Finite Group Databasebeta

← C31×#13057 #21469816

Subgroups — C31×#13057 #21469816

Order 7936.

700 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#454C211yesmaximal, characteristic
2C31×#2372C211yesmaximal, characteristic, complemented
3C2×C31×#355C211yesmaximal, characteristic, complemented
4C31×#1793C212yesmaximal, complemented
5C31×#1394C212yesmaximal, complemented
6C2×C31×2−1+4C2211yescharacteristic
7C2×C31×AGL(1,ℤ/8)—21no—
8C2×C31×#124C2211yescharacteristic
9C31×#454—22no—
10C31×#454C414yescomplemented
11C2×C31×#104—21no—
12C2×C31×#104—22no—
13C31×#417—22no—
14C2×C31×#101C2212yes—
15C2×C31×#101C2211yescharacteristic
16C31×#391C2212yes—
17C31×#388—22no—
18C31×#386—21no—
19C4×C22⋊C4×C31—22no—
20C31×#376—21no—
21C31×#355—22no—
22C31×2−1+4—21no—
23C31×2−1+4—22no—
24C31×2−1+4—42no—
25C31×2−1+4C2×C412yescomplemented
26C2×#41×C31—22no—
27C2×#41×C31—21no—
28C2×#41×C31—41no—
29C22×Q8×C31—41no—
30C22×Q8×C31C2311yescharacteristic
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C31×AGL(1,ℤ/8)—24no—
34C31×AGL(1,ℤ/8)—42no—
35C2×SD16×C31—41no—
36C2×D8×C31—41no—
37C2×M16×C31—21no—
38C2×M16×C31—22no—
39C31×#124—22no—
40C31×#124C2×C412yes—
41C31×#124D412yes—
42C2×C4⋊C4×C31D412yes—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31—22no—
47C2×C42×C31D411yescharacteristic
48C2×C42×C31D412yes—
49C31×#104—24no—
50C31×#104—44no—
51C31×#103—42no—
52C31×#102—42no—
53C31×#101—21no—
54C31×#101—22no—
55C31×#101—22no—
56C31×#101C2×C412yescomplemented
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—22no—
60C31×#95—22no—
61C24×C31—21no—
62#41×C31—24no—
63#41×C31—22no—
64#41×C31—22no—
65#41×C31—24no—
66#41×C31—42no—
67#41×C31—41no—
68#41×C31—42no—
69#41×C31—42no—
70#41×C31—44no—
71#41×C31—42no—
72#41×C31—41no—
73C2×Q8×C31—22no—
74C2×Q8×C31—21no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—42no—
78C2×Q8×C31—42no—
79C2×Q8×C31—44no—
80C2×Q8×C31C22×C411yescharacteristic
81C2×Q8×C31C2×D411yescharacteristic
82C2×Q8×C31#4112yes—
83C2×D4×C31—22no—
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31C22⋊C412yes—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
93C22×C4×C31—42no—
94C22×C4×C31—41no—
95C22×C4×C31—44no—
96C22×C4×C31—81no—
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D412yes—
99SD16×C31—44no—
100D8×C31—44no—
101M16×C31—22no—
102M16×C31—41no—
103M16×C31—42no—
104M16×C31—42no—
105C2×C8×C31—41no—
106C2×C8×C31—42no—
107C4⋊C4×C31—42no—
108C22⋊C4×C31—24no—
109C22⋊C4×C31—24no—
110C22⋊C4×C31—44no—
111C22⋊C4×C31—81no—
112C42×C31—21no—
113C42×C31—44no—
114C42×C31—42no—
115C42×C31C22⋊C412yes—
116#13057C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—22no—
118C23×C31—42no—
119C23×C31—42no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31#12011yescharacteristic
124Q8×C31—22no—
125Q8×C31—24no—
126Q8×C31—41no—
127Q8×C31—41no—
128Q8×C31—42no—
129Q8×C31—42no—
130Q8×C31—41no—
131D4×C31—22no—
132D4×C31—24no—
133D4×C31—42no—
134D4×C31—42no—
135D4×C31—41no—
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—81no—
139C2×C4×C31—22no—
140C2×C4×C31—21no—
141C2×C4×C31—22no—
142C2×C4×C31—21no—
143C2×C4×C31—24no—
144C2×C4×C31—22no—
145C2×C4×C31—42no—
146C2×C4×C31—44no—
147C2×C4×C31—44no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—84no—
157C2×C4×C31C2×C22⋊C411yescharacteristic
158C2×C4×C31C4×D412yes—
159C2×C4×C31#12112yes—
160C2×C4×C31#12311yescharacteristic
161C8×C31—42no—
162C8×C31—82no—
163C2×#454C2×C3111yescharacteristic
164#2372C2×C3111yescharacteristic, complemented
165C2×#355C2×C3111yescharacteristic, complemented
166#1793C2×C3112yescomplemented
167#1394C2×C3112yescomplemented
168C22×C31—22no—
169C22×C31—44no—
170C22×C31—42no—
171C22×C31—42no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—82no—
175C22×C31—82no—
176C22×C31#38611yescharacteristic
177C22×C31B(4,𝔽2)11yescharacteristic
178C22×C31B(4,𝔽2)11yescharacteristic
179C4×C31—21no—
180C4×C31—22no—
181C4×C31—22no—
182C4×C31—21no—
183C4×C31—42no—
184C4×C31—44no—
185C4×C31—42no—
186C4×C31—82no—
187C2×2−1+4C22×C3111yescharacteristic
188C2×AGL(1,ℤ/8)—21no—
189C2×#124C22×C3111yescharacteristic
190#454—22no—
191#454C4×C3114yescomplemented
192C2×#104—21no—
193C2×#104—22no—
194#417—22no—
195C2×#101C22×C3112yes—
196C2×#101C22×C3111yescharacteristic
197#391C22×C3112yes—
198#388—22no—
199#386—21no—
200C4×C22⋊C4—22no—
201#376—21no—
202#355—22no—
203C2×C31—21no—
204C2×C31—21no—
205C2×C31—82no—
206C2×C31—82no—
207C2×C31#188911yescharacteristic
208C2×C31#220412yes—
2092−1+4—21no—
2102−1+4—22no—
2112−1+4—42no—
2122−1+4C2×C4×C3112yescomplemented
213C2×#41—21no—
214C2×#41—22no—
215C2×#41—41no—
216C22×Q8—41no—
217C22×Q8C23×C3111yescharacteristic
218C22×D4D4×C3111yescharacteristic
219C23×C4—21no—
220AGL(1,ℤ/8)—24no—
221AGL(1,ℤ/8)—42no—
222C2×SD16—41no—
223C2×D8—41no—
224C2×M16—21no—
225C2×M16—22no—
226#124—22no—
227#124C2×C4×C3112yes—
228#124D4×C3112yes—
229C2×C4⋊C4D4×C3112yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C42—22no—
234C2×C42D4×C3111yescharacteristic
235C2×C42D4×C3112yes—
236#104—24no—
237#104—44no—
238#103—42no—
239#102—42no—
240#101—21no—
241#101—22no—
242#101—22no—
243#101C2×C4×C3112yescomplemented
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248C31#1305711yescharacteristic, direct product factor, Sylow
249C24—21no—
250#41—24no—
251#41—22no—
252#41—22no—
253#41—24no—
254#41—42no—
255#41—41no—
256#41—42no—
257#41—42no—
258#41—44no—
259#41—42no—
260#41—41no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—44no—
268C2×Q8C22×C4×C3111yescharacteristic
269C2×Q8C2×D4×C3111yescharacteristic
270C2×Q8#41×C3112yes—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C22⋊C4×C3112yes—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—81no—
285C22×C4C2×D4×C3111yescharacteristic
286C22×C4C2×D4×C3112yes—
287SD16—44no—
288D8—44no—
289M16—22no—
290M16—41no—
291M16—42no—
292M16—42no—
293C2×C8—41no—
294C2×C8—42no—
295C4⋊C4—42no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—44no—
299C22⋊C4—81no—
300C42—21no—
301C42—42no—
302C42—44no—
303C42C22⋊C4×C3112yes—
304C23—22no—
305C23—42no—
306C23—42no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C31×#12011yescharacteristic
311Q8—22no—
312Q8—24no—
313Q8—41no—
314Q8—41no—
315Q8—42no—
316Q8—42no—
317Q8—41no—
318D4—22no—
319D4—24no—
320D4—42no—
321D4—42no—
322D4—41no—
323D4—42no—
324D4—42no—
325D4—81no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4C2×C22⋊C4×C3111yescharacteristic
345C2×C4C4×D4×C3112yes—
346C2×C4C31×#12112yes—
347C2×C4C31×#12311yescharacteristic
348C8—42no—
349C8—82no—
350C22—22no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22C31×#38611yescharacteristic
359C22C31×B(4,𝔽2)11yescharacteristic
360C22C31×B(4,𝔽2)11yescharacteristic
361C4—21no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—44no—
366C4—42no—
367C4—42no—
368C4—82no—
369C2—21no—
370C2—21no—
371C2—82no—
372C2—82no—
373C2C31×#188911yescharacteristic
374C2C31×#220412yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×#454C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2372C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1793C2semidirect
5. C31×#1394C2semidirect
10. C31×#454C4semidirect
25. C31×2−1+4C2×C4semidirect
56. C31×#101C2×C4semidirect
116. #13057C31direct (N × H)
164. #2372C2×C31semidirect
165. C2×#355C2×C31semidirect
166. #1793C2×C31semidirect
167. #1394C2×C31semidirect
191. #454C4×C31semidirect
212. 2−1+4C2×C4×C31semidirect
243. #101C2×C4×C31semidirect
248. C31#13057direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4542
C31×#23722
C2×C31×#3552
C31×#17932
C31×#13942
#1305731