Finite Group Databasebeta

← C31×#10200 #21466959

Subgroups — C31×#10200 #21466959

Order 7936.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3245C211yesmaximal, characteristic
2C31×#3227C211yesmaximal, characteristic, complemented
3C31×#3141C211yesmaximal, characteristic, complemented
4C31×#1981C211yesmaximal, characteristic, complemented
5C31×#1980C211yesmaximal, characteristic, complemented
6C31×#1502C211yesmaximal, characteristic, complemented
7C31×#1283C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C31×#498—21no—
12C31×#489—22no—
13C31×#487—21no—
14C31×#486—21no—
15C31×#486—21no—
16C31×#481—22no—
17C31×#476—22no—
18C31×#474—21no—
19C31×#474—21no—
20C31×#466—22no—
21C31×#459—21no—
22C31×#459—21no—
23C31×#431C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#104—22no—
26C31×#417—22no—
27C2×C31×#103—22no—
28C4×M16×C31C2211yescharacteristic
29C4×C4⋊C4×C31C2211yescharacteristic
30C31×#327—22no—
31C2×#41×C31—21no—
32C2×#41×C31—21no—
33C22×Q8×C31—21no—
34C2×D8×C31—22no—
35C2×M16×C31D412yes—
36C22×C8×C31D412yes—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8D411yescharacteristic
41C31×C4⋊Q8D411yescharacteristic
42C31×C4⋊Q8D412yes—
43C31×#127D411yescharacteristic
44C31×#127D411yescharacteristic
45C31×#124—21no—
46C31×#124—21no—
47C31×#121—21no—
48C31×#121—21no—
49C31×#121—21no—
50C31×#121—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C42×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×#107—22no—
61C31×#106—24no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×#104—44no—
66C31×#103—24no—
67C31×#103—44no—
68C31×#102—24no—
69C31×#102—22no—
70C31×#102—22no—
71C31×#102—24no—
72C31×C22⋊C8D412yes—
73C31×#97—21no—
74C4×C8×C31—21no—
75C4×C8×C31—21no—
76C31×#95—21no—
77#41×C31—42no—
78#41×C31—42no—
79#41×C31—42no—
80#41×C31—42no—
81C2×Q8×C31—21no—
82C2×Q8×C31—21no—
83C2×Q8×C31—22no—
84C2×Q8×C31—42no—
85C2×Q8×C31—42no—
86C2×Q8×C31—41no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31—41no—
98C22×C4×C31—41no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102D8×C31—82no—
103M16×C31—22no—
104M16×C31—22no—
105C2×C8×C31—22no—
106C2×C8×C31—22no—
107C2×C8×C31—22no—
108C2×C8×C31—42no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—22no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C42×C31—42no—
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31C2×D411yescharacteristic
127C42×C31#4111yescharacteristic
128#10200C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—41no—
130C23×C31—41no—
131C23×C31#12711yescharacteristic
132Q8×C31—44no—
133Q8×C31—42no—
134Q8×C31—42no—
135Q8×C31—81no—
136Q8×C31—81no—
137D4×C31—42no—
138D4×C31—42no—
139D4×C31—41no—
140D4×C31—41no—
141D4×C31—41no—
142D4×C31—41no—
143D4×C31—42no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148D4×C31—81no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—41no—
152C2×C4×C31—42no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—81no—
157C2×C4×C31—82no—
158C2×C4×C31—81no—
159C2×C4×C31—82no—
160C2×C4×C31—82no—
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12011yescharacteristic
163C2×C4×C31#12011yescharacteristic
164C2×C4×C31#12111yescharacteristic
165C2×C4×C31#12111yescharacteristic
166C2×C4×C31#12111yescharacteristic
167C2×C4×C31#13512yes—
168C2×C4×C31AGL(1,ℤ/8)12yes—
169C8×C31—42no—
170C8×C31—42no—
171C8×C31—42no—
172#3245C2×C3111yescharacteristic
173#3227C2×C3111yescharacteristic, complemented
174#3141C2×C3111yescharacteristic, complemented
175#1981C2×C3111yescharacteristic, complemented
176#1980C2×C3111yescharacteristic, complemented
177#1502C2×C3111yescharacteristic, complemented
178#1283C2×C3111yescharacteristic, complemented
179C22×C31—21no—
180C22×C31—21no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31#39211yescharacteristic
188C22×C31#49511yescharacteristic
189C22×C31#49611yescharacteristic
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—82no—
194C4×C31—82no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31#44912yes—
198C4×C31#45311yescharacteristic
199C4×C31#45411yescharacteristic
200C4×C31#46312yes—
201#532C22×C3111yescharacteristic
202#532C22×C3111yescharacteristic
203C2×C4⋊Q8C22×C3111yescharacteristic
204#498—21no—
205#489—22no—
206#487—21no—
207#486—21no—
208#486—21no—
209#481—22no—
210#476—22no—
211#474—21no—
212#474—21no—
213#466—22no—
214#459—21no—
215#459—21no—
216#431C22×C3111yescharacteristic
217C2×C4⋊C8C22×C3111yescharacteristic
218C2×#104—22no—
219#417—22no—
220C2×#103—22no—
221C4×M16C22×C3111yescharacteristic
222C4×C4⋊C4C22×C3111yescharacteristic
223#327—22no—
224C2×C31—21no—
225C2×C31—21no—
226C2×C31—161no—
227C2×C31—161no—
228C2×C31#200611yescharacteristic
229C2×C31#200911yescharacteristic
230C2×C31#201211yescharacteristic
231C2×#41—21no—
232C2×#41—21no—
233C22×Q8—21no—
234C2×D8—22no—
235C2×M16D4×C3112yes—
236C22×C8D4×C3112yes—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8D4×C3112yes—
241C4⋊Q8D4×C3111yescharacteristic
242C4⋊Q8D4×C3111yescharacteristic
243#127D4×C3111yescharacteristic
244#127D4×C3111yescharacteristic
245#124—21no—
246#124—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C3111yescharacteristic
260#107—22no—
261#106—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265#104—44no—
266#103—24no—
267#103—44no—
268#102—24no—
269#102—22no—
270#102—22no—
271#102—24no—
272C22⋊C8D4×C3112yes—
273#97—21no—
274C4×C8—21no—
275C4×C8—21no—
276#95—21no—
277C31#1020011yescharacteristic, direct product factor, Sylow
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4C2×D4×C3111yescharacteristic
302C22×C4C2×D4×C3111yescharacteristic
303D8—82no—
304M16—22no—
305M16—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—42no—
325C42C2×D4×C3111yescharacteristic
326C42C2×D4×C3111yescharacteristic
327C42C2×D4×C3111yescharacteristic
328C42#41×C3111yescharacteristic
329C23—41no—
330C23—41no—
331C23C31×#12711yescharacteristic
332Q8—44no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—42no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4C31×#12011yescharacteristic
362C2×C4C31×#12011yescharacteristic
363C2×C4C31×#12011yescharacteristic
364C2×C4C31×#12111yescharacteristic
365C2×C4C31×#12111yescharacteristic
366C2×C4C31×#12111yescharacteristic
367C2×C4C31×#13512yes—
368C2×C4C31×AGL(1,ℤ/8)12yes—
369C8—42no—
370C8—42no—
371C8—42no—
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C31×#39211yescharacteristic
381C22C31×#49511yescharacteristic
382C22C31×#49611yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4C31×#44912yes—
391C4C31×#45311yescharacteristic
392C4C31×#45411yescharacteristic
393C4C31×#46312yes—
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2C31×#200611yescharacteristic
399C2C31×#200911yescharacteristic
400C2C31×#201211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3245C2
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×#3227C2semidirect
3. C31×#3141C2semidirect
4. C31×#1981C2semidirect
5. C31×#1980C2semidirect
6. C31×#1502C2semidirect
7. C31×#1283C2semidirect
128. #10200C31direct (N × H)
173. #3227C2×C31semidirect
174. #3141C2×C31semidirect
175. #1981C2×C31semidirect
176. #1980C2×C31semidirect
177. #1502C2×C31semidirect
178. #1283C2×C31semidirect
277. C31#10200direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32452
C31×#32272
C31×#31412
C31×#19812
C31×#19802
C31×#15022
C31×#12832
#1020031