Finite Group Databasebeta

← C31×#10549 #21467308

Subgroups — C31×#10549 #21467308

Order 7936.

502 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3151C211yesmaximal, characteristic
2C31×#1950C211yesmaximal, characteristic, complemented
3C31×#1547C211yesmaximal, characteristic, complemented
4C31×#1545C211yesmaximal, characteristic, complemented
5C2×C31×#331C211yesmaximal, characteristic, complemented
6C31×#1480C211yesmaximal, characteristic, complemented
7C31×#1283C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C2×C31×AGL(1,ℤ/8)—21no—
10C31×#532C2211yescharacteristic
11C2×C31×#127C2211yescharacteristic
12C31×#496—22no—
13C31×#493—21no—
14C31×#492—21no—
15C31×#431C2211yescharacteristic
16C2×C31×#108—21no—
17C31×#423C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×C4⋊C8C2211yescharacteristic
20C4×M16×C31C2211yescharacteristic
21C31×#341—21no—
22C31×#332—21no—
23C31×#331—22no—
24C31×#331—21no—
25C31×#331—22no—
26C31×D4⋊C8—22no—
27C2×#41×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C31×AGL(1,ℤ/8)—42no—
32C31×AGL(1,ℤ/8)—42no—
33C31×AGL(1,ℤ/8)—42no—
34C31×AGL(1,ℤ/8)—42no—
35C2×SD16×C31—22no—
36C2×SD16×C31—22no—
37C2×D8×C31—22no—
38C2×D8×C31—22no—
39C2×M16×C31—21no—
40C2×M16×C31D411yescharacteristic
41C2×M16×C31D411yescharacteristic
42C22×C8×C31—21no—
43C22×C8×C31—21no—
44C31×C4⋊Q8D411yescharacteristic
45C31×#127—21no—
46C31×#127—21no—
47C31×#127—21no—
48C31×#127D411yescharacteristic
49C31×#127D412yes—
50C31×#124—21no—
51C31×#121—21no—
52C31×#121—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C2×C42×C31C2311yescharacteristic
56C31×#108—42no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8D412yes—
64C31×C22⋊C8—21no—
65C31×#97—21no—
66C4×C8×C31—21no—
67C4×C8×C31—21no—
68C24×C31—41no—
69#41×C31—41no—
70#41×C31—41no—
71#41×C31—41no—
72#41×C31—41no—
73C2×Q8×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—22no—
76C2×D4×C31—21no—
77C2×D4×C31—22no—
78C2×D4×C31—21no—
79C2×D4×C31—22no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—81no—
89C2×D4×C31—81no—
90C2×D4×C31—82no—
91C22×C4×C31—41no—
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31#4111yescharacteristic
94C22×C4×C31#4111yescharacteristic
95SD16×C31—44no—
96SD16×C31—44no—
97D8×C31—44no—
98D8×C31—44no—
99M16×C31—21no—
100M16×C31—21no—
101M16×C31—41no—
102M16×C31—41no—
103M16×C31D812yes—
104M16×C31SD1612yes—
105C2×C8×C31—21no—
106C2×C8×C31—21no—
107C2×C8×C31—22no—
108C2×C8×C31—41no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C2×C8×C31—41no—
113C2×C8×C31—81no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C22⋊C4×C31—41no—
117C22⋊C4×C31—41no—
118C42×C31C2×D411yescharacteristic
119C42×C31C2×D411yescharacteristic
120C42×C31C2×D411yescharacteristic
121C42×C31#4111yescharacteristic
122#10549C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—41no—
124C23×C31—42no—
125C23×C31—82no—
126C23×C31—82no—
127C23×C31—81no—
128C23×C31—81no—
129C23×C31#12411yescharacteristic
130Q8×C31—41no—
131Q8×C31—41no—
132D4×C31—41no—
133D4×C31—42no—
134D4×C31—41no—
135D4×C31—42no—
136D4×C31—41no—
137D4×C31—41no—
138D4×C31—81no—
139D4×C31—82no—
140D4×C31—82no—
141D4×C31—82no—
142D4×C31—81no—
143D4×C31—81no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—81no—
153C2×C4×C31—81no—
154C2×C4×C31#12011yescharacteristic
155C2×C4×C31#12111yescharacteristic
156C2×C4×C31#12111yescharacteristic
157C2×C4×C31#12111yescharacteristic
158C2×C4×C31#12311yescharacteristic
159C2×C4×C31#12311yescharacteristic
160C2×C4×C31C2×D811yescharacteristic
161C2×C4×C31C2×SD1611yescharacteristic
162C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
164C8×C31—22no—
165C8×C31—22no—
166C8×C31—42no—
167C8×C31—81no—
168C8×C31—81no—
169C8×C31—81no—
170#3151C2×C3111yescharacteristic
171#1950C2×C3111yescharacteristic, complemented
172#1547C2×C3111yescharacteristic, complemented
173#1545C2×C3111yescharacteristic, complemented
174C2×#331C2×C3111yescharacteristic, complemented
175#1480C2×C3111yescharacteristic, complemented
176#1283C2×C3111yescharacteristic, complemented
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—82no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—82no—
183C22×C31—81no—
184C22×C31—82no—
185C22×C31—162no—
186C22×C31—162no—
187C22×C31#39411yescharacteristic
188C22×C31#48611yescharacteristic
189C22×C31#49011yescharacteristic
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—81no—
194C4×C31#44711yescharacteristic
195C4×C31#45011yescharacteristic
196C4×C31#45911yescharacteristic
197C4×C31#46011yescharacteristic
198C4×C31#47111yescharacteristic
199C4×C31#47211yescharacteristic
200C2×AGL(1,ℤ/8)—21no—
201C2×AGL(1,ℤ/8)—21no—
202#532C22×C3111yescharacteristic
203C2×#127C22×C3111yescharacteristic
204#496—22no—
205#493—21no—
206#492—21no—
207#431C22×C3111yescharacteristic
208C2×#108—21no—
209#423C22×C3111yescharacteristic
210C2×C4⋊C8C22×C3111yescharacteristic
211C2×C4⋊C8C22×C3111yescharacteristic
212C4×M16C22×C3111yescharacteristic
213#341—21no—
214#332—21no—
215#331—22no—
216#331—21no—
217#331—22no—
218D4⋊C8—22no—
219C2×C31—21no—
220C2×C31—21no—
221C2×C31—161no—
222C2×C31—162no—
223C2×C31#204111yescharacteristic
224C2×C31#204811yescharacteristic
225C2×C31#205111yescharacteristic
226C2×#41—21no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230AGL(1,ℤ/8)—42no—
231AGL(1,ℤ/8)—42no—
232AGL(1,ℤ/8)—42no—
233AGL(1,ℤ/8)—42no—
234C2×SD16—22no—
235C2×SD16—22no—
236C2×D8—22no—
237C2×D8—22no—
238C2×M16—21no—
239C2×M16D4×C3111yescharacteristic
240C2×M16D4×C3111yescharacteristic
241C22×C8—21no—
242C22×C8—21no—
243C4⋊Q8D4×C3111yescharacteristic
244#127—21no—
245#127—21no—
246#127—21no—
247#127D4×C3112yes—
248#127D4×C3111yescharacteristic
249#124—21no—
250#121—21no—
251#121—21no—
252C4×D4—21no—
253C4×D4—21no—
254C2×C42C23×C3111yescharacteristic
255#108—42no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8D4×C3112yes—
263C22⋊C8—21no—
264#97—21no—
265C4×C8—21no—
266C4×C8—21no—
267C31#1054911yescharacteristic, direct product factor, Sylow
268C24—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4—82no—
291C22×C4—41no—
292C22×C4C2×D4×C3111yescharacteristic
293C22×C4#41×C3111yescharacteristic
294C22×C4#41×C3111yescharacteristic
295SD16—44no—
296SD16—44no—
297D8—44no—
298D8—44no—
299M16—21no—
300M16—21no—
301M16—41no—
302M16—41no—
303M16D8×C3112yes—
304M16SD16×C3112yes—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—81no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42C2×D4×C3111yescharacteristic
319C42C2×D4×C3111yescharacteristic
320C42C2×D4×C3111yescharacteristic
321C42#41×C3111yescharacteristic
322C23—41no—
323C23—42no—
324C23—82no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23C31×#12411yescharacteristic
329Q8—41no—
330Q8—41no—
331D4—41no—
332D4—42no—
333D4—41no—
334D4—42no—
335D4—41no—
336D4—41no—
337D4—81no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C31×#12011yescharacteristic
354C2×C4C31×#12111yescharacteristic
355C2×C4C31×#12111yescharacteristic
356C2×C4C31×#12111yescharacteristic
357C2×C4C31×#12311yescharacteristic
358C2×C4C31×#12311yescharacteristic
359C2×C4C2×D8×C3111yescharacteristic
360C2×C4C2×SD16×C3111yescharacteristic
361C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
362C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
363C8—22no—
364C8—22no—
365C8—42no—
366C8—81no—
367C8—81no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—82no—
373C22—81no—
374C22—82no—
375C22—82no—
376C22—81no—
377C22—162no—
378C22—162no—
379C22C31×#39411yescharacteristic
380C22C31×#48611yescharacteristic
381C22C31×#49011yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4C31×#44711yescharacteristic
387C4C31×#45011yescharacteristic
388C4C31×#45911yescharacteristic
389C4C31×#46011yescharacteristic
390C4C31×#47111yescharacteristic
391C4C31×#47211yescharacteristic
392C2—21no—
393C2—21no—
394C2—162no—
395C2—161no—
396C2C31×#204111yescharacteristic
397C2C31×#204811yescharacteristic
398C2C31×#205111yescharacteristic
subgroupG/H
center, socleC22×C31#394
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3151C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1950C2semidirect
3. C31×#1547C2semidirect
4. C31×#1545C2semidirect
5. C2×C31×#331C2semidirect
6. C31×#1480C2semidirect
7. C31×#1283C2semidirect
122. #10549C31direct (N × H)
171. #1950C2×C31semidirect
172. #1547C2×C31semidirect
173. #1545C2×C31semidirect
174. C2×#331C2×C31semidirect
175. #1480C2×C31semidirect
176. #1283C2×C31semidirect
267. C31#10549direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31512
C31×#19502
C31×#15472
C31×#15452
C2×C31×#3312
C31×#14802
C31×#12832
#1054931