Subgroups — C₃₁×#21306 #21478065

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 380 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×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1986	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1733	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#533	—	2	1	no	—
8	C₃₁×#533	—	2	1	no	—
9	C₃₁×#533	—	2	1	no	—
10	C₃₁×#533	—	2	1	no	—
11	C₃₁×#533	C₄	1	4	yes	complemented
12	C₃₁×#533	C₂²	1	4	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	complemented
14	C₂×C₃₁×#125	C₄	1	2	yes	complemented
15	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
16	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
17	C₂×C₃₁×#117	C₂²	1	2	yes	—
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₃₁	—	2	2	no	—
21	C₄×M₁₆×C₃₁	C₂²	1	4	yes	—
22	C₂×C₃₁×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	—	2	1	no	—
24	C₃₁×#384	—	2	1	no	—
25	C₃₁×#382	—	2	1	no	—
26	C₃₁×#380	—	2	2	no	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
28	C₃₁×#339	C₂²	1	16	yes	—
29	C₂²×Q₈×C₃₁	—	2	2	no	—
30	C₂³×C₄×C₃₁	D₄	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	—	2	4	no	—
34	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
35	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
40	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
41	C₃₁×#125	—	2	2	no	—
42	C₃₁×#125	—	2	2	no	—
43	C₃₁×#125	C₂×C₄	1	8	yes	—
44	C₃₁×C₂²⋊Q₈	—	2	4	no	—
45	C₃₁×C₂²⋊Q₈	—	2	4	no	—
46	C₃₁×C₂²⋊Q₈	—	2	4	no	—
47	C₃₁×C₂²⋊Q₈	—	2	4	no	—
48	C₄×Q₈×C₃₁	—	2	4	no	—
49	C₄×Q₈×C₃₁	—	2	4	no	—
50	C₄×Q₈×C₃₁	—	2	4	no	—
51	C₄×Q₈×C₃₁	—	2	4	no	—
52	C₃₁×#117	—	2	2	no	—
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	8	no	—
55	C₃₁×#117	C₂×C₄	1	4	yes	—
56	C₃₁×#117	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
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	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	2	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#97	—	2	4	no	—
76	C₄×C₈×C₃₁	—	2	4	no	—
77	C₃₁×#95	—	2	4	no	—
78	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	4	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	4	8	no	—
82	C₂²×C₄×C₃₁	—	2	4	no	—
83	C₂²×C₄×C₃₁	—	2	4	no	—
84	C₂²×C₄×C₃₁	—	4	4	no	—
85	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
97	C₂²×C₄×C₃₁	#41	1	4	yes	—
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₃₁	—	2	8	no	—
100	M₁₆×C₃₁	—	2	4	no	—
101	M₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
102	C₂×C₈×C₃₁	—	2	8	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₄⋊C₄×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	—	4	8	no	—
110	C₄⋊C₄×C₃₁	—	4	4	no	—
111	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
112	C₂²⋊C₄×C₃₁	—	2	4	no	—
113	C₂²⋊C₄×C₃₁	—	2	4	no	—
114	C₂²⋊C₄×C₃₁	—	4	4	no	—
115	C₄²×C₃₁	—	2	4	no	—
116	C₄²×C₃₁	—	2	4	no	—
117	C₄²×C₃₁	—	2	8	no	—
118	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
119	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
120	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
122	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
125	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
126	C₄²×C₃₁	C₂×D₄	1	2	yes	—
127	C₄²×C₃₁	C₂×D₄	1	2	yes	—
128	#21306	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	4	2	no	—
130	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂³×C₃₁	#124	1	2	yes	—
133	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
134	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	Q₈×C₃₁	—	4	8	no	—
136	Q₈×C₃₁	—	4	8	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	8	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	8	no	—
145	C₂×C₄×C₃₁	—	4	8	no	—
146	C₂×C₄×C₃₁	—	4	4	no	—
147	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
154	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
155	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
157	C₂×C₄×C₃₁	#124	1	2	yes	—
158	C₂×C₄×C₃₁	C₄⋊Q₈	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	C₂²×Q₈	1	2	yes	—
162	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
163	C₈×C₃₁	—	2	8	no	—
164	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
165	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
166	#2285	C₂×C₃₁	1	1	yes	characteristic
167	#1986	C₂×C₃₁	1	8	yes	complemented
168	#1733	C₂×C₃₁	1	4	yes	—
169	C₂²×C₃₁	—	2	2	no	—
170	C₂²×C₃₁	—	2	2	no	—
171	C₂²×C₃₁	—	2	2	no	—
172	C₂²×C₃₁	—	4	2	no	—
173	C₂²×C₃₁	—	4	2	no	—
174	C₂²×C₃₁	#391	1	2	yes	—
175	C₂²×C₃₁	#391	1	2	yes	—
176	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
178	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	4	8	no	—
183	C₄×C₃₁	—	8	4	no	—
184	C₄×C₃₁	#391	1	2	yes	—
185	C₄×C₃₁	#391	1	2	yes	—
186	C₄×C₃₁	#421	1	4	yes	—
187	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
188	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
189	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
190	#533	—	2	1	no	—
191	#533	—	2	1	no	—
192	#533	—	2	1	no	—
193	#533	—	2	1	no	—
194	#533	C₄×C₃₁	1	4	yes	complemented
195	#533	C₂²×C₃₁	1	4	yes	—
196	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	complemented
197	C₂×#125	C₄×C₃₁	1	2	yes	complemented
198	C₂×C₂²⋊Q₈	—	2	2	no	—
199	C₂×C₄×Q₈	—	2	2	no	—
200	C₂×#117	C₂²×C₃₁	1	2	yes	—
201	C₂×#117	C₂²×C₃₁	1	2	yes	—
202	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
203	C₄×M₁₆	—	2	2	no	—
204	C₄×M₁₆	C₂²×C₃₁	1	4	yes	—
205	C₂×#97	—	2	1	no	—
206	C₂×C₄×C₈	—	2	1	no	—
207	#384	—	2	1	no	—
208	#382	—	2	1	no	—
209	#380	—	2	2	no	—
210	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
211	#339	C₂²×C₃₁	1	16	yes	—
212	C₂×C₃₁	—	2	1	no	—
213	C₂×C₃₁	—	2	1	no	—
214	C₂×C₃₁	—	2	2	no	—
215	C₂×C₃₁	—	2	2	no	—
216	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
217	C₂×C₃₁	C₂×#421	1	2	yes	—
218	C₂²×Q₈	—	2	2	no	—
219	C₂³×C₄	D₄×C₃₁	1	2	yes	—
220	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	4	no	—
223	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
224	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
225	C₂²×C₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
229	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
230	#125	—	2	2	no	—
231	#125	—	2	2	no	—
232	#125	C₂×C₄×C₃₁	1	8	yes	—
233	C₂²⋊Q₈	—	2	4	no	—
234	C₂²⋊Q₈	—	2	4	no	—
235	C₂²⋊Q₈	—	2	4	no	—
236	C₂²⋊Q₈	—	2	4	no	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	—	2	4	no	—
239	C₄×Q₈	—	2	4	no	—
240	C₄×Q₈	—	2	4	no	—
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	#117	—	2	8	no	—
244	#117	C₂×C₄×C₃₁	1	4	yes	—
245	#117	C₂³×C₃₁	1	4	yes	—
246	C₂×C₄⋊C₄	—	2	4	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	—	2	2	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
257	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₃₁	1	2	yes	—
260	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
263	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
264	#97	—	2	4	no	—
265	C₄×C₈	—	2	4	no	—
266	#95	—	2	4	no	—
267	C₃₁	#21306	1	1	yes	characteristic, direct product factor, Sylow
268	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	2	4	no	—
271	C₂×Q₈	—	4	8	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	4	4	no	—
275	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
276	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
282	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
283	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
287	C₂²×C₄	#41×C₃₁	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
289	M₁₆	—	2	8	no	—
290	M₁₆	—	2	4	no	—
291	M₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
292	C₂×C₈	—	2	8	no	—
293	C₂×C₈	—	2	4	no	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	4	8	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
302	C₂²⋊C₄	—	2	4	no	—
303	C₂²⋊C₄	—	2	4	no	—
304	C₂²⋊C₄	—	4	4	no	—
305	C₄²	—	2	4	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	8	no	—
308	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
309	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
310	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
311	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
312	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
313	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
314	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
315	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
316	C₄²	C₂×D₄×C₃₁	1	2	yes	—
317	C₄²	C₂×D₄×C₃₁	1	2	yes	—
318	C₂³	—	4	2	no	—
319	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
320	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
321	C₂³	C₃₁×#124	1	2	yes	—
322	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
323	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	Q₈	—	4	8	no	—
325	Q₈	—	4	8	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	8	no	—
334	C₂×C₄	—	4	8	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
337	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
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	1	yes	characteristic
341	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
343	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
344	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
345	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
346	C₂×C₄	C₃₁×#124	1	2	yes	—
347	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
348	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
351	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
352	C₈	—	2	8	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	C₃₁×#391	1	2	yes	—
359	C₂²	C₃₁×#391	1	2	yes	—
360	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
361	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
362	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	8	no	—
367	C₄	—	8	4	no	—
368	C₄	C₃₁×#391	1	2	yes	—
369	C₄	C₃₁×#391	1	2	yes	—
370	C₄	C₃₁×#421	1	4	yes	—
371	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
372	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	2	no	—
376	C₂	—	2	2	no	—
377	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
378	C₂	C₂×C₃₁×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1986	C₂	semidirect
11. C₃₁×#533	C₄	semidirect
13. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
14. C₂×C₃₁×#125	C₄	semidirect
34. C₂×M₁₆×C₃₁	Q₈	semidirect
101. M₁₆×C₃₁	C₂×Q₈	semidirect
128. #21306	C₃₁	direct (N × H)
167. #1986	C₂×C₃₁	semidirect
194. #533	C₄×C₃₁	semidirect
196. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
197. C₂×#125	C₄×C₃₁	semidirect
223. C₂×M₁₆	Q₈×C₃₁	semidirect
267. C₃₁	#21306	direct (N × H)
291. M₁₆	C₂×Q₈×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×#533	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2285	2
C₃₁×#1986	2
C₃₁×#1733	2
#21306	31

Subgroups — C31×#21306 #21478065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21306 #21478065