Subgroups — C₃₁×#2417 #16842350

Order 3968.

648 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#398	C₂	1	2	yes	maximal, complemented
6	C₃₁×#396	C₂	1	2	yes	maximal, complemented
7	C₃₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#386	C₂	1	2	yes	maximal
10	C₃₁×#380	C₂	1	2	yes	maximal, complemented
11	C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
16	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₂²⋊Q₈	—	2	4	no	—
18	C₃₁×C₂²⋊Q₈	—	2	4	no	—
19	C₄×D₄×C₃₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₃₁×#95	C₂²	1	2	yes	—
38	C₃₁×#95	C₂²	1	2	yes	—
39	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	2	no	—
42	C₂×Q₈×C₃₁	—	2	2	no	—
43	C₂×Q₈×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	—	2	2	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	Q₈	1	4	yes	complemented
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	D₄	1	4	yes	—
56	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	4	2	no	—
72	C₄⋊C₄×C₃₁	D₄	1	4	yes	complemented
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	4	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	4	no	—
77	C₂²⋊C₄×C₃₁	—	4	2	no	—
78	C₄²×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	1	no	—
80	C₂³×C₃₁	—	2	1	no	—
81	C₂³×C₃₁	—	2	1	no	—
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	C₂×D₄	1	2	yes	—
90	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
91	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
92	C₂³×C₃₁	#41	1	2	yes	—
93	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
94	Q₈×C₃₁	—	4	4	no	—
95	D₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	8	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	8	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
122	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	#41	1	2	yes	—
126	#2417	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	4	1	no	—
137	C₂²×C₃₁	—	4	1	no	—
138	C₂²×C₃₁	—	4	1	no	—
139	C₂²×C₃₁	—	4	1	no	—
140	C₂²×C₃₁	C₂²⋊Q₈	1	4	yes	complemented
141	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
145	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
146	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	—	4	2	no	—
156	C₄×C₃₁	#121	1	4	yes	—
157	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#398	C₂×C₃₁	1	2	yes	complemented
162	#396	C₂×C₃₁	1	2	yes	complemented
163	#391	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#386	C₂×C₃₁	1	2	yes	—
166	#380	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#380	C₂×C₃₁	1	2	yes	complemented
168	C₂×C₃₁	—	2	4	no	—
169	C₂×C₃₁	—	4	2	no	—
170	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
171	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₃₁	#537	1	1	yes	characteristic
173	C₂×C₃₁	#546	1	1	yes	characteristic
174	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
175	C₂×C₃₁	#552	1	1	yes	characteristic
176	C₂×C₃₁	#554	1	1	yes	characteristic
177	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
178	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
179	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
180	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
181	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	4	no	—
183	C₂²⋊Q₈	—	2	4	no	—
184	C₄×D₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
198	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
199	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
200	#95	C₂²×C₃₁	1	2	yes	—
201	#95	C₂²×C₃₁	1	2	yes	—
202	#95	C₂²×C₃₁	1	2	yes	—
203	#95	C₂²×C₃₁	1	2	yes	—
204	C₃₁	#2417	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
206	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	Q₈×C₃₁	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	D₄×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	D₄×C₃₁	1	4	yes	complemented
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	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	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	C₂×D₄×C₃₁	1	2	yes	—
256	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
257	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
258	C₂³	#41×C₃₁	1	2	yes	—
259	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	2	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
287	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
288	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
289	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
291	C₂×C₄	#41×C₃₁	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	C₃₁×C₂²⋊Q₈	1	4	yes	complemented
306	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
309	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
310	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
311	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
312	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	C₃₁×#121	1	4	yes	—
322	C₂	—	2	4	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
325	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
326	C₂	C₃₁×#537	1	1	yes	characteristic
327	C₂	C₃₁×#546	1	1	yes	characteristic
328	C₂	D₄×Q₈×C₃₁	1	1	yes	characteristic
329	C₂	C₃₁×#552	1	1	yes	characteristic
330	C₂	C₃₁×#554	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
2. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
4. C₂²×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#398	C₂	semidirect
6. C₃₁×#396	C₂	semidirect
7. C₃₁×#391	C₂	semidirect
8. C₃₁×#389	C₂	semidirect
10. C₃₁×#380	C₂	semidirect
11. C₃₁×#380	C₂	semidirect
46. C₂×D₄×C₃₁	Q₈	semidirect
72. C₄⋊C₄×C₃₁	D₄	semidirect
126. #2417	C₃₁	direct (N × H)
140. C₂²×C₃₁	C₂²⋊Q₈	semidirect
157. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
158. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
160. C₂²×C₄⋊C₄	C₂×C₃₁	semidirect
161. #398	C₂×C₃₁	semidirect
162. #396	C₂×C₃₁	semidirect
163. #391	C₂×C₃₁	semidirect
164. #389	C₂×C₃₁	semidirect
166. #380	C₂×C₃₁	semidirect
167. #380	C₂×C₃₁	semidirect
204. C₃₁	#2417	direct (N × H)
212. C₂×D₄	Q₈×C₃₁	semidirect
238. C₄⋊C₄	D₄×C₃₁	semidirect
305. C₂²	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₄×D₄×C₃₁	2
C₂²×C₄⋊C₄×C₃₁	2
C₃₁×#398	2
C₃₁×#396	2
C₃₁×#391	2
C₃₁×#389	2
C₃₁×#386	2
C₃₁×#380	2
#2417	31

Subgroups — C31×#2417 #16842350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2417 #16842350