Finite Group Databasebeta

← C31×#9958 #21466717

Subgroups — C31×#9958 #21466717

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3187C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C31×#2504C211yesmaximal, characteristic
4C31×#1996C211yesmaximal, characteristic, complemented
5C31×#1824C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic
7C31×#1282C211yesmaximal, characteristic, complemented
8C2×C31×C22⋊Q8C2211yescharacteristic
9C2×C31×#121C2211yescharacteristic
10C22×C4⋊C4×C31C2211yescharacteristic
11C31×#484—22no—
12C31×#484—22no—
13C31×#483—21no—
14C31×#481—21no—
15C31×#465—22no—
16C31×#450—21no—
17C31×#448—21no—
18C2×C31×#107—21no—
19C2×C31×#103—21no—
20C2×C31×C22⋊C8C2211yescharacteristic
21C2×C31×C22⋊C8C2211yescharacteristic
22C31×#397—22no—
23C31×#390—21no—
24C31×#386—22no—
25C31×#382—21no—
26C31×#380C2211yescharacteristic
27C4×C22⋊C4×C31C2211yescharacteristic
28C31×#342—22no—
29C31×#340—22no—
30C31×#327—22no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C23×C4×C31C2311yescharacteristic
36C2×SD16×C31—22no—
37C22×C8×C31—21no—
38C22×C8×C31D412yes—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×#121—21no—
44C31×#121—21no—
45C31×#121—21no—
46C31×#121—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31D412yes—
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31D412yes—
62C2×C42×C31—21no—
63C31×#107—24no—
64C31×#106—22no—
65C31×#103—24no—
66C31×#103—22no—
67C31×#102—22no—
68C31×C22⋊C8—21no—
69C31×C22⋊C8—21no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8D412yes—
72C31×#95—24no—
73C31×#95—21no—
74C31×#95—21no—
75C24×C31—21no—
76C24×C31C2×D411yescharacteristic
77C2×Q8×C31—21no—
78C2×Q8×C31—21no—
79C2×Q8×C31—42no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—42no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—42no—
86C2×D4×C31—42no—
87C22×C4×C31—21no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—42no—
93C22×C4×C31—41no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—82no—
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102C22×C4×C31#4111yescharacteristic
103SD16×C31—82no—
104C2×C8×C31—22no—
105C2×C8×C31—22no—
106C2×C8×C31—42no—
107C2×C8×C31—42no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—22no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—42no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—22no—
120C22⋊C4×C31—22no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—42no—
126C22⋊C4×C31—84no—
127C42×C31—42no—
128#9958C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—42no—
132C23×C31—42no—
133C23×C31—41no—
134C23×C31—42no—
135C23×C31#12011yescharacteristic
136C23×C31#12111yescharacteristic
137C23×C31#12711yescharacteristic
138C23×C31#13512yes—
139C23×C31#13712yes—
140Q8×C31—42no—
141Q8×C31—42no—
142D4×C31—42no—
143D4×C31—81no—
144D4×C31—82no—
145D4×C31—81no—
146D4×C31—81no—
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—42no—
155C2×C4×C31—44no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—82no—
163C2×C4×C31—82no—
164C2×C4×C31—81no—
165C2×C4×C31—81no—
166C2×C4×C31—81no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C8×C31—42no—
174C8×C31—42no—
175#3187C2×C3111yescharacteristic
176C2×#484C2×C3111yescharacteristic, complemented
177#2504C2×C3111yescharacteristic
178#1996C2×C3111yescharacteristic, complemented
179#1824C2×C3111yescharacteristic, complemented
180#1519C2×C3111yescharacteristic
181#1282C2×C3111yescharacteristic, complemented
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—42no—
189C22×C31—42no—
190C22×C31—84no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31#39211yescharacteristic
195C22×C31#45212yes—
196C22×C31#46412yes—
197C22×C31#49511yescharacteristic
198C22×C31#49711yescharacteristic
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—82no—
203C4×C31—82no—
204C4×C31—81no—
205C4×C31—81no—
206C4×C31—81no—
207C4×C31—161no—
208C4×C31#45812yes—
209C2×C22⋊Q8C22×C3111yescharacteristic
210C2×#121C22×C3111yescharacteristic
211C22×C4⋊C4C22×C3111yescharacteristic
212#484—22no—
213#484—22no—
214#483—21no—
215#481—21no—
216#465—22no—
217#450—21no—
218#448—21no—
219C2×#107—21no—
220C2×#103—21no—
221C2×C22⋊C8C22×C3111yescharacteristic
222C2×C22⋊C8C22×C3111yescharacteristic
223#397—22no—
224#390—21no—
225#386—22no—
226#382—21no—
227#380C22×C3111yescharacteristic
228C4×C22⋊C4C22×C3111yescharacteristic
229#342—22no—
230#340—22no—
231#327—22no—
232C2×C31—22no—
233C2×C31—21no—
234C2×C31—22no—
235C2×C31—21no—
236C2×C31—161no—
237C2×C31#200811yescharacteristic
238C2×C31#201511yescharacteristic
239C2×C31#202211yescharacteristic
240C22×Q8D4×C3111yescharacteristic
241C22×D4—21no—
242C22×D4D4×C3111yescharacteristic
243C23×C4—21no—
244C23×C4C23×C3111yescharacteristic
245C2×SD16—22no—
246C22×C8—21no—
247C22×C8D4×C3112yes—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4D4×C3112yes—
264C2×C4⋊C4D4×C3111yescharacteristic
265C2×C4⋊C4D4×C3111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4D4×C3112yes—
271C2×C42—21no—
272#107—24no—
273#106—22no—
274#103—24no—
275#103—22no—
276#102—22no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8D4×C3112yes—
281#95—24no—
282#95—21no—
283#95—21no—
284C31#995811yescharacteristic, direct product factor, Sylow
285C24—21no—
286C24C2×D4×C3111yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4C2×D4×C3111yescharacteristic
308C22×C4C2×D4×C3111yescharacteristic
309C22×C4C2×D4×C3111yescharacteristic
310C22×C4C2×D4×C3111yescharacteristic
311C22×C4C2×D4×C3111yescharacteristic
312C22×C4#41×C3111yescharacteristic
313SD16—82no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—84no—
337C42—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23C31×#12011yescharacteristic
345C23C31×#12111yescharacteristic
346C23C31×#12711yescharacteristic
347C23C31×#13512yes—
348C23C31×#13712yes—
349Q8—42no—
350Q8—42no—
351D4—42no—
352D4—81no—
353D4—82no—
354D4—81no—
355D4—81no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4C31×#12011yescharacteristic
379C2×C4C31×#12011yescharacteristic
380C2×C4C31×#12111yescharacteristic
381C2×C4C31×#12111yescharacteristic
382C8—42no—
383C8—42no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C31×#39211yescharacteristic
397C22C31×#45212yes—
398C22C31×#46412yes—
399C22C31×#49511yescharacteristic
400C22C31×#49711yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—161no—
410C4C31×#45812yes—
411C2—22no—
412C2—21no—
413C2—22no—
414C2—21no—
415C2—161no—
416C2C31×#200811yescharacteristic
417C2C31×#201511yescharacteristic
418C2C31×#202211yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#484C2semidirect
4. C31×#1996C2semidirect
5. C31×#1824C2semidirect
7. C31×#1282C2semidirect
128. #9958C31direct (N × H)
176. C2×#484C2×C31semidirect
178. #1996C2×C31semidirect
179. #1824C2×C31semidirect
181. #1282C2×C31semidirect
284. C31#9958direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31872
C2×C31×#4842
C31×#25042
C31×#19962
C31×#18242
C31×#15192
C31×#12822
#995831