Subgroups — C₃₁×#20671 #21477430

Order 7936.

1324 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#568	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2911	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#410	C₂	1	2	yes	maximal
4	C₃₁×#1812	C₂	1	8	yes	maximal, complemented
5	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#568	—	2	8	no	—
8	C₂×C₃₁×#131	C₄	1	4	yes	complemented
9	C₂×C₃₁×#131	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₃₁	C₄	1	4	yes	complemented
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	4	yes	—
14	C₃₁×#428	—	2	8	no	—
15	C₃₁×#427	C₂²	1	8	yes	complemented
16	C₃₁×#410	—	2	4	no	—
17	C₃₁×#410	C₂²	1	8	yes	—
18	C₂×C₃₁×#99	—	2	4	no	—
19	C₃₁×H(3,ℤ/4)	C₂²	1	8	yes	complemented
20	C₂×#41×C₃₁	—	2	4	no	—
21	C₂×#41×C₃₁	—	2	4	no	—
22	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×#41×C₃₁	D₄	1	2	yes	—
24	C₂×#41×C₃₁	Q₈	1	2	yes	—
25	C₂×#41×C₃₁	C₂³	1	2	yes	—
26	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	D₄	1	2	yes	—
28	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#131	—	2	16	no	—
32	C₃₁×#131	—	2	16	no	—
33	C₂×M₁₆×C₃₁	—	2	4	no	—
34	C₂×M₁₆×C₃₁	—	2	16	no	—
35	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
39	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
40	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#117	—	2	4	no	—
45	C₃₁×#117	—	2	4	no	—
46	C₃₁×#117	—	2	4	no	—
47	C₃₁×#117	C₂³	1	8	yes	—
48	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	—	2	4	no	—
51	C₃₁×#107	—	2	8	no	—
52	C₃₁×#106	—	2	8	no	—
53	C₃₁×#99	—	2	16	no	—
54	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
56	#41×C₃₁	—	2	4	no	—
57	#41×C₃₁	—	2	4	no	—
58	#41×C₃₁	—	2	8	no	—
59	#41×C₃₁	—	2	8	no	—
60	#41×C₃₁	—	2	4	no	—
61	#41×C₃₁	—	2	4	no	—
62	C₂×Q₈×C₃₁	—	2	4	no	—
63	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
65	C₂×D₄×C₃₁	—	2	8	no	—
66	C₂×D₄×C₃₁	—	2	4	no	—
67	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
68	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	4	4	no	—
74	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	#41	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₃₁	—	2	4	no	—
86	M₁₆×C₃₁	—	2	4	no	—
87	M₁₆×C₃₁	—	2	16	no	—
88	M₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
89	M₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	16	no	—
92	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
95	C₄⋊C₄×C₃₁	—	2	8	no	—
96	C₄⋊C₄×C₃₁	—	4	4	no	—
97	C₂²⋊C₄×C₃₁	—	2	8	no	—
98	C₂²⋊C₄×C₃₁	—	4	4	no	—
99	C₄²×C₃₁	—	2	8	no	—
100	C₄²×C₃₁	—	4	4	no	—
101	#20671	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	—	8	1	no	—
104	C₂³×C₃₁	—	8	2	no	—
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₃₁	#121	1	4	yes	—
109	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	4	no	—
111	Q₈×C₃₁	—	2	4	no	—
112	D₄×C₃₁	—	2	8	no	—
113	D₄×C₃₁	—	2	8	no	—
114	D₄×C₃₁	—	2	4	no	—
115	D₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	8	no	—
120	C₂×C₄×C₃₁	—	4	8	no	—
121	C₂×C₄×C₃₁	—	8	8	no	—
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#117	1	2	yes	—
128	C₂×C₄×C₃₁	#117	1	2	yes	—
129	C₂×C₄×C₃₁	#121	1	4	yes	—
130	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
136	C₈×C₃₁	—	2	4	no	—
137	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
138	C₂×#568	C₂×C₃₁	1	1	yes	characteristic
139	#2911	C₂×C₃₁	1	2	yes	complemented
140	C₂×#410	C₂×C₃₁	1	2	yes	—
141	#1812	C₂×C₃₁	1	8	yes	complemented
142	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	complemented
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	—	4	4	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	4	no	—
149	C₂²×C₃₁	—	8	4	no	—
150	C₂²×C₃₁	#380	1	2	yes	—
151	C₂²×C₃₁	#380	1	2	yes	—
152	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₃₁	C₂×#121	1	2	yes	—
154	C₄×C₃₁	—	2	4	no	—
155	C₄×C₃₁	—	8	8	no	—
156	C₄×C₃₁	#380	1	2	yes	—
157	C₄×C₃₁	#380	1	2	yes	—
158	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
159	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
160	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
162	#568	—	2	8	no	—
163	C₂×#131	C₄×C₃₁	1	4	yes	complemented
164	C₂×#131	C₂²×C₃₁	1	4	yes	—
165	C₂²×M₁₆	C₄×C₃₁	1	4	yes	complemented
166	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×#117	C₂²×C₃₁	1	4	yes	—
169	#428	—	2	8	no	—
170	#427	C₂²×C₃₁	1	8	yes	complemented
171	#410	—	2	4	no	—
172	#410	C₂²×C₃₁	1	8	yes	—
173	C₂×#99	—	2	4	no	—
174	H(3,ℤ/4)	C₂²×C₃₁	1	8	yes	complemented
175	C₂×C₃₁	—	2	2	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	2	no	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	4	4	no	—
180	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
181	C₂×#41	—	2	4	no	—
182	C₂×#41	—	2	4	no	—
183	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×#41	D₄×C₃₁	1	2	yes	—
185	C₂×#41	Q₈×C₃₁	1	2	yes	—
186	C₂×#41	C₂³×C₃₁	1	2	yes	—
187	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
188	C₂²×D₄	D₄×C₃₁	1	2	yes	—
189	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
191	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	#131	—	2	16	no	—
193	#131	—	2	16	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	16	no	—
196	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
198	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
199	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
200	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
201	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
203	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
204	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
205	#117	—	2	4	no	—
206	#117	—	2	4	no	—
207	#117	—	2	4	no	—
208	#117	C₂³×C₃₁	1	8	yes	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₄²	—	2	4	no	—
212	#107	—	2	8	no	—
213	#106	—	2	8	no	—
214	#99	—	2	16	no	—
215	C₃₁	#20671	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	#41	—	2	4	no	—
220	#41	—	2	8	no	—
221	#41	—	2	8	no	—
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
226	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
230	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
231	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
232	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
237	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
239	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	#41×C₃₁	1	4	yes	—
246	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	16	no	—
250	M₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
251	M₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	16	no	—
254	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
255	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
256	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	4	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	2	no	—
266	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
267	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂³	C₃₁×#121	1	4	yes	—
270	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	4	no	—
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	8	8	no	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
286	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#117	1	2	yes	—
289	C₂×C₄	C₃₁×#117	1	2	yes	—
290	C₂×C₄	C₃₁×#121	1	4	yes	—
291	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
292	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
295	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	4	no	—
306	C₂²	C₃₁×#380	1	2	yes	—
307	C₂²	C₃₁×#380	1	2	yes	—
308	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
309	C₂²	C₂×C₃₁×#121	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	8	8	no	—
312	C₄	C₃₁×#380	1	2	yes	—
313	C₄	C₃₁×#380	1	2	yes	—
314	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
315	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
316	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	4	no	—
322	C₂	C₂×C₃₁×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#380
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2911	C₂	semidirect
4. C₃₁×#1812	C₂	semidirect
5. C₂×C₃₁×H(3,ℤ/4)	C₂	semidirect
8. C₂×C₃₁×#131	C₄	semidirect
10. C₂²×M₁₆×C₃₁	C₄	semidirect
15. C₃₁×#427	C₂²	semidirect
19. C₃₁×H(3,ℤ/4)	C₂²	semidirect
38. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
42. C₂²×C₈×C₃₁	C₂×C₄	semidirect
101. #20671	C₃₁	direct (N × H)
139. #2911	C₂×C₃₁	semidirect
141. #1812	C₂×C₃₁	semidirect
142. C₂×H(3,ℤ/4)	C₂×C₃₁	semidirect
163. C₂×#131	C₄×C₃₁	semidirect
165. C₂²×M₁₆	C₄×C₃₁	semidirect
170. #427	C₂²×C₃₁	semidirect
174. H(3,ℤ/4)	C₂²×C₃₁	semidirect
199. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
203. C₂²×C₈	C₂×C₄×C₃₁	semidirect
215. C₃₁	#20671	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#568	2
C₃₁×#2911	2
C₂×C₃₁×#410	2
C₃₁×#1812	2
C₂×C₃₁×H(3,ℤ/4)	2
#20671	31

Subgroups — C31×#20671 #21477430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20671 #21477430