Finite Group Databasebeta

← C31×#10500 #21467259

Subgroups — C31×#10500 #21467259

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3205C211yesmaximal, characteristic
2C31×#1982C211yesmaximal, characteristic, complemented
3C31×#1980C211yesmaximal, characteristic, complemented
4C31×#1965C211yesmaximal, characteristic, complemented
5C31×#1565C211yesmaximal, characteristic, complemented
6C31×#1547C211yesmaximal, characteristic, complemented
7C31×#1303C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C31×#127C2211yescharacteristic
10C31×#493—21no—
11C31×#486—21no—
12C31×#459—21no—
13C31×#459—21no—
14C31×#447—22no—
15C31×#431C2211yescharacteristic
16C31×#423C2211yescharacteristic
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×#104—21no—
19C2×C31×#104—21no—
20C2×C31×#102—22no—
21C2×C31×#100—21no—
22C2×C31×#100—21no—
23C4×M16×C31C2211yescharacteristic
24C4×C4⋊C4×C31C2211yescharacteristic
25C31×#335—21no—
26C31×#334—21no—
27C31×#332—21no—
28C31×#331—21no—
29C2×#41×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31D412yes—
33C2×D8×C31—22no—
34C2×D8×C31—22no—
35C2×M16×C31—21no—
36C2×M16×C31D411yescharacteristic
37C2×M16×C31D411yescharacteristic
38C22×C8×C31—21no—
39C31×C4⋊Q8D411yescharacteristic
40C31×#127—21no—
41C31×#127—21no—
42C31×#127—21no—
43C31×#127—21no—
44C31×#127D411yescharacteristic
45C31×#124—21no—
46C31×#121—21no—
47C31×#121—21no—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×C4⋊C8—21no—
54C31×C4⋊C8—21no—
55C31×C4⋊C8—21no—
56C31×C4⋊C8—21no—
57C31×C4⋊C8D411yescharacteristic
58C31×C4⋊C8D411yescharacteristic
59C31×#104—42no—
60C31×#104—42no—
61C31×#102—22no—
62C31×#102—22no—
63C31×#102—44no—
64C31×#100—42no—
65C31×#100—42no—
66C31×C22⋊C8—21no—
67C31×#97—21no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C31×#95—21no—
71C24×C31—22no—
72#41×C31—41no—
73#41×C31—41no—
74#41×C31—41no—
75#41×C31—41no—
76C2×Q8×C31—21no—
77C2×D4×C31—22no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—22no—
81C2×D4×C31—21no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C2×D4×C31—82no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31#4111yescharacteristic
99D8×C31—82no—
100D8×C31—82no—
101M16×C31—21no—
102M16×C31—21no—
103M16×C31—21no—
104M16×C31—41no—
105M16×C31—41no—
106M16×C31D812yescomplemented
107C2×C8×C31—21no—
108C2×C8×C31—21no—
109C2×C8×C31—21no—
110C2×C8×C31—21no—
111C2×C8×C31—42no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—41no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C42×C31—41no—
121C42×C31—41no—
122C42×C31C2×D411yescharacteristic
123C42×C31C2×D411yescharacteristic
124C42×C31#4111yescharacteristic
125C42×C31#4111yescharacteristic
126#10500C3111yesmaximal, characteristic, direct product factor, Sylow
127C23×C31—41no—
128C23×C31—42no—
129C23×C31—42no—
130C23×C31—42no—
131C23×C31—82no—
132C23×C31—82no—
133C23×C31#12111yescharacteristic
134Q8×C31—41no—
135Q8×C31—41no—
136D4×C31—44no—
137D4×C31—41no—
138D4×C31—42no—
139D4×C31—41no—
140D4×C31—42no—
141D4×C31—41no—
142D4×C31—41no—
143D4×C31—81no—
144D4×C31—82no—
145D4×C31—82no—
146D4×C31—81no—
147D4×C31—81no—
148D4×C31—81no—
149D4×C31—81no—
150D4×C31—81no—
151D4×C31—81no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—41no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12111yescharacteristic
163C2×C4×C31#12111yescharacteristic
164C2×C4×C31#12311yescharacteristic
165C2×C4×C31#12311yescharacteristic
166C2×C4×C31#12411yescharacteristic
167C2×C4×C31C2×D811yescharacteristic
168C2×C4×C31#13511yescharacteristic
169C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
171C8×C31—22no—
172C8×C31—41no—
173C8×C31—41no—
174C8×C31—41no—
175C8×C31—81no—
176C8×C31—81no—
177#3205C2×C3111yescharacteristic
178#1982C2×C3111yescharacteristic, complemented
179#1980C2×C3111yescharacteristic, complemented
180#1965C2×C3111yescharacteristic, complemented
181#1565C2×C3111yescharacteristic, complemented
182#1547C2×C3111yescharacteristic, complemented
183#1303C2×C3111yescharacteristic, complemented
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—82no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—82no—
190C22×C31—82no—
191C22×C31—81no—
192C22×C31—162no—
193C22×C31—162no—
194C22×C31#39411yescharacteristic
195C22×C31#45911yescharacteristic
196C22×C31#46311yescharacteristic
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—82no—
201C4×C31—81no—
202C4×C31#45311yescharacteristic
203C4×C31#45511yescharacteristic
204C4×C31#45911yescharacteristic
205C4×C31#46311yescharacteristic
206C4×C31#46611yescharacteristic
207C4×C31#46911yescharacteristic
208#532C22×C3111yescharacteristic
209C2×#127C22×C3111yescharacteristic
210#493—21no—
211#486—21no—
212#459—21no—
213#459—21no—
214#447—22no—
215#431C22×C3111yescharacteristic
216#423C22×C3111yescharacteristic
217C2×C4⋊C8C22×C3111yescharacteristic
218C2×#104—21no—
219C2×#104—21no—
220C2×#102—22no—
221C2×#100—21no—
222C2×#100—21no—
223C4×M16C22×C3111yescharacteristic
224C4×C4⋊C4C22×C3111yescharacteristic
225#335—21no—
226#334—21no—
227#332—21no—
228#331—21no—
229C2×C31—21no—
230C2×C31—21no—
231C2×C31—161no—
232C2×C31—162no—
233C2×C31#203711yescharacteristic
234C2×C31#204811yescharacteristic
235C2×C31#205411yescharacteristic
236C2×#41—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4D4×C3112yes—
240C2×D8—22no—
241C2×D8—22no—
242C2×M16—21no—
243C2×M16D4×C3111yescharacteristic
244C2×M16D4×C3111yescharacteristic
245C22×C8—21no—
246C4⋊Q8D4×C3111yescharacteristic
247#127—21no—
248#127—21no—
249#127—21no—
250#127—21no—
251#127D4×C3111yescharacteristic
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C2×C4⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C3111yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8D4×C3111yescharacteristic
265C4⋊C8D4×C3111yescharacteristic
266#104—42no—
267#104—42no—
268#102—22no—
269#102—22no—
270#102—44no—
271#100—42no—
272#100—42no—
273C22⋊C8—21no—
274#97—21no—
275C4×C8—21no—
276C4×C8—21no—
277#95—21no—
278C31#1050011yescharacteristic, direct product factor, Sylow
279C24—22no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×D4×C3111yescharacteristic
305C22×C4C2×D4×C3111yescharacteristic
306C22×C4#41×C3111yescharacteristic
307D8—82no—
308D8—82no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16D8×C3112yescomplemented
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—41no—
329C42—41no—
330C42C2×D4×C3111yescharacteristic
331C42C2×D4×C3111yescharacteristic
332C42#41×C3111yescharacteristic
333C42#41×C3111yescharacteristic
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23C31×#12111yescharacteristic
341Q8—41no—
342Q8—41no—
343D4—44no—
344D4—41no—
345D4—42no—
346D4—41no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—81no—
351D4—82no—
352D4—82no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C31×#12011yescharacteristic
369C2×C4C31×#12111yescharacteristic
370C2×C4C31×#12111yescharacteristic
371C2×C4C31×#12311yescharacteristic
372C2×C4C31×#12311yescharacteristic
373C2×C4C31×#12411yescharacteristic
374C2×C4C2×D8×C3111yescharacteristic
375C2×C4C31×#13511yescharacteristic
376C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
378C8—22no—
379C8—41no—
380C8—41no—
381C8—41no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—81no—
387C22—82no—
388C22—82no—
389C22—81no—
390C22—82no—
391C22—81no—
392C22—162no—
393C22—162no—
394C22C31×#39411yescharacteristic
395C22C31×#45911yescharacteristic
396C22C31×#46311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4C31×#45311yescharacteristic
403C4C31×#45511yescharacteristic
404C4C31×#45911yescharacteristic
405C4C31×#46311yescharacteristic
406C4C31×#46611yescharacteristic
407C4C31×#46911yescharacteristic
408C2—21no—
409C2—21no—
410C2—162no—
411C2—161no—
412C2C31×#203711yescharacteristic
413C2C31×#204811yescharacteristic
414C2C31×#205411yescharacteristic
subgroupG/H
center, socleC22×C31#394
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3205C2
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×#1982C2semidirect
3. C31×#1980C2semidirect
4. C31×#1965C2semidirect
5. C31×#1565C2semidirect
6. C31×#1547C2semidirect
7. C31×#1303C2semidirect
106. M16×C31D8semidirect
126. #10500C31direct (N × H)
178. #1982C2×C31semidirect
179. #1980C2×C31semidirect
180. #1965C2×C31semidirect
181. #1565C2×C31semidirect
182. #1547C2×C31semidirect
183. #1303C2×C31semidirect
278. C31#10500direct (N × H)
314. M16D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32052
C31×#19822
C31×#19802
C31×#19652
C31×#15652
C31×#15472
C31×#13032
#1050031