Finite Group Databasebeta

← C31×#9957 #21466716

Subgroups — C31×#9957 #21466716

Order 7936.

744 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3189C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic
3C31×#2434C211yesmaximal, characteristic
4C31×#1996C211yesmaximal, characteristic
5C31×#1787C211yesmaximal, characteristic
6C31×#1520C211yesmaximal, characteristic
7C31×#1282C211yesmaximal, characteristic
8C23×Q8×C31C2211yescharacteristic
9C22×Q16×C31—21no—
10C2×C31×C4⋊Q8—21no—
11C2×C31×C22⋊Q8C2211yescharacteristic
12C2×C31×C22⋊Q8C2211yescharacteristic
13C31×#484—21no—
14C31×#484—21no—
15C31×#467—22no—
16C31×#451—21no—
17C31×#451—21no—
18C31×#451—22no—
19C31×#451—22no—
20C2×C31×#103—21no—
21C2×C31×#103—22no—
22C2×C31×C22⋊C8C2211yescharacteristic
23C2×C31×C22⋊C8C2211yescharacteristic
24C31×#395—22no—
25C31×#391—21no—
26C31×#385—22no—
27C31×#380C2211yescharacteristic
28C4×C22⋊C4×C31C2211yescharacteristic
29C31×#342—22no—
30C31×#328—22no—
31C22×Q8×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—44no—
37C22×Q8×C31D412yes—
38C22×Q8×C31D411yescharacteristic
39C22×Q8×C31D411yescharacteristic
40C23×C4×C31—21no—
41C23×C4×C31C2311yescharacteristic
42C2×Q16×C31—24no—
43C2×Q16×C31—22no—
44C2×Q16×C31—44no—
45C22×C8×C31—21no—
46C22×C8×C31D412yes—
47C31×C4⋊Q8—44no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—21no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—21no—
54C31×C22⋊Q8—21no—
55C31×C22⋊Q8—21no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31D412yes—
66C2×C42×C31—21no—
67C31×#107—22no—
68C31×#103—24no—
69C31×#103—22no—
70C31×#103—22no—
71C31×#103—44no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8D412yes—
76C31×#95—24no—
77C31×#95—21no—
78C31×#95—21no—
79C24×C31C2×D411yescharacteristic
80C2×Q8×C31—24no—
81C2×Q8×C31—24no—
82C2×Q8×C31—24no—
83C2×Q8×C31—22no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—22no—
88C2×Q8×C31—21no—
89C2×Q8×C31—44no—
90C2×Q8×C31—42no—
91C2×Q8×C31—42no—
92C2×Q8×C31—42no—
93C2×Q8×C31—42no—
94C2×Q8×C31—42no—
95C2×Q8×C31—44no—
96C2×Q8×C31—44no—
97C2×Q8×C31—41no—
98C2×Q8×C31—41no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—42no—
106C22×C4×C31—44no—
107C22×C4×C31—41no—
108C22×C4×C31—41no—
109C22×C4×C31—41no—
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2×D411yescharacteristic
112C22×C4×C31C2×D411yescharacteristic
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31#4111yescharacteristic
116Q16×C31—48no—
117Q16×C31—82no—
118C2×C8×C31—22no—
119C2×C8×C31—22no—
120C2×C8×C31—42no—
121C2×C8×C31—42no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—48no—
127C4⋊C4×C31—42no—
128C4⋊C4×C31—41no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31—41no—
131C4⋊C4×C31—41no—
132C22⋊C4×C31—22no—
133C22⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—42no—
138C22⋊C4×C31Q1614yescomplemented
139C42×C31—42no—
140#9957C3111yesmaximal, characteristic, direct product factor, Sylow
141C23×C31—41no—
142C23×C31—41no—
143C23×C31#12011yescharacteristic
144C23×C31#12111yescharacteristic
145C23×C31#12711yescharacteristic
146C23×C31C2×Q1612yes—
147C23×C31#13712yes—
148Q8×C31—24no—
149Q8×C31—24no—
150Q8×C31—42no—
151Q8×C31—42no—
152Q8×C31—44no—
153Q8×C31—42no—
154Q8×C31—44no—
155Q8×C31—42no—
156Q8×C31—42no—
157Q8×C31—81no—
158C2×C4×C31—22no—
159C2×C4×C31—22no—
160C2×C4×C31—22no—
161C2×C4×C31—24no—
162C2×C4×C31—24no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—44no—
170C2×C4×C31—44no—
171C2×C4×C31—42no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31—81no—
181C2×C4×C31—81no—
182C2×C4×C31—81no—
183C2×C4×C31#12011yescharacteristic
184C2×C4×C31#12011yescharacteristic
185C2×C4×C31#12111yescharacteristic
186C2×C4×C31#12111yescharacteristic
187C8×C31—42no—
188C8×C31—42no—
189#3189C2×C3111yescharacteristic
190C2×#451C2×C3111yescharacteristic
191#2434C2×C3111yescharacteristic
192#1996C2×C3111yescharacteristic
193#1787C2×C3111yescharacteristic
194#1520C2×C3111yescharacteristic
195#1282C2×C3111yescharacteristic
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—42no—
203C22×C31—42no—
204C22×C31#39211yescharacteristic
205C22×C31#45112yes—
206C22×C31#46212yes—
207C22×C31#49411yescharacteristic
208C22×C31#49711yescharacteristic
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—21no—
212C4×C31—44no—
213C4×C31—44no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31—161no—
219C4×C31B(4,𝔽2)12yes—
220C23×Q8C22×C3111yescharacteristic
221C22×Q16—21no—
222C2×C4⋊Q8—21no—
223C2×C22⋊Q8C22×C3111yescharacteristic
224C2×C22⋊Q8C22×C3111yescharacteristic
225#484—21no—
226#484—21no—
227#467—22no—
228#451—21no—
229#451—21no—
230#451—22no—
231#451—22no—
232C2×#103—21no—
233C2×#103—22no—
234C2×C22⋊C8C22×C3111yescharacteristic
235C2×C22⋊C8C22×C3111yescharacteristic
236#395—22no—
237#391—21no—
238#385—22no—
239#380C22×C3111yescharacteristic
240C4×C22⋊C4C22×C3111yescharacteristic
241#342—22no—
242#328—22no—
243C2×C31—22no—
244C2×C31—21no—
245C2×C31—22no—
246C2×C31—21no—
247C2×C31#200911yescharacteristic
248C2×C31#201611yescharacteristic
249C2×C31#202111yescharacteristic
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—44no—
256C22×Q8D4×C3111yescharacteristic
257C22×Q8D4×C3111yescharacteristic
258C22×Q8D4×C3112yes—
259C23×C4—21no—
260C23×C4C23×C3111yescharacteristic
261C2×Q16—24no—
262C2×Q16—22no—
263C2×Q16—44no—
264C22×C8—21no—
265C22×C8D4×C3112yes—
266C4⋊Q8—44no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4D4×C3111yescharacteristic
280C2×C4⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4D4×C3112yes—
285C2×C42—21no—
286#107—22no—
287#103—24no—
288#103—22no—
289#103—22no—
290#103—44no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8D4×C3112yes—
295#95—24no—
296#95—21no—
297#95—21no—
298C31#995711yescharacteristic, direct product factor, Sylow
299C24C2×D4×C3111yescharacteristic
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—41no—
318C2×Q8—41no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333C22×C4C2×D4×C3111yescharacteristic
334C22×C4C2×D4×C3111yescharacteristic
335C22×C4#41×C3111yescharacteristic
336Q16—48no—
337Q16—82no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C22⋊C4Q16×C3114yescomplemented
359C42—42no—
360C23—41no—
361C23—41no—
362C23C31×#12011yescharacteristic
363C23C31×#12111yescharacteristic
364C23C31×#12711yescharacteristic
365C23C2×Q16×C3112yes—
366C23C31×#13712yes—
367Q8—24no—
368Q8—24no—
369Q8—42no—
370Q8—42no—
371Q8—44no—
372Q8—42no—
373Q8—44no—
374Q8—42no—
375Q8—42no—
376Q8—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4C31×#12011yescharacteristic
403C2×C4C31×#12011yescharacteristic
404C2×C4C31×#12111yescharacteristic
405C2×C4C31×#12111yescharacteristic
406C8—42no—
407C8—42no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22C31×#39211yescharacteristic
417C22C31×#45112yes—
418C22C31×#46212yes—
419C22C31×#49411yescharacteristic
420C22C31×#49711yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—44no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—161no—
431C4C31×B(4,𝔽2)12yes—
432C2—22no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2C31×#200911yescharacteristic
437C2C31×#201611yescharacteristic
438C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
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
138. C22⋊C4×C31Q16semidirect
140. #9957C31direct (N × H)
298. C31#9957direct (N × H)
358. C22⋊C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31892
C2×C31×#4512
C31×#24342
C31×#19962
C31×#17872
C31×#15202
C31×#12822
#995731