Subgroups — C₃₁×#11245 #21468004

Order 7936.

624 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₂×C₃₁×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#428	—	2	4	no	—
12	C₃₁×#427	—	2	2	no	—
13	C₃₁×#424	—	2	2	no	—
14	C₃₁×#424	—	2	2	no	—
15	C₃₁×#423	C₄	1	4	yes	complemented
16	C₃₁×#344	—	2	4	no	—
17	C₃₁×#338	—	2	2	no	—
18	C₃₁×H(3,ℤ/4)	—	2	4	no	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	1	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	D₄	1	2	yes	—
36	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
37	C₃₁×#117	—	2	1	no	—
38	C₃₁×#117	—	2	1	no	—
39	C₃₁×#117	—	2	2	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	D₄	1	4	yes	—
43	C₃₁×#117	D₄	1	2	yes	—
44	C₃₁×#117	Q₈	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
51	C₃₁×#107	—	2	4	no	—
52	C₃₁×#106	—	2	4	no	—
53	C₃₁×C₄⋊C₈	—	2	8	no	—
54	C₂⁴×C₃₁	#41	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₃₁	#41	1	2	yes	—
65	C₂²×C₄×C₃₁	#41	1	2	yes	—
66	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
67	M₁₆×C₃₁	—	2	4	no	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	4	1	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	—	4	1	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	—	2	4	no	—
91	C₂²⋊C₄×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	—	4	2	no	—
95	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
96	#11245	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	8	1	no	—
98	C₂³×C₃₁	#117	1	1	yes	characteristic
99	C₂³×C₃₁	#121	1	1	yes	characteristic
100	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₃₁	#123	1	2	yes	—
102	C₂³×C₃₁	#125	1	2	yes	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	8	2	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#121	1	2	yes	—
113	C₂×C₄×C₃₁	#121	1	2	yes	—
114	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#125	1	2	yes	—
122	C₂×C₄×C₃₁	#126	1	2	yes	—
123	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
125	C₈×C₃₁	—	2	4	no	—
126	C₈×C₃₁	—	4	2	no	—
127	C₈×C₃₁	—	4	4	no	—
128	#2911	C₂×C₃₁	1	1	yes	characteristic
129	#2905	C₂×C₃₁	1	1	yes	characteristic
130	C₂×#344	C₂×C₃₁	1	2	yes	—
131	C₂×#338	C₂×C₃₁	1	1	yes	characteristic
132	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	—
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	1	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	#380	1	1	yes	characteristic
139	C₂²×C₃₁	#387	1	2	yes	—
140	C₂²×C₃₁	#394	1	1	yes	characteristic
141	C₂²×C₃₁	#398	1	2	yes	—
142	C₂²×C₃₁	#398	1	1	yes	characteristic
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#386	1	1	yes	characteristic
146	C₄×C₃₁	#389	1	2	yes	—
147	C₄×C₃₁	#391	1	1	yes	characteristic
148	C₄×C₃₁	#395	1	1	yes	characteristic
149	C₄×C₃₁	#396	1	2	yes	—
150	C₄×C₃₁	#400	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
152	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
154	C₂×#117	C₂²×C₃₁	1	2	yes	—
155	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
156	#428	—	2	4	no	—
157	#427	—	2	2	no	—
158	#424	—	2	2	no	—
159	#424	—	2	2	no	—
160	#423	C₄×C₃₁	1	4	yes	complemented
161	#344	—	2	4	no	—
162	#338	—	2	2	no	—
163	H(3,ℤ/4)	—	2	4	no	—
164	C₂×C₃₁	—	2	1	no	—
165	C₂×C₃₁	—	2	1	no	—
166	C₂×C₃₁	—	2	2	no	—
167	C₂×C₃₁	—	2	2	no	—
168	C₂×C₃₁	—	2	1	no	—
169	C₂×C₃₁	#1441	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	D₄×C₃₁	1	2	yes	—
187	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	D₄×C₃₁	1	4	yes	—
194	#117	D₄×C₃₁	1	2	yes	—
195	#117	Q₈×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
202	#107	—	2	4	no	—
203	#106	—	2	4	no	—
204	C₄⋊C₈	—	2	8	no	—
205	C₃₁	#11245	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	#41×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
216	C₂²×C₄	#41×C₃₁	1	2	yes	—
217	C₂²×C₄	#41×C₃₁	1	2	yes	—
218	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	4	2	no	—
247	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
248	C₂³	—	8	1	no	—
249	C₂³	C₃₁×#117	1	1	yes	characteristic
250	C₂³	C₃₁×#121	1	1	yes	characteristic
251	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂³	C₃₁×#123	1	2	yes	—
253	C₂³	C₃₁×#125	1	2	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
262	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
263	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
264	C₂×C₄	C₃₁×#121	1	2	yes	—
265	C₂×C₄	C₃₁×#121	1	2	yes	—
266	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
267	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
269	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
272	C₂×C₄	C₃₁×#125	1	2	yes	—
273	C₂×C₄	C₃₁×#126	1	2	yes	—
274	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
275	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₃₁×#380	1	1	yes	characteristic
285	C₂²	C₃₁×#387	1	2	yes	—
286	C₂²	C₃₁×#394	1	1	yes	characteristic
287	C₂²	C₃₁×#398	1	2	yes	—
288	C₂²	C₃₁×#398	1	1	yes	characteristic
289	C₄	—	4	4	no	—
290	C₄	—	8	4	no	—
291	C₄	C₃₁×#386	1	1	yes	characteristic
292	C₄	C₃₁×#389	1	2	yes	—
293	C₄	C₃₁×#391	1	1	yes	characteristic
294	C₄	C₃₁×#395	1	1	yes	characteristic
295	C₄	C₃₁×#396	1	2	yes	—
296	C₄	C₃₁×#400	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	C₃₁×#1441	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#391
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂×C₃₁	#1441
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₁×#423	C₄	semidirect
96. #11245	C₃₁	direct (N × H)
160. #423	C₄×C₃₁	semidirect
205. C₃₁	#11245	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2911	2
C₃₁×#2905	2
C₂×C₃₁×#344	2
C₂×C₃₁×#338	2
C₂×C₃₁×H(3,ℤ/4)	2
#11245	31

Subgroups — C31×#11245 #21468004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11245 #21468004