Subgroups — C₃₁×#22268 #21479027

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1880	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1877	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#492	C₄	1	8	yes	complemented
18	C₂×C₃₁×#103	C₂²	1	4	yes	—
19	C₂×C₃₁×#102	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#391	C₂²	1	2	yes	—
23	C₃₁×#390	—	2	1	no	—
24	C₃₁×#390	—	2	1	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	—	2	1	no	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	4	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
40	C₂³×C₄×C₃₁	—	2	2	no	—
41	C₂×SD₁₆×C₃₁	—	4	4	no	—
42	C₂×SD₁₆×C₃₁	—	4	4	no	—
43	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₃₁	—	4	4	no	—
51	C₄×D₄×C₃₁	—	4	4	no	—
52	C₄×D₄×C₃₁	—	4	4	no	—
53	C₄×D₄×C₃₁	—	4	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#103	—	4	4	no	—
75	C₃₁×#102	—	4	4	no	—
76	C₄×C₈×C₃₁	—	2	2	no	—
77	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₂⁴×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	4	no	—
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂×D₄×C₃₁	—	2	4	no	—
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	4	no	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	4	4	no	—
99	C₂²×C₄×C₃₁	—	4	4	no	—
100	C₂²×C₄×C₃₁	—	4	4	no	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	SD₁₆×C₃₁	—	4	16	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	2	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	—	4	4	no	—
125	C₂²⋊C₄×C₃₁	—	4	4	no	—
126	C₂²⋊C₄×C₃₁	—	4	4	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	4	4	no	—
133	C₄²×C₃₁	SD₁₆	1	4	yes	complemented
134	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
135	C₄²×C₃₁	#41	1	2	yes	—
136	#22268	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	8	no	—
146	Q₈×C₃₁	—	4	4	no	—
147	D₄×C₃₁	—	2	8	no	—
148	D₄×C₃₁	—	4	4	no	—
149	D₄×C₃₁	—	4	8	no	—
150	D₄×C₃₁	—	4	8	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	8	8	no	—
171	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₃₁	#135	1	2	yes	—
177	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₃₁	#137	1	2	yes	—
179	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
183	C₈×C₃₁	—	2	4	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
186	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
188	#1957	C₂×C₃₁	1	1	yes	characteristic
189	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1880	C₂×C₃₁	1	4	yes	complemented
191	#1877	C₂×C₃₁	1	4	yes	complemented
192	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	4	4	no	—
195	C₂²×C₃₁	—	4	4	no	—
196	C₂²×C₃₁	—	4	4	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₃₁	#535	1	1	yes	characteristic
203	C₂²×C₃₁	#535	1	1	yes	characteristic
204	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
206	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	C₄×SD₁₆	1	4	yes	—
216	C₄×C₃₁	#440	1	4	yes	—
217	C₄×C₃₁	#518	1	2	yes	—
218	C₄×C₃₁	#540	1	2	yes	—
219	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
220	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
223	C₂×C₄×D₄	—	2	1	no	—
224	C₂×C₄×D₄	—	2	1	no	—
225	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
226	#492	C₄×C₃₁	1	8	yes	complemented
227	C₂×#103	C₂²×C₃₁	1	4	yes	—
228	C₂×#102	C₂²×C₃₁	1	4	yes	—
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
231	#391	C₂²×C₃₁	1	2	yes	—
232	#390	—	2	1	no	—
233	#390	—	2	1	no	—
234	#390	C₂²×C₃₁	1	2	yes	—
235	#389	—	2	1	no	—
236	#389	—	2	1	no	—
237	#384	C₂²×C₃₁	1	1	yes	characteristic
238	#384	C₂²×C₃₁	1	1	yes	characteristic
239	C₄×C₄⋊C₄	—	2	1	no	—
240	C₄×C₄⋊C₄	—	2	1	no	—
241	C₄³	C₂²×C₃₁	1	1	yes	characteristic
242	#340	C₂²×C₃₁	1	4	yes	—
243	#336	C₂²×C₃₁	1	2	yes	—
244	#336	C₂²×C₃₁	1	2	yes	—
245	C₂×C₃₁	—	8	4	no	—
246	C₂×C₃₁	#2328	1	1	yes	characteristic
247	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
248	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
249	C₂×C₃₁	#3206	1	1	yes	characteristic
250	C₂×C₃₁	#3207	1	1	yes	characteristic
251	C₂×C₃₁	#3208	1	1	yes	characteristic
252	C₂×C₃₁	#3210	1	1	yes	characteristic
253	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
254	C₂²×D₄	—	2	1	no	—
255	C₂²×D₄	—	2	1	no	—
256	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
257	C₂³×C₄	—	2	2	no	—
258	C₂×SD₁₆	—	4	4	no	—
259	C₂×SD₁₆	—	4	4	no	—
260	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
261	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
263	C₄⋊Q₈	—	2	2	no	—
264	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
265	#127	—	2	2	no	—
266	#127	C₂×C₄×C₃₁	1	4	yes	—
267	C₄×Q₈	—	4	4	no	—
268	C₄×D₄	—	4	4	no	—
269	C₄×D₄	—	4	4	no	—
270	C₄×D₄	—	4	4	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	#103	—	4	4	no	—
292	#102	—	4	4	no	—
293	C₄×C₈	—	2	2	no	—
294	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	C₃₁	#22268	1	1	yes	characteristic, direct product factor, Sylow
298	C₂⁴	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	4	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	4	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	4	no	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
326	SD₁₆	—	4	16	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	4	4	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₂²⋊C₄	—	4	4	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	4	4	no	—
351	C₄²	SD₁₆×C₃₁	1	4	yes	complemented
352	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
353	C₄²	#41×C₃₁	1	2	yes	—
354	C₂³	—	2	4	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	Q₈	—	2	8	no	—
363	Q₈	—	4	4	no	—
364	D₄	—	2	8	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	8	no	—
367	D₄	—	4	4	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	8	no	—
388	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
391	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
393	C₂×C₄	C₃₁×#135	1	2	yes	—
394	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
395	C₂×C₄	C₃₁×#137	1	2	yes	—
396	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
399	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
400	C₈	—	2	4	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	4	4	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
410	C₂²	C₃₁×#535	1	1	yes	characteristic
411	C₂²	C₃₁×#535	1	1	yes	characteristic
412	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
413	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
414	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
415	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	4	no	—
419	C₄	—	4	1	no	—
420	C₄	—	4	1	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	C₄×SD₁₆×C₃₁	1	4	yes	—
424	C₄	C₃₁×#440	1	4	yes	—
425	C₄	C₃₁×#518	1	2	yes	—
426	C₄	C₃₁×#540	1	2	yes	—
427	C₂	—	8	4	no	—
428	C₂	C₃₁×#2328	1	1	yes	characteristic
429	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
430	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
431	C₂	C₃₁×#3206	1	1	yes	characteristic
432	C₂	C₃₁×#3207	1	1	yes	characteristic
433	C₂	C₃₁×#3208	1	1	yes	characteristic
434	C₂	C₃₁×#3210	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	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
5. C₃₁×#1956	C₂	semidirect
6. C₃₁×#1880	C₂	semidirect
7. C₃₁×#1877	C₂	semidirect
8. C₃₁×#1834	C₂	semidirect
9. C₃₁×#1744	C₂	semidirect
17. C₃₁×#492	C₄	semidirect
43. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
47. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
77. C₄×C₈×C₃₁	C₂×C₄	semidirect
133. C₄²×C₃₁	SD₁₆	semidirect
136. #22268	C₃₁	direct (N × H)
186. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
189. #1956	C₂×C₃₁	semidirect
190. #1880	C₂×C₃₁	semidirect
191. #1877	C₂×C₃₁	semidirect
192. #1834	C₂×C₃₁	semidirect
193. #1744	C₂×C₃₁	semidirect
226. #492	C₄×C₃₁	semidirect
260. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
264. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
294. C₄×C₈	C₂×C₄×C₃₁	semidirect
297. C₃₁	#22268	direct (N × H)
351. C₄²	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#492	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#127	2
C₃₁×#1957	2
C₃₁×#1956	2
C₃₁×#1880	2
C₃₁×#1877	2
C₃₁×#1834	2
C₃₁×#1744	2
#22268	31

Subgroups — C31×#22268 #21479027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22268 #21479027