Subgroups — C₃₁×#21949 #21478708

Order 7936.

1010 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#407	C₂	1	2	yes	maximal
2	C₂×C₃₁×#407	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1855	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1811	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#108	C₂²	1	8	yes	complemented
9	C₃₁×#407	—	2	2	no	—
10	C₃₁×#407	—	2	2	no	—
11	C₃₁×#407	C₄	1	2	yes	—
12	C₃₁×#407	C₄	1	4	yes	—
13	C₃₁×#407	C₄	1	2	yes	—
14	C₃₁×#407	C₂²	1	4	yes	—
15	C₃₁×#407	C₂²	1	4	yes	—
16	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₃₁×#341	C₂²	1	8	yes	complemented
20	C₂⁵×C₃₁	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	4	4	no	—
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
29	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	—	4	4	no	—
34	C₂×M₁₆×C₃₁	—	4	8	no	—
35	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
38	C₂²×C₈×C₃₁	—	2	4	no	—
39	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
40	C₃₁×#108	—	4	8	no	—
41	C₃₁×C₂²⋊C₈	—	2	4	no	—
42	C₃₁×C₂²⋊C₈	—	2	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	4	no	—
44	C₃₁×C₂²⋊C₈	—	2	4	no	—
45	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
46	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
47	C₂⁴×C₃₁	—	4	2	no	—
48	C₂⁴×C₃₁	—	4	4	no	—
49	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
53	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	8	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₃₁	—	2	16	no	—
83	M₁₆×C₃₁	—	4	4	no	—
84	M₁₆×C₃₁	—	4	4	no	—
85	M₁₆×C₃₁	—	4	8	no	—
86	C₂×C₈×C₃₁	—	2	8	no	—
87	C₂×C₈×C₃₁	—	2	8	no	—
88	C₂×C₈×C₃₁	—	4	8	no	—
89	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
90	#21949	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	4	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	2	4	no	—
94	C₂³×C₃₁	—	2	4	no	—
95	C₂³×C₃₁	—	2	4	no	—
96	C₂³×C₃₁	—	4	1	no	—
97	C₂³×C₃₁	—	4	2	no	—
98	C₂³×C₃₁	—	4	1	no	—
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	2	no	—
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	—	4	16	no	—
104	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	1	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
133	C₈×C₃₁	—	2	8	no	—
134	C₈×C₃₁	—	4	8	no	—
135	C₂×#407	C₂×C₃₁	1	2	yes	—
136	C₂×#407	C₂×C₃₁	1	1	yes	characteristic
137	#1855	C₂×C₃₁	1	8	yes	complemented
138	#1811	C₂×C₃₁	1	4	yes	complemented
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	2	no	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	2	2	no	—
149	C₂²×C₃₁	—	4	8	no	—
150	C₂²×C₃₁	—	4	4	no	—
151	C₂²×C₃₁	—	4	4	no	—
152	C₂²×C₃₁	—	4	8	no	—
153	C₂²×C₃₁	—	4	4	no	—
154	C₂²×C₃₁	#430	1	4	yes	—
155	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₃₁	#543	1	2	yes	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	#518	1	2	yes	—
162	C₄×C₃₁	#534	1	2	yes	—
163	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
165	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
166	C₂×#108	C₂²×C₃₁	1	8	yes	complemented
167	#407	—	2	2	no	—
168	#407	—	2	2	no	—
169	#407	C₄×C₃₁	1	2	yes	—
170	#407	C₄×C₃₁	1	4	yes	—
171	#407	C₄×C₃₁	1	2	yes	—
172	#407	C₂²×C₃₁	1	4	yes	—
173	#407	C₂²×C₃₁	1	4	yes	—
174	C₂×C₂²⋊C₈	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
177	#341	C₂²×C₃₁	1	8	yes	complemented
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	2	4	no	—
180	C₂×C₃₁	—	4	1	no	—
181	C₂×C₃₁	—	4	2	no	—
182	C₂×C₃₁	—	4	1	no	—
183	C₂×C₃₁	#2315	1	1	yes	characteristic
184	C₂×C₃₁	C₂×#430	1	2	yes	—
185	C₂⁵	Q₈×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	—	4	4	no	—
187	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
189	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₃₁	1	2	yes	—
192	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
194	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
196	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	4	4	no	—
199	C₂×M₁₆	—	4	8	no	—
200	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
201	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
202	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
205	#108	—	4	8	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
211	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
212	C₃₁	#21949	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	—	4	2	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
219	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
235	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₃₁	1	2	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	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
245	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
248	M₁₆	—	2	16	no	—
249	M₁₆	—	4	4	no	—
250	M₁₆	—	4	4	no	—
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	4	8	no	—
255	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	16	no	—
269	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
270	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
273	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
298	C₈	—	2	8	no	—
299	C₈	—	4	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	C₃₁×#430	1	4	yes	—
316	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
317	C₂²	C₃₁×#543	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	C₃₁×#518	1	2	yes	—
323	C₄	C₃₁×#534	1	2	yes	—
324	C₂	—	2	4	no	—
325	C₂	—	2	2	no	—
326	C₂	—	4	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	C₃₁×#2315	1	1	yes	characteristic
330	C₂	C₂×C₃₁×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₃₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1855	C₂	semidirect
4. C₃₁×#1811	C₂	semidirect
8. C₂×C₃₁×#108	C₂²	semidirect
19. C₃₁×#341	C₂²	semidirect
90. #21949	C₃₁	direct (N × H)
137. #1855	C₂×C₃₁	semidirect
138. #1811	C₂×C₃₁	semidirect
166. C₂×#108	C₂²×C₃₁	semidirect
177. #341	C₂²×C₃₁	semidirect
212. C₃₁	#21949	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×#407	2
C₃₁×#1855	2
C₃₁×#1811	2
#21949	31

Subgroups — C31×#21949 #21478708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21949 #21478708