Finite Group Databasebeta

← C31×#19515 #21476274

Subgroups — C31×#19515 #21476274

Order 7936.

576 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3235C211yesmaximal, characteristic
2C31×#3230C211yesmaximal, characteristic, complemented
3C31×#3227C211yesmaximal, characteristic
4C31×#2981C211yesmaximal, characteristic
5C31×#1703C212yesmaximal, complemented
6C31×#1695C212yesmaximal, complemented
7C31×#1683C212yesmaximal, complemented
8C31×#1675C212yesmaximal, complemented
9C31×#1572C211yesmaximal, characteristic, complemented
10C31×#1558C211yesmaximal, characteristic, complemented
11C31×#1554C211yesmaximal, characteristic, complemented
12C31×#533C2211yescharacteristic
13C31×#532C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#501—21no—
16C31×#500C2211yescharacteristic
17C31×#500C2211yescharacteristic
18C31×#498—21no—
19C31×#490—21no—
20C31×#487C2211yescharacteristic
21C31×#487C2211yescharacteristic
22C31×#486—21no—
23C31×#484—22no—
24C31×#482—22no—
25C31×#481—22no—
26C31×#477—22no—
27C31×#476C2212yes—
28C31×#476C2212yes—
29C31×#476C2212yes—
30C31×#474C2212yes—
31C31×#436C2212yes—
32C31×#435—22no—
33C8×D4×C31C2212yes—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#408—22no—
39C31×#335C2212yes—
40C31×#334C2212yes—
41C31×#333C2212yes—
42C31×#333C2212yes—
43C31×#333C2212yes—
44C31×#332C2212yes—
45C2×#41×C31—21no—
46C22×Q8×C31—21no—
47C2×M16×C31—22no—
48C22×C8×C31—22no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8D411yescharacteristic
53C31×C4⋊Q8C2311yescharacteristic
54C31×C4⋊Q8C2311yescharacteristic
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2311yescharacteristic
57C31×#127D411yescharacteristic
58C31×#125—21no—
59C31×#124—21no—
60C31×C22⋊Q8—21no—
61C31×C22⋊Q8—21no—
62C31×#121—21no—
63C31×#121D412yes—
64C4×Q8×C31—21no—
65C4×Q8×C31—21no—
66C4×D4×C31—21no—
67C4×D4×C31C2312yes—
68C31×#117—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C42×C31C2311yescharacteristic
72C31×#107—22no—
73C31×#107—22no—
74C31×#107—22no—
75C31×#106—22no—
76C31×#106—22no—
77C31×#106—22no—
78C31×C4⋊C8D412yes—
79C31×C4⋊C8C2312yes—
80C31×C4⋊C8C2312yes—
81C31×C4⋊C8C2312yes—
82C31×#103—22no—
83C31×#103—22no—
84C31×#102—22no—
85C31×#102—22no—
86C31×#102—22no—
87C31×#102—22no—
88C31×C22⋊C8—21no—
89C31×C22⋊C8—21no—
90C31×C22⋊C8—21no—
91C31×C22⋊C8—22no—
92C31×C22⋊C8—22no—
93C31×C22⋊C8D412yes—
94C31×#97—21no—
95C31×#97—21no—
96C4×C8×C31—21no—
97C4×C8×C31C2311yescharacteristic
98C4×C8×C31C2311yescharacteristic
99#41×C31—82no—
100C2×Q8×C31—21no—
101C2×Q8×C31—21no—
102C2×Q8×C31—21no—
103C2×Q8×C31—21no—
104C2×Q8×C31—21no—
105C2×Q8×C31—41no—
106C2×Q8×C31—82no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31C2×D412yes—
111C22×C4×C31—22no—
112C22×C4×C31—41no—
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31#4111yescharacteristic
116M16×C31—82no—
117C2×C8×C31—22no—
118C2×C8×C31—22no—
119C2×C8×C31—22no—
120C2×C8×C31—82no—
121C2×C8×C31C2×D412yes—
122C2×C8×C31C2×D412yes—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—41no—
133C4⋊C4×C31—41no—
134C4⋊C4×C31—41no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31C2×D412yes—
137C22⋊C4×C31—22no—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—41no—
140C42×C31—41no—
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31#4111yescharacteristic
144C42×C31C2411yescharacteristic
145#19515C3111yesmaximal, characteristic, direct product factor, Sylow
146C23×C31—22no—
147C23×C31#12111yescharacteristic
148Q8×C31—42no—
149Q8×C31—81no—
150Q8×C31—81no—
151Q8×C31—81no—
152Q8×C31—81no—
153Q8×C31—82no—
154D4×C31—24no—
155D4×C31—81no—
156D4×C31—81no—
157D4×C31—82no—
158C2×C4×C31—22no—
159C2×C4×C31—22no—
160C2×C4×C31—22no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—82no—
169C2×C4×C31#12111yescharacteristic
170C2×C4×C31#12111yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31C22×D411yescharacteristic
173C2×C4×C31C22×D411yescharacteristic
174C2×C4×C31C2×#4111yescharacteristic
175C8×C31—22no—
176C8×C31—22no—
177C8×C31—42no—
178C8×C31—42no—
179C8×C31—42no—
180#3235C2×C3111yescharacteristic
181#3230C2×C3111yescharacteristic, complemented
182#3227C2×C3111yescharacteristic
183#2981C2×C3111yescharacteristic
184#1703C2×C3112yescomplemented
185#1695C2×C3112yescomplemented
186#1683C2×C3112yescomplemented
187#1675C2×C3112yescomplemented
188#1572C2×C3111yescharacteristic, complemented
189#1558C2×C3111yescharacteristic, complemented
190#1554C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—42no—
197C22×C31C2×#12111yescharacteristic
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—42no—
203C4×C31—81no—
204C4×C31—81no—
205C4×C31—81no—
206C4×C31—81no—
207C4×C31#57712yes—
208C4×C31#57812yes—
209#533C22×C3111yescharacteristic
210#532C22×C3111yescharacteristic
211C2×C4⋊Q8C22×C3111yescharacteristic
212#501—21no—
213#500C22×C3111yescharacteristic
214#500C22×C3111yescharacteristic
215#498—21no—
216#490—21no—
217#487C22×C3111yescharacteristic
218#487C22×C3111yescharacteristic
219#486—21no—
220#484—22no—
221#482—22no—
222#481—22no—
223#477—22no—
224#476C22×C3112yes—
225#476C22×C3112yes—
226#476C22×C3112yes—
227#474C22×C3112yes—
228#436C22×C3112yes—
229#435—22no—
230C8×D4C22×C3112yes—
231#432C22×C3111yescharacteristic
232#432C22×C3111yescharacteristic
233#431C22×C3111yescharacteristic
234#431C22×C3111yescharacteristic
235#408—22no—
236#335C22×C3112yes—
237#334C22×C3112yes—
238#333C22×C3112yes—
239#333C22×C3112yes—
240#333C22×C3112yes—
241#332C22×C3112yes—
242C2×C31—41no—
243C2×C31—82no—
244C2×C31#304211yescharacteristic
245C2×C31#305911yescharacteristic
246C2×C31#306811yescharacteristic
247C2×#41—21no—
248C22×Q8—21no—
249C2×M16—22no—
250C22×C8—22no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8D4×C3111yescharacteristic
255C4⋊Q8C23×C3111yescharacteristic
256C4⋊Q8C23×C3111yescharacteristic
257C4⋊Q8C23×C3111yescharacteristic
258C4⋊Q8C23×C3111yescharacteristic
259#127D4×C3111yescharacteristic
260#125—21no—
261#124—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264#121—21no—
265#121D4×C3112yes—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×D4—21no—
269C4×D4C23×C3112yes—
270#117—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C42C23×C3111yescharacteristic
274#107—22no—
275#107—22no—
276#107—22no—
277#106—22no—
278#106—22no—
279#106—22no—
280C4⋊C8D4×C3112yes—
281C4⋊C8C23×C3112yes—
282C4⋊C8C23×C3112yes—
283C4⋊C8C23×C3112yes—
284#103—22no—
285#103—22no—
286#102—22no—
287#102—22no—
288#102—22no—
289#102—22no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8D4×C3112yes—
296#97—21no—
297#97—21no—
298C4×C8—21no—
299C4×C8C23×C3111yescharacteristic
300C4×C8C23×C3111yescharacteristic
301C31#1951511yescharacteristic, direct product factor, Sylow
302#41—82no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4C2×D4×C3112yes—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4#41×C3111yescharacteristic
319M16—82no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—82no—
324C2×C8C2×D4×C3112yes—
325C2×C8C2×D4×C3112yes—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4C2×D4×C3112yes—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—41no—
343C42—41no—
344C42C2×D4×C3111yescharacteristic
345C42C2×D4×C3111yescharacteristic
346C42#41×C3111yescharacteristic
347C42C24×C3111yescharacteristic
348C23—22no—
349C23C31×#12111yescharacteristic
350Q8—42no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354Q8—81no—
355Q8—82no—
356D4—24no—
357D4—81no—
358D4—81no—
359D4—82no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4C31×#12111yescharacteristic
372C2×C4C31×#12111yescharacteristic
373C2×C4C31×#12111yescharacteristic
374C2×C4C22×D4×C3111yescharacteristic
375C2×C4C22×D4×C3111yescharacteristic
376C2×C4C2×#41×C3111yescharacteristic
377C8—22no—
378C8—22no—
379C8—42no—
380C8—42no—
381C8—42no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22C2×C31×#12111yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4C31×#57712yes—
399C4C31×#57812yes—
400C2—41no—
401C2—82no—
402C2C31×#304211yescharacteristic
403C2C31×#305911yescharacteristic
404C2C31×#306811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#532C22
LCSG ▷ 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×#3230C2semidirect
5. C31×#1703C2semidirect
6. C31×#1695C2semidirect
7. C31×#1683C2semidirect
8. C31×#1675C2semidirect
9. C31×#1572C2semidirect
10. C31×#1558C2semidirect
11. C31×#1554C2semidirect
145. #19515C31direct (N × H)
181. #3230C2×C31semidirect
184. #1703C2×C31semidirect
185. #1695C2×C31semidirect
186. #1683C2×C31semidirect
187. #1675C2×C31semidirect
188. #1572C2×C31semidirect
189. #1558C2×C31semidirect
190. #1554C2×C31semidirect
301. C31#19515direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32352
C31×#32302
C31×#32272
C31×#29812
C31×#17032
C31×#16952
C31×#16832
C31×#16752
C31×#15722
C31×#15582
C31×#15542
#1951531