Finite Group Databasebeta

← C31×#10178 #21466937

Subgroups — C31×#10178 #21466937

Order 7936.

592 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
1C31×#3208C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C2×C31×#459C211yesmaximal, characteristic
4C31×#1545C211yesmaximal, characteristic, complemented
5C31×#1481C211yesmaximal, characteristic, complemented
6C31×#1460C211yesmaximal, characteristic, complemented
7C31×#1301C211yesmaximal, characteristic, complemented
8C22×D8×C31—21no—
9C31×#533C2211yescharacteristic
10C2×C31×#127C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#492—21no—
13C31×#486—21no—
14C31×#478—22no—
15C31×#476—21no—
16C31×#474—21no—
17C31×#460—22no—
18C31×#459—22no—
19C31×#459—22no—
20C31×#450—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#428—21no—
24C31×#423C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C31×#418—21no—
27C2×C31×#102—21no—
28C31×#331—22no—
29C31×Q8⋊C8—22no—
30C31×#323—22no—
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C2×SD16×C31—24no—
37C2×D8×C31—24no—
38C2×D8×C31—42no—
39C2×D8×C31—42no—
40C2×M16×C31—21no—
41C22×C8×C31—21no—
42C31×C4⋊Q8D411yescharacteristic
43C31×C4⋊Q8D411yescharacteristic
44C31×#127—21no—
45C31×#127—21no—
46C31×#127—21no—
47C31×#127D412yes—
48C31×#125—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C31×#117—21no—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—21no—
60C2×C42×C31C2311yescharacteristic
61C31×#107—22no—
62C31×#106—22no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8D412yes—
68C31×C4⋊C8D412yes—
69C31×#102—24no—
70C31×#102—22no—
71C31×#102—22no—
72C31×#102—24no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8D412yes—
75C4×C8×C31—21no—
76C4×C8×C31—21no—
77C24×C31—21no—
78C24×C31—41no—
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—22no—
84C2×D4×C31—22no—
85C2×D4×C31—22no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C2×D4×C31—82no—
97C2×D4×C31—82no—
98C2×D4×C31SD1612yescomplemented
99C22×C4×C31—21no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—82no—
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31#4111yescharacteristic
106SD16×C31—84no—
107D8×C31—48no—
108M16×C31—41no—
109M16×C31—41no—
110C2×C8×C31—22no—
111C2×C8×C31—22no—
112C2×C8×C31—22no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—41no—
123C4⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C42×C31—41no—
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132#10178C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—22no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—42no—
137C23×C31—81no—
138C23×C31—81no—
139C23×C31—82no—
140C23×C31—82no—
141C23×C31—81no—
142C23×C31—81no—
143C23×C31#12111yescharacteristic
144Q8×C31—42no—
145Q8×C31—42no—
146D4×C31—22no—
147D4×C31—22no—
148D4×C31—44no—
149D4×C31—41no—
150D4×C31—42no—
151D4×C31—41no—
152D4×C31—42no—
153D4×C31—82no—
154D4×C31—82no—
155D4×C31—81no—
156D4×C31—81no—
157D4×C31—82no—
158C2×C4×C31—22no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—82no—
164C2×C4×C31—82no—
165C2×C4×C31—81no—
166C2×C4×C31#12011yescharacteristic
167C2×C4×C31#12011yescharacteristic
168C2×C4×C31#12011yescharacteristic
169C2×C4×C31#12111yescharacteristic
170C2×C4×C31#12111yescharacteristic
171C2×C4×C31#12711yescharacteristic
172C2×C4×C31C2×SD1611yescharacteristic
173C2×C4×C31#13511yescharacteristic
174C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C31#13711yescharacteristic
176C8×C31—42no—
177C8×C31—42no—
178C8×C31—42no—
179C8×C31—81no—
180#3208C2×C3111yescharacteristic, complemented
181#3077C2×C3111yescharacteristic, complemented
182C2×#459C2×C3111yescharacteristic
183#1545C2×C3111yescharacteristic, complemented
184#1481C2×C3111yescharacteristic, complemented
185#1460C2×C3111yescharacteristic, complemented
186#1301C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—42no—
190C22×C31—82no—
191C22×C31—82no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—82no—
195C22×C31—82no—
196C22×C31—82no—
197C22×C31—162no—
198C22×C31—162no—
199C22×C31#39211yescharacteristic
200C22×C31#46511yescharacteristic
201C22×C31#46811yescharacteristic
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—81no—
206C4×C31—81no—
207C4×C31—81no—
208C4×C31#44811yescharacteristic
209C4×C31#44911yescharacteristic
210C4×C31#45011yescharacteristic
211C4×C31#45211yescharacteristic
212C4×C31#47212yes—
213C22×D8—21no—
214#533C22×C3111yescharacteristic
215C2×#127C22×C3111yescharacteristic
216C2×C4×D4C22×C3111yescharacteristic
217#492—21no—
218#486—21no—
219#478—22no—
220#476—21no—
221#474—21no—
222#460—22no—
223#459—22no—
224#459—22no—
225#450—22no—
226#431C22×C3111yescharacteristic
227#431C22×C3111yescharacteristic
228#428—21no—
229#423C22×C3111yescharacteristic
230C2×C4⋊C8C22×C3111yescharacteristic
231#418—21no—
232C2×#102—21no—
233#331—22no—
234Q8⋊C8—22no—
235#323—22no—
236C2×C31—21no—
237C2×C31—21no—
238C2×C31—82no—
239C2×C31—162no—
240C2×C31#200011yescharacteristic
241C2×C31#200811yescharacteristic
242C2×C31#201811yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4D4×C3111yescharacteristic
247C23×C4—21no—
248C2×SD16—24no—
249C2×D8—24no—
250C2×D8—42no—
251C2×D8—42no—
252C2×M16—21no—
253C22×C8—21no—
254C4⋊Q8D4×C3111yescharacteristic
255C4⋊Q8D4×C3111yescharacteristic
256#127—21no—
257#127—21no—
258#127—21no—
259#127D4×C3112yes—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117—21no—
270C2×C4⋊C4D4×C3111yescharacteristic
271C2×C22⋊C4—21no—
272C2×C42C23×C3111yescharacteristic
273#107—22no—
274#106—22no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8D4×C3112yes—
280C4⋊C8D4×C3112yes—
281#102—24no—
282#102—22no—
283#102—22no—
284#102—24no—
285C22⋊C8—21no—
286C22⋊C8D4×C3112yes—
287C4×C8—21no—
288C4×C8—21no—
289C31#1017811yescharacteristic, direct product factor, Sylow
290C24—21no—
291C24—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4SD16×C3112yescomplemented
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4#41×C3111yescharacteristic
319SD16—84no—
320D8—48no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—41no—
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42C2×D4×C3111yescharacteristic
344C42C2×D4×C3111yescharacteristic
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23—82no—
353C23—81no—
354C23—81no—
355C23C31×#12111yescharacteristic
356Q8—42no—
357Q8—42no—
358D4—22no—
359D4—22no—
360D4—44no—
361D4—41no—
362D4—42no—
363D4—41no—
364D4—42no—
365D4—82no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—82no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4C31×#12011yescharacteristic
379C2×C4C31×#12011yescharacteristic
380C2×C4C31×#12011yescharacteristic
381C2×C4C31×#12111yescharacteristic
382C2×C4C31×#12111yescharacteristic
383C2×C4C31×#12711yescharacteristic
384C2×C4C2×SD16×C3111yescharacteristic
385C2×C4C31×#13511yescharacteristic
386C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C31×#13711yescharacteristic
388C8—42no—
389C8—42no—
390C8—42no—
391C8—81no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—162no—
403C22—162no—
404C22C31×#39211yescharacteristic
405C22C31×#46511yescharacteristic
406C22C31×#46811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4C31×#44811yescharacteristic
414C4C31×#44911yescharacteristic
415C4C31×#45011yescharacteristic
416C4C31×#45211yescharacteristic
417C4C31×#47212yes—
418C2—21no—
419C2—21no—
420C2—82no—
421C2—162no—
422C2C31×#200011yescharacteristic
423C2C31×#200811yescharacteristic
424C2C31×#201811yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#459C2
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
1. C31×#3208C2semidirect
2. C31×#3077C2semidirect
4. C31×#1545C2semidirect
5. C31×#1481C2semidirect
6. C31×#1460C2semidirect
7. C31×#1301C2semidirect
98. C2×D4×C31SD16semidirect
132. #10178C31direct (N × H)
180. #3208C2×C31semidirect
181. #3077C2×C31semidirect
183. #1545C2×C31semidirect
184. #1481C2×C31semidirect
185. #1460C2×C31semidirect
186. #1301C2×C31semidirect
289. C31#10178direct (N × H)
311. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32082
C31×#30772
C2×C31×#4592
C31×#15452
C31×#14812
C31×#14602
C31×#13012
#1017831