Subgroups — C₃₁×#21134 #21477893

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2319	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal
7	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₃₁	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#388	—	2	2	no	—
28	C₃₁×#381	—	2	1	no	—
29	C₃₁×#380	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
35	C₃₁×#376	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#95	—	2	1	no	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	—	2	4	no	—
48	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₃₁	—	2	4	no	—
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	C₂×C₄	1	4	yes	complemented
74	C₃₁×#106	—	2	4	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#102	C₂×C₄	1	4	yes	—
78	C₃₁×#102	C₂×C₄	1	4	yes	complemented
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	D₄	1	1	yes	characteristic
86	C₃₁×#95	D₄	1	1	yes	characteristic
87	C₃₁×#95	D₄	1	2	yes	—
88	C₂⁴×C₃₁	—	2	1	no	—
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₃₁	#41	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	8	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	8	2	no	—
106	C₂²×C₄×C₃₁	—	8	2	no	—
107	C₂²×C₄×C₃₁	—	8	2	no	—
108	C₂²×C₄×C₃₁	—	8	2	no	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	D₈×C₃₁	—	2	8	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	8	2	no	—
128	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
130	C₄⋊C₄×C₃₁	#41	1	4	yes	—
131	C₂²⋊C₄×C₃₁	—	2	8	no	—
132	C₂²⋊C₄×C₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂²⋊C₄×C₃₁	—	8	2	no	—
135	C₂²⋊C₄×C₃₁	—	8	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	8	1	no	—
139	C₄²×C₃₁	—	8	1	no	—
140	C₄²×C₃₁	—	8	2	no	—
141	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
142	C₄²×C₃₁	#41	1	2	yes	—
143	#21134	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	#117	1	4	yes	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	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	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	4	no	—
181	C₂×C₄×C₃₁	—	8	4	no	—
182	C₂×C₄×C₃₁	—	8	8	no	—
183	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
193	#2319	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2275	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1926	C₂×C₃₁	1	2	yes	complemented
196	#1918	C₂×C₃₁	1	2	yes	complemented
197	#1879	C₂×C₃₁	1	2	yes	—
198	#1864	C₂×C₃₁	1	2	yes	complemented
199	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1762	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1762	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×C₃₁	#525	1	2	yes	—
215	C₂²×C₃₁	#576	1	2	yes	—
216	C₂²×C₃₁	#577	1	1	yes	characteristic
217	C₂²×C₃₁	#577	1	1	yes	characteristic
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	4	4	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
227	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
228	C₄×C₃₁	#529	1	2	yes	—
229	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×D₈	C₄×C₃₁	1	8	yes	complemented
234	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#106	C₂²×C₃₁	1	2	yes	—
236	C₂×#102	C₂²×C₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₃₁	1	2	yes	—
238	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	#390	C₂²×C₃₁	1	2	yes	—
243	#389	C₂²×C₃₁	1	2	yes	—
244	#388	—	2	2	no	—
245	#381	—	2	1	no	—
246	#380	—	2	1	no	—
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	C₄×C₂²⋊C₄	—	2	1	no	—
251	C₄×C₂²⋊C₄	—	2	2	no	—
252	#376	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#95	—	2	1	no	—
254	#340	C₂²×C₃₁	1	2	yes	—
255	#340	C₂²×C₃₁	1	2	yes	—
256	#340	C₂²×C₃₁	1	2	yes	—
257	#336	C₂²×C₃₁	1	2	yes	—
258	#336	C₂²×C₃₁	1	2	yes	—
259	C₂×C₃₁	—	4	4	no	—
260	C₂×C₃₁	—	8	2	no	—
261	C₂×C₃₁	#2290	1	1	yes	characteristic
262	C₂×C₃₁	#2948	1	1	yes	characteristic
263	C₂×C₃₁	#2952	1	1	yes	characteristic
264	C₂×C₃₁	#3118	1	2	yes	—
265	C₂×C₃₁	#3119	1	2	yes	—
266	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	—	2	1	no	—
270	C₂³×C₄	—	2	1	no	—
271	C₂×D₈	—	2	4	no	—
272	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
273	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
274	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄×D₄	—	2	4	no	—
277	C₄×D₄	—	2	4	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
281	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	C₂×C₄×C₃₁	1	4	yes	complemented
298	#106	—	2	4	no	—
299	#102	—	2	4	no	—
300	#102	—	2	4	no	—
301	#102	C₂×C₄×C₃₁	1	4	yes	—
302	#102	C₂×C₄×C₃₁	1	4	yes	complemented
303	#97	—	2	2	no	—
304	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
305	#95	—	2	2	no	—
306	#95	—	2	1	no	—
307	#95	—	2	1	no	—
308	#95	—	2	2	no	—
309	#95	D₄×C₃₁	1	1	yes	characteristic
310	#95	D₄×C₃₁	1	1	yes	characteristic
311	#95	D₄×C₃₁	1	2	yes	—
312	C₃₁	#21134	1	1	yes	characteristic, direct product factor, Sylow
313	C₂⁴	—	2	1	no	—
314	C₂⁴	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂×D₄	#41×C₃₁	1	4	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	—	8	2	no	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	D₈	—	2	8	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
346	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	8	2	no	—
353	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
354	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
355	C₄⋊C₄	#41×C₃₁	1	4	yes	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	8	2	no	—
360	C₂²⋊C₄	—	8	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	8	1	no	—
364	C₄²	—	8	1	no	—
365	C₄²	—	8	2	no	—
366	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
367	C₄²	#41×C₃₁	1	2	yes	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	1	no	—
378	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	4	no	—
383	D₄	—	4	2	no	—
384	D₄	C₃₁×#117	1	4	yes	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	8	4	no	—
405	C₂×C₄	—	8	4	no	—
406	C₂×C₄	—	8	8	no	—
407	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
408	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
409	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
411	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
412	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
413	C₈	—	2	2	no	—
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	4	2	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
427	C₂²	C₃₁×#525	1	2	yes	—
428	C₂²	C₃₁×#576	1	2	yes	—
429	C₂²	C₃₁×#577	1	1	yes	characteristic
430	C₂²	C₃₁×#577	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	4	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	2	no	—
439	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
440	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
441	C₄	C₃₁×#529	1	2	yes	—
442	C₂	—	4	4	no	—
443	C₂	—	8	2	no	—
444	C₂	C₃₁×#2290	1	1	yes	characteristic
445	C₂	C₃₁×#2948	1	1	yes	characteristic
446	C₂	C₃₁×#2952	1	1	yes	characteristic
447	C₂	C₃₁×#3118	1	2	yes	—
448	C₂	C₃₁×#3119	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2319	C₂	semidirect
3. C₃₁×#2275	C₂	semidirect
4. C₃₁×#1926	C₂	semidirect
5. C₃₁×#1918	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₄×C₃₁×#106	C₂	semidirect
9. C₃₁×#1762	C₂	semidirect
10. C₃₁×#1762	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
16. C₄×D₈×C₃₁	C₄	semidirect
57. C₄×D₄×C₃₁	C₂×C₄	semidirect
73. C₃₁×#107	C₂×C₄	semidirect
78. C₃₁×#102	C₂×C₄	semidirect
80. C₄×C₈×C₃₁	C₂×C₄	semidirect
143. #21134	C₃₁	direct (N × H)
193. #2319	C₂×C₃₁	semidirect
194. #2275	C₂×C₃₁	semidirect
195. #1926	C₂×C₃₁	semidirect
196. #1918	C₂×C₃₁	semidirect
198. #1864	C₂×C₃₁	semidirect
199. C₄×#106	C₂×C₃₁	semidirect
200. #1762	C₂×C₃₁	semidirect
201. #1762	C₂×C₃₁	semidirect
202. #1746	C₂×C₃₁	semidirect
233. C₄×D₈	C₄×C₃₁	semidirect
281. C₄×D₄	C₂×C₄×C₃₁	semidirect
297. #107	C₂×C₄×C₃₁	semidirect
302. #102	C₂×C₄×C₃₁	semidirect
304. C₄×C₈	C₂×C₄×C₃₁	semidirect
312. C₃₁	#21134	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₃₁	2
C₃₁×#2319	2
C₃₁×#2275	2
C₃₁×#1926	2
C₃₁×#1918	2
C₃₁×#1879	2
C₃₁×#1864	2
C₄×C₃₁×#106	2
C₃₁×#1762	2
C₃₁×#1746	2
#21134	31

Subgroups — C31×#21134 #21477893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21134 #21477893