Finite Group Databasebeta

← C2×AGL(1,ℤ/32) #72216318

Subgroups — C2×AGL(1,ℤ/32) #72216318

Order 1024.

649 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
1C2×#13658C211yesmaximal, characteristic
2C2×#12068C211yesmaximal, characteristic, complemented
3C2×#12067C211yesmaximal, characteristic, complemented
4AGL(1,ℤ/32)C218yesmaximal, direct product factor
5C2×#7457C211yesmaximal, characteristic, complemented
6C2×#7390C211yesmaximal, characteristic, complemented
7C2×#7389C211yesmaximal, characteristic, complemented
8C2×#7333C211yesmaximal, characteristic, complemented
9C2×#2263C411yescharacteristic
10C2×#2179C2211yescharacteristic
11C2×#2177C2211yescharacteristic
12C2×#2171—21no—
13C2×#1580—21no—
14#13658C414yes—
15#13658C2214yes—
16C2×#1426C411yescharacteristic
17C2×#1419C411yescharacteristic
18C2×#1418C411yescharacteristic
19C2×#1398C2211yescharacteristic
20C2×#1398C2211yescharacteristic
21#12068—22no—
22#12068C2214yescomplemented
23#12067—22no—
24#12067C2214yescomplemented
25C2×#1369C2211yescharacteristic
26C2×#1368C2211yescharacteristic
27C2×#1336—21no—
28C2×#1333—21no—
29C2×#1313C2211yescharacteristic
30#7457C2214yescomplemented
31#7390C2214yescomplemented
32#7389C2214yescomplemented
33#7333C2214yescomplemented
34C2×#508C2×C411yescharacteristic
35C22×D16C2×C411yescharacteristic
36C2×#504—41no—
37C2×#443—21no—
38C2×C4×D8—21no—
39C2×C8×D4—41no—
40#2263C814yescomplemented
41#2263C2×C414yes—
42C2×SD64C812yescomplemented
43C2×D32C812yescomplemented
44C2×M64—21no—
45C2×M64C2×C411yescharacteristic
46C2×M64C2×C411yescharacteristic
47#2179—21no—
48#2179—21no—
49#2179C2×C412yes—
50#2179C2312yes—
51#2177—21no—
52#2177—21no—
53#2177C2×C412yes—
54#2177C2312yes—
55#2171—24no—
56#2171—42no—
57C2×#368C2×C411yescharacteristic
58C2×#367C2×C411yescharacteristic
59C2×#364—21no—
60C2×#359—21no—
61C2×#357—21no—
62C2×#350—41no—
63C2×#346D411yescharacteristic
64C2×#346C2311yescharacteristic
65#1580—24no—
66#1580—42no—
67C2×#334—21no—
68C2×D4⋊C8—41no—
69C2×#322D411yescharacteristic
70#1426C2×C414yes—
71#1419C2×C414yes—
72#1418C2×C414yes—
73#1398—21no—
74#1398C2×C412yes—
75#1398C2312yes—
76#1398C2312yes—
77#1369—21no—
78#1369C2312yes—
79#1368—21no—
80#1368C2312yes—
81#1336—24no—
82#1333—24no—
83#1313D412yes—
84#1313C2312yes—
85C2×#135—21no—
86C22×D8—21no—
87C2×#131—41no—
88C23×C8—81no—
89C2×C4×D4—41no—
90#508—21no—
91#508—21no—
92#508C2×C812yescomplemented
93#508C22×C412yes—
94C2×Q32C2×C811yescharacteristic, complemented
95C2×SD32—21no—
96C2×D16—21no—
97C2×D16—21no—
98C2×D16—21no—
99C2×D16C2×C812yes—
100C2×D16C2×C812yes—
101C2×D16C2×C811yescharacteristic, complemented
102C2×D16C22×C412yes—
103#504—44no—
104#504—82no—
105C2×M32—41no—
106C2×M32—81no—
107C22×C16—21no—
108C22×C16—81no—
109C22×C16C22×C411yescharacteristic
110C22×C16#4111yescharacteristic
111#443—22no—
112#443—22no—
113#443—41no—
114#443—41no—
115C4×D8—22no—
116C4×D8—22no—
117C4×D8—41no—
118C4×D8—41no—
119C8×D4—44no—
120C8×D4—82no—
121C2×#108—21no—
122C2×#107—21no—
123C2×C4⋊C8—41no—
124C2×#104—41no—
125C2×#102—41no—
126C2×C22⋊C8—81no—
127C2×C4×C8—21no—
128C2×C4×C8C2×D411yescharacteristic
129SD64C2×C818yescomplemented
130D32C2×C818yescomplemented
131M64—21no—
132M64—22no—
133M64—41no—
134M64C2×C812yescomplemented
135M64C22×C412yes—
136M64C22×C412yes—
137C2×C32—21no—
138C2×C32—41no—
139C2×C32C2×C812yescomplemented
140#368—21no—
141#368C22×C412yes—
142#367—21no—
143#367C22×C412yes—
144#364—22no—
145#364—41no—
146#359—22no—
147#359—22no—
148#357—22no—
149#357—22no—
150#350—44no—
151#346—21no—
152#346C22×C411yescharacteristic
153#346C2×D412yes—
154#346C2×D411yescharacteristic
155#346#4111yescharacteristic
156#346C2411yescharacteristic
157#334—22no—
158#334—41no—
159D4⋊C8—44no—
160#322—21no—
161#322C2×D412yes—
162C2×#41—41no—
163C22×D4—41no—
164C23×C4—81no—
165#135—22no—
166#135—22no—
167#135—41no—
168#135—41no—
169C2×Q16—21no—
170C2×SD16—41no—
171C2×D8—22no—
172C2×D8—22no—
173C2×D8—22no—
174C2×D8—21no—
175C2×D8—41no—
176C2×D8—41no—
177C2×D8—41no—
178#131—42no—
179#131—42no—
180#131—81no—
181#131—81no—
182C2×M16—41no—
183C2×M16—81no—
184C22×C8—41no—
185C22×C8—81no—
186C22×C8—84no—
187C22×C8—81no—
188C22×C8—164no—
189C22×C8C4×D411yescharacteristic
190C22×C8AGL(1,ℤ/8)11yescharacteristic
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—81no—
194C4×D4—81no—
195C2×C4⋊C4—41no—
196C2×C22⋊C4—81no—
197C2×C42AGL(1,ℤ/8)11yescharacteristic
198Q32—21no—
199Q32C22×C812yescomplemented
200SD32—22no—
201SD32—22no—
202D16—22no—
203D16—22no—
204D16—22no—
205D16—21no—
206D16C22×C814yes—
207D16C22×C812yescomplemented
208M32—42no—
209M32—84no—
210M32—81no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—22no—
214C2×C16—41no—
215C2×C16—41no—
216C2×C16—84no—
217C2×C16—81no—
218C2×C16C4×D411yescharacteristic
219C2×C16C4×D411yescharacteristic
220C2×C16C22×C811yescharacteristic
221C2×C16C22×C811yescharacteristic
222C2×C16#13111yescharacteristic
223C2×C16#13111yescharacteristic
224C2×C16C23×C411yescharacteristic
225C2×C16C2×#4111yescharacteristic
226#108—22no—
227#108—41no—
228#107—22no—
229#107—41no—
230C4⋊C8—42no—
231C4⋊C8—81no—
232#104—42no—
233#104—42no—
234#102—42no—
235#102—42no—
236C22⋊C8—84no—
237C4×C8—22no—
238C4×C8—41no—
239C4×C8C4×D411yescharacteristic
240C4×C8AGL(1,ℤ/8)11yescharacteristic
241C4×C8AGL(1,ℤ/8)11yescharacteristic
242C4×C8C22×D411yescharacteristic
243C32—22no—
244C32—42no—
245C32C22×C814yescomplemented
246C24—81no—
247#41—42no—
248#41—42no—
249#41—81no—
250#41—81no—
251C2×Q8—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4—81no—
258C2×D4—81no—
259C2×D4—81no—
260C22×C4—21no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—162no—
266C22×C4—162no—
267C22×C4#44211yescharacteristic
268Q16—22no—
269Q16—41no—
270SD16—42no—
271SD16—42no—
272D8—24no—
273D8—22no—
274D8—42no—
275D8—42no—
276D8—41no—
277D8—42no—
278M16—42no—
279M16—82no—
280M16—82no—
281M16—81no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—42no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C2×C8—81no—
289C2×C8—82no—
290C2×C8—168no—
291C2×C8—162no—
292C2×C8C8×D411yescharacteristic
293C2×C8C8×D411yescharacteristic
294C2×C8#44211yescharacteristic
295C2×C8#44211yescharacteristic
296C2×C8#44411yescharacteristic
297C2×C8#44411yescharacteristic
298C2×C8C2×C4×D411yescharacteristic
299C2×C8C2×AGL(1,ℤ/8)11yescharacteristic
300C4⋊C4—42no—
301C4⋊C4—81no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—21no—
305C42#44211yescharacteristic
306C42C2×AGL(1,ℤ/8)11yescharacteristic
307C16—21no—
308C16—21no—
309C16—42no—
310C16—82no—
311C16C8×D411yescharacteristic
312C16C8×D411yescharacteristic
313C16C23×C811yescharacteristic
314C16C2×#13111yescharacteristic
315C23—82no—
316C23—81no—
317C23—82no—
318C23—82no—
319C23—162no—
320C23—162no—
321C23AGL(1,ℤ/16)11yescharacteristic
322Q8—42no—
323Q8—81no—
324D4—44no—
325D4—42no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—81no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—162no—
340C2×C4—162no—
341C2×C4—164no—
342C2×C4—162no—
343C2×C4#158011yescharacteristic
344C2×C4#158011yescharacteristic
345C2×C4AGL(1,ℤ/16)11yescharacteristic
346C2×C4AGL(1,ℤ/16)11yescharacteristic
347C2×C4AGL(1,ℤ/16)11yescharacteristic
348C2×C4C2×#44211yescharacteristic
349C8—21no—
350C8—21no—
351C8—82no—
352C8—82no—
353C8—162no—
354C8#158011yescharacteristic
355C8#158011yescharacteristic
356C8C2×C8×D411yescharacteristic
357C8C2×#44411yescharacteristic
358C22—22no—
359C22—82no—
360C22—84no—
361C22—162no—
362C22—162no—
363C22—164no—
364C22—162no—
365C22—164no—
366C22#1206711yescharacteristic
367C22#1206711yescharacteristic
368C22C2×AGL(1,ℤ/16)11yescharacteristic
369C4—41no—
370C4—41no—
371C4—82no—
372C4—162no—
373C4#1206711yescharacteristic
374C4#1206711yescharacteristic
375C4C2×#158011yescharacteristic
376C4C2×AGL(1,ℤ/16)11yescharacteristic
377C2—21no—
378C2—21no—
379C2—162no—
380C2—164no—
381C2AGL(1,ℤ/32)12yesdirect product factor
382C2C2×#1206711yescharacteristic
subgroupG/H
center, socleC22#12067
commutatorC16C23×C8
Frattini, cosocle#346C24
prime radicalC2×#2263C4
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#1313 ◁ G
LECSG ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#12068C2semidirect
3. C2×#12067C2semidirect
4. AGL(1,ℤ/32)C2direct (N × H)
5. C2×#7457C2semidirect
6. C2×#7390C2semidirect
7. C2×#7389C2semidirect
8. C2×#7333C2semidirect
22. #12068C22semidirect
24. #12067C22semidirect
30. #7457C22semidirect
31. #7390C22semidirect
32. #7389C22semidirect
33. #7333C22semidirect
40. #2263C8semidirect
42. C2×SD64C8semidirect
43. C2×D32C8semidirect
92. #508C2×C8semidirect
94. C2×Q32C2×C8semidirect
101. C2×D16C2×C8semidirect
129. SD64C2×C8semidirect
130. D32C2×C8semidirect
134. M64C2×C8semidirect
139. C2×C32C2×C8semidirect
199. Q32C22×C8semidirect
207. D16C22×C8semidirect
245. C32C22×C8semidirect
381. C2AGL(1,ℤ/32)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#136582
C2×#120682
C2×#120672
AGL(1,ℤ/32)2
C2×#74572
C2×#73902
C2×#73892
C2×#73332