Subgroups — C₃₁×#21385 #21478144

Order 7936.

1258 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1987	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1981	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1872	C₂	1	4	yes	maximal
7	C₄×C₃₁×#103	C₂	1	4	yes	maximal, complemented
8	C₃₁×#533	—	2	4	no	—
9	C₃₁×#533	C₄	1	8	yes	complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	—	2	1	no	—
13	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₄²×C₃₁	—	2	1	no	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#431	—	2	4	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	8	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₃₁×#95	—	2	1	no	—
31	C₃₁×#340	C₂²	1	8	yes	—
32	C₂²×Q₈×C₃₁	—	2	1	no	—
33	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₁	—	2	2	no	—
35	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
42	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
43	C₃₁×#125	—	2	8	no	—
44	C₃₁×C₂²⋊Q₈	—	2	8	no	—
45	C₃₁×C₂²⋊Q₈	—	2	4	no	—
46	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
47	C₄×Q₈×C₃₁	—	2	8	no	—
48	C₄×Q₈×C₃₁	—	2	4	no	—
49	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₃₁×#117	—	2	8	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄²×C₃₁	—	2	2	no	—
62	C₂×C₄²×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	4	4	no	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	D₄	1	4	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	—	2	4	no	—
72	C₃₁×#103	—	2	16	no	—
73	C₃₁×C₂²⋊C₈	—	2	4	no	—
74	C₄×C₈×C₃₁	—	2	4	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×Q₈×C₃₁	—	2	4	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	8	no	—
81	C₂×Q₈×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₃₁	—	2	4	no	—
84	C₂²×C₄×C₃₁	—	2	8	no	—
85	C₂²×C₄×C₃₁	—	2	4	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	4	no	—
90	C₂²×C₄×C₃₁	—	4	4	no	—
91	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
93	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
94	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	2	8	no	—
105	C₄⋊C₄×C₃₁	—	2	8	no	—
106	C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	2	8	no	—
110	C₄⋊C₄×C₃₁	—	2	8	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
114	C₄⋊C₄×C₃₁	#41	1	8	yes	—
115	C₂²⋊C₄×C₃₁	—	2	8	no	—
116	C₄²×C₃₁	—	2	8	no	—
117	C₄²×C₃₁	—	4	4	no	—
118	C₄²×C₃₁	—	4	4	no	—
119	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
120	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
121	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
122	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
123	C₄²×C₃₁	C₂×D₄	1	2	yes	—
124	C₄²×C₃₁	C₂×D₄	1	2	yes	—
125	#21385	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₃₁	—	2	2	no	—
127	C₂³×C₃₁	—	2	2	no	—
128	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
130	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
131	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₃₁	—	2	8	no	—
133	Q₈×C₃₁	—	4	8	no	—
134	Q₈×C₃₁	—	4	4	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	4	no	—
142	C₂×C₄×C₃₁	—	2	8	no	—
143	C₂×C₄×C₃₁	—	4	4	no	—
144	C₂×C₄×C₃₁	—	4	16	no	—
145	C₂×C₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	4	4	no	—
147	C₂×C₄×C₃₁	—	4	4	no	—
148	C₂×C₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	#103	1	16	yes	complemented
150	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
156	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
158	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
159	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
162	C₈×C₃₁	—	4	4	no	—
163	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
164	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
165	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
166	#1987	C₂×C₃₁	1	2	yes	complemented
167	#1981	C₂×C₃₁	1	2	yes	complemented
168	#1872	C₂×C₃₁	1	4	yes	—
169	C₄×#103	C₂×C₃₁	1	4	yes	complemented
170	C₂²×C₃₁	—	2	4	no	—
171	C₂²×C₃₁	—	2	4	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	C₂×#103	1	4	yes	—
175	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²×C₃₁	#525	1	2	yes	—
177	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
178	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
179	C₂²×C₃₁	#575	1	2	yes	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	—	4	8	no	—
184	C₄×C₃₁	—	4	4	no	—
185	C₄×C₃₁	—	4	4	no	—
186	C₄×C₃₁	C₂×#103	1	4	yes	—
187	C₄×C₃₁	C₂×#103	1	4	yes	—
188	C₄×C₃₁	C₄×#41	1	2	yes	—
189	C₄×C₃₁	#532	1	2	yes	—
190	#533	—	2	4	no	—
191	#533	C₄×C₃₁	1	8	yes	complemented
192	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×#125	—	2	1	no	—
195	C₂×C₂²⋊Q₈	—	2	1	no	—
196	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄×Q₈	—	2	1	no	—
198	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
199	C₂×#117	—	2	1	no	—
200	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
201	C₂²×C₄²	—	2	1	no	—
202	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
203	#431	—	2	4	no	—
204	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
205	C₂×#103	C₂²×C₃₁	1	8	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
208	C₄×C₄⋊C₄	—	2	1	no	—
209	C₄×C₄⋊C₄	—	2	1	no	—
210	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
212	C₂×#95	—	2	1	no	—
213	#340	C₂²×C₃₁	1	8	yes	—
214	C₂×C₃₁	—	2	4	no	—
215	C₂×C₃₁	#2297	1	1	yes	characteristic
216	C₂×C₃₁	C₂²×#103	1	1	yes	characteristic
217	C₂×C₃₁	#2896	1	1	yes	characteristic
218	C₂×C₃₁	#3101	1	2	yes	—
219	C₂×C₃₁	#3104	1	2	yes	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
222	C₂³×C₄	—	2	2	no	—
223	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
224	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
225	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
230	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
231	#125	—	2	8	no	—
232	C₂²⋊Q₈	—	2	8	no	—
233	C₂²⋊Q₈	—	2	4	no	—
234	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
235	C₄×Q₈	—	2	8	no	—
236	C₄×Q₈	—	2	4	no	—
237	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
238	#117	—	2	8	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	8	no	—
246	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
247	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	—	2	2	no	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	4	4	no	—
253	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
254	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄²	D₄×C₃₁	1	4	yes	—
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
259	C₄⋊C₈	—	2	4	no	—
260	#103	—	2	16	no	—
261	C₂²⋊C₈	—	2	4	no	—
262	C₄×C₈	—	2	4	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	C₃₁	#21385	1	1	yes	characteristic, direct product factor, Sylow
266	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	8	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	8	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	—	4	4	no	—
280	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
281	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
282	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
283	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
284	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
289	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	8	no	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	8	no	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
303	C₄⋊C₄	#41×C₃₁	1	8	yes	—
304	C₂²⋊C₄	—	2	8	no	—
305	C₄²	—	2	8	no	—
306	C₄²	—	4	4	no	—
307	C₄²	—	4	4	no	—
308	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
309	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
310	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
311	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
312	C₄²	C₂×D₄×C₃₁	1	2	yes	—
313	C₄²	C₂×D₄×C₃₁	1	2	yes	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
317	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
318	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
319	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
320	Q₈	—	2	8	no	—
321	Q₈	—	4	8	no	—
322	Q₈	—	4	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	16	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	C₃₁×#103	1	16	yes	complemented
338	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
339	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
340	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
341	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
342	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
343	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
344	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
345	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
346	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
350	C₈	—	4	4	no	—
351	C₂²	—	2	4	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	C₂×C₃₁×#103	1	4	yes	—
356	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
357	C₂²	C₃₁×#525	1	2	yes	—
358	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
359	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
360	C₂²	C₃₁×#575	1	2	yes	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	4	8	no	—
365	C₄	—	4	4	no	—
366	C₄	—	4	4	no	—
367	C₄	C₂×C₃₁×#103	1	4	yes	—
368	C₄	C₂×C₃₁×#103	1	4	yes	—
369	C₄	C₄×#41×C₃₁	1	2	yes	—
370	C₄	C₃₁×#532	1	2	yes	—
371	C₂	—	2	4	no	—
372	C₂	C₃₁×#2297	1	1	yes	characteristic
373	C₂	C₂²×C₃₁×#103	1	1	yes	characteristic
374	C₂	C₃₁×#2896	1	1	yes	characteristic
375	C₂	C₃₁×#3101	1	2	yes	—
376	C₂	C₃₁×#3104	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1987	C₂	semidirect
5. C₃₁×#1981	C₂	semidirect
7. C₄×C₃₁×#103	C₂	semidirect
9. C₃₁×#533	C₄	semidirect
41. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
42. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
49. C₄×Q₈×C₃₁	C₂×C₄	semidirect
125. #21385	C₃₁	direct (N × H)
149. C₂×C₄×C₃₁	#103	semidirect
166. #1987	C₂×C₃₁	semidirect
167. #1981	C₂×C₃₁	semidirect
169. C₄×#103	C₂×C₃₁	semidirect
191. #533	C₄×C₃₁	semidirect
229. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
230. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
237. C₄×Q₈	C₂×C₄×C₃₁	semidirect
265. C₃₁	#21385	direct (N × H)
337. C₂×C₄	C₃₁×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#533	2
C₂×C₃₁×#431	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1987	2
C₃₁×#1981	2
C₃₁×#1872	2
C₄×C₃₁×#103	2
#21385	31

Subgroups — C31×#21385 #21478144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21385 #21478144