Finite Group Databasebeta

← C31×#13569 #21470328

Subgroups — C31×#13569 #21470328

Order 7936.

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

normal subgroups only

This group has 492 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×#454C211yesmaximal, characteristic
2C2×C31×#354C211yesmaximal, characteristic, complemented
3C31×#1533C211yesmaximal, characteristic, complemented
4C31×#1404C214yesmaximal, complemented
5C2×C31×2−1+4C411yescharacteristic, complemented
6C2×C31×AGL(1,ℤ/8)—21no—
7C2×C31×#124C2211yescharacteristic
8C31×#454—21no—
9C31×#454—21no—
10C31×#454C412yescomplemented
11C31×#454C2212yes—
12C31×#432—21no—
13C2×C31×#104—21no—
14C2×C31×#101C411yescharacteristic, complemented
15C2×C31×#99—21no—
16C31×#354—21no—
17C31×#354C2212yescomplemented
18C31×#329—21no—
19C31×#329C2212yescomplemented
20C31×2−1+4—21no—
21C31×2−1+4—41no—
22C31×2−1+4—42no—
23C31×2−1+4C2×C412yescomplemented
24C2×#41×C31—21no—
25C2×#41×C31—41no—
26C2×#41×C31—41no—
27C22×Q8×C31—41no—
28C22×Q8×C31C2×C411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C31×AGL(1,ℤ/8)—22no—
31C31×AGL(1,ℤ/8)—22no—
32C31×AGL(1,ℤ/8)—42no—
33C2×SD16×C31—41no—
34C2×D8×C31—41no—
35C2×M16×C31—21no—
36C31×#124—41no—
37C31×#124C2×C411yescharacteristic
38C31×#124D411yescharacteristic
39C31×#124D411yescharacteristic
40C31×#124C2311yescharacteristic
41C2×C22⋊C4×C31—41no—
42C2×C22⋊C4×C31—41no—
43C2×C42×C31D411yescharacteristic
44C31×C4⋊C8—41no—
45C31×#104—22no—
46C31×#104—22no—
47C31×#101—21no—
48C31×#101C2×C412yescomplemented
49C31×#99—22no—
50C31×#99—41no—
51C31×C22⋊C8—41no—
52C31×#97—22no—
53C24×C31—21no—
54#41×C31—22no—
55#41×C31—22no—
56#41×C31—41no—
57#41×C31—42no—
58#41×C31—42no—
59#41×C31—42no—
60#41×C31—42no—
61#41×C31—44no—
62#41×C31—42no—
63#41×C31—42no—
64#41×C31—41no—
65C2×Q8×C31—21no—
66C2×Q8×C31—21no—
67C2×Q8×C31—41no—
68C2×Q8×C31—41no—
69C2×Q8×C31—42no—
70C2×Q8×C31—42no—
71C2×Q8×C31—44no—
72C2×Q8×C31—41no—
73C2×Q8×C31C22⋊C411yescharacteristic
74C2×Q8×C31C22×C411yescharacteristic
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
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—41no—
86C22×C4×C31—42no—
87C22×C4×C31—81no—
88C22×C4×C31C22⋊C411yescharacteristic
89SD16×C31—42no—
90SD16×C31—42no—
91D8×C31—42no—
92D8×C31—42no—
93M16×C31—22no—
94M16×C31—41no—
95C2×C8×C31—41no—
96C2×C8×C31—42no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—81no—
103C42×C31—21no—
104C42×C31C22⋊C411yescharacteristic
105C42×C31C2×D411yescharacteristic
106#13569C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—82no—
113C23×C31#9911yescharacteristic
114Q8×C31—22no—
115Q8×C31—41no—
116Q8×C31—41no—
117Q8×C31—42no—
118Q8×C31—41no—
119Q8×C31—42no—
120Q8×C31—42no—
121D4×C31—22no—
122D4×C31—41no—
123D4×C31—41no—
124D4×C31—41no—
125D4×C31—42no—
126D4×C31—42no—
127D4×C31—42no—
128D4×C31—42no—
129D4×C31—81no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—42no—
137C2×C4×C31—42no—
138C2×C4×C31—44no—
139C2×C4×C31—44no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—81no—
144C2×C4×C31—82no—
145C2×C4×C31—162no—
146C2×C4×C31#9911yescharacteristic
147C2×C4×C31C2×C22⋊C411yescharacteristic
148C8×C31—42no—
149C8×C31—82no—
150C2×#454C2×C3111yescharacteristic
151C2×#354C2×C3111yescharacteristic, complemented
152#1533C2×C3111yescharacteristic, complemented
153#1404C2×C3114yescomplemented
154C22×C31—41no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31—82no—
161C22×C31—82no—
162C22×C31—161no—
163C22×C31AΣL(1,𝔽16)11yescharacteristic
164C22×C31AΣL(1,𝔽16)11yescharacteristic
165C22×C31C2×#9911yescharacteristic
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—41no—
169C4×C31—41no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31—42no—
173C4×C31—162no—
174C2×2−1+4C4×C3111yescharacteristic, complemented
175C2×AGL(1,ℤ/8)—21no—
176C2×#124C22×C3111yescharacteristic
177#454—21no—
178#454—21no—
179#454C4×C3112yescomplemented
180#454C22×C3112yes—
181#432—21no—
182C2×#104—21no—
183C2×#101C4×C3111yescharacteristic, complemented
184C2×#99—21no—
185#354—21no—
186#354C22×C3112yescomplemented
187#329—21no—
188#329C22×C3112yescomplemented
189C2×C31—21no—
190C2×C31—21no—
191C2×C31—21no—
192C2×C31—81no—
193C2×C31—82no—
194C2×C31—81no—
195C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
1962−1+4—21no—
1972−1+4—41no—
1982−1+4—42no—
1992−1+4C2×C4×C3112yescomplemented
200C2×#41—21no—
201C2×#41—41no—
202C2×#41—41no—
203C22×Q8—41no—
204C22×Q8C2×C4×C3111yescharacteristic
205C22×D4D4×C3111yescharacteristic
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—22no—
208AGL(1,ℤ/8)—42no—
209C2×SD16—41no—
210C2×D8—41no—
211C2×M16—21no—
212#124—41no—
213#124C2×C4×C3111yescharacteristic
214#124D4×C3111yescharacteristic
215#124D4×C3111yescharacteristic
216#124C23×C3111yescharacteristic
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42D4×C3111yescharacteristic
220C4⋊C8—41no—
221#104—22no—
222#104—22no—
223#101—21no—
224#101C2×C4×C3112yescomplemented
225#99—22no—
226#99—41no—
227C22⋊C8—41no—
228#97—22no—
229C31#1356911yescharacteristic, direct product factor, Sylow
230C24—21no—
231#41—22no—
232#41—22no—
233#41—41no—
234#41—42no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—44no—
239#41—42no—
240#41—42no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—41no—
250C2×Q8C22⋊C4×C3111yescharacteristic
251C2×Q8C22×C4×C3111yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4C22⋊C4×C3111yescharacteristic
259C2×D4C2×D4×C3111yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—81no—
265C22×C4C22⋊C4×C3111yescharacteristic
266SD16—42no—
267SD16—42no—
268D8—42no—
269D8—42no—
270M16—22no—
271M16—41no—
272C2×C8—41no—
273C2×C8—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C42—21no—
281C42C22⋊C4×C3111yescharacteristic
282C42C2×D4×C3111yescharacteristic
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—82no—
289C23C31×#9911yescharacteristic
290Q8—22no—
291Q8—41no—
292Q8—41no—
293Q8—42no—
294Q8—41no—
295Q8—42no—
296Q8—42no—
297D4—22no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—162no—
322C2×C4C31×#9911yescharacteristic
323C2×C4C2×C22⋊C4×C3111yescharacteristic
324C8—42no—
325C8—82no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22—82no—
333C22—82no—
334C22—161no—
335C22C31×AΣL(1,𝔽16)11yescharacteristic
336C22C31×AΣL(1,𝔽16)11yescharacteristic
337C22C2×C31×#9911yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—162no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—81no—
350C2—81no—
351C2—82no—
352C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×AΣL(1,𝔽16)
commutatorC2×Q8C22×C4×C31
Frattini, cosocle#124C23×C31
prime radicalC2×C31×#454C2
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×C4×C31 ◁ C2×C31×#124 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#354C2semidirect
3. C31×#1533C2semidirect
4. C31×#1404C2semidirect
5. C2×C31×2−1+4C4semidirect
10. C31×#454C4semidirect
14. C2×C31×#101C4semidirect
17. C31×#354C22semidirect
19. C31×#329C22semidirect
23. C31×2−1+4C2×C4semidirect
48. C31×#101C2×C4semidirect
106. #13569C31direct (N × H)
151. C2×#354C2×C31semidirect
152. #1533C2×C31semidirect
153. #1404C2×C31semidirect
174. C2×2−1+4C4×C31semidirect
179. #454C4×C31semidirect
183. C2×#101C4×C31semidirect
186. #354C22×C31semidirect
188. #329C22×C31semidirect
199. 2−1+4C2×C4×C31semidirect
224. #101C2×C4×C31semidirect
229. C31#13569direct (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
C2×C31×#3542
C31×#15332
C31×#14042
#1356931