7×Q8—412no—
175C2×C4×C7—24no—
176C2×C4×C7—24no—
177C2×C4×C7—412no—
178C2×C4×C7—412no—
179C2×C4×C7#121112yes—
180C2×C4×C7C22×D413yes—
181C2×C4×C7C22×D416yes—
182C2×C4×C7C2×#4116yes—
183C2×C4×C7C2511yescharacteristic
184C2×Dic7—212no—
185C2×Dic7—212no—
186C2×Dic7C22×D414yes—
187C2×Dic7C22×D414yes—
188D28—224no—
189D28—24no—
190D28—24no—
191D28—24no—
192D28—412no—
193D28C22×D418yes—
194C4×D7—432no—
195C7×C8—24no—
196C7⋊C8—24no—
197C25—141no—
198C25—141no—
199C22×Q8—23no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8C22×D716yes—
203C22×D4—78no—
204C22×D4—76no—
205C22×D4—76no—
206C22×D4—144no—
207C22×D4—144no—
208C22×D4—143no—
209C22×D4—144no—
210C22×D4—143no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—71no—
214C23×C4—71no—
215C23×C4—1412no—
216C23×C4—288no—
217C23×C4C22×D711yescharacteristic
218C2×SD16—716no—
219C2×SD16—1448no—
220C2×D8—716no—
221C2×D8—1448no—
222C22×C8—24no—
223C22×C8—76no—
224C22×C8—144no—
225C22×C8C22×D716yes—
226#124—716no—
227#124—1424no—
228C4×D4—716no—
229C4×D4—1424no—
230C2×C4⋊C4—23no—
231C2×C4⋊C4C22×D716yes—
232C2×C22⋊C4—1412no—
233C2×C22⋊C4—1412no—
234C2×C22⋊C4—1412no—
235C2×C42—76no—
236C2×C42—76no—
237C4⋊C8—716no—
238#103C22×D7116yescomplemented
239#102—716no—
240C22×C7#463116yescomplemented
241C22×C7C2×#12116yes—
242C22×C7C2×#13516yes—
243C22×C7C2×AGL(1,ℤ/8)16yes—
244C22×C7C23×D411yescharacteristic
245C2×D7—28no—
246C2×D7—448no—
247C2×D7—44no—
248C2×D7—44no—
249C2×D7—44no—
250C2×D7—44no—
251C2×D7—44no—
252C2×D7—824no—
253C4×C7—44no—
254C4×C7—44no—
255C4×C7C2×#12116yes—
256C4×C7C23×D411yescharacteristic
257C4×C7C22×#4111yescharacteristic
258Dic7—24no—
259Dic7—24no—
260C24—148no—
261C24—146no—
262C24—146no—
263C24—284no—
264C24—284no—
265C24—284no—
266C24—284no—
267C24—284no—
268C24D4×D711yescharacteristic
269C2×Q8—26no—
270C2×Q8—212no—
271C2×Q8—212no—
272C2×Q8—412no—
273C2×Q8C23×D714yes—
274C2×D4—724no—
275C2×D4—74no—
276C2×D4—74no—
277C2×D4—1424no—
278C2×D4—1412no—
279C2×D4—1412no—
280C2×D4—146no—
281C2×D4—1412no—
282C2×D4—146no—
283C2×D4—2812no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—76no—
289C22×C4—76no—
290C22×C4—1416no—
291C22×C4—144no—
292C22×C4—144no—
293C22×C4—2848no—
294C22×C4#115714yes—
295C22×C4D4×D713yes—
296C22×C4D4×D714yes—
297C22×C4C23×D713yes—
298SD16—1432no—
299D8—1432no—
300C2×C8—212no—
301C2×C8—74no—
302C2×C8—1412no—
303C2×C8C23×D714yes—
304C4⋊C4—26no—
305C4⋊C4C23×D714yes—
306C22⋊C4—1416no—
307C22⋊C4—1416no—
308C22⋊C4—1416no—
309C42—712no—
310C42—74no—
311C2×C7C2×#463112yescomplemented
312C2×C7C22×#12111yescharacteristic
313C2×C7C22×#13511yescharacteristic
314C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
315D7—48no—
316D7—84no—
317C23—1424no—
318C23—144no—
319C23—144no—
320C23—2812no—
321C23—2812no—
322C23—2832no—
323C23—2812no—
324C23—2812no—
325C23—2812no—
326C23—5616no—
327C23#644714yes—
328C23#647714yes—
329C23#648214yes—
330C23C2×D4×D713yes—
331Q8—24no—
332Q8—24no—
333Q8—412no—
334D4—78no—
335D4—1424no—
336D4—144no—
337D4—144no—
338D4—144no—
339D4—2812no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—412no—
343C2×C4—412no—
344C2×C4—74no—
345C2×C4—74no—
346C2×C4—1412no—
347C2×C4—1412no—
348C2×C4—2832no—
349C2×C4#6447112yes—
350C2×C4C2×#115716yes—
351C2×C4C2×D4×D713yes—
352C2×C4C2×D4×D716yes—
353C2×C4C24×D711yescharacteristic
354C8—24no—
355C8—144no—
356C7C22×#46311yescharacteristic, complemented, Sylow
357C22—148no—
358C22—2848no—
359C22—284no—
360C22—284no—
361C22—284no—
362C22—284no—
363C22—284no—
364C22—5624no—
365C22#89591116yesdirect product factor
366C22C2×#644716yes—
367C22C2×#647716yes—
368C22C2×#648216yes—
369C22C22×D4×D711yescharacteristic
370C4—44no—
371C4—44no—
372C4—144no—
373C4—144no—
374C4C2×#644716yes—
375C4C22×#115711yescharacteristic
376C4C22×D4×D711yescharacteristic
377C2—288no—
378C2—564no—
379C2C2×#89591112yesdirect product factor
380C2C22×#644711yescharacteristic
381C2C22×#647711yescharacteristic
382C2C22×#648211yescharacteristic
subgroupG/H
centerC24D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC22×C7×#103C2
socleC24×C7C2×D4
prime radicalC23×D56C2
hypercenterC22×#103D7
hypocenterC7C22×#463
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C22×#6510C2semidirect
2. C23×#1143C2semidirect
4. C22×#6456C2semidirect
5. C22×C7×#103C2semidirect
6. C22×Dic7⋊C8C2semidirect
7. C22×#6383C2semidirect
8. C2×#89591C2direct (N × H)
16. C2×#6510C22semidirect
18. C22×#1143C22semidirect
27. C2×#6456C22semidirect
28. C2×C7×#103C22semidirect
29. C2×Dic7⋊C8C22semidirect
30. C2×#6383C22semidirect
31. #89591C22direct (N × H)
59. #6510C23semidirect
61. C2×#1143C23semidirect
72. #6456C23semidirect
73. C7×#103C23semidirect
74. Dic7⋊C8C23semidirect
75. #6383C23semidirect
82. C22×#103D7semidirect
156. C2×#103C2×D7semidirect
238. #103C22×D7semidirect
240. C22×C7#463semidirect
311. C2×C7C2×#463semidirect
356. C7C22×#463semidirect
365. C22#89591direct (N × H)
379. C2C2×#89591direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#65102
C23×#11432
C23×D562
C22×#64562
C22×C7×#1032
C22×Dic7⋊C82
C22×#63832
C2×#895912
C22×#4637