Subgroups — C₃₁×#18062 #21474821

Order 7936.

618 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	C₃₁×#2984	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1593	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1584	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1579	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1491	C₂	1	2	yes	maximal
8	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#446	—	2	4	no	—
12	C₃₁×#446	C₂²	1	4	yes	—
13	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₃₁	C₄	1	2	yes	—
15	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
16	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
17	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#432	—	2	2	no	—
21	C₃₁×#432	C₂²	1	2	yes	—
22	C₃₁×#431	—	2	2	no	—
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
25	C₂×C₃₁×#103	C₄	1	2	yes	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#335	C₂²	1	2	yes	—
29	C₃₁×#334	C₂²	1	2	yes	—
30	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
32	C₃₁×#322	C₂²	1	2	yes	—
33	C₃₁×#322	C₂²	1	2	yes	—
34	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₃₁	—	2	4	no	—
36	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₃₁	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂×C₄	1	2	yes	—
53	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
55	C₃₁×C₄⋊C₈	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	4	no	—
57	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
58	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
59	C₃₁×#103	C₂×C₄	1	2	yes	—
60	C₃₁×#103	C₂×C₄	1	2	yes	—
61	C₃₁×#103	C₂×C₄	1	2	yes	—
62	C₃₁×#103	C₂×C₄	1	2	yes	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
65	C₃₁×#97	—	2	4	no	—
66	C₃₁×#97	—	2	2	no	—
67	C₃₁×#97	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	4	no	—
69	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	2	yes	—
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	4	2	no	—
78	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
85	Q₁₆×C₃₁	—	2	4	no	—
86	Q₁₆×C₃₁	—	2	2	no	—
87	Q₁₆×C₃₁	M₁₆	1	4	yes	complemented
88	M₁₆×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	4	1	no	—
93	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	#41	1	1	yes	characteristic
112	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
113	#18062	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	2	no	—
116	Q₈×C₃₁	—	2	2	no	—
117	Q₈×C₃₁	—	2	2	no	—
118	Q₈×C₃₁	—	4	4	no	—
119	Q₈×C₃₁	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	—	4	4	no	—
142	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
143	#2984	C₂×C₃₁	1	2	yes	—
144	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
145	#1593	C₂×C₃₁	1	2	yes	complemented
146	#1584	C₂×C₃₁	1	4	yes	complemented
147	#1579	C₂×C₃₁	1	2	yes	complemented
148	#1568	C₂×C₃₁	1	1	yes	characteristic
149	#1491	C₂×C₃₁	1	2	yes	—
150	#1451	C₂×C₃₁	1	1	yes	characteristic
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	#444	1	2	yes	—
154	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	4	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	4	2	no	—
161	C₄×C₃₁	—	4	2	no	—
162	C₄×C₃₁	—	4	2	no	—
163	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₃₁	#568	1	1	yes	characteristic
165	C₄×C₃₁	#578	1	2	yes	—
166	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
168	#446	—	2	4	no	—
169	#446	C₂²×C₃₁	1	4	yes	—
170	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
171	C₄×Q₁₆	C₄×C₃₁	1	2	yes	—
172	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
173	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
174	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
175	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
176	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
177	#432	—	2	2	no	—
178	#432	C₂²×C₃₁	1	2	yes	—
179	#431	—	2	2	no	—
180	#431	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
182	C₂×#103	C₄×C₃₁	1	2	yes	—
183	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	#335	C₂²×C₃₁	1	2	yes	—
186	#334	C₂²×C₃₁	1	2	yes	—
187	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
188	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
189	#322	C₂²×C₃₁	1	2	yes	—
190	#322	C₂²×C₃₁	1	2	yes	—
191	C₂×C₃₁	—	2	2	no	—
192	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
193	C₂×C₃₁	#2950	1	1	yes	characteristic
194	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
195	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
198	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂×M₁₆	—	2	1	no	—
201	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
206	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
207	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
208	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
213	#107	C₂×C₄×C₃₁	1	2	yes	—
214	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
215	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
219	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
220	#103	C₂×C₄×C₃₁	1	2	yes	—
221	#103	C₂×C₄×C₃₁	1	2	yes	—
222	#103	C₂×C₄×C₃₁	1	2	yes	—
223	#103	C₂×C₄×C₃₁	1	2	yes	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
226	#97	—	2	4	no	—
227	#97	—	2	2	no	—
228	#97	—	2	1	no	—
229	C₄×C₈	—	2	4	no	—
230	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₃₁	1	2	yes	—
233	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	2	yes	—
236	C₃₁	#18062	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
241	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	M₁₆×C₃₁	1	4	yes	complemented
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
266	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄²	#41×C₃₁	1	1	yes	characteristic
274	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
275	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	4	4	no	—
280	Q₈	C₂×M₁₆×C₃₁	1	4	yes	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	4	no	—
303	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	C₃₁×#444	1	2	yes	—
307	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
317	C₄	C₃₁×#568	1	1	yes	characteristic
318	C₄	C₃₁×#578	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
321	C₂	C₃₁×#2950	1	1	yes	characteristic
322	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1593	C₂	semidirect
4. C₃₁×#1584	C₂	semidirect
5. C₃₁×#1579	C₂	semidirect
87. Q₁₆×C₃₁	M₁₆	semidirect
113. #18062	C₃₁	direct (N × H)
145. #1593	C₂×C₃₁	semidirect
146. #1584	C₂×C₃₁	semidirect
147. #1579	C₂×C₃₁	semidirect
236. C₃₁	#18062	direct (N × H)
249. Q₁₆	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2984	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#1593	2
C₃₁×#1584	2
C₃₁×#1579	2
C₃₁×#1568	2
C₃₁×#1491	2
C₃₁×#1451	2
#18062	31

Subgroups — C31×#18062 #21474821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18062 #21474821