Finite Group Databasebeta

← C31×#13561 #21470320

Subgroups — C31×#13561 #21470320

Order 7936.

550 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 90 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×#453C211yesmaximal, characteristic
2C2×C31×#353C211yesmaximal, characteristic, complemented
3C31×#1555C211yesmaximal, characteristic, complemented
4C31×#1402C214yesmaximal, complemented
5C2×C31×H(5,𝔽2)C411yescharacteristic, complemented
6C2×C31×AGL(1,ℤ/8)—21no—
7C2×C31×#127C2211yescharacteristic
8C31×#453—21no—
9C31×#453—21no—
10C31×#453C412yescomplemented
11C31×#453C2212yes—
12C31×#432—21no—
13C2×C31×#104—21no—
14C2×C31×#100C411yescharacteristic, complemented
15C2×C31×#99—21no—
16C31×#353—21no—
17C31×#353C2212yescomplemented
18C31×#331—21no—
19C31×#331C2212yescomplemented
20C31×H(5,𝔽2)—21no—
21C31×H(5,𝔽2)—41no—
22C31×H(5,𝔽2)—42no—
23C31×H(5,𝔽2)C2×C412yescomplemented
24C2×#41×C31—21no—
25C2×#41×C31—41no—
26C22×D4×C31—41no—
27C22×D4×C31—41no—
28C22×D4×C31—41no—
29C22×D4×C31C2×C411yescharacteristic
30C22×D4×C31D411yescharacteristic
31C31×AGL(1,ℤ/8)—22no—
32C31×AGL(1,ℤ/8)—22no—
33C31×AGL(1,ℤ/8)—42no—
34C2×SD16×C31—41no—
35C2×D8×C31—41no—
36C2×M16×C31—21no—
37C31×#127—41no—
38C31×#127C2×C411yescharacteristic
39C31×#127D411yescharacteristic
40C31×#127D411yescharacteristic
41C31×#127C2311yescharacteristic
42C2×C22⋊C4×C31—41no—
43C2×C42×C31D411yescharacteristic
44C31×C4⋊C8—22no—
45C31×#104—22no—
46C31×#104—22no—
47C31×#100—21no—
48C31×#100C2×C412yescomplemented
49C31×#99—22no—
50C31×#99—41no—
51C31×C22⋊C8—41no—
52C31×#97—41no—
53C24×C31—21no—
54C24×C31—21no—
55C24×C31—41no—
56#41×C31—22no—
57#41×C31—22no—
58#41×C31—41no—
59#41×C31—42no—
60#41×C31—44no—
61#41×C31—41no—
62#41×C31—42no—
63C2×Q8×C31—21no—
64C2×D4×C31—21no—
65C2×D4×C31—21no—
66C2×D4×C31—21no—
67C2×D4×C31—41no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—42no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—42no—
79C2×D4×C31—44no—
80C2×D4×C31—42no—
81C2×D4×C31—41no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—84no—
85C2×D4×C31C22⋊C411yescharacteristic
86C2×D4×C31C22⋊C411yescharacteristic
87C2×D4×C31C22×C411yescharacteristic
88C2×D4×C31C2×D411yescharacteristic
89C22×C4×C31—21no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—81no—
93C22×C4×C31C22⋊C411yescharacteristic
94SD16×C31—42no—
95SD16×C31—42no—
96D8×C31—42no—
97D8×C31—42no—
98M16×C31—22no—
99M16×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—42no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—81no—
104C42×C31—21no—
105C42×C31C22⋊C411yescharacteristic
106C42×C31C2×D411yescharacteristic
107#13561C3111yesmaximal, characteristic, direct product factor, Sylow
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—42no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—42no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—82no—
120C23×C31—82no—
121C23×C31—162no—
122C23×C31#9911yescharacteristic
123Q8×C31—22no—
124Q8×C31—41no—
125D4×C31—22no—
126D4×C31—41no—
127D4×C31—41no—
128D4×C31—41no—
129D4×C31—41no—
130D4×C31—41no—
131D4×C31—42no—
132D4×C31—44no—
133D4×C31—44no—
134D4×C31—42no—
135D4×C31—42no—
136D4×C31—81no—
137D4×C31—82no—
138D4×C31—82no—
139D4×C31—82no—
140D4×C31—82no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—41no—
150C2×C4×C31—42no—
151C2×C4×C31—82no—
152C2×C4×C31—162no—
153C2×C4×C31#9911yescharacteristic
154C2×C4×C31C2×C22⋊C411yescharacteristic
155C8×C31—42no—
156C8×C31—82no—
157C2×#453C2×C3111yescharacteristic
158C2×#353C2×C3111yescharacteristic, complemented
159#1555C2×C3111yescharacteristic, complemented
160#1402C2×C3114yescomplemented
161C22×C31—41no—
162C22×C31—42no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—81no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—82no—
172C22×C31—82no—
173C22×C31—82no—
174C22×C31—161no—
175C22×C31—161no—
176C22×C31—162no—
177C22×C31—162no—
178C22×C31AΣL(1,𝔽16)11yescharacteristic
179C22×C31AΣL(1,𝔽16)11yescharacteristic
180C22×C31C2×#9911yescharacteristic
181C4×C31—22no—
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—41no—
185C4×C31—42no—
186C4×C31—162no—
187C2×H(5,𝔽2)C4×C3111yescharacteristic, complemented
188C2×AGL(1,ℤ/8)—21no—
189C2×#127C22×C3111yescharacteristic
190#453—21no—
191#453—21no—
192#453C4×C3112yescomplemented
193#453C22×C3112yes—
194#432—21no—
195C2×#104—21no—
196C2×#100C4×C3111yescharacteristic, complemented
197C2×#99—21no—
198#353—21no—
199#353C22×C3112yescomplemented
200#331—21no—
201#331C22×C3112yescomplemented
202C2×C31—21no—
203C2×C31—21no—
204C2×C31—21no—
205C2×C31—81no—
206C2×C31—81no—
207C2×C31—81no—
208C2×C31—81no—
209C2×C31—82no—
210C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
211H(5,𝔽2)—21no—
212H(5,𝔽2)—41no—
213H(5,𝔽2)—42no—
214H(5,𝔽2)C2×C4×C3112yescomplemented
215C2×#41—21no—
216C2×#41—41no—
217C22×D4—41no—
218C22×D4—41no—
219C22×D4—41no—
220C22×D4C2×C4×C3111yescharacteristic
221C22×D4D4×C3111yescharacteristic
222AGL(1,ℤ/8)—22no—
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—42no—
225C2×SD16—41no—
226C2×D8—41no—
227C2×M16—21no—
228#127—41no—
229#127C2×C4×C3111yescharacteristic
230#127D4×C3111yescharacteristic
231#127D4×C3111yescharacteristic
232#127C23×C3111yescharacteristic
233C2×C22⋊C4—41no—
234C2×C42D4×C3111yescharacteristic
235C4⋊C8—22no—
236#104—22no—
237#104—22no—
238#100—21no—
239#100C2×C4×C3112yescomplemented
240#99—22no—
241#99—41no—
242C22⋊C8—41no—
243#97—41no—
244C31#1356111yescharacteristic, direct product factor, Sylow
245C24—21no—
246C24—21no—
247C24—41no—
248#41—22no—
249#41—22no—
250#41—41no—
251#41—42no—
252#41—44no—
253#41—41no—
254#41—42no—
255C2×Q8—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—84no—
277C2×D4C22⋊C4×C3111yescharacteristic
278C2×D4C22⋊C4×C3111yescharacteristic
279C2×D4C22×C4×C3111yescharacteristic
280C2×D4C2×D4×C3111yescharacteristic
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—81no—
285C22×C4C22⋊C4×C3111yescharacteristic
286SD16—42no—
287SD16—42no—
288D8—42no—
289D8—42no—
290M16—22no—
291M16—41no—
292C2×C8—41no—
293C2×C8—42no—
294C22⋊C4—42no—
295C22⋊C4—81no—
296C42—21no—
297C42C22⋊C4×C3111yescharacteristic
298C42C2×D4×C3111yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—42no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—82no—
311C23—82no—
312C23—162no—
313C23C31×#9911yescharacteristic
314Q8—22no—
315Q8—41no—
316D4—22no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—44no—
324D4—44no—
325D4—42no—
326D4—42no—
327D4—81no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—162no—
344C2×C4C31×#9911yescharacteristic
345C2×C4C2×C22⋊C4×C3111yescharacteristic
346C8—42no—
347C8—82no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—161no—
362C22—161no—
363C22—162no—
364C22—162no—
365C22C31×AΣL(1,𝔽16)11yescharacteristic
366C22C31×AΣL(1,𝔽16)11yescharacteristic
367C22C2×C31×#9911yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—42no—
373C4—162no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—81no—
378C2—81no—
379C2—81no—
380C2—81no—
381C2—82no—
382C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×AΣL(1,𝔽16)
commutatorC2×D4C22×C4×C31
Frattini, cosocle#127C23×C31
prime radicalC2×C31×#453C2
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×C4×C31 ◁ C2×C31×#127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#353C2semidirect
3. C31×#1555C2semidirect
4. C31×#1402C2semidirect
5. C2×C31×H(5,𝔽2)C4semidirect
10. C31×#453C4semidirect
14. C2×C31×#100C4semidirect
17. C31×#353C22semidirect
19. C31×#331C22semidirect
23. C31×H(5,𝔽2)C2×C4semidirect
48. C31×#100C2×C4semidirect
107. #13561C31direct (N × H)
158. C2×#353C2×C31semidirect
159. #1555C2×C31semidirect
160. #1402C2×C31semidirect
187. C2×H(5,𝔽2)C4×C31semidirect
192. #453C4×C31semidirect
196. C2×#100C4×C31semidirect
199. #353C22×C31semidirect
201. #331C22×C31semidirect
214. H(5,𝔽2)C2×C4×C31semidirect
239. #100C2×C4×C31semidirect
244. C31#13561direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4532
C2×C31×#3532
C31×#15552
C31×#14022
#1356131