Finite Group Databasebeta

← C31×#13568 #21470327

Subgroups — C31×#13568 #21470327

Order 7936.

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

normal subgroups only

This group has 498 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
1C31×#3018C211yesmaximal, characteristic
2C31×#2126C211yesmaximal, characteristic, complemented
3C31×#1525C211yesmaximal, characteristic, complemented
4C31×#1404C214yesmaximal, complemented
5C31×#585C411yescharacteristic, complemented
6C31×#575—21no—
7C31×#532C2211yescharacteristic
8C31×#456C412yescomplemented
9C31×#455—21no—
10C31×#454C2212yes—
11C31×#453—21no—
12C2×C31×#104—21no—
13C31×#413C411yescharacteristic, complemented
14C31×#410—21no—
15C4×M16×C31—21no—
16C31×#354C2212yescomplemented
17C31×#353—21no—
18C31×#329—21no—
19C31×#329C2212yescomplemented
20C31×2−1+4—41no—
21C31×2−1+4C2×C412yes—
22C31×H(5,𝔽2)—21no—
23C31×H(5,𝔽2)—41no—
24C31×H(5,𝔽2)—41no—
25C2×#41×C31—21no—
26C2×#41×C31—41no—
27C2×#41×C31—42no—
28C2×#41×C31C2×C411yescharacteristic
29C2×#41×C31D411yescharacteristic
30C31×#137—22no—
31C31×#137—41no—
32C31×AGL(1,ℤ/8)—22no—
33C31×AGL(1,ℤ/8)—41no—
34C31×#135—42no—
35C2×M16×C31—21no—
36C2×M16×C31—41no—
37C31×C4⋊Q8C2×C411yescharacteristic
38C31×#127D411yescharacteristic
39C31×#124D411yescharacteristic
40C31×#124C2311yescharacteristic
41C31×#121—41no—
42C4×D4×C31—41no—
43C31×#117—41no—
44C2×C42×C31D411yescharacteristic
45C31×#104—22no—
46C31×#104—22no—
47C31×#101C2×C412yes—
48C31×#100—21no—
49C31×#99—22no—
50C31×#99—41no—
51C31×#97—22no—
52C4×C8×C31—41no—
53#41×C31—22no—
54#41×C31—22no—
55#41×C31—41no—
56#41×C31—42no—
57#41×C31—41no—
58#41×C31—41no—
59#41×C31—41no—
60#41×C31—41no—
61#41×C31—42no—
62#41×C31—42no—
63#41×C31—44no—
64#41×C31—42no—
65#41×C31—41no—
66#41×C31—42no—
67C2×Q8×C31—21no—
68C2×Q8×C31—42no—
69C2×Q8×C31—41no—
70C2×Q8×C31C22⋊C411yescharacteristic
71C2×Q8×C31C22×C411yescharacteristic
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—41no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—44no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31C22⋊C411yescharacteristic
82C2×D4×C31C2×D411yescharacteristic
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31—42no—
88C22×C4×C31C22⋊C411yescharacteristic
89Q16×C31—42no—
90SD16×C31—42no—
91SD16×C31—42no—
92D8×C31—42no—
93M16×C31—22no—
94M16×C31—41no—
95M16×C31—44no—
96C2×C8×C31—42no—
97C2×C8×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—81no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—42no—
103C42×C31—21no—
104C42×C31—81no—
105C42×C31C22⋊C411yescharacteristic
106C42×C31C2×D411yescharacteristic
107#13568C3111yesmaximal, characteristic, direct product factor, Sylow
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—41no—
111C23×C31—42no—
112C23×C31#9911yescharacteristic
113Q8×C31—22no—
114Q8×C31—41no—
115Q8×C31—41no—
116Q8×C31—42no—
117Q8×C31—41no—
118D4×C31—22no—
119D4×C31—41no—
120D4×C31—41no—
121D4×C31—42no—
122D4×C31—42no—
123D4×C31—42no—
124D4×C31—44no—
125D4×C31—42no—
126D4×C31—41no—
127D4×C31—82no—
128D4×C31—81no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—42no—
136C2×C4×C31—42no—
137C2×C4×C31—41no—
138C2×C4×C31—44no—
139C2×C4×C31—42no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—42no—
145C2×C4×C31—82no—
146C2×C4×C31#9911yescharacteristic
147C2×C4×C31C2×C22⋊C411yescharacteristic
148C8×C31—42no—
149C8×C31—44no—
150#3018C2×C3111yescharacteristic
151#2126C2×C3111yescharacteristic, complemented
152#1525C2×C3111yescharacteristic, complemented
153#1404C2×C3114yescomplemented
154C22×C31—21no—
155C22×C31—41no—
156C22×C31—42no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31—161no—
162C22×C31—162no—
163C22×C31C2×#9911yescharacteristic
164C4×C31—22no—
165C4×C31—21no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—42no—
170C4×C31—162no—
171C4×C31AΣL(1,𝔽16)11yescharacteristic
172C4×C31AΣL(1,𝔽16)11yescharacteristic
173#585C4×C3111yescharacteristic, complemented
174#575—21no—
175#532C22×C3111yescharacteristic
176#456C4×C3112yescomplemented
177#455—21no—
178#454C22×C3112yes—
179#453—21no—
180C2×#104—21no—
181#413C4×C3111yescharacteristic, complemented
182#410—21no—
183C4×M16—21no—
184#354C22×C3112yescomplemented
185#353—21no—
186#329—21no—
187#329C22×C3112yescomplemented
188C2×C31—21no—
189C2×C31—41no—
190C2×C31—81no—
191C2×C31—81no—
192C2×C31—82no—
193C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
1942−1+4—41no—
1952−1+4C2×C4×C3112yes—
196H(5,𝔽2)—21no—
197H(5,𝔽2)—41no—
198H(5,𝔽2)—41no—
199C2×#41—21no—
200C2×#41—41no—
201C2×#41—42no—
202C2×#41C2×C4×C3111yescharacteristic
203C2×#41D4×C3111yescharacteristic
204#137—22no—
205#137—41no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—41no—
208#135—42no—
209C2×M16—21no—
210C2×M16—41no—
211C4⋊Q8C2×C4×C3111yescharacteristic
212#127D4×C3111yescharacteristic
213#124D4×C3111yescharacteristic
214#124C23×C3111yescharacteristic
215#121—41no—
216C4×D4—41no—
217#117—41no—
218C2×C42D4×C3111yescharacteristic
219#104—22no—
220#104—22no—
221#101C2×C4×C3112yes—
222#100—21no—
223#99—22no—
224#99—41no—
225#97—22no—
226C4×C8—41no—
227C31#1356811yescharacteristic, direct product factor, Sylow
228#41—22no—
229#41—22no—
230#41—41no—
231#41—42no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236#41—42no—
237#41—42no—
238#41—44no—
239#41—42no—
240#41—41no—
241#41—42no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8C22⋊C4×C3111yescharacteristic
246C2×Q8C22×C4×C3111yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4C22⋊C4×C3111yescharacteristic
257C2×D4C2×D4×C3111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4C22⋊C4×C3111yescharacteristic
264Q16—42no—
265SD16—42no—
266SD16—42no—
267D8—42no—
268M16—22no—
269M16—41no—
270M16—44no—
271C2×C8—42no—
272C2×C8—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C42—21no—
279C42—81no—
280C42C22⋊C4×C3111yescharacteristic
281C42C2×D4×C3111yescharacteristic
282C23—21no—
283C23—21no—
284C23—41no—
285C23—42no—
286C23C31×#9911yescharacteristic
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—42no—
291Q8—41no—
292D4—22no—
293D4—41no—
294D4—41no—
295D4—42no—
296D4—42no—
297D4—42no—
298D4—44no—
299D4—42no—
300D4—41no—
301D4—82no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4C31×#9911yescharacteristic
321C2×C4C2×C22⋊C4×C3111yescharacteristic
322C8—42no—
323C8—44no—
324C22—21no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—161no—
332C22—162no—
333C22C2×C31×#9911yescharacteristic
334C4—22no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—42no—
340C4—162no—
341C4C31×AΣL(1,𝔽16)11yescharacteristic
342C4C31×AΣL(1,𝔽16)11yescharacteristic
343C2—21no—
344C2—41no—
345C2—81no—
346C2—81no—
347C2—82no—
348C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC4×C31AΣL(1,𝔽16)
commutatorC2×Q8C22×C4×C31
Frattini, cosocle#124C23×C31
socleC2×C31C2×AΣL(1,𝔽16)
prime radicalC31×#3018C2
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ C31×#532 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2126C2semidirect
3. C31×#1525C2semidirect
4. C31×#1404C2semidirect
5. C31×#585C4semidirect
8. C31×#456C4semidirect
13. C31×#413C4semidirect
16. C31×#354C22semidirect
19. C31×#329C22semidirect
107. #13568C31direct (N × H)
151. #2126C2×C31semidirect
152. #1525C2×C31semidirect
153. #1404C2×C31semidirect
173. #585C4×C31semidirect
176. #456C4×C31semidirect
181. #413C4×C31semidirect
184. #354C22×C31semidirect
187. #329C22×C31semidirect
227. C31#13568direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30182
C31×#21262
C31×#15252
C31×#14042
#1356831