Subgroups — C₃₁×#21815 #21478574

Order 7936.

1444 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1980	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1962	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1789	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#532	—	2	1	no	—
9	C₃₁×#532	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	C₄	1	2	yes	complemented
13	C₃₁×#532	C₄	1	2	yes	complemented
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₃₁×#532	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#127	C₄	1	1	yes	characteristic, complemented
18	C₂×C₃₁×#124	C₄	1	2	yes	complemented
19	C₂×C₃₁×#121	—	2	1	no	—
20	C₂×C₃₁×#121	—	2	1	no	—
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₃₁×#117	—	2	1	no	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#432	—	2	1	no	—
26	C₃₁×#432	—	2	1	no	—
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#423	C₂²	1	2	yes	complemented
30	C₂×C₃₁×#104	C₂²	1	16	yes	complemented
31	C₃₁×#407	C₂²	1	2	yes	—
32	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#376	—	2	1	no	—
38	C₄³×C₃₁	C₂²	1	2	yes	complemented
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	4	8	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×#41×C₃₁	C₂³	1	4	yes	—
44	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₃₁×#127	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#127	C₂×C₄	1	2	yes	—
61	C₃₁×#127	C₂×C₄	1	2	yes	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#124	C₂×C₄	1	4	yes	—
65	C₃₁×#124	C₂×C₄	1	4	yes	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#121	—	2	2	no	—
70	C₃₁×#121	—	2	2	no	—
71	C₃₁×#121	—	2	2	no	—
72	C₃₁×#121	—	2	2	no	—
73	C₃₁×#121	—	2	2	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₃₁×#117	—	2	8	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×C₄⋊C₈	—	2	2	no	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×#104	—	4	16	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×#97	—	2	2	no	—
110	C₄×C₈×C₃₁	—	2	2	no	—
111	C₃₁×#95	—	2	2	no	—
112	C₃₁×#95	—	2	2	no	—
113	C₂⁴×C₃₁	—	2	2	no	—
114	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
115	#41×C₃₁	—	2	16	no	—
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	—	4	4	no	—
121	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	4	4	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	8	no	—
132	C₂×D₄×C₃₁	—	4	4	no	—
133	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	—	4	8	no	—
142	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	M₁₆×C₃₁	—	2	8	no	—
161	C₂×C₈×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	8	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	—	2	8	no	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₄²×C₃₁	—	2	8	no	—
173	C₄²×C₃₁	—	2	8	no	—
174	C₄²×C₃₁	—	4	4	no	—
175	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	#21815	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	4	no	—
195	C₂³×C₃₁	—	4	1	no	—
196	C₂³×C₃₁	—	4	4	no	—
197	C₂³×C₃₁	—	4	4	no	—
198	C₂³×C₃₁	—	4	4	no	—
199	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₃₁	—	2	8	no	—
203	Q₈×C₃₁	—	4	4	no	—
204	D₄×C₃₁	—	2	8	no	—
205	D₄×C₃₁	—	4	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	8	no	—
209	D₄×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	4	4	no	—
211	D₄×C₃₁	—	4	4	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	8	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	4	no	—
234	C₂×C₄×C₃₁	—	4	8	no	—
235	C₂×C₄×C₃₁	#104	1	4	yes	complemented
236	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
249	C₈×C₃₁	—	4	4	no	—
250	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
251	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1980	C₂×C₃₁	1	4	yes	complemented
254	#1962	C₂×C₃₁	1	4	yes	complemented
255	#1789	C₂×C₃₁	1	4	yes	complemented
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	2	1	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	4	no	—
267	C₂²×C₃₁	—	8	8	no	—
268	C₂²×C₃₁	C₂×#104	1	2	yes	—
269	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²×C₃₁	#535	1	2	yes	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
277	C₄×C₃₁	—	2	1	no	—
278	C₄×C₃₁	—	4	8	no	—
279	C₄×C₃₁	—	4	4	no	—
280	C₄×C₃₁	C₂×#104	1	2	yes	—
281	C₄×C₃₁	C₂×#104	1	2	yes	—
282	C₄×C₃₁	#421	1	2	yes	—
283	C₄×C₃₁	#518	1	2	yes	—
284	C₄×C₃₁	#535	1	2	yes	—
285	C₂²×#41	C₂²×C₃₁	1	2	yes	—
286	#532	—	2	1	no	—
287	#532	—	2	1	no	—
288	#532	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	C₄×C₃₁	1	2	yes	complemented
291	#532	C₄×C₃₁	1	2	yes	complemented
292	#532	C₂²×C₃₁	1	2	yes	—
293	#532	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
295	C₂×#127	C₄×C₃₁	1	1	yes	characteristic, complemented
296	C₂×#124	C₄×C₃₁	1	2	yes	complemented
297	C₂×#121	—	2	1	no	—
298	C₂×#121	—	2	1	no	—
299	C₂×C₄×D₄	—	2	1	no	—
300	C₂×C₄×D₄	—	2	1	no	—
301	C₂×#117	—	2	1	no	—
302	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
303	#432	—	2	1	no	—
304	#432	—	2	1	no	—
305	#431	—	2	1	no	—
306	#431	—	2	1	no	—
307	#423	C₂²×C₃₁	1	2	yes	complemented
308	C₂×#104	C₂²×C₃₁	1	16	yes	complemented
309	#407	C₂²×C₃₁	1	2	yes	—
310	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
311	C₄×C₄⋊C₄	—	2	1	no	—
312	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
313	C₄×C₂²⋊C₄	—	2	1	no	—
314	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
315	#376	—	2	1	no	—
316	C₄³	C₂²×C₃₁	1	2	yes	complemented
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	2	2	no	—
319	C₂×C₃₁	—	4	1	no	—
320	C₂×C₃₁	—	8	4	no	—
321	C₂×C₃₁	#2309	1	1	yes	characteristic
322	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
323	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	4	8	no	—
327	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
328	C₂×#41	C₂³×C₃₁	1	4	yes	—
329	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
334	C₂³×C₄	—	2	2	no	—
335	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	2	1	no	—
341	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
342	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
343	#127	—	2	1	no	—
344	#127	—	2	1	no	—
345	#127	C₂×C₄×C₃₁	1	2	yes	—
346	#127	C₂×C₄×C₃₁	1	2	yes	—
347	#124	—	2	2	no	—
348	#124	—	2	2	no	—
349	#124	C₂×C₄×C₃₁	1	4	yes	—
350	#124	C₂×C₄×C₃₁	1	4	yes	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	—	2	2	no	—
357	#121	—	2	2	no	—
358	#121	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	2	no	—
367	#117	—	2	8	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₄⋊C₈	—	2	2	no	—
390	C₄⋊C₈	—	2	2	no	—
391	#104	—	4	16	no	—
392	C₂²⋊C₈	—	2	2	no	—
393	C₂²⋊C₈	—	2	2	no	—
394	#97	—	2	2	no	—
395	C₄×C₈	—	2	2	no	—
396	#95	—	2	2	no	—
397	#95	—	2	2	no	—
398	C₃₁	#21815	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	—	2	2	no	—
400	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
401	#41	—	2	16	no	—
402	#41	—	4	8	no	—
403	#41	—	4	8	no	—
404	#41	—	4	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	4	4	no	—
407	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	8	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	8	no	—
428	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	M₁₆	—	2	8	no	—
447	C₂×C₈	—	2	4	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	—	2	8	no	—
460	C₄²	—	4	4	no	—
461	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	4	1	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	4	no	—
483	C₂³	—	4	4	no	—
484	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
486	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	Q₈	—	2	8	no	—
488	Q₈	—	4	4	no	—
489	D₄	—	2	8	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	4	8	no	—
493	D₄	—	4	8	no	—
494	D₄	—	4	4	no	—
495	D₄	—	4	4	no	—
496	D₄	—	4	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	8	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	C₃₁×#104	1	4	yes	complemented
521	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
531	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
533	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
534	C₈	—	4	4	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	8	8	no	—
547	C₂²	C₂×C₃₁×#104	1	2	yes	—
548	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
549	C₂²	C₃₁×#535	1	2	yes	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	2	1	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	8	no	—
558	C₄	—	4	4	no	—
559	C₄	C₂×C₃₁×#104	1	2	yes	—
560	C₄	C₂×C₃₁×#104	1	2	yes	—
561	C₄	C₃₁×#421	1	2	yes	—
562	C₄	C₃₁×#518	1	2	yes	—
563	C₄	C₃₁×#535	1	2	yes	—
564	C₂	—	2	2	no	—
565	C₂	—	2	2	no	—
566	C₂	—	4	1	no	—
567	C₂	—	8	4	no	—
568	C₂	C₃₁×#2309	1	1	yes	characteristic
569	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
570	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	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×C₂²⋊C₄
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₄×C₃₁×#117	C₂	semidirect
4. C₃₁×#1980	C₂	semidirect
5. C₃₁×#1962	C₂	semidirect
6. C₃₁×#1789	C₂	semidirect
12. C₃₁×#532	C₄	semidirect
13. C₃₁×#532	C₄	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
17. C₂×C₃₁×#127	C₄	semidirect
18. C₂×C₃₁×#124	C₄	semidirect
29. C₃₁×#423	C₂²	semidirect
30. C₂×C₃₁×#104	C₂²	semidirect
32. C₄×M₁₆×C₃₁	C₂²	semidirect
34. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
38. C₄³×C₃₁	C₂²	semidirect
191. #21815	C₃₁	direct (N × H)
235. C₂×C₄×C₃₁	#104	semidirect
251. #2920	C₂×C₃₁	semidirect
252. C₄×#117	C₂×C₃₁	semidirect
253. #1980	C₂×C₃₁	semidirect
254. #1962	C₂×C₃₁	semidirect
255. #1789	C₂×C₃₁	semidirect
290. #532	C₄×C₃₁	semidirect
291. #532	C₄×C₃₁	semidirect
294. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
295. C₂×#127	C₄×C₃₁	semidirect
296. C₂×#124	C₄×C₃₁	semidirect
307. #423	C₂²×C₃₁	semidirect
308. C₂×#104	C₂²×C₃₁	semidirect
310. C₄×M₁₆	C₂²×C₃₁	semidirect
312. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
316. C₄³	C₂²×C₃₁	semidirect
398. C₃₁	#21815	direct (N × H)
520. C₂×C₄	C₃₁×#104	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₃₁×#2920	2
C₄×C₃₁×#117	2
C₃₁×#1980	2
C₃₁×#1962	2
C₃₁×#1789	2
#21815	31

Subgroups — C31×#21815 #21478574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21815 #21478574