Subgroups — C₃₁×#22812 #21479571

Order 7936.

802 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2357	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2356	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1973	C₂	1	2	yes	maximal
5	C₃₁×#1927	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1926	C₂	1	2	yes	maximal
7	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1776	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×SD₁₆×C₃₁	C₄	1	8	yes	complemented
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#391	C₂²	1	2	yes	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#388	—	2	2	no	—
29	C₃₁×#387	—	2	2	no	—
30	C₃₁×#383	—	2	1	no	—
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	1	yes	characteristic
33	C₃₁×#381	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₃₁×#376	—	2	1	no	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#107	—	2	4	no	—
70	C₃₁×#106	—	4	2	no	—
71	C₃₁×#106	C₂×C₄	1	4	yes	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×#103	—	4	2	no	—
74	C₃₁×#103	C₂×C₄	1	4	yes	—
75	C₃₁×#102	—	4	2	no	—
76	C₃₁×#102	C₂×C₄	1	4	yes	—
77	C₃₁×#97	—	2	4	no	—
78	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	D₄	1	1	yes	characteristic
88	C₃₁×#95	D₄	1	1	yes	characteristic
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	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	SD₁₆×C₃₁	—	2	8	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	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	C₂×C₈×C₃₁	#41	1	2	yes	—
123	C₂×C₈×C₃₁	#41	1	2	yes	—
124	C₄⋊C₄×C₃₁	—	2	4	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂²⋊C₄×C₃₁	—	2	4	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	8	2	no	—
137	C₄²×C₃₁	—	2	4	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	4	4	no	—
141	C₄²×C₃₁	—	8	2	no	—
142	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
143	C₄²×C₃₁	#41	1	2	yes	—
144	#22812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₃₁	—	2	2	no	—
152	Q₈×C₃₁	—	2	2	no	—
153	Q₈×C₃₁	—	4	1	no	—
154	Q₈×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	4	1	no	—
158	D₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	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	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	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	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	#117	1	4	yes	—
192	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
193	#2357	C₂×C₃₁	1	1	yes	characteristic
194	#2356	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1973	C₂×C₃₁	1	2	yes	—
196	#1927	C₂×C₃₁	1	2	yes	complemented
197	#1926	C₂×C₃₁	1	2	yes	—
198	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1804	C₂×C₃₁	1	1	yes	characteristic
201	#1776	C₂×C₃₁	1	2	yes	complemented
202	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	4	2	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₃₁	#547	1	1	yes	characteristic
210	C₂²×C₃₁	#550	1	1	yes	characteristic
211	C₂²×C₃₁	#575	1	2	yes	—
212	C₂²×C₃₁	#577	1	2	yes	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
223	C₄×C₃₁	#520	1	1	yes	characteristic
224	C₄×C₃₁	#555	1	1	yes	characteristic
225	C₄×C₃₁	#556	1	1	yes	characteristic
226	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
229	C₄×SD₁₆	C₄×C₃₁	1	8	yes	complemented
230	C₂×#107	C₂²×C₃₁	1	2	yes	—
231	C₂×#106	C₂²×C₃₁	1	2	yes	—
232	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#103	C₂²×C₃₁	1	2	yes	—
235	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#102	C₂²×C₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#97	C₂²×C₃₁	1	2	yes	—
239	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
240	#391	C₂²×C₃₁	1	2	yes	—
241	#390	C₂²×C₃₁	1	2	yes	—
242	#388	—	2	2	no	—
243	#387	—	2	2	no	—
244	#383	—	2	1	no	—
245	#382	C₂²×C₃₁	1	1	yes	characteristic
246	#382	C₂²×C₃₁	1	1	yes	characteristic
247	#381	—	2	1	no	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
250	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₄×C₂²⋊C₄	—	2	1	no	—
252	#376	—	2	1	no	—
253	#340	C₂²×C₃₁	1	2	yes	—
254	#340	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	2	yes	—
256	C₂×C₃₁	—	8	2	no	—
257	C₂×C₃₁	#2354	1	1	yes	characteristic
258	C₂×C₃₁	#2943	1	1	yes	characteristic
259	C₂×C₃₁	#2949	1	1	yes	characteristic
260	C₂×C₃₁	#3351	1	2	yes	—
261	C₂×C₃₁	#3401	1	2	yes	—
262	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
264	C₂³×C₄	—	2	1	no	—
265	C₂×SD₁₆	—	2	2	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
268	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄×Q₈	—	2	2	no	—
272	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
273	C₄×D₄	—	2	2	no	—
274	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×C₂²⋊C₄	—	2	1	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	#107	—	2	4	no	—
290	#106	—	4	2	no	—
291	#106	C₂×C₄×C₃₁	1	4	yes	—
292	C₄⋊C₈	—	2	2	no	—
293	#103	—	4	2	no	—
294	#103	C₂×C₄×C₃₁	1	4	yes	—
295	#102	—	4	2	no	—
296	#102	C₂×C₄×C₃₁	1	4	yes	—
297	#97	—	2	4	no	—
298	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
299	#95	—	2	1	no	—
300	#95	—	2	2	no	—
301	#95	—	2	1	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	1	no	—
305	#95	—	2	1	no	—
306	#95	—	2	2	no	—
307	#95	D₄×C₃₁	1	1	yes	characteristic
308	#95	D₄×C₃₁	1	1	yes	characteristic
309	C₃₁	#22812	1	1	yes	characteristic, direct product factor, Sylow
310	C₂⁴	—	2	1	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	8	2	no	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	SD₁₆	—	2	8	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
342	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
343	C₂×C₈	#41×C₃₁	1	2	yes	—
344	C₂×C₈	#41×C₃₁	1	2	yes	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
354	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
355	C₂²⋊C₄	—	2	4	no	—
356	C₂²⋊C₄	—	4	4	no	—
357	C₂²⋊C₄	—	8	2	no	—
358	C₄²	—	2	4	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	4	no	—
362	C₄²	—	8	2	no	—
363	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
364	C₄²	#41×C₃₁	1	2	yes	—
365	C₂³	—	2	2	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
371	Q₈	—	2	2	no	—
372	Q₈	—	2	2	no	—
373	Q₈	—	4	1	no	—
374	Q₈	—	4	1	no	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	1	no	—
378	D₄	—	4	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	4	no	—
401	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
402	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
403	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₈	C₃₁×#117	1	4	yes	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
418	C₂²	C₃₁×#547	1	1	yes	characteristic
419	C₂²	C₃₁×#550	1	1	yes	characteristic
420	C₂²	C₃₁×#575	1	2	yes	—
421	C₂²	C₃₁×#577	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	4	2	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	8	2	no	—
430	C₄	—	8	2	no	—
431	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
432	C₄	C₃₁×#520	1	1	yes	characteristic
433	C₄	C₃₁×#555	1	1	yes	characteristic
434	C₄	C₃₁×#556	1	1	yes	characteristic
435	C₂	—	8	2	no	—
436	C₂	C₃₁×#2354	1	1	yes	characteristic
437	C₂	C₃₁×#2943	1	1	yes	characteristic
438	C₂	C₃₁×#2949	1	1	yes	characteristic
439	C₂	C₃₁×#3351	1	2	yes	—
440	C₂	C₃₁×#3401	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×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2356	C₂	semidirect
5. C₃₁×#1927	C₂	semidirect
7. C₃₁×#1846	C₂	semidirect
8. C₃₁×#1805	C₂	semidirect
10. C₃₁×#1776	C₂	semidirect
11. C₃₁×#1770	C₂	semidirect
15. C₄×SD₁₆×C₃₁	C₄	semidirect
144. #22812	C₃₁	direct (N × H)
194. #2356	C₂×C₃₁	semidirect
196. #1927	C₂×C₃₁	semidirect
198. #1846	C₂×C₃₁	semidirect
199. #1805	C₂×C₃₁	semidirect
201. #1776	C₂×C₃₁	semidirect
202. #1770	C₂×C₃₁	semidirect
229. C₄×SD₁₆	C₄×C₃₁	semidirect
309. C₃₁	#22812	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×SD₁₆×C₃₁	2
C₃₁×#2357	2
C₃₁×#2356	2
C₃₁×#1973	2
C₃₁×#1927	2
C₃₁×#1926	2
C₃₁×#1846	2
C₃₁×#1805	2
C₃₁×#1804	2
C₃₁×#1776	2
C₃₁×#1770	2
#22812	31

Subgroups — C31×#22812 #21479571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22812 #21479571