Subgroups — C₃₁×#17742 #21474501

Order 7936.

1178 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1556	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1549	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1546	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₃₁	—	2	1	no	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
15	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#432	—	2	1	no	—
24	C₃₁×#432	—	2	1	no	—
25	C₃₁×#432	C₂²	1	2	yes	—
26	C₃₁×#432	C₂²	1	2	yes	—
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	C₂²	1	2	yes	—
30	C₃₁×#431	C₂²	1	2	yes	—
31	C₃₁×#423	C₂²	1	2	yes	—
32	C₃₁×#423	C₂²	1	2	yes	—
33	C₃₁×#407	—	2	1	no	—
34	C₃₁×#407	—	2	1	no	—
35	C₄×M₁₆×C₃₁	—	2	1	no	—
36	C₄×M₁₆×C₃₁	—	2	1	no	—
37	C₃₁×#332	C₂²	1	16	yes	complemented
38	C₂²×Q₈×C₃₁	—	2	2	no	—
39	C₂²×Q₈×C₃₁	—	2	2	no	—
40	C₂²×Q₈×C₃₁	—	2	2	no	—
41	C₂²×Q₈×C₃₁	—	2	2	no	—
42	C₂²×Q₈×C₃₁	—	2	2	no	—
43	C₂²×Q₈×C₃₁	—	2	2	no	—
44	C₂²×Q₈×C₃₁	—	4	8	no	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
65	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
79	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	—	2	2	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
94	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×Q₈×C₃₁	—	2	4	no	—
105	C₂×Q₈×C₃₁	—	2	4	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	4	no	—
108	C₂×Q₈×C₃₁	—	4	8	no	—
109	C₂×Q₈×C₃₁	—	4	8	no	—
110	C₂×Q₈×C₃₁	—	4	8	no	—
111	C₂×Q₈×C₃₁	—	4	8	no	—
112	C₂×Q₈×C₃₁	—	4	8	no	—
113	C₂×Q₈×C₃₁	—	4	8	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	4	8	no	—
121	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	M₁₆×C₃₁	—	4	4	no	—
140	M₁₆×C₃₁	—	4	4	no	—
141	C₂×C₈×C₃₁	—	2	4	no	—
142	C₂×C₈×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	—	2	4	no	—
144	C₄⋊C₄×C₃₁	—	2	4	no	—
145	C₄⋊C₄×C₃₁	—	2	4	no	—
146	C₄⋊C₄×C₃₁	—	2	4	no	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
151	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
167	#17742	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	Q₈×C₃₁	—	4	8	no	—
175	Q₈×C₃₁	—	4	8	no	—
176	Q₈×C₃₁	—	4	8	no	—
177	Q₈×C₃₁	—	4	8	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	8	no	—
194	C₂×C₄×C₃₁	—	4	8	no	—
195	C₂×C₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	4	8	no	—
197	C₂×C₄×C₃₁	#101	1	4	yes	—
198	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
208	C₂×C₄×C₃₁	#137	1	4	yes	—
209	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₈×C₃₁	—	4	4	no	—
213	C₈×C₃₁	—	4	4	no	—
214	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
215	#2920	C₂×C₃₁	1	1	yes	characteristic
216	#2920	C₂×C₃₁	1	1	yes	characteristic
217	#1556	C₂×C₃₁	1	4	yes	complemented
218	#1549	C₂×C₃₁	1	4	yes	complemented
219	#1546	C₂×C₃₁	1	4	yes	complemented
220	C₂²×C₃₁	—	2	8	no	—
221	C₂²×C₃₁	—	2	1	no	—
222	C₂²×C₃₁	—	2	1	no	—
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	2	1	no	—
225	C₂²×C₃₁	—	2	1	no	—
226	C₂²×C₃₁	—	2	1	no	—
227	C₂²×C₃₁	C₂×#101	1	2	yes	—
228	C₂²×C₃₁	#418	1	2	yes	—
229	C₂²×C₃₁	#419	1	2	yes	—
230	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
231	C₄×C₃₁	—	2	1	no	—
232	C₄×C₃₁	—	2	1	no	—
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	2	1	no	—
236	C₄×C₃₁	—	2	1	no	—
237	C₄×C₃₁	—	4	8	no	—
238	C₄×C₃₁	C₂×#101	1	2	yes	—
239	C₄×C₃₁	C₂×#101	1	2	yes	—
240	C₄×C₃₁	#418	1	2	yes	—
241	C₄×C₃₁	#419	1	2	yes	—
242	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
243	C₄×C₃₁	C₂×#137	1	2	yes	—
244	C₂³×Q₈	—	2	1	no	—
245	C₂×C₄⋊Q₈	—	2	1	no	—
246	C₂×C₄⋊Q₈	—	2	1	no	—
247	C₂×C₄⋊Q₈	—	2	1	no	—
248	C₂×C₄⋊Q₈	—	2	1	no	—
249	C₂×C₄⋊Q₈	—	2	1	no	—
250	C₂×C₄⋊Q₈	—	2	1	no	—
251	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
252	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
253	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
254	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
257	C₂²×C₄⋊C₄	—	2	1	no	—
258	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
259	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
260	#432	—	2	1	no	—
261	#432	—	2	1	no	—
262	#432	C₂²×C₃₁	1	2	yes	—
263	#432	C₂²×C₃₁	1	2	yes	—
264	#431	—	2	1	no	—
265	#431	—	2	1	no	—
266	#431	C₂²×C₃₁	1	2	yes	—
267	#431	C₂²×C₃₁	1	2	yes	—
268	#423	C₂²×C₃₁	1	2	yes	—
269	#423	C₂²×C₃₁	1	2	yes	—
270	#407	—	2	1	no	—
271	#407	—	2	1	no	—
272	C₄×M₁₆	—	2	1	no	—
273	C₄×M₁₆	—	2	1	no	—
274	#332	C₂²×C₃₁	1	16	yes	complemented
275	C₂×C₃₁	—	2	2	no	—
276	C₂×C₃₁	—	2	2	no	—
277	C₂×C₃₁	—	2	2	no	—
278	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
279	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
280	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
281	C₂²×Q₈	—	2	2	no	—
282	C₂²×Q₈	—	2	2	no	—
283	C₂²×Q₈	—	2	2	no	—
284	C₂²×Q₈	—	2	2	no	—
285	C₂²×Q₈	—	2	2	no	—
286	C₂²×Q₈	—	2	2	no	—
287	C₂²×Q₈	—	4	8	no	—
288	C₂³×C₄	—	2	2	no	—
289	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
290	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	2	no	—
296	C₄⋊Q₈	—	2	2	no	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	—	2	2	no	—
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
307	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
308	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
309	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
320	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
321	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
322	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
337	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	—	2	2	no	—
341	C₂²⋊C₈	—	2	2	no	—
342	#97	—	2	2	no	—
343	#97	—	2	2	no	—
344	C₄×C₈	—	2	2	no	—
345	C₄×C₈	—	2	2	no	—
346	C₃₁	#17742	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
348	C₂×Q₈	—	2	4	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	2	4	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	4	8	no	—
353	C₂×Q₈	—	4	8	no	—
354	C₂×Q₈	—	4	8	no	—
355	C₂×Q₈	—	4	8	no	—
356	C₂×Q₈	—	4	8	no	—
357	C₂×Q₈	—	4	8	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	4	8	no	—
365	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	M₁₆	—	4	4	no	—
384	M₁₆	—	4	4	no	—
385	C₂×C₈	—	2	4	no	—
386	C₂×C₈	—	2	4	no	—
387	C₄⋊C₄	—	2	4	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
395	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
396	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
397	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
398	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
399	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
400	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
401	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
411	C₂³	—	2	2	no	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
415	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
416	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
417	Q₈	—	4	8	no	—
418	Q₈	—	4	8	no	—
419	Q₈	—	4	8	no	—
420	Q₈	—	4	8	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	4	8	no	—
438	C₂×C₄	—	4	8	no	—
439	C₂×C₄	—	4	8	no	—
440	C₂×C₄	C₃₁×#101	1	4	yes	—
441	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
451	C₂×C₄	C₃₁×#137	1	4	yes	—
452	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	C₈	—	4	4	no	—
456	C₈	—	4	4	no	—
457	C₂²	—	2	8	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	2	1	no	—
464	C₂²	C₂×C₃₁×#101	1	2	yes	—
465	C₂²	C₃₁×#418	1	2	yes	—
466	C₂²	C₃₁×#419	1	2	yes	—
467	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₄	—	2	1	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	4	8	no	—
475	C₄	C₂×C₃₁×#101	1	2	yes	—
476	C₄	C₂×C₃₁×#101	1	2	yes	—
477	C₄	C₃₁×#418	1	2	yes	—
478	C₄	C₃₁×#419	1	2	yes	—
479	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
480	C₄	C₂×C₃₁×#137	1	2	yes	—
481	C₂	—	2	2	no	—
482	C₂	—	2	2	no	—
483	C₂	—	2	2	no	—
484	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
485	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
486	C₂	C₂×C₃₁×#419	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1556	C₂	semidirect
5. C₃₁×#1549	C₂	semidirect
6. C₃₁×#1546	C₂	semidirect
37. C₃₁×#332	C₂²	semidirect
167. #17742	C₃₁	direct (N × H)
217. #1556	C₂×C₃₁	semidirect
218. #1549	C₂×C₃₁	semidirect
219. #1546	C₂×C₃₁	semidirect
274. #332	C₂²×C₃₁	semidirect
346. C₃₁	#17742	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₃₁×#2920	2
C₃₁×#1556	2
C₃₁×#1549	2
C₃₁×#1546	2
#17742	31

Subgroups — C31×#17742 #21474501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17742 #21474501