Subgroups — C₂×C₃₁×#1981 #21478013

Order 7936.

1686 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₁×#103	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1981	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	—	2	1	no	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₃₁×#103	—	2	16	no	—
20	C₂×C₃₁×#103	C₂²	1	16	yes	—
21	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₂×C₃₁×#95	—	2	1	no	—
24	C₂²×Q₈×C₃₁	—	2	4	no	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	8	no	—
27	C₂²×Q₈×C₃₁	—	2	8	no	—
28	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
29	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₃₁	—	2	2	no	—
31	C₂³×C₄×C₃₁	—	2	2	no	—
32	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	—	2	8	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
38	C₃₁×C₄⋊Q₈	—	2	16	no	—
39	C₃₁×C₄⋊Q₈	—	2	8	no	—
40	C₃₁×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
41	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
49	C₂×C₄²×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	—	4	4	no	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	D₄	1	4	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₃₁×C₄⋊C₈	—	2	8	no	—
56	C₃₁×#103	—	2	32	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	8	no	—
60	C₂×Q₈×C₃₁	—	2	4	no	—
61	C₂×Q₈×C₃₁	—	2	32	no	—
62	C₂×Q₈×C₃₁	—	2	8	no	—
63	C₂×Q₈×C₃₁	—	2	8	no	—
64	C₂×Q₈×C₃₁	—	4	8	no	—
65	C₂×Q₈×C₃₁	—	4	8	no	—
66	C₂×Q₈×C₃₁	—	4	8	no	—
67	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	2	8	no	—
70	C₂²×C₄×C₃₁	—	2	8	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	4	8	no	—
74	C₂²×C₄×C₃₁	—	4	4	no	—
75	C₂²×C₄×C₃₁	—	4	4	no	—
76	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
79	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₃₁	#41	1	4	yes	—
87	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₃₁	—	2	8	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	4	8	no	—
91	C₄⋊C₄×C₃₁	—	2	16	no	—
92	C₄⋊C₄×C₃₁	—	2	16	no	—
93	C₄⋊C₄×C₃₁	—	2	8	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
96	C₄²×C₃₁	—	4	8	no	—
97	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
98	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
99	C₄²×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×#1981	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₃₁	#121	1	4	yes	—
103	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
104	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
105	C₂³×C₃₁	#137	1	4	yes	—
106	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₁	—	2	16	no	—
108	Q₈×C₃₁	—	4	4	no	—
109	Q₈×C₃₁	—	4	4	no	—
110	Q₈×C₃₁	—	4	16	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	8	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	2	8	no	—
115	C₂×C₄×C₃₁	—	4	8	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	16	no	—
119	C₂×C₄×C₃₁	—	4	16	no	—
120	C₂×C₄×C₃₁	#103	1	16	yes	complemented
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
125	C₂×C₄×C₃₁	#121	1	4	yes	—
126	C₂×C₄×C₃₁	#124	1	4	yes	—
127	C₂×C₄×C₃₁	#127	1	4	yes	—
128	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
130	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₈×C₃₁	—	4	4	no	—
135	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
136	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
137	C₂²×#103	C₂×C₃₁	1	4	yes	—
138	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
139	#1981	C₂×C₃₁	1	8	yes	direct product factor
140	C₂²×C₃₁	#390	1	4	yes	—
141	C₂²×C₃₁	C₂×#103	1	4	yes	—
142	C₂²×C₃₁	#419	1	4	yes	—
143	C₂²×C₃₁	#461	1	8	yes	—
144	C₂²×C₃₁	#462	1	8	yes	—
145	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#121	1	2	yes	—
147	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×#137	1	2	yes	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	4	8	no	—
152	C₄×C₃₁	—	4	8	no	—
153	C₄×C₃₁	#390	1	4	yes	—
154	C₄×C₃₁	C₂×#103	1	8	yes	complemented
155	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
156	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
158	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
159	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
160	C₂×C₄⋊Q₈	—	2	2	no	—
161	C₂×C₄⋊Q₈	—	2	4	no	—
162	C₂×C₄⋊Q₈	C₄×C₃₁	1	8	yes	complemented
163	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
164	C₂²×C₄⋊C₄	—	2	1	no	—
165	C₂²×C₄⋊C₄	—	2	1	no	—
166	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
167	C₂²×C₄²	—	2	1	no	—
168	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	2	4	no	—
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
171	C₂×#103	—	2	16	no	—
172	C₂×#103	C₂²×C₃₁	1	16	yes	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
175	C₂×#95	—	2	1	no	—
176	C₂×C₃₁	#1981	1	8	yes	direct product factor
177	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
178	C₂×C₃₁	C₂²×#103	1	1	yes	characteristic
179	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#461	1	2	yes	—
181	C₂×C₃₁	C₂×#462	1	2	yes	—
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	2	8	no	—
185	C₂²×Q₈	—	2	8	no	—
186	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
187	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	—	2	2	no	—
190	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	—	2	8	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
196	C₄⋊Q₈	—	2	16	no	—
197	C₄⋊Q₈	—	2	8	no	—
198	C₄⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
199	C₂×C₄⋊C₄	—	2	8	no	—
200	C₂×C₄⋊C₄	—	2	8	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
206	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	—	4	4	no	—
209	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
210	C₂×C₄²	D₄×C₃₁	1	2	yes	—
211	C₂×C₄²	D₄×C₃₁	1	4	yes	—
212	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
213	C₄⋊C₈	—	2	8	no	—
214	#103	—	2	32	no	—
215	#95	—	2	4	no	—
216	C₃₁	C₂×#1981	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	32	no	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	4	8	no	—
226	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	8	no	—
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
236	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
237	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
238	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
239	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
245	C₂²×C₄	#41×C₃₁	1	4	yes	—
246	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	8	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	16	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
255	C₄²	—	4	8	no	—
256	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
257	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
258	C₄²	C₂×D₄×C₃₁	1	4	yes	—
259	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
260	C₂³	C₃₁×#121	1	4	yes	—
261	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
262	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
263	C₂³	C₃₁×#137	1	4	yes	—
264	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	Q₈	—	2	16	no	—
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	16	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	16	no	—
278	C₂×C₄	C₃₁×#103	1	16	yes	complemented
279	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
283	C₂×C₄	C₃₁×#121	1	4	yes	—
284	C₂×C₄	C₃₁×#124	1	4	yes	—
285	C₂×C₄	C₃₁×#127	1	4	yes	—
286	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
287	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
288	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
292	C₈	—	4	4	no	—
293	C₂²	C₃₁×#390	1	4	yes	—
294	C₂²	C₂×C₃₁×#103	1	4	yes	—
295	C₂²	C₃₁×#419	1	4	yes	—
296	C₂²	C₃₁×#461	1	8	yes	—
297	C₂²	C₃₁×#462	1	8	yes	—
298	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
299	C₂²	C₂×C₃₁×#121	1	2	yes	—
300	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
301	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
302	C₂²	C₂×C₃₁×#137	1	2	yes	—
303	C₄	—	2	4	no	—
304	C₄	—	4	8	no	—
305	C₄	—	4	8	no	—
306	C₄	C₃₁×#390	1	4	yes	—
307	C₄	C₂×C₃₁×#103	1	8	yes	complemented
308	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
309	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
310	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
311	C₂	C₃₁×#1981	1	8	yes	direct product factor
312	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
313	C₂	C₂²×C₃₁×#103	1	1	yes	characteristic
314	C₂	C₂×C₃₁×#419	1	1	yes	characteristic
315	C₂	C₂×C₃₁×#461	1	2	yes	—
316	C₂	C₂×C₃₁×#462	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1981	C₂	direct (N × H)
10. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
40. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
100. C₂×#1981	C₃₁	direct (N × H)
120. C₂×C₄×C₃₁	#103	semidirect
139. #1981	C₂×C₃₁	direct (N × H)
154. C₄×C₃₁	C₂×#103	semidirect
162. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
176. C₂×C₃₁	#1981	direct (N × H)
198. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
216. C₃₁	C₂×#1981	direct (N × H)
278. C₂×C₄	C₃₁×#103	semidirect
307. C₄	C₂×C₃₁×#103	semidirect
311. C₂	C₃₁×#1981	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×C₄⋊C₈	2
C₂²×C₃₁×#103	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1981	2
C₂×#1981	31

Subgroups — C2×C31×#1981 #21478013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1981 #21478013