Finite Group Databasebeta

← C2×C31×#1610 #21474874

Subgroups — C2×C31×#1610 #21474874

Order 7936.

1308 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C2×C31×#480C212yesmaximal, complemented
3C2×C31×#398C211yesmaximal, characteristic, complemented
4C31×#1610C218yesmaximal, direct product factor
5C2×C31×#327C212yesmaximal, complemented
6C2×C31×#324C211yesmaximal, characteristic, complemented
7C22×#41×C31—21no—
8C31×#537—24no—
9C31×#537—22no—
10C31×#537C2214yes—
11C2×C31×C22⋊Q8—22no—
12C2×C31×#121—22no—
13C2×C31×#121—21no—
14C2×C31×#121C2212yes—
15C22×C4⋊C4×C31—21no—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#480—24no—
19C31×#480C2218yescomplemented
20C2×C31×#107—22no—
21C2×C31×#102—22no—
22C2×C31×C22⋊C8C2212yes—
23C31×#398—22no—
24C31×#398C2214yescomplemented
25C2×C31×#95—21no—
26C2×C31×#95C2211yescharacteristic
27C31×#327C2218yescomplemented
28C31×#324C2214yescomplemented
29C2×#41×C31—24no—
30C2×#41×C31—24no—
31C2×#41×C31—44no—
32C2×#41×C31—44no—
33C22×Q8×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31—22no—
36C22×D4×C31—21no—
37C22×D4×C31D412yes—
38C23×C4×C31—22no—
39C23×C4×C31—21no—
40C23×C4×C31—21no—
41C23×C4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C22×C8×C31—22no—
45C31×C22⋊Q8—28no—
46C31×C22⋊Q8—28no—
47C31×#121—24no—
48C31×#121—28no—
49C31×#121—28no—
50C31×#121—24no—
51C31×#121—22no—
52C31×#121—22no—
53C31×#121D414yes—
54C31×#121C2314yes—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—44no—
63C2×C4⋊C4×C31D412yes—
64C2×C4⋊C4×C31D412yes—
65C2×C4⋊C4×C31C2312yes—
66C2×C4⋊C4×C31C2312yes—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C31×#107—28no—
70C31×#102—28no—
71C31×C22⋊C8—22no—
72C31×C22⋊C8C2314yes—
73C31×#95—21no—
74C31×#95—24no—
75C31×#95C2312yes—
76C24×C31—22no—
77C24×C31C2×D411yescharacteristic
78#41×C31—48no—
79#41×C31—416no—
80#41×C31—48no—
81C2×Q8×C31—24no—
82C2×Q8×C31—44no—
83C2×D4×C31—24no—
84C2×D4×C31—24no—
85C2×D4×C31—24no—
86C2×D4×C31—24no—
87C2×D4×C31—22no—
88C2×D4×C31—44no—
89C2×D4×C31—44no—
90C2×D4×C31—44no—
91C2×D4×C31—44no—
92C2×D4×C31—82no—
93C2×D4×C31C2×D414yes—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—22no—
97C22×C4×C31—28no—
98C22×C4×C31—24no—
99C22×C4×C31—21no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—21no—
103C22×C4×C31—22no—
104C22×C4×C31—42no—
105C22×C4×C31—48no—
106C22×C4×C31—42no—
107C22×C4×C31—42no—
108C22×C4×C31—42no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31—42no—
112C22×C4×C31—42no—
113C22×C4×C31D814yes—
114C22×C4×C31C2×D412yes—
115C22×C4×C31C2×D412yes—
116C22×C4×C31C2×D412yes—
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2×D412yes—
119C22×C4×C31C2411yescharacteristic
120C2×C8×C31—24no—
121C2×C8×C31—44no—
122C4⋊C4×C31—24no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—24no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—42no—
128C4⋊C4×C31—48no—
129C4⋊C4×C31C2×D414yes—
130C22⋊C4×C31—28no—
131C22⋊C4×C31—24no—
132C22⋊C4×C31—24no—
133C2×#1610C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—22no—
135C23×C31—22no—
136C23×C31—24no—
137C23×C31—44no—
138C23×C31—44no—
139C23×C31—44no—
140C23×C31#12011yescharacteristic
141C23×C31#12011yescharacteristic
142C23×C31C2×D812yes—
143C23×C31AGL(1,ℤ/8)12yes—
144C23×C31C22×D411yescharacteristic
145Q8×C31—48no—
146D4×C31—44no—
147D4×C31—48no—
148D4×C31—44no—
149D4×C31—44no—
150D4×C31—48no—
151D4×C31—82no—
152D4×C31—84no—
153C2×C4×C31—22no—
154C2×C4×C31—22no—
155C2×C4×C31—24no—
156C2×C4×C31—24no—
157C2×C4×C31—24no—
158C2×C4×C31—24no—
159C2×C4×C31—22no—
160C2×C4×C31—24no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—416no—
165C2×C4×C31—48no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—44no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—81no—
175C2×C4×C31—82no—
176C2×C4×C31—82no—
177C2×C4×C31#12012yes—
178C2×C4×C31C2×D814yes—
179C2×C4×C31C22×D412yes—
180C8×C31—44no—
181C2×#537C2×C3111yescharacteristic
182C2×#480C2×C3112yescomplemented
183C2×#398C2×C3111yescharacteristic, complemented
184#1610C2×C3118yesdirect product factor
185C2×#327C2×C3112yescomplemented
186C2×#324C2×C3111yescharacteristic, complemented
187C22×C31—24no—
188C22×C31—24no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—44no—
194C22×C31—44no—
195C22×C31—44no—
196C22×C31—88no—
197C22×C31#44712yes—
198C22×C31#44712yes—
199C22×C31#45612yes—
200C22×C31#45812yes—
201C22×C31C2×#12011yescharacteristic
202C22×C31C22×D811yescharacteristic
203C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C31—24no—
205C4×C31—22no—
206C4×C31—22no—
207C4×C31—44no—
208C4×C31—44no—
209C4×C31—42no—
210C4×C31—82no—
211C22×#41—21no—
212#537—24no—
213#537—22no—
214#537C22×C3114yes—
215C2×C22⋊Q8—22no—
216C2×#121—22no—
217C2×#121—21no—
218C2×#121C22×C3112yes—
219C22×C4⋊C4—21no—
220C22×C4⋊C4C22×C3111yescharacteristic
221C22×C4⋊C4C22×C3111yescharacteristic
222#480—24no—
223#480C22×C3118yescomplemented
224C2×#107—22no—
225C2×#102—22no—
226C2×C22⋊C8C22×C3112yes—
227#398—22no—
228#398C22×C3114yescomplemented
229C2×#95—21no—
230C2×#95C22×C3111yescharacteristic
231#327C22×C3118yescomplemented
232#324C22×C3114yescomplemented
233C2×C31—21no—
234C2×C31—21no—
235C2×C31—22no—
236C2×C31—84no—
237C2×C31#161014yesdirect product factor
238C2×C31C2×#44711yescharacteristic
239C2×C31C2×#45611yescharacteristic
240C2×C31C2×#45811yescharacteristic
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—44no—
244C2×#41—44no—
245C22×Q8—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4D4×C3112yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4D4×C3111yescharacteristic
254C23×C4D4×C3111yescharacteristic
255C23×C4C23×C3111yescharacteristic
256C22×C8—22no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259#121—24no—
260#121—28no—
261#121—28no—
262#121—24no—
263#121—22no—
264#121—22no—
265#121D4×C3114yes—
266#121C23×C3114yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4D4×C3112yes—
276C2×C4⋊C4D4×C3112yes—
277C2×C4⋊C4C23×C3112yes—
278C2×C4⋊C4C23×C3112yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281#107—28no—
282#102—28no—
283C22⋊C8—22no—
284C22⋊C8C23×C3114yes—
285#95—24no—
286#95—21no—
287#95C23×C3112yes—
288C31C2×#161011yescharacteristic, direct product factor, Sylow
289C24—22no—
290C24C2×D4×C3111yescharacteristic
291#41—416no—
292#41—48no—
293#41—48no—
294C2×Q8—24no—
295C2×Q8—44no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—22no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—82no—
306C2×D4C2×D4×C3114yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4D8×C3114yes—
327C22×C4C2×D4×C3112yes—
328C22×C4C2×D4×C3112yes—
329C22×C4C2×D4×C3112yes—
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3112yes—
332C22×C4C24×C3111yescharacteristic
333C2×C8—24no—
334C2×C8—44no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—48no—
342C4⋊C4C2×D4×C3114yes—
343C22⋊C4—28no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23C31×#12011yescharacteristic
353C23C31×#12011yescharacteristic
354C23C2×D8×C3112yes—
355C23C31×AGL(1,ℤ/8)12yes—
356C23C22×D4×C3111yescharacteristic
357Q8—48no—
358D4—44no—
359D4—48no—
360D4—48no—
361D4—44no—
362D4—44no—
363D4—82no—
364D4—84no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—416no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4C31×#12012yes—
390C2×C4C2×D8×C3114yes—
391C2×C4C22×D4×C3112yes—
392C8—44no—
393C22—24no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—88no—
403C22C31×#44712yes—
404C22C31×#44712yes—
405C22C31×#45612yes—
406C22C31×#45812yes—
407C22C2×C31×#12011yescharacteristic
408C22C22×D8×C3111yescharacteristic
409C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
410C4—24no—
411C4—22no—
412C4—22no—
413C4—44no—
414C4—44no—
415C4—42no—
416C4—82no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—84no—
421C2C31×#161014yesdirect product factor
422C2C2×C31×#44711yescharacteristic
423C2C2×C31×#45611yescharacteristic
424C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#480C2semidirect
3. C2×C31×#398C2semidirect
4. C31×#1610C2direct (N × H)
5. C2×C31×#327C2semidirect
6. C2×C31×#324C2semidirect
19. C31×#480C22semidirect
24. C31×#398C22semidirect
27. C31×#327C22semidirect
28. C31×#324C22semidirect
133. C2×#1610C31direct (N × H)
182. C2×#480C2×C31semidirect
183. C2×#398C2×C31semidirect
184. #1610C2×C31direct (N × H)
185. C2×#327C2×C31semidirect
186. C2×#324C2×C31semidirect
223. #480C22×C31semidirect
228. #398C22×C31semidirect
231. #327C22×C31semidirect
232. #324C22×C31semidirect
237. C2×C31#1610direct (N × H)
288. C31C2×#1610direct (N × H)
421. C2C31×#1610direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C2×C31×#4802
C2×C31×#3982
C31×#16102
C2×C31×#3272
C2×C31×#3242
C2×#161031