Subgroups — C₃₁×#21252 #21478011

Order 7936.

1610 conjugacy classes of subgroups.

The subgroup lattice has 456 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	C₂×C₃₁×#421	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1980	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×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	4	yes	complemented
13	C₃₁×#532	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×#127	C₄	1	1	yes	characteristic, complemented
16	C₂×C₃₁×#124	C₄	1	2	yes	complemented
17	C₂×C₃₁×#121	—	2	2	no	—
18	C₂×C₄×D₄×C₃₁	—	2	2	no	—
19	C₂×C₃₁×#117	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#421	—	2	16	no	—
22	C₂×C₃₁×#104	C₂²	1	16	yes	complemented
23	C₄×M₁₆×C₃₁	—	2	2	no	—
24	C₄×M₁₆×C₃₁	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#97	—	2	1	no	—
26	C₂×C₄×C₈×C₃₁	—	2	1	no	—
27	C₃₁×#384	—	2	1	no	—
28	C₃₁×#382	—	2	1	no	—
29	C₃₁×#380	—	2	2	no	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	—	2	8	no	—
34	C₂×#41×C₃₁	—	2	8	no	—
35	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×#41×C₃₁	C₂³	1	4	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂³×C₄×C₃₁	D₄	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	D₄	1	8	yes	complemented
47	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×C₄⋊Q₈	—	2	1	no	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#127	C₂×C₄	1	4	yes	—
55	C₃₁×#124	—	2	2	no	—
56	C₃₁×#124	—	2	2	no	—
57	C₃₁×#124	C₂×C₄	1	8	yes	—
58	C₃₁×#121	—	2	4	no	—
59	C₃₁×#121	—	2	4	no	—
60	C₃₁×#121	—	2	4	no	—
61	C₃₁×#121	—	2	4	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₄×D₄×C₃₁	—	2	4	no	—
66	C₃₁×#117	—	2	8	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#104	—	2	32	no	—
83	C₃₁×#97	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	4	no	—
85	C₃₁×#95	—	2	4	no	—
86	C₂⁴×C₃₁	—	2	2	no	—
87	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
88	#41×C₃₁	—	2	16	no	—
89	#41×C₃₁	—	2	8	no	—
90	#41×C₃₁	—	2	8	no	—
91	#41×C₃₁	—	4	8	no	—
92	#41×C₃₁	—	4	4	no	—
93	#41×C₃₁	—	4	4	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	8	no	—
96	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	4	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	4	8	no	—
104	C₂×D₄×C₃₁	—	4	4	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	4	4	no	—
112	C₂²×C₄×C₃₁	—	4	8	no	—
113	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₃₁	#41	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₃₁	—	2	8	no	—
128	M₁₆×C₃₁	—	2	4	no	—
129	M₁₆×C₃₁	C₂×D₄	1	8	yes	complemented
130	C₂×C₈×C₃₁	—	2	8	no	—
131	C₂×C₈×C₃₁	—	2	4	no	—
132	C₄⋊C₄×C₃₁	—	2	4	no	—
133	C₄⋊C₄×C₃₁	—	2	4	no	—
134	C₄⋊C₄×C₃₁	—	4	8	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₂²⋊C₄×C₃₁	—	2	8	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₄²×C₃₁	—	2	8	no	—
140	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
141	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
142	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
143	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
148	C₄²×C₃₁	C₂×D₄	1	2	yes	—
149	C₄²×C₃₁	C₂×D₄	1	2	yes	—
150	#21252	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	4	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	8	no	—
154	C₂³×C₃₁	—	4	4	no	—
155	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂³×C₃₁	#124	1	2	yes	—
158	C₂³×C₃₁	#127	1	2	yes	—
159	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	8	no	—
161	Q₈×C₃₁	—	4	4	no	—
162	D₄×C₃₁	—	2	8	no	—
163	D₄×C₃₁	—	4	8	no	—
164	D₄×C₃₁	—	4	4	no	—
165	D₄×C₃₁	—	4	4	no	—
166	D₄×C₃₁	—	4	8	no	—
167	D₄×C₃₁	—	4	4	no	—
168	D₄×C₃₁	—	4	8	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	4	8	no	—
178	C₂×C₄×C₃₁	—	4	8	no	—
179	C₂×C₄×C₃₁	—	4	8	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
188	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
189	C₂×C₄×C₃₁	#121	1	4	yes	—
190	C₂×C₄×C₃₁	#121	1	4	yes	—
191	C₂×C₄×C₃₁	#124	1	2	yes	—
192	C₂×C₄×C₃₁	#127	1	2	yes	—
193	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₈×C₃₁	—	2	8	no	—
198	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
199	C₂×#421	C₂×C₃₁	1	4	yes	complemented
200	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1980	C₂×C₃₁	1	8	yes	complemented
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	8	no	—
209	C₂²×C₃₁	—	4	4	no	—
210	C₂²×C₃₁	—	8	8	no	—
211	C₂²×C₃₁	#390	1	2	yes	—
212	C₂²×C₃₁	#390	1	2	yes	—
213	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	4	4	no	—
220	C₄×C₃₁	—	8	4	no	—
221	C₄×C₃₁	#390	1	2	yes	—
222	C₄×C₃₁	#390	1	2	yes	—
223	C₄×C₃₁	#421	1	4	yes	complemented
224	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
225	C₄×C₃₁	C₂×#121	1	2	yes	—
226	C₂²×#41	C₂²×C₃₁	1	2	yes	—
227	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
228	#532	—	2	1	no	—
229	#532	—	2	1	no	—
230	#532	—	2	1	no	—
231	#532	—	2	1	no	—
232	#532	C₄×C₃₁	1	4	yes	complemented
233	#532	C₂²×C₃₁	1	4	yes	—
234	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
235	C₂×#127	C₄×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#124	C₄×C₃₁	1	2	yes	complemented
237	C₂×#121	—	2	2	no	—
238	C₂×C₄×D₄	—	2	2	no	—
239	C₂×#117	C₂²×C₃₁	1	2	yes	—
240	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
241	#421	—	2	16	no	—
242	C₂×#104	C₂²×C₃₁	1	16	yes	complemented
243	C₄×M₁₆	—	2	2	no	—
244	C₄×M₁₆	C₂²×C₃₁	1	4	yes	complemented
245	C₂×#97	—	2	1	no	—
246	C₂×C₄×C₈	—	2	1	no	—
247	#384	—	2	1	no	—
248	#382	—	2	1	no	—
249	#380	—	2	2	no	—
250	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
251	C₂×C₃₁	—	2	1	no	—
252	C₂×C₃₁	—	2	2	no	—
253	C₂×C₃₁	—	2	1	no	—
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	8	4	no	—
256	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
257	C₂×C₃₁	C₂×#421	1	2	yes	—
258	C₂×#41	—	2	2	no	—
259	C₂×#41	—	2	2	no	—
260	C₂×#41	—	2	8	no	—
261	C₂×#41	—	2	8	no	—
262	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
263	C₂×#41	C₂³×C₃₁	1	4	yes	—
264	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
265	C₂²×D₄	—	2	2	no	—
266	C₂²×D₄	—	2	2	no	—
267	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	D₄×C₃₁	1	2	yes	—
270	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂×M₁₆	D₄×C₃₁	1	8	yes	complemented
274	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
275	C₂²×C₈	—	2	2	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
279	#127	—	2	1	no	—
280	#127	—	2	1	no	—
281	#127	C₂×C₄×C₃₁	1	4	yes	—
282	#124	—	2	2	no	—
283	#124	—	2	2	no	—
284	#124	C₂×C₄×C₃₁	1	8	yes	—
285	#121	—	2	4	no	—
286	#121	—	2	4	no	—
287	#121	—	2	4	no	—
288	#121	—	2	4	no	—
289	C₄×D₄	—	2	4	no	—
290	C₄×D₄	—	2	4	no	—
291	C₄×D₄	—	2	4	no	—
292	C₄×D₄	—	2	4	no	—
293	#117	—	2	8	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	4	no	—
297	C₂×C₂²⋊C₄	—	2	4	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
302	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄²	D₄×C₃₁	1	2	yes	—
305	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
308	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
309	#104	—	2	32	no	—
310	#97	—	2	4	no	—
311	C₄×C₈	—	2	4	no	—
312	#95	—	2	4	no	—
313	C₃₁	#21252	1	1	yes	characteristic, direct product factor, Sylow
314	C₂⁴	—	2	2	no	—
315	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
316	#41	—	2	16	no	—
317	#41	—	2	8	no	—
318	#41	—	2	8	no	—
319	#41	—	4	8	no	—
320	#41	—	4	4	no	—
321	#41	—	4	4	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	8	no	—
324	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	4	8	no	—
332	C₂×D₄	—	4	4	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	4	8	no	—
341	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
342	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
343	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
344	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
353	C₂²×C₄	#41×C₃₁	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
355	M₁₆	—	2	8	no	—
356	M₁₆	—	2	4	no	—
357	M₁₆	C₂×D₄×C₃₁	1	8	yes	complemented
358	C₂×C₈	—	2	8	no	—
359	C₂×C₈	—	2	4	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	4	8	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	2	8	no	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₄²	—	2	8	no	—
368	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
369	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
370	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
371	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
372	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
373	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
374	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
375	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
376	C₄²	C₂×D₄×C₃₁	1	2	yes	—
377	C₄²	C₂×D₄×C₃₁	1	2	yes	—
378	C₂³	—	2	4	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	8	no	—
381	C₂³	—	4	4	no	—
382	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
383	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
384	C₂³	C₃₁×#124	1	2	yes	—
385	C₂³	C₃₁×#127	1	2	yes	—
386	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	Q₈	—	2	8	no	—
388	Q₈	—	4	4	no	—
389	D₄	—	2	8	no	—
390	D₄	—	4	8	no	—
391	D₄	—	4	4	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	8	no	—
395	D₄	—	4	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	8	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	4	no	—
408	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
409	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
410	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
411	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
412	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
415	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
416	C₂×C₄	C₃₁×#121	1	4	yes	—
417	C₂×C₄	C₃₁×#121	1	4	yes	—
418	C₂×C₄	C₃₁×#124	1	2	yes	—
419	C₂×C₄	C₃₁×#127	1	2	yes	—
420	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
424	C₈	—	2	8	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	8	no	—
431	C₂²	—	4	4	no	—
432	C₂²	—	8	8	no	—
433	C₂²	C₃₁×#390	1	2	yes	—
434	C₂²	C₃₁×#390	1	2	yes	—
435	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
436	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
437	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	8	4	no	—
443	C₄	C₃₁×#390	1	2	yes	—
444	C₄	C₃₁×#390	1	2	yes	—
445	C₄	C₃₁×#421	1	4	yes	complemented
446	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
447	C₄	C₂×C₃₁×#121	1	2	yes	—
448	C₂	—	2	1	no	—
449	C₂	—	2	2	no	—
450	C₂	—	2	1	no	—
451	C₂	—	2	2	no	—
452	C₂	—	8	4	no	—
453	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
454	C₂	C₂×C₃₁×#421	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×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. C₂×C₃₁×#421	C₂	semidirect
3. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2285	C₂	semidirect
5. C₃₁×#1980	C₂	semidirect
12. C₃₁×#532	C₄	semidirect
14. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
15. C₂×C₃₁×#127	C₄	semidirect
16. C₂×C₃₁×#124	C₄	semidirect
22. C₂×C₃₁×#104	C₂²	semidirect
24. C₄×M₁₆×C₃₁	C₂²	semidirect
30. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
46. C₂×M₁₆×C₃₁	D₄	semidirect
129. M₁₆×C₃₁	C₂×D₄	semidirect
150. #21252	C₃₁	direct (N × H)
199. C₂×#421	C₂×C₃₁	semidirect
200. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
201. #2285	C₂×C₃₁	semidirect
202. #1980	C₂×C₃₁	semidirect
223. C₄×C₃₁	#421	semidirect
232. #532	C₄×C₃₁	semidirect
234. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
235. C₂×#127	C₄×C₃₁	semidirect
236. C₂×#124	C₄×C₃₁	semidirect
242. C₂×#104	C₂²×C₃₁	semidirect
244. C₄×M₁₆	C₂²×C₃₁	semidirect
250. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
273. C₂×M₁₆	D₄×C₃₁	semidirect
313. C₃₁	#21252	direct (N × H)
357. M₁₆	C₂×D₄×C₃₁	semidirect
445. C₄	C₃₁×#421	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
C₂×C₃₁×#421	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2285	2
C₃₁×#1980	2
#21252	31

Subgroups — C31×#21252 #21478011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21252 #21478011