Subgroups — C₃₁×#23026 #21479785

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2362	C₂	1	2	yes	maximal
5	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1927	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1873	C₂	1	2	yes	maximal
9	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₃₁×#467	C₄	1	8	yes	complemented
18	C₄×Q₁₆×C₃₁	—	2	4	no	—
19	C₃₁×#427	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#106	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#388	—	2	2	no	—
33	C₃₁×#388	—	2	2	no	—
34	C₃₁×#387	—	2	2	no	—
35	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×C₂²⋊Q₈	—	2	4	no	—
55	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	4	2	no	—
58	C₃₁×#117	—	2	4	no	—
59	C₃₁×#117	—	4	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	C₂×C₄	1	4	yes	—
77	C₃₁×#107	D₄	1	4	yes	—
78	C₃₁×#106	—	2	4	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	4	2	no	—
81	C₃₁×#103	C₂×C₄	1	8	yes	—
82	C₃₁×C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	1	yes	characteristic
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	#41	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₃₁	—	2	8	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₃₁	#41	1	2	yes	—
128	C₄⋊C₄×C₃₁	—	2	8	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	—	4	4	no	—
135	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
137	C₂²⋊C₄×C₃₁	—	2	8	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	—	4	4	no	—
147	C₄²×C₃₁	—	4	4	no	—
148	#23026	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	C₄×D₄	1	2	yes	—
156	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	4	no	—
158	Q₈×C₃₁	—	2	4	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₃₁	#135	1	2	yes	—
191	C₂×C₄×C₃₁	#135	1	2	yes	—
192	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	—	4	1	no	—
201	C₈×C₃₁	C₄×D₄	1	4	yes	—
202	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
203	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
205	#2362	C₂×C₃₁	1	2	yes	—
206	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1927	C₂×C₃₁	1	2	yes	complemented
209	#1873	C₂×C₃₁	1	2	yes	—
210	#1865	C₂×C₃₁	1	2	yes	complemented
211	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	—	4	1	no	—
220	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₃₁	D₄²	1	1	yes	characteristic
222	C₂²×C₃₁	#547	1	1	yes	characteristic
223	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
225	C₂²×C₃₁	#578	1	1	yes	characteristic
226	C₂²×C₃₁	#578	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	4	4	no	—
232	C₄×C₃₁	—	4	4	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	4	4	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
237	C₄×C₃₁	#520	1	1	yes	characteristic
238	C₄×C₃₁	#548	1	1	yes	characteristic
239	C₄×C₃₁	#553	1	1	yes	characteristic
240	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
241	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
243	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
244	C₂×#117	C₂²×C₃₁	1	2	yes	—
245	#467	C₄×C₃₁	1	8	yes	complemented
246	C₄×Q₁₆	—	2	4	no	—
247	#427	—	2	4	no	—
248	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#106	C₂²×C₃₁	1	2	yes	—
252	C₂×#103	C₂²×C₃₁	1	2	yes	—
253	C₂×#103	C₂²×C₃₁	1	2	yes	—
254	C₂×#103	C₂²×C₃₁	1	2	yes	—
255	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	#391	C₂²×C₃₁	1	2	yes	—
259	#389	C₂²×C₃₁	1	2	yes	—
260	#388	—	2	2	no	—
261	#388	—	2	2	no	—
262	#387	—	2	2	no	—
263	C₄×C₄⋊C₄	—	2	2	no	—
264	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
265	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	1	yes	characteristic
269	#336	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₃₁	—	4	2	no	—
271	C₂×C₃₁	#2361	1	1	yes	characteristic
272	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
273	C₂×C₃₁	#2950	1	1	yes	characteristic
274	C₂×C₃₁	#3283	1	1	yes	characteristic
275	C₂×C₃₁	#3285	1	1	yes	characteristic
276	C₂×C₃₁	#3336	1	1	yes	characteristic
277	C₂×C₃₁	#3342	1	1	yes	characteristic
278	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×Q₁₆	—	2	2	no	—
281	C₂×Q₁₆	—	2	2	no	—
282	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
283	C₂²×C₈	—	2	2	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²⋊Q₈	—	2	4	no	—
291	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
292	C₄×Q₈	—	2	4	no	—
293	C₄×Q₈	—	4	2	no	—
294	#117	—	2	4	no	—
295	#117	—	4	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#107	—	2	2	no	—
312	#107	C₂×C₄×C₃₁	1	4	yes	—
313	#107	D₄×C₃₁	1	4	yes	—
314	#106	—	2	4	no	—
315	#103	—	2	4	no	—
316	#103	—	4	2	no	—
317	#103	C₂×C₄×C₃₁	1	8	yes	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	D₄×C₃₁	1	1	yes	characteristic
326	#95	D₄×C₃₁	1	1	yes	characteristic
327	C₃₁	#23026	1	1	yes	characteristic, direct product factor, Sylow
328	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	#41×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	Q₁₆	—	2	8	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	8	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₂×C₈	#41×C₃₁	1	2	yes	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
373	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	4	no	—
383	C₄²	—	4	4	no	—
384	C₄²	—	4	4	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	C₄×D₄×C₃₁	1	2	yes	—
392	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	Q₈	—	2	4	no	—
394	Q₈	—	2	4	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	4	2	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₃₁×#135	1	2	yes	—
427	C₂×C₄	C₃₁×#135	1	2	yes	—
428	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
433	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	4	1	no	—
436	C₈	—	4	1	no	—
437	C₈	C₄×D₄×C₃₁	1	4	yes	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
446	C₂²	D₄²×C₃₁	1	1	yes	characteristic
447	C₂²	C₃₁×#547	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
449	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
450	C₂²	C₃₁×#578	1	1	yes	characteristic
451	C₂²	C₃₁×#578	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	4	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	4	no	—
460	C₄	—	8	2	no	—
461	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
462	C₄	C₃₁×#520	1	1	yes	characteristic
463	C₄	C₃₁×#548	1	1	yes	characteristic
464	C₄	C₃₁×#553	1	1	yes	characteristic
465	C₂	—	4	2	no	—
466	C₂	C₃₁×#2361	1	1	yes	characteristic
467	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
468	C₂	C₃₁×#2950	1	1	yes	characteristic
469	C₂	C₃₁×#3283	1	1	yes	characteristic
470	C₂	C₃₁×#3285	1	1	yes	characteristic
471	C₂	C₃₁×#3336	1	1	yes	characteristic
472	C₂	C₃₁×#3342	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1969	C₂	semidirect
6. C₃₁×#1942	C₂	semidirect
7. C₃₁×#1927	C₂	semidirect
9. C₃₁×#1865	C₂	semidirect
10. C₄×C₃₁×#107	C₂	semidirect
17. C₃₁×#467	C₄	semidirect
148. #23026	C₃₁	direct (N × H)
203. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
206. #1969	C₂×C₃₁	semidirect
207. #1942	C₂×C₃₁	semidirect
208. #1927	C₂×C₃₁	semidirect
210. #1865	C₂×C₃₁	semidirect
211. C₄×#107	C₂×C₃₁	semidirect
245. #467	C₄×C₃₁	semidirect
327. C₃₁	#23026	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×C₃₁×#467	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₃₁×#427	2
C₃₁×#2362	2
C₃₁×#1969	2
C₃₁×#1942	2
C₃₁×#1927	2
C₃₁×#1873	2
C₃₁×#1865	2
C₄×C₃₁×#107	2
C₈×C₂²⋊C₄×C₃₁	2
#23026	31

Subgroups — C31×#23026 #21479785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23026 #21479785